University of Southern California Dissertations and Theses

Photoexcitation and nonradiative relaxation in molecular systems: methodology, optical properties, and dynamics

(USC Thesis Other)

Photoexcitation and nonradiative relaxation in molecular systems: methodology, optical properties, and dynamics

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content UNIVERSITY OF SOUTHERN CALIFORNIA
DOCTORAL DISSERTATION
Photoexcitation and Nonradiative
Relaxation in Molecular Systems:
Methodology, Optical Properties, and
Dynamics
Author:
Andrew E. SIFAIN
Supervisor:
Dr. Oleg V . PREZHDO
A dissertation submitted in fulfillment of the requirements
for the degree of Doctor of Philosophy
in the
Department of Physics and Astronomy
USC Dornsife College of Letters, Arts and Sciences
Conferred by the Faculty of the USC Graduate School
May 2019

iii
Declaration of Authorship
I, Andrew E. SIFAIN, declare that this dissertation titled, “Photoexcitation and Nonra-
diative Relaxation in Molecular Systems: Methodology, Optical Properties, and Dynam-
ics” and the work presented in it are my own. I confirm that:
• This work was done wholly or mainly while in candidature for a research degree
at this University.
• Where any part of this dissertation has previously been submitted for a degree
or any other qualification at this University or any other institution, this has been
clearly stated.
• Where I have consulted the published work of others, this is always clearly at-
tributed.
• Where I have quoted from the work of others, the source is always given. With
the exception of such quotations, this dissertation is entirely my own work.
• I have acknowledged all main sources of help.
• Where the dissertation is based on work done by myself jointly with others, I have
made clear exactly what was done by others and what I have contributed myself.
Signed:
Date:

v
“It is not knowledge, but the act of learning, not possession but the act of getting there, which
grants the greatest enjoyment.”
Carl Friedrich Gauss

vii
UNIVERSITY OF SOUTHERN CALIFORNIA
Abstract
Department of Physics and Astronomy
USC Dornsife College of Letters, Arts and Sciences
Doctor of Philosophy
Photoexcitation and Nonradiative Relaxation in Molecular Systems: Methodology,
Optical Properties, and Dynamics
by Andrew E. SIFAIN
Computational modeling of photoexcited molecules provides a fundamental under-
standing of processes such as photodissociation, photoisomerization, and charge and
energy transfer. To model these processes requires the development and utilization of
theories in electronic structure and quantum dynamics. One of the most popular meth-
ods for modeling nuclear-electronic dynamics is surface hopping. In this dissertation,
surface hopping will be presented in different spotlights. We describe the method along
with its advantages, disadvantages, and the contributions we make to improve its re-
liability and applicability. These efforts are carried out with simple Hamiltonians that
can be solved exactly, thus allowing for an assessment of this approximate method.
While this is an important step in methods development, modeling the large number
of degrees of freedom in realistic systems requires efficient software. We present a soft-
ware that combines surface hopping with numerically efficient methods for calculating
ground and excited state potential energy surfaces–a necessary feature for modeling
systems made up of hundreds of atoms and processes lasting up to tens of picoseconds.
As part of our contribution to this software, we implement and benchmark an implicit
solvent model, including investigating its effects during the nonradiative relaxation in
organic conjugated molecules. The ultimate goal is to improve and assess the accuracy
of these theoretical tools with the intention of progressing real-life applications. Having
said that, a portion of this dissertation explores the early stages of an application called
photoactive energetic materials–a field seeking to discover mechanically and electri-
cally insensitive materials that undergo detonation through optical initiation. Our work
investigates the optical properties in energetic materials, identified through an exper-
imental collaboration, and is aimed at unveiling design principles to enhance control
over the initiation threshold. In summary, we provide a comprehensive view of atom-
istic simulations of photoexcited molecules, starting from the methods used to describe
electronic transitions through a manifold of excited states as a result of photoexcitation,
followed by the development of software used to model realistic systems, and finally
the use of such tools to discover how desired photophysical properties can be attained
for practical use. The final part of this dissertation explores a new and booming field of
research in the computational sciences called machine learning. Machine learning is a
clever way of attaining calculations of ab initio accuracy at a tiny fraction of the compu-
tational cost through the use of statistically trained models. We present an application
viii
using machine learning and discuss how the use of this method can change the scope of
nonadiabatic molecular dynamics.
ix
Acknowledgements
I would like to express my appreciation to the many people that supported me through-
out my education, culminating with this dissertation. First, I would like to thank my
doctoral advisers Oleg Prezhdo (professor, USC) and Sergei Tretiak (staff scientist, LANL)
for their continuous support during this journey both pedagogically and financially. I
am indebted to their mentorship which has shaped me into the critical thinker that I am
today.
I would like to thank the following people (in alphabetic order) with whom I closely
collaborated with. They are Kipton Barros (staff scientist, LANL), Josiah Bjorgaard
(postdoc, LANL), David Chavez (staff scientist, LANL), Sebastian Fernandez-Alberti
(professor, Universidad Nacional de
Quilmes/CONICET), Brendan Gifford (postdoc, LANL), Olexandr Isayev (professor,
UNC Chapel Hill), Nicholas Lubbers (staff scientist, LANL), Thomas Myers (staff sci-
entist, LLNL), Benjamin Nebgen (staff scientist, LANL), Tammie Nelson (staff scientist,
LANL), Adrian Roitberg (professor, UF), Justin Smith (postdoc, LANL), Linjun Wang
(professor, Zhejiang University), and Alexander White (staff scientist, LANL). Their in-
sightful discussions, exchange of creative ideas, and constructive criticisms have made
this dissertation possible.
I am grateful to my teachers at the University of Rochester who sparked my interest
in physics. Professor Frank Wolfs was my first physics teacher, and it was his intro-
ductory classical mechanics course that encouraged me to major in physics. Professor
Douglass Cline’s course in advanced classical mechanics furthered my interest in this
subject through his enthralling lectures. Professor Carl Hagen’s courses in quantum me-
chanics were both conceptually and mathematically stimulating. His assigned problem
sets were quite challenging and took many hours to complete, but that made his courses
all the more gratifying. The positive experiences that I had with these professors, among
others, welcomed the idea of graduate school.
I would like to thank the people that helped ensure a smooth learning experience.
Undergraduate and graduate school were fulfilling because of all the behind-the-scenes
effort of Janet Fogg and Laura Blumkin. I thank you both. Your effort does not go un-
noticed and will not be forgotten. The same is true of all the staff at the Center for Non-
linear Studies (CNLS), who have fostered a collaborative and intellectually stimulating
working environment during the two plus years that I was with them. In particular, I
would like to thank Angel Garcia, Enrique Batista, Kacy Hopwood, Amanda Martinez,
Donald Thompson, and Jennie Harvey.
My friends during the last decade have also been central to my success. Thank
you for all your help during the late-night study sessions and for your own aspirations
which have encouraged me to push myself. Special thanks to Dev Ashish Khaitan,
Timothy Dehaas, and Brendan Gifford for many things, whether it be intriguing con-
versation, countless gourmet meals, or opening your home to me.
Finally, I am grateful to the NSF and DOE for financially supporting the research
presented in this dissertation.

xi
Contents
Declaration of Authorship iii
Abstract vii
Acknowledgements ix
1 Introduction 1
1.0.1 Methods Development . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.0.2 Modeling Molecular Systems . . . . . . . . . . . . . . . . . . . . . . 4
1.0.3 Photoactive Energetic Materials . . . . . . . . . . . . . . . . . . . . 5
1.0.4 Machine Learning in Computational Chemistry . . . . . . . . . . . 6
1.0.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Electronic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.2 Nuclear Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Surface Hopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Fewest Switches Surface Hopping . . . . . . . . . . . . . . . . . . . 12
2.3.2 Global Flux Surface Hopping . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.1 Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5.2 Equilibration Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Momentum Reversal Following Classically-Forbidden Electronic Transitions
Improves Detailed Balance in Surface Hopping 21
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.1 Recap of Fewest-Switches Surface Hopping . . . . . . . . . . . . . . 23
3.2.2 Two-Level Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2.3 Three-Level Subsystem . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4 Numerical Tests of Coherence-Corrected Surface Hopping Methods 31
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Overcoherence Problem of Surface Hopping . . . . . . . . . . . . . . . . . 33
xii
4.3 Superexchange Theory and Reaction Rates in Donor-Bridge-Acceptor Sys-
tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Methods and Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . 36
4.4.1 Global Flux Surface Hopping . . . . . . . . . . . . . . . . . . . . . . 36
4.4.2 Surface Hopping Dynamics and Decoherence Corrections . . . . . 36
Truhlar’s Decay-of-Mixing . . . . . . . . . . . . . . . . . . . . . . . 37
Decay-of-Mixing Dephasing-Informed . . . . . . . . . . . . . . . . 37
Subotnik’s Augmented Surface Hopping . . . . . . . . . . . . . . . 38
4.4.3 DbA Model Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers 45
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.1 The Nonadiabatic EXcited-state Molecular Dynamics (NEXMD)
Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2.2 Studied PPVO Derivatives . . . . . . . . . . . . . . . . . . . . . . . 47
5.2.3 Nonadiabatic Dynamics Simulations . . . . . . . . . . . . . . . . . . 48
5.2.4 Natural Transition Orbitals (NTOs) and Transition Density (TD)
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.3.1 Optical Spectra and Initial Excitation . . . . . . . . . . . . . . . . . 51
5.3.2 Potential Energy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3.3 Excited-State Populations and Lifetimes . . . . . . . . . . . . . . . . 53
5.3.4 Transition Density (TD) Analysis . . . . . . . . . . . . . . . . . . . . 54
5.3.5 Excited State Dipole Moments . . . . . . . . . . . . . . . . . . . . . 55
5.3.6 Exciton Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3.7 Bond Length Alternation (BLA) . . . . . . . . . . . . . . . . . . . . . 57
5.3.8 Relative Computational Time . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6 NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline 63
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 Computational Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.1 Ground State Sampling . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.2 Optical Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . 64
6.2.3 Photoexcitation to S
1
– Adiabatic Dynamics . . . . . . . . . . . . . . 65
6.2.4 Analysis of NEXMD of Geometries . . . . . . . . . . . . . . . . . . . 65
6.2.5 Isomerization Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2.6 High-Energy Photoexcitation – Nonadiabatic Dynamics . . . . . . 66
6.2.7 Constant Energy Dynamics . . . . . . . . . . . . . . . . . . . . . . . 66
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.1 4-styrylquinoline (SQ) . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.3.2 Optical Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . 68
6.3.3 Photoisomerization Reaction Pathway . . . . . . . . . . . . . . . . . 69
6.3.4 Trans-to-Cis Photoisomerization Dynamics . . . . . . . . . . . . . . 70
xiii
6.3.5 Nonradiative Relaxation Following High-Energy Photoexcitation . 71
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
7 Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A Time-
Dependent Density Functional Theory Study 81
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2.1 Computational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.2.2 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.3.1 Benchmarking Model Quantum Chemistry . . . . . . . . . . . . . . 84
7.3.2 Optical Absorption of Explosive Compounds . . . . . . . . . . . . . 86
7.3.3 Oxygen-Containing Compounds . . . . . . . . . . . . . . . . . . . . 91
7.3.4 Characterizing MLCT Bands . . . . . . . . . . . . . . . . . . . . . . 93
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8 Cooperative Enhancement of the Nonlinear Optical Response in Conjugated
Energetic Materials: A TD-DFT Study 99
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.2 Conjugated Energetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.3 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.3.1 Optical Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . 102
8.3.2 Benchmarking Model Quantum Chemistry . . . . . . . . . . . . . . 103
8.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9 Discovering a Transferable Charge Assignment Model using Machine Learn-
ing 111
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9.2 HIP-NN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.3 Predicting Molecular Dipole Moment . . . . . . . . . . . . . . . . . . . . . 113
9.4 Assessing ACA Charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.5 Machine Learned Infrared Spectra with ACA Charges . . . . . . . . . . . . 117
9.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10 Closing Remarks 121
A Supporting Information for Chapter 4 125
A.1 Global Flux Versus Fewest Switches Surface Hopping in the Diabatic Rep-
resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
A.2 Scattering Problems: Surface Hopping Versus Exact Quantum Mechanics 127
A.2.1 Two-Level Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
A.2.2 Three-Level Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.3 Subotnik’s Augmented Surface Hopping . . . . . . . . . . . . . . . . . . . 130
xiv
B Supporting Information for Chapter 5 135
B.1 Nonadiabatic Modeling Procedure . . . . . . . . . . . . . . . . . . . . . . . 135
B.2 Optical Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.3 Natural Transition Orbitals (NTOs) . . . . . . . . . . . . . . . . . . . . . . . 137
B.4 Energy Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.5 Exponential Rate and Time Decay Constants of S
1
. . . . . . . . . . . . . . 141
B.6 Transition Density (TD) Analysis . . . . . . . . . . . . . . . . . . . . . . . . 142
B.7 Excited-State Dipole Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 144
B.8 Charge Transfer Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
B.9 Solvent-Stable Resonance Structures (Zwitterions) . . . . . . . . . . . . . . 147
B.10 Relative Computational Time . . . . . . . . . . . . . . . . . . . . . . . . . . 148
C Supporting Information for Chapter 6 149
C.1 Comparing Absorption Spectra from DFT and Semiempirical Levels of
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.2 Potential Energy Surface Scan . . . . . . . . . . . . . . . . . . . . . . . . . . 149
C.3 Molecular Orbitals of SQ Conformations during Isomerization . . . . . . . 151
C.4 Time-Domain Isomerization Data from Experiment . . . . . . . . . . . . . 151
C.5 NEXMD Simulations in Vacuum . . . . . . . . . . . . . . . . . . . . . . . . 152
C.6 Vibrational Spectra During Nonadiabatic Dynamics . . . . . . . . . . . . . 154
D Supporting Information for Chapter 7 157
D.1 Optical Absorption: Functionals and Basis Sets . . . . . . . . . . . . . . . . 157
D.2 Optical Absorption: Polarization Functions . . . . . . . . . . . . . . . . . . 158
D.3 Natural Transition Orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
E Supporting Information for Chapter 9 163
E.1 HIP-NN Architecture and Training Details . . . . . . . . . . . . . . . . . . 163
E.1.1 HIP-NN Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . 163
E.1.2 Training Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
E.2 Details of Charge Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . 163
E.3 Additional Data of Dipole Prediction . . . . . . . . . . . . . . . . . . . . . . 164
Bibliography 171
xv
List of Figures
1.1 A trajectory passing through a region of strong nonadiabatic coupling results in
a transition between electronic states. . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Diagram of the quantum-classical model. . . . . . . . . . . . . . . . . . . . 15
2.2 State populations at 500 K, computed with FSSH and GFSH. . . . . . . . . 16
2.3 Equilibrated population of states 1, 2, and 3 at various temperatures. . . . 17
2.4 State populations at 600 K, computed with FSSH and GFSH. . . . . . . . . 19
3.1 Equilibrated populations for a two-level quantum subsystem with nu-
clear velocity reversed and not reversed after a frustrated hop. . . . . . . 24
3.2 Equilibrated populations for a three-level superexchange model. . . . . . 25
3.3 Normalized energy distribution of the atom coupled to the quantum sub-
system as a function of state occupation. . . . . . . . . . . . . . . . . . . . . 26
3.4 Normalized distribution of the target state’s population for thej1i!j2i
andj2i!j1i transition, along with respective coherences. . . . . . . . . . 27
3.5 Normalized energy distributions given that thej1i!j2i andj3i!j2i
transitions have been attempted. . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Liouville pathways from the donor to acceptor in the diabatic representation.
Pathway A is that of superexchange, whereas B and C are sequential pathways
that pass through the population of bridge statej2i. . . . . . . . . . . . . . . . . 35
4.2 (A) Cartoon schematic of the DbA model represented as a linear chain of di-
atomic molecules. (B) Adiabatic potential energy surfaces of the DbA model
with respect to the reaction coordinate. Donor, bridge, and acceptor diabats are
labeled by 1, 2, and 3, respectively. The tunneling energy gap is labeled byDe. . . 40
4.3 Donor-to-acceptor reaction rates as a function of driving force (e
0
), diabatic cou-
pling (V), and tunneling energy gap (De). Reaction rates were computed with
surface hopping (red circles) and Marcus theory (black line). Decay-Of-Mixing
(DOM) data include C = 1 (orange triangles) and C = 10 (red circles) (see
Eq. 4.13). Curve fits of the surface hopping data are shown in light grey. . . . . . 43
5.1 Chemical structures of PPVO derivatives. . . . . . . . . . . . . . . . . . . . 48
5.2 Calculated linear optical absorption spectra of the studied PPVO chro-
mophores. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.3 Histograms of the PESs over the ensemble of trajectories. . . . . . . . . . . 53
5.4 Excited state populations during the 1 ps dynamics and their associated
relaxation rates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Fraction of transition density (TD) on the specified molecular fragment
during the 1 ps dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.6 Dipole magnitude and direction during the 1 ps dynamics. . . . . . . . . . 57
xvi
5.7 Excited state dipole moment versus exciton localization L
D
. . . . . . . . . 58
5.8 Bond length alternation (BLA) between adjacent carbon-carbon atoms
during the 1 ps dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.1 Conformations of SQ from t-SQ to its finite stable product, BP . . . . . . . . 67
6.2 Optical absorption spectra of SQ conformations. . . . . . . . . . . . . . . . 68
6.3 Cartoon schematic of NEXMD simulations of t-SQ! c-SQ photoisomer-
ization dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 Labeling of different SQ conformations on Ramachandran plots as a func-
tion off
C=C
andf
CC
dihedral angles. . . . . . . . . . . . . . . . . . . . . 75
6.5 Ramachandran diagram of the low-energy simulation (i.e., excitation to
S
1
) and labeling of the different reaction pathways. . . . . . . . . . . . . . . 76
6.6 Energies of the ground and first excited state as the molecule evolves from
t-SQ to p-SQ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.7 Fraction of trajectories evolving on S
1
that encountered an energy gap
E
1
E
0
< 1.0 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.8 Excited state populations and bond length alternation (BLA) between ad-
jacent carbon-carbon atoms during the 1 ps nonadiabatic dynamics. . . . . 79
7.1 Octahedral geometry and ligands of Fe (II) complexes 1 through 3. . . . . . . . . 84
7.2 Optical absorption of complexes 1 through 3 computed with different lev-
els of theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.3 Optical absorption of complexes 1 through 4. . . . . . . . . . . . . . . . . . 86
7.4 Optical absorption of complexes 5 through 8. . . . . . . . . . . . . . . . . . 87
7.5 Quadrupole moment of the ligand scaffold in its ground state geometry
versus wavelength of the strongest photoactive excited state within the
low-energy CT band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.6 Comparison of the absorption spectra in complexes 1(A) and 10(A). . . . . 91
7.7 Optical absorption of complexes with and without oxygen substituents. . 92
7.8 MLCT, in units of number of electrons, versus wavelength of the strongest pho-
toactive excited states within the low-energy CT band. . . . . . . . . . . . . . . 94
8.1 Chemical structures of the CEMs. . . . . . . . . . . . . . . . . . . . . . . . . . 100
8.2 Optical absorption of compounds A through C. . . . . . . . . . . . . . . . . 104
8.3 OPA and TPA spectra of the monomer and dimers of material A. . . . . . 106
8.4 OPA and TPA spectra of the monomer and dimers of materials B and C. . 107
8.5 Changes in m
ge
at the optically excited state from monomer A and dimer
5A, along with NTOs of the excitation in 5A. . . . . . . . . . . . . . . . . . 109
9.1 Abstract schematic of HIP-NN in the context of dipole prediction, illus-
trated for a water molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.2 Size distributions of molecules in three datasets: ANI-1x, DrugBank, and
Tripeptides. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.3 2D histograms showing the correlation between predicted (ACA) and
reference (QM) electrostatic moments using three test datasets: ANI-1x,
DrugBank, and Tripeptides. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.4 2D histograms showing correlations between all pairs of charge models. . 116
9.5 Infrared spectra of select molecules, computed with ACA charges. . . . . . 119
xvii
A.1 Three-level model in the diabatic representation. . . . . . . . . . . . . . . . . . 125
A.2 Scattering probabilities on diabatj3i as a function of initial nuclear momentum. . 126
A.3 Two-level scattering models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
A.4 Fraction of trajectories that reflect or transmit in two-level models. . . . . 129
A.5 Model X: three-level scattering model. . . . . . . . . . . . . . . . . . . . . . . . 130
A.6 Fraction of trajectories that reflect or transmit in model X. . . . . . . . . . . . . . 131
B.1 Comparison between theoretical and experimental absorption and emis-
sion spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.2 Histograms of energy gaps over the entire 1 ps ensemble of trajectories
withe=f1, 2, 5, 20g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
B.3 Fraction of the maximum number of small energy gaps as a function di-
electric constant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
B.4 Fraction of TD due to ground to excited state transitions. . . . . . . . . . . 143
B.5 Calculated permanent excited state dipole moment as a function of time. . 144
B.6 Difference of dipole moments in the ground and excited state. . . . . . . . 146
B.7 Resonance structures offH, NO
2
g andfNH
2
, NO
2
g. . . . . . . . . . . . . 147
B.8 Bar graphs showing the relative CPU times of thefH, Hg,fH, NO
2
g,
andfNH
2
, NO
2
g nonadiabatic ensembles computed with (e= 20) and
without(e= 1) the solvent model. . . . . . . . . . . . . . . . . . . . . . . . 148
C.1 Absorption spectra computed with DFT and semiempirical levels of theory.150
C.2 PES scan along the IRC using DFT and semiempirical levels of theory. . . 151
C.3 Comparison between PES scan and NEXMD geometries obtained from
molecular dynamics simulations . . . . . . . . . . . . . . . . . . . . . . . . 152
C.4 HOMO and LUMO of different conformations of SQ during isomerization. . . . 153
C.5 Experimental decay in the absorbance of t-SQ as a function of time in
n-hexane and ethanol. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
C.6 Ramachandran plots of NEXMD simulations in vacuum. . . . . . . . . . . 154
C.7 PES scan off
C=C
for fixedf
CC
. . . . . . . . . . . . . . . . . . . . . . . . . 155
C.8 Vibrational spectra from the nonadiabatic simulations in different envi-
ronments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
D.1 Optical absorption of complexes 1 through 3, computed with various
density functionals and basis sets. . . . . . . . . . . . . . . . . . . . . . . . 157
D.2 Optical absorption of complexes 1 through 3, computed with the TPSSh
density functional and various basis sets with and without polarization
functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
E.1 Pair correlation plots of charge predictions from four neural networks
constituting the entire ensemble. . . . . . . . . . . . . . . . . . . . . . . . . 165
E.2 Bar charts showing RMSE and MAE in dipole prediction for each molecule
size in ANI-1x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
E.3 Bar charts showing RMSE and MAE in dipole prediction for each molecule
size in DrugBank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
E.4 Bar charts showing RMSE and MAE in dipole prediction for each molecule
in Tripeptide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
xviii
E.5 2D histograms showing correlations between predicted and reference elec-
trostatic moments using five different charge models and the GDB-5 test
dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
E.6 2D histograms showing correlations between predicted and reference elec-
trostatic moments using three different charge models and a random sub-
set of the ANI-1x dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
E.7 Infrared spectra of small molecules (ranging between 6 to 15 total atoms)
calculated using ACA and QM. . . . . . . . . . . . . . . . . . . . . . . . . . 170
xix
List of Tables
2.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Total number of frustrated and successfully-invoked hops within 10 ps at
1600 K using 300 trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1 Fitting parameters of the donor-to-acceptor reaction rates of Fig. 4.3. Data
were fit to functions (shown in the column labels) representing the scal-
ing behavior of the Marcus theory expression of Eq. 4.7 with respect to
driving force (e
0
), coupling (V), and tunneling energy (De). . . . . . . . . . 41
6.1 Relaxation rates of the trans-to-cis isomerization process in different en-
vironments from different initial conditions. . . . . . . . . . . . . . . . . . 70
7.1 Measured and calculated bond lengths (Å) of complexes 1 through 3. . . . 84
7.2 NTOs of the photoactive excited states within the low-energy CT band of
6(A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.3 OB% of several compounds with and without oxygen substituents in
their ligand scaffolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.4 NTOs of photoactive excited states within the low-energy CT bands of
1(B) and 1(C). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
7.5 NTOs of photoactive excited states within the low-energy CT band of 8(B). 96
8.1 Sensitivity data of materials A through C, PETN, and RDX. . . . . . . . . . 101
8.2 Properties of OPA and TPA spectra in materials A through C. . . . . . . . 108
B.1 Natural transition orbitals (NTOs) of the first excited state S
1
withe= 1. . 138
B.2 Natural transition orbitals (NTOs) of photoactive excited states in the
vicinity of 4.30 eV withe= 1. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
B.3 Exponential rate and time decay constant of S
1
across all molecules and
dielectric constants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
D.1 Natural transition orbitals (NTOs) of photoactive excited states within the
charge transfer bands of complexes 5(A), 8(B), and 10(A). . . . . . . . . . . 159
E.1 Summary of test and extensibility datasets along with statistical measures
for dipole and quadrupole prediction. . . . . . . . . . . . . . . . . . . . . . 164

xxi
List of Abbreviations
ACA Affordable Charge Assignment
AU Arbitrary Units
AM1 Austin Model 1
AO Atomic Orbital
B3LYP Becke 3-parameter Lee-Yang-Parr – density functional
BLA Bond Length Alternation
BP BenzoPhenanthridine
c-SQ cis-SQ
CEM Conjugated Energetic Molecule
CEO Collective Electronic Oscillator
CIS Configuration Iinteraction Singles
CM5 Charge Model 5
CPCM Conductor-like Polarizable Continuum Model
CPU Core Processing Unit
CT Charge Transfer
DFT Density Functional Theory
DHBP DiHydroBenzoPhenanthridine
DNAZ 3,3-DiNitroAZetidine
DNN Deep Neural Network
FSSH Fewest-Switches Surface Hopping
FSTU Fewest-Switches with Time Uncertainty
FWHM Full Width at Half Maximum
GFSH Global Flux Surface Hopping
HIP-NN Hierarchically Interacting Particle Neural Network
LANL08 Los Alamos National Laboratory ‘08 – basis set
LANL2DZ Los Alamos National Laboratory 2 Double Zeta – basis set
HMX octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine
HOMO Hightest Occupied Molecular Orbital
IRC Intrinsic Reaction Coordinate
LUMO Lowest Unoccupied Molecular Orbital
MAE Mean Absolute Error
MGGA Meta-Generalized Gradient Approximation
ML Machine Learning
MLCT Metal-to-Ligand Charge Transfer
MSK Merz-Singh-Kollman
MO Molecular Orbital
NAO Natural Atomic Orbital
NBO Natural Bond Orbital
NEXMD Nonadiabatic EXcited-State Molecular Dynamics
xxii
NIR Near InfraRed
NPA Natural Population Analysis
NTO Natural Transition Orbital
OB Oxygen Balance
OPA One-Photon Absorption
p-SQ perp-SQ
PCM Polarizable Continuum Model
PDF Probability Density Function
PES Potential Energy Surface
PETN PentaErythritol TetraNitrate
PPV Para-Phenylene Vinylene
PPVO Oligo(Para-Phenylene Vinylene)
QM Quantum Mechanical
QM/MM Quantum Mechanics / Molecular Mechanics
RDX 1,3,5-trinitroperhydro-1,3,5-triazine
RECP Relativistic Effective Core Potential
RMSE Root Mean Square Error
SH Surface Hopping
SQ 4-StyrylQuinoline
SQM Semiempirical Quantum Mechanics
t-SQ trans-SQ
TD Transition Density
TD-DFT Time-Dependent Density Functional Theory
TD-HF Time-Dependent Hartree Fock
TPA Two-Photon Absorption
TPSS Tao Perdew Staroverov Scuseria – density functional
TPSSh Tao Perdew Staroverov Scuseria hybrid – density functional
UV-Vis UltraViolet-Visible
VMD Visual Molecular Dynamics
xxiii
List of Symbols
A(W) absorption spectrum –
b
i
bond length m
c expansion coefficient –
d
ij
nonadiabatic coupling vector between adiabatic states m
1
e electron C
f (e) coefficient of induced polarization –
f
ge
oscillator strength –
F
L
Langevin force kgms
2
F
R
random force kgms
2
H Hamiltonian kgm
2
s
2
H
e
electronic Hamiltonian kgm
2
s
2
k relaxation rate s
1
k
B
Boltzmann’s constant m
2
kgs
2
K
1
L
D
electronic delocalization –
m nuclear or atomic mass kg
P nuclear or atomic momentum kgms
1
P
i
fraction of TD on the i-th fragment –
r electronic displacement m
R nuclear or atomic displacement m
˙
R nuclear or atomic velocity ms
1
S
i
singlet excited state –
t time s
T temperature K
V potential energy kgm
2
s
2
V
ij
(i6= j) energetic coupling between diabatic states kgm
2
s
2
V
M
Morse potential kgm
2
s
2
g Langevin friction parameter s
1
De
ij
energy splitting between adiabatic states kgm
2
s
2
Dt time step s
e dielectric constant –
e
i
, E
i
, or V
ii
adiabatic energy kgm
2
s
2
l wavelength m
1
m
ii
or~ m
ii
permanent electric dipole moment C m
m
ij
or~ m
ij
(i6= j) transition electric dipole moment C m
r density matrix –
s
2
TPA intensity m
4
s(GM)
t time constant second
f(r; R) adiabatic eigenfunction m
N/2
(N is dimension)
xxiv
f
CC
CC dihedral angle –
f
C=C
C=C dihedral angle –
y(r; R) wavefunction m
N/2
(N is dimension)
W vertical excitation energy kgm
2
s
2
W
OPA
OPA vertical excitation energy kgm
2
s
2
W
TPA
TPA vertical excitation energy kgm
2
s
2
¯ h Planck’s constant m
2
kgs
1
jii adiabatic eigenstate –
r
R
gradient with respect to nuclear reaction coordinates m
1
xxv
Dedicated to my parents for their love and support throughout
my life, and my wife for being my best friend ever since our first
days of college.

1
Chapter 1
Introduction
Modeling at an atomistic level not only aids in our first-principles understanding of
materials but it can also help elucidate materials’ interaction with external stimuli. Such
knowledge brings us closer to controlling chemical outcomes that can ultimately lead
to harnessing useful photophysical properties. Computational modeling is particularly
useful because processes occurring on the time and length scales of atoms are diffi-
cult to understand from experiments alone. Experiments are also difficult and costly to
perform, but due to rapidly growing advancements in computational resources, experi-
ments now have the benefit of being theoretically informed. For instance, modeling can
be used to identify and tailor design materials, thus making it possible to control the
properties (electronic, optical, etc.) that are important for a given application. Modeling
plays a pivotal role in this field as it both motivates and explains and is integral for the
efficient development of materials for societal benefits.
One area of materials research is light’s interaction with matter. A molecule can
absorb a photon and become quantum mechanically excited. This is not a natural state
for the molecule which is why it will proceed to find its way back to the ground state.
This relaxation occurs in a variety of flavors such as emission of a photon or conversion
of electronic energy into heat. Different paths of relaxation also occur on different time
scales. Understanding how a molecule loses its energy at an atomistic level involves
modeling the interactions between electrons and nuclei. Due to the complex nature
of these atomic constituents, describing them in mathematical form to be solved by a
computer requires approximations. The most fundamental approximation in quantum
chemistry is the Born–Oppenheimer approximation.
The Born–Oppenheimer approximation is based on the assumption that nuclei evolve
on much slower time scales than electrons.[1] Within this approximation, eigenvalues
of the electronic Hamiltonian (all terms of the total nuclear-electronic Hamiltonian mi-
nus the nuclear kinetic energy), which are known as adiabatic potential energy surfaces
(PESs), are functions of fixed nuclear geometries. Electronic dynamics are governed by
the underlying Coulombic interactions whereas nuclear dynamics are governed by the
forces generated by PESs. Throughout the dynamics, PESs may become closely spaced,
where small changes in nuclear geometry can give rise to large changes in the electronic
wavefunction. The Born–Oppenheimer approximation is no longer valid at these ge-
ometries and a nuclear trajectory may experience an electronic transition due to strong
electron-nuclear (or nonadiabatic) coupling (Figure 1.1). This is what causes a molecule
to relax nonradiatively into lower-lying excited states. Several methods have been devel-
oped to simulate nonadiabatic dynamics, including wavepacket-based methods such
2 Chapter 1. Introduction
FIGURE 1.1: A trajectory passing through a region of strong nonadiabatic
coupling results in a transition between electronic states.
ab initio multiple spawning,[2] Herman-Kluk frozen Gaussians,[3] and coupled Gaus-
sian wavepackets.[4] Due to its simplicity and low computational cost, Tully’s surface
hopping[5, 6] has been the most popular method to date.
1.0.1 Methods Development
In a fully rigorous treatment, molecular processes should be simulated at a fully
quantum mechanical level of theory, but due to the complexity of the Schrödinger equa-
tion and the number of degrees of freedom in a realistic simulation, this is not com-
putationally feasible. This introduces the second approximation, which is a classical
representation of the nuclear degrees of freedom. As opposed to wavepacket dynam-
ics, Tully’s fewest-switches surface hopping (FSSH) uses a swarm of independent and
classical nuclear trajectories, where at any given time, each trajectory is propagated
along a single PES corresponding to an active electronically-excited state. Stochastic
hops between PESs are determined by an ad hoc algorithm that depends on popula-
tion flux. The minimization criterion of FSSH is the key ingredient of the method since
a single PES describes the underlying reaction mechanism.[5–9] This important con-
cept improves upon traditional Ehrenfest dynamics which assumes nuclei evolve on
a mean PES. The surface hopping method has been used to study scattering at metal
surfaces,[10] proton-coupled electron transfer,[11, 12] photoinduced dynamics of conju-
gated molecules,[13–15] conversion efficiency in solar energy harvesting materials,[16,
17] and charge transport in organic materials.[18] Despite these advantages, the use of
independent and classical trajectories has well-known shortcomings including lack of
zero-point energy, tunneling, representation dependence, and overcoherence.[19] We
address a few of these issues in this thesis.
Among surface hopping’s weaknesses is its performance in the diabatic represen-
tation.[20] This is an unfortunate circumstance since diabatic electronic states offer an
Chapter 1. Introduction 3
intuitive understanding of chemical phenomena. Unlike adiabatic states, diabatic states
are independent of the reaction coordinate and correspond to well-defined product and
reactant channels. A simple example is the bonding character of the diabats making
up NaCl’s resonance structures (i.e., NaCl and Na
+
Cl

) which are independent of the
Na-Cl bond length. Conversely, the adiabatic ground state changes in character from co-
valent to ionic as the Na-Cl bond length decreases (and the inverse relationship is true
for the excited state); the adiabatic state is dependent on reaction coordinate in order
to remain an eigenstate of the electronic Hamiltonian. Thus, adiabatic states are convo-
luted in their bonding character whereas diabatic states fully decompose the wavefunc-
tion into ionic and covalent contributions. Besides bonding, diabatic states are useful
for electron transfer dynamics and predicting chemical reaction rates. Although true
diabatic states do not exist for molecules,[21] approximate diabats are both attainable
and elucidating.[21] Moreover, the methods used to construct approximate diabats are
becoming computationally efficient, to the extent that “on the fly” diabatization is pos-
sible.[21] Merging such diabatization techniques with surface hopping is a natural step
for excited state dynamics, and therefore improvements to the surface hopping algo-
rithm that make it viable in the diabatic representation are desirable.
Recent work using a model Hamiltonian has shown that a simple modification of
the original surface hopping algorithm can improve its performance in the diabatic
representation for a class of systems exhibiting a superexchange-like mechanism.[22]
In these systems, the quantum subsystem is reduced to three states, where two of the
states, which are lower in energy than the third state, are indirectly coupled through
the high-energy intermediate. The fraction of trajectories that asymptotically reflect or
transmit on their associated surfaces, upon exiting the interaction region, are calculated
and compared to the probabilities given by the Schrödinger equation. The modified
surface hopping method was shown to give more accurate scattering probabilities in the
diabatic representation compared to FSSH.[22] While these results are encouraging, fur-
ther tests ensuring the validity of the method are important. For example, a respectable
excited state dynamics theory must describe system-bath interactions in order to model
electron-vibrational relaxation and phonon-assisted transport.[23] At thermal equilib-
rium, a quantum subsystem must attain the correct thermal populations according to
the Maxwell-Boltzmann distribution.[1] Ehrenfest dynamics suffers in this regard since
a single effective surface inhibits accurate branching ratios.[24] In surface hopping, in-
dependent trajectories of an ensemble evolve on a single surface associated to some
basis state at any given time and provide a more accurate representation of nuclear-
electronic dynamics, branching ratios, and detailed balance.[6, 20, 25, 26] Using a multi-
atom model with interatomic interactions given by the Morse potential, along with cou-
pling to a Langevin thermostat, the modified surface hopping algorithm is tested to
determine if it obeys the Boltzmann populations at thermal equilibrium (Chapter 2).[27]
While surface hopping has the benefit of attaining detailed balance, it is found that
the theory only approximately obeys the Boltzmann populations at thermal equilib-
rium. This problem has been noticed in model studies where a quantum subsystem is
coupled to an implicit bath (i.e., a Langevin thermostat). Excited states are generally
overpopulated, but that the deviation from the Boltzmann populations diminishes in
the case of small adiabatic splitting and/or strong nonadiabatic coupling.[26, 28] Iso-
lating the source of the problem and finding minor corrections to the nuclear-electronic
4 Chapter 1. Introduction
dynamics would be beneficial. Without delving into details, it is shown that the prob-
lem is related to classically forbidden (or frustrated) hops. These hops are electronic
transitions that are directed upward in energy, but which are energetically inaccessible
due to energy conservation. Frustrated hops are known to obstruct internal consistency,
i.e., the fraction of trajectories on any given surface does not match its corresponding
population computed with the Liouville equation.[29] Using model systems, tests are
carried out with simple ad hoc corrections in the treatment of nuclear degrees of free-
dom to reduce the inconsistencies of frustrated hops such that both internal consistency
and the Boltzmann populations are accurately attained (Chapter 3).[30]
The problem with surface hopping that is addressed more often than others is over-
coherence. Overcoherence is a consequence of a coherent propagation of the electronic
wavefunction along a nuclear trajectory.[31] In exact calculations, however, a nuclear
wavepacket bifurcates and the daughter wavepacket on one surface eventually moves
independently from the daughter wavepacket on any other surface. Long-time surface
hopping dynamics[29] and systems with several regions of nonadiabatic coupling[32]
are susceptible to spurious results without the inclusion of decoherence. A hierarchy of
methods have been designed to decohere the electronic wavefunction.[33–40] The gen-
eral formulation is to define a collapse rate indicative of wavepacket separation. How-
ever, even instantaneous collapse of the wavefunction following surface hops has been
shown to obey internal consistency better than more rigorous methods.[41] Ultimately,
further investigation is needed since rigorous methods are generally more expensive;
something theoreticians like to avoid. Understanding when certain decoherence meth-
ods are more appropriate than others is valuable information, but this is a challenging
task since the boundary at which ad hoc approaches become insufficient is imprecisely
defined, especially for molecules where approximations are almost always necessary.
Alternatively, testing a variety of decoherence methods against well established theories
might be useful for understanding their limitations. The theory we use to benchmark
decoherence methods is the Marcus theory of electron transfer (Chapter 4).[1]
1.0.2 Modeling Molecular Systems
Modeling photoexcited molecular systems requires the development of software
that merges the tools of electronic structure and quantum dynamics. The field of theo-
retical nonadiabatic excited state molecular dynamics is now mature enough to provide
a thorough description of the evolution of a molecular system through multiple excited
states following photoexcitation. However, providing an accurate description of excited
state processes at numerically efficient costs remains a challenge, particularly in the case
of large systems or long time scales. The Non-adiabatic EXcited-state Molecular Dynam-
ics (NEXMD) software is an efficient framework for excited state modeling.[13, 14] The
code calculates electronically excited states using optimized semiempirical Hamiltoni-
ans. This level of theory, merged with coherence-corrected surface hopping methods
for electronic transitions, makes it feasible to simulate the nonadiabatic dynamics of
molecules with sizes on the order of hundreds of atoms and for time scales lasting up
to tens of picoseconds. This is a signature property of the code enabling the simula-
tion of large molecular systems, where more elaborate ab initio approaches would be
numerically prohibitive. But numerical efficiency must be combined with realistic sim-
ulation for reliable results and predictions. For example, molecules should be modeled
Chapter 1. Introduction 5
in realistic environments since polarization can affect both optical properties and dy-
namics.[42] A contribution to NEXMD’s capabilities presented herein is to implement a
numerically efficient implicit solvent model and analyze its effects on dynamics (Chap-
ter 5).[43] Further assessment of the NEXMD software is carried out with a case study
using experimental evidence as the gold standard. The molecular dynamics model-
ing of photoisomerization presents various challenges requiring not only an accurate
description of ground and excited state PESs, but also geometries, barriers, and coni-
cal intersections.[44, 45] NEXMD’s performance of describing the photoisomerization
dynamics of a representative molecule is evaluated and compared to experimental ob-
servation (Chapter 6).[46]
1.0.3 Photoactive Energetic Materials
We have discussed the tools for modeling excited state dynamics, but developing
a photoactive application starts from understanding the material’s optical properties.
There has been an effort to design energetic materials that are less sensitive to external
stimuli to reduce the risk to accidental initiation and increase control over the initia-
tion process.[47, 48] Optical initiation of explosives is a new area of research that has
gained worldwide interest as it offers enhanced safety and quantum control.[49] Conju-
gated energetic coordination complexes are secondary explosives that are less sensitive
to electrical and mechanical stimuli than traditional explosives, such as pentaerythri-
tol tetranitrate (PETN), and possess the optical properties of carbon-based systems. An
experimental study has detailed the synthesis and characterization of this class of explo-
sives,[50, 51] but further investigation into what simple molecular modifications can be
made to tune the optical window of absorption such that it coincides with conventional
lasers is important for economic reasons. Chemical modeling may provide a cheap route
to discover compounds that meet experimental and engineering needs. We carry out a
theoretical investigation to predict the optical response in a large set of novel coordina-
tion complexes with a diverse set of ligand architectures to tune the absorption of the
low-energy charge transfer (CT) band (Chapter 7).[52]
Conjugated energetic molecules (CEMs) is another class of explosive compounds
that possess similar electrical, mechanical, and optical properties. A theoretical study
has detailed the relationship between their structure and optical response as isolated
molecules.[53] However, the effect of intermolecular interactions on optical response
should not be ignored as crystal structures are ultimately used in application.[54] A
study that models multichromophore interactions in these materials is necessary for re-
liable characterization of their optical properties. For quantitative accuracy, the level of
theory used to predict the optical response is time-dependent density functional the-
ory (TD-DFT).[55] While TD-DFT provides a more ab initio description of the optical
response compared to the semiempirical Hamiltonians used for dynamics, it limits the
sizes of the systems used for modeling. Nevertheless, nearest-neighbor interactions
likely provide a sufficient first-order approximation. Our approach is to extract dimers
from recently obtained crystallized CEMs with various orientations (Chapter 8).[56] The
use of dimers allows us to compute multichromophore optical properties, such as the
cooperative enhancement in the two-photon cross section. Two-photon absorption de-
pends on the degree of electronic delocalization spanning the chromophore units. This
process may be tuned by conjugation, the relative orientation of the chromophores, and
6 Chapter 1. Introduction
the addition of functional groups.[57] Design principles that can both increase the in-
tensity of nonlinear absorption and tune the absorption window to be within the limits
of conventional and widely-available lasers is valuable information. The excited state
dynamics of all energetic materials presented in this thesis is not the focus of our work,
but a followup study on their photodissociation process is necessary for their complete
understanding as well as improving design and performance.
1.0.4 Machine Learning in Computational Chemistry
The scientific contributions introduced thus far have focused on the methods used
to describe nonradiative relaxation through multiple excited states, the development of
software used to model molecular systems including their solute-solvent interactions,
and the formulation of a photoinduced application that requires the use of theoretical
tools to predict the absorption properties of energetic compounds. The remainder of
this thesis focuses on a new area of research in the computational sciences: machine
learning.
Machine learning has boomed in popularity over the past several years because it
can potentially accelerate molecular simulations.[58] In computational quantum chem-
istry, the most prominent discovery related to accelerating molecular simulation is prob-
ably density functional theory (DFT), which has reduced computational cost from ex-
ponential to cubic scaling with the number of constituent atoms.[59] This speedup has
made DFT the most popular method for chemical modeling and was the subject of the
1998 Nobel Prize in Chemistry. As a result of these achievements, determining the
ground and excited state properties of chemical systems hundreds to even thousands
of atoms in size has become computationally feasible.[60] But even the computational
scaling of DFT poses a problem for applications such as molecular dynamics and mate-
rials discovery. Alternatively, one may sacrifice ab initio theory and resort to semiem-
pirical quantum chemistry. In semiempirical Hamiltonians, computationally expensive
interactions, such as two-electron integrals, are parameterized to match experimental
data or other ab initio calculations.[61] Unfortunately, these methods are generally not
transferable to chemical environments that differ from the small subset used in the fit-
ting process. To bypass the costs of ab initio calculations, but still produce results of ab
initio quality, is the overarching goal of machine learning.
Machine learning is a potential game changer for nonadiabatic molecular dynamics
if excited states and their couplings, computed with a high-level ab initio theory, can be
attained in a tiny fraction of the computational cost. Acquiring the appropriate reference
data and preparing machine learning algorithms to handle such complex data is still in
its infancy, but the signs suggest a fruitful future ahead.[62] For the sake of getting our
foot in the door, we carried out a simpler application using machine learning: partial
atomic charge assignments.
Atom-centered partial charges are used in computational methods and applications
such as parametrization of force fields, solvation free energies, molecular docking, and
cheminformatics.[63–65] There are several known charge models to date,[66–68] but
partitioning electron density is not well-defined; charge models can give different and
often conflicting results. Among the most respected models are those that reproduce
electrostatic quantities. For example, charge model 5 (CM5) stands out as a robust
Chapter 1. Introduction 7
method that is parametrized to approximately reproduce quantum mechanical molec-
ular dipole moment.[68] A drawback of conventional charge models is that ab initio
calculation is still required, thus limiting their use for applications such as dynamic
charges for force field parametrization of large biomolecules. Our work formulates a
new charge model using machine learning that follows in the footsteps of CM5 but that
is trained to reproduce the dipoles of very large reference datasets (Chapter 9).[69]
1.0.5 Thesis Organization
This thesis broadly contributes to the field of computational chemistry with empha-
sis on excited state chemistry. It is organized as follows and each chapter is, more or less,
self-contained. First, we contribute to the surface hopping method; a mixed quantum-
classical theory that describes the coupling between electrons and nuclei, specifically
how nuclear configuration and nuclear motion facilitates nonadiabatic transitions be-
tween electronic states (Chapters 2, 3, and 4). Next, we contribute to the development
and evaluation of the software needed to model molecular systems in their polarizable
environments, including benchmarking theoretical predictions with experimental ob-
servation (Chapters 5 and 6). Besides preparation of the theoretical tools for excited
state modeling, we also lay out the foundation for an application that bridges the gap
between molecular design and optical properties in novel energetic compounds (Chap-
ters 7 and 8). Finally, with the emergence of artificial intelligence in the natural sciences
being stronger than ever, we too share in advancing this field within computational
chemistry (Chapter 9). The application studied is not rooted in excited state chemistry,
but still highlights the advantages machine learning can have on the field in general.

9
Chapter 2
Mixed Quantum-Classical
Equilibrium in Global Flux Surface
Hopping
Reprinted with permission from J. Chem. Phys. 142, 224102 (2015). Copyright 2018
American Institute of Physics.
2.1 Introduction
Physical processes on atomic and subatomic length scales are accurately described
by quantum mechanics. Most systems of interest, however, are commonly composed of
a large number of degrees of freedom, which are computationally expensive. Preserv-
ing all dominant quantum effects, while not compromising the underlying physics, be-
comes highly desirable. One of the most successful solutions is mixed quantum-classical
dynamics,[23, 70, 71] where highly quantum particles, i.e., subatomic particles, are re-
served to a quantum mechanical description, while larger constituents, such as nuclei,
evolve classically. These guidelines change, however, depending on the nature of the
study; for example, select nuclear motions describe tunneling and zero-point effects
in proton transfer,[12, 72–74] thus, for such nuclei, a quantum mechanical description
would be more appropriate.
Having proved to be accurate in many applications, fewest switches surface hop-
ping[5] (FSSH) has retained its popularity within mixed quantum-classical dynamics.
Electrons, for example, are treated quantum mechanically, to incorporate nonadiabatic
transitions between states,[8] while nuclei evolve classically, on potential energy sur-
faces. The particular surface, on which nuclei evolve, depends on the active electronic
state of the quantum subsystem. At each time step, the electron remains in its current
state or switches to a state with positive population flux; ergo, the “fewest switches"
criterion.[5] Despite its success, FSSH has been subject to many setbacks, some of which
have been resolved, i.e., phase correction[75] and decoherence.[35] In this manuscript,
we highlight FSSH’s inability of realizing superexchange.[22]
Superexchange is a class of dynamical processes where two electronic states are in-
directly coupled by an intermediate state of higher energy. A typical example is singlet
fission, where two singlet excitations convert to a triplet pair state by means of charge
transfer states.[76–79] Singlet fission is a spin-allowed process, which can potentially
10 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
double the number of electron-hole pairs, and thus largely increase the efficiency of or-
ganic photovoltaics.[80] To access an uncoupled state in the diabatic representation, a
switch to the highest-energy state is required. When the energy needed to occupy an
excited state is larger than the dispensable nuclear kinetic energy, the switch is not in-
voked. Such classically forbidden transitions are source of disagreement between FSSH
and exact quantum results (obtained from wave propagation), where transmission (on
the indirectly-coupled excited state) was found to be strongly underestimated, in the re-
gion where nuclear kinetic energy is less than the energy gap between the ground and
intermediate, highest-energy state.[22]
An alternative surface hopping strategy has been recently proposed called global
flux surface hopping (GFSH),[22] which is based on population flux flowing to and
from all states. Its fundamental difference to FSSH is highlighted when states are not
directly coupled. As opposed to FSSH, which requires direct coupling for single-hop
transitions, GFSH does not. When tested against the superexchange mechanism in the
diabatic representation, GFSH showed a dramatic qualitative improvement over FSSH.
GFSH closely followed the exact results in the superexchange regime, while FSSH gave
zero transmission.[22]
Auger processes fall within the realm of superexchange in the diabatic representa-
tion since transitions into intermediate, higher-energy states are involved. Both surface
hopping strategies have been applied to study the Auger mechanism in a quantum
dot.[22] The Auger process is a two-particle reaction, involving simultaneous electron
and hole transitions. As a consequence of its ability to permit two-particle transfer in
a single time step, GFSH gave a faster estimate of the Auger rate. FSSH was hindered
by the relatively large energy barriers of the intermediate states, so classically forbidden
transitions were frequently encountered, producing a slower, less accurate estimate of
the Auger rate.
Internal consistency ensures that the fraction of trajectories on each potential en-
ergy surface is equivalent to the corresponding average probability determined by a
coherent propagation of quantum amplitudes. Both FSSH and GFSH satisfy internal
consistency.[22] This property is violated, however, when trivial crossings are encoun-
tered,[81–86] which occur when the energy gaps between potential energy surfaces are
negligible. As a result, the finite time step cannot ensure an accurate evaluation of the
surface hopping probabilities. A solution to this problem has been proposed, where
probabilities are corrected by a self-consistency check.[82] GFSH automatically satisfies
this criteria, so unlike FSSH, it needs no additional correction.[22]
Until now, it has not been shown whether GFSH satisfies detailed balance at thermal
equilibrium. It is well known that when total quantum-classical energy conservation is
not accounted for, a quantum system in contact with a classical bath does not attain
the correct equilibrium behavior, but rather transitions between states occur with equal
probability, i.e., a system at infinite temperature.[87] Moreover, the self-consistent field
(Ehrenfest) method also fails at thermal equilibrium, strongly limiting its applicability.
In fact, studies have shown that Ehrenfest deviates dramatically from Boltzmann when
k
B
T is much smaller than the energy gap in a two-level system.[25] This is not the case
for surface hopping, which does attain the Boltzmann populations,[25] thereby making
it a more reliable method for mixed quantum-classical simulations at thermal equilib-
rium. GFSH is equivalent to FSSH for two-state systems, so it automatically attains the
2.2. Equations of Motion 11
correct equilibrium behavior in this limit. In this chapter, using a three-level superex-
change model, we test GFSH (and compare it to FSSH).
Furthermore, we compare relaxation rates due to an abrupt change in temperature
using both methods. Relaxation dynamics have been a common interest within the
last couple decades; an example is vibrational energy relaxation, where an excited vi-
brational mode releases excess energy to a bath of intermolecular and/or intramolecu-
lar accepting modes.[88] This phenomena has been studied in context of the hydrogen
stretch in a moderately-strong hydrogen-bonded complex dissolved in polar liquid.[88]
2.2 Equations of Motion
2.2.1 Electronic Dynamics
Without loss of generality, we consider a quantized electron, coupled to its classically
governed atom. The combined Hamiltonian is,
H =
P
2
2m
+ H
e
(r, R) , (2.1)
where r and R represent the electronic and nuclear coordinates, respectively, and P, the
nuclear momentum. The time-dependent Schrödinger equation is,
i¯ h
¶y(r, R)
¶t
= H
e
(r, R)y(r, R) . (2.2)
The wavefunctiony(r, R) can be expanded in terms of an orthonormal basis set

f
j
(r, R)

,
y(r, R)=
å
j
c
j
f
j
(r, R) , (2.3)
where c
j
is the complex-valued amplitude associated to basis function f
j
(r, R). Substi-
tuting Eq. (2.3) into Eq. (2.2), multiplying on the left by f
i
(r, R), and integrating over
all electronic coordinates gives,
i¯ h ˙ c
i
=
å
j
c
j

V
ij
i¯ h
˙
R d
ij

, (2.4)
where V
ij
=hf
i
(r, R)j H
e
(r, R)jf
j
(r, R)i
e
and d
ij
=hf
i
(r, R)jr
R
jf
j
(r, R)i
e
. The latter
is the nonadiabatic coupling vector. State probabilities c
i
c

i
are determined by integrat-
ing Eq. (2.4). Using Eq. (2.4),
i¯ h ˙ r
ij
=
å
l
r
lj

V
il
i¯ h
˙
R d
il

r
il

V
lj
i¯ h
˙
R d
lj

, (2.5)
wherer
ij
= c
i
c

j
. Deriving Eq. (2.5) requires the use of,
d

ij
=d
ji
(2.6a)
and
d
ii
= 0, (2.6b)
12 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
which are simply consequences of the orthonormality of the basis set. From Eq. (2.5),
the time-evolution of state probabilities directly follows,
˙ r
ii
=
å
j6=i
b
ij
, (2.7)
where b
ij
= 2¯ h
1
Im

r

ij
V
ij

2Re

r

ij
˙
R d
ij

.
2.2.2 Nuclear Dynamics
As shown in Eq. (2.1), the electronic Hamiltonian H
e
(r, R) acts as the potential
energy of the atom. The atom evolves on the potential energy surface associated to the
occupied state of its quantum subsystem. When the electron is in state i, the atom is
governed by,
˙
P
j
=
¶V
ii
¶R
j
, (2.8)
where j is a cartesian coordinate. In the adiabatic representation, V
ii
is the i
th
eigenvalue
of H
e
(r, R). The occupied state of the quantum subsystem at each consecutive time step
is determined by the surface hopping algorithm used in the simulation. Eq. (2.8) must
be modified if the system were subjected to any additional nuclear forces.
2.3 Surface Hopping
2.3.1 Fewest Switches Surface Hopping
At every time step, the probability of a transition from state i to j for i6= j is calcu-
lated,
g
ij
= max

Dt b
ij
r
ii
, 0

, (2.9)
whereDt b
ij
is the flux of population, transferred from state i to j in timeDt. A switch
from state i to k is invoked when,
j=k1
å
j=1
g
ij
< x <
j=k
å
j=1
g
ij
, (2.10)
wherex is a random number between 0 and 1. Whenx >å
j
g
ij
, the electron remains in
its current state.
If the electron transitions to a state of different energy, the nuclear velocity is rescaled
to conserve energy. Rescaling typically occurs in the direction of the nonadiabatic cou-
pling vector,[5]
˙
R(t+Dt)=
˙
R(t)g
ij
d
ij
m
, (2.11)
whereg
ij
is the scaling factor determined by energy conservation,
g
ij
=
(
g
+
ij
for
˙
R d
ij
< 0
g

ij
for
˙
R d
ij
> 0
, (2.12a)
2.4. Model 13
where
g

ij
=
m
d
2
ij
0
@ ˙
R d
ij

s

˙
R d
ij

2
2
d
2
ij
m
De
ij
1
A
(2.12b)
and
De
ij
= V
jj
(R) V
ii
(R) . (2.12c)
Eq. (2.11) ensuresD
˙
R is in the direction of d
ij
ord
ij
; the sign is determined by Eq.
(2.12a). In simple terms, forDe
ij
< 0,D
˙
R is along the direction of nonadiabatic coupling
that increases
˙
R, and vice versa forDe
ij
> 0. If
1
2
m

˙
Rd
ij
d
ij

2
< De
ij
, the quantity under
the square root, in Eq. (2.12b), is negative and so the switch is not invoked and
˙
R is
unchanged. This is known as a classically forbidden transition. The process is repeated
for as many time steps needed to fulfill the objective of the study.
2.3.2 Global Flux Surface Hopping
At each time step, the change in state population is calculated,
Dr
ii
= r
ii
(t+Dt)r
ii
(t) . (2.13)
The states are then classified into one of two groups; one with reduced population
(group A) and the other with increased population (group B). Only state transitions
from group A to B are considered. The probability of a switch from state i2 A to j2 B
is,
g
ij
=
Dr
jj
r
ii
Dr
ii
å
k2A
Dr
kk
. (2.14)
The probability increase of state j,Dr
jj
, is attributed to the reduction of probability from
all states in group A. To limit this quantity to the exiting states’s contribution, i.e., state
i, one multiplies by
Dr
ii
å
k2A
Dr
kk
, which is the ratio of state i’s probability change to the total
probability change in group A. Finally, Eq. (2.14) is normalized by the probability of
the exiting state. For two-level systems, Eq. (2.14) reduces to the FSSH algorithm in
Eq. (2.9). All subsequent steps in FSSH, i.e. random number generation and velocity
rescaling, are then implemented for GFSH.
2.4 Model
Numerical calculations are carried out, with an N-atom linear chain, governed clas-
sically via the nearest-neighbor, attractive Morse potential,[89]
V =
N
å
i=1
V
M
(R
i
R
i+1
) , (2.15a)
where
V
M
(q)= V
0

a
2
R
2
a
3
R
3
+ .58a
4
R
4

(2.15b)
14 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
and R
i
is the position of the i
th
atom. The term R
N+1
is held constant. A three-level
superexchange model, V
ij
, is coupled to atom 1,
H
1
=
P
2
1
2m
+ V
M
+ V
ij
. (2.16)
The electronic state amplitudes evolve according to Eq. (2.4), with constant state ener-
gies, i.e., V
ii
, and constant nonadiabatic coupling vectors, shown in Table 2.1a. Atom N
is governed by the Langevin equation,
˙
P
N
=
¶V
¶R
N
gm
˙
R
N
+ F
R
, (2.17)
where g is the Langevin friction parameter and F
R
is a Gaussian random force with
mean zero and standard deviation
q
2gmk
B
TDt
1
,[90] with k
B
being the Boltzmann
constant. The integration time step,Dt, is chosen to be 0.01 fs. All parameters regarding
the linear chain are shown in Table 2.1b. The coefficients in Eq. (3.3b) represent those
of an attractive Morse potential. The purpose of Langevin dynamics is to introduce a
heat bath, from which the quantum subsystem can equilibrate to the Boltzmann state
populations,
population
i
=
exp(e
i
/k
B
T)
å
i
exp(e
i
/k
B
T)
, (2.18)
where e
i
is the energy of the i
th
state. States 1 and 3 are not coupled since V
13
=
V
31
= 0 and d
13
= d
31
= 0.
1
An intermediate hop to state 2 is required to connect states
1 and 3. GFSH, however, allows for direct transitions.
2.5 Results and Discussion
2.5.1 Detailed Balance
The fraction of trajectories on each state has been calculated at every time step, as
the system approaches thermal equilibrium. All trajectories were 50 ps (5 10
6
time
1
The model was constructed to inhibit single-step transitionsj1i!j3i using FSSH. It remains an open
question if such a model can exist in the adiabatic representation.
TABLE 2.1: Simulation Parameters
Parameter Value Units
V
11
0.0 a.u.
V
22
0.01 a.u.
V
33
0.005 a.u.
d
12
-6.5 a.u.
d
13
0.0 a.u.
d
23
8.0 a.u.
(A) Quantum subsystem.
Parameter Value Units
N 20
m 12 Da
V
0
175 kJ/mol
a 4 Å
1
g 10
14
s
1
(B) Classical linear-chain.
2.5. Results and Discussion 15
FIGURE 2.1: Diagram of the quantum-classical model. Atom 1 (leftmost)
is coupled to the quantum subsystem, in which, states 1 and 3 are uncou-
pled. Atom 20 (rightmost) is governed by a Langevin process. Nuclear
motion is driven by a nearest-neighbor, attractive Morse potential.
steps) in length, with the quantum subsystem initially prepared in the ground state. To
show detailed balance, the fraction of trajectories were averaged every 1 ps (10
5
time
steps). Averaged results are referred to as state populations. Differential equations were
integrated using fourth-order Runge-Kutta (RK4).[91] The stochastic Langevin equation
of Eq. (2.17) was also solved at the RK4 level.[81, 92] A sample, at 500 K, of the time-
dependent state populations is shown in Fig. 2.2.
All transitions within FSSH require an intermediate switch to the highest-energy
state. A switch is prohibited when energy conservation is not satisfied. This is evi-
denced in multiple locations, shown in Fig. 2.2a, where state 3 is overpopulated for
relatively long time compared to the 1 ps sampling interval. As a result, detailed bal-
ance is not accurately attained. Fig. 2.2c shows the corresponding number of classically
forbidden transitions, while occupying either state 1 or 3. There is a clear correlation
between Figs. 2.2a and 2.2c; as the population of state 1 or 3 increases, the number of
forbidden transitions increases as well. The magnitude of the y-axis in Fig. 2.2d is two
orders smaller than that shown in Fig. 2.2c; so even the slightest increase in forbidden
transitions could affect detailed balance.
By equipartition, nuclear kinetic energy has mean value k
B
T/2 at thermal equilib-
rium. At 500 K, k
B
T/2 is about 13 times smaller than the largest energy gap. Therefore,
it is common for state 3 to be overpopulated, until enough dispensable kinetic energy
allows for a transition to state 2. A similar situation arises when state 1 is occupied.
GFSH bypasses the intermediate, highest-energy state; hence, transitions are less
restricted by energy conservation, particularly between states 1 and 3, as shown in Fig.
2.2d. Unlike FSSH, which requires state 1 to access state 3 by way of state 2, GFSH allows
for a direct transition, requiring only V
33
V
11
= .005 a.u., as opposed to V
22
V
11
= .01
a.u. Moreover, transitions from state 3 to 1 are not restricted by energy conservation. In
Fig. 2.2b, state populations are steadily maintained after 30 ps, and with less-apparent
occurrences of overpopulating states 1 and 3. GFSH would perform even better than
FSSH for smaller sampling intervals. Thus, for superexchange systems, GFSH is a more
accurate description of, not only, thermal equilibrium, but also physical processes that
occur on short time-scales (where both methods give significantly different results).
Detailed balance has been tested at multiple temperatures, the highest being 2000
16 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
æ
æ
æ
æ
æ
æ
æ
ææææ
æææ
ææææ
ææææ
ææ
æ
ææ
æææ
æææææ
ææææææææææ
ææææææææ
ææ ææ
ææææææææææææææææææææææææææ
ææææææææææ
ææ
æææææ
æ
æ
æ
ææ
æææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
æ
æ
æ
æ
æææ
ææææ
æææ
ææææ
ææææ
ææ
æ
ææ
æææ
æææææ
æææææææ
æææ
ææææææææ
ææææ
ææææææææææææææææææææææææææ ææææææææææ
æææææææ
æ state 1
æ state 2
æ state 3
(a)
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
time HpsL
population
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
ææææ
æ
ææææ
æ
æ
ææææææææææææ
æ
æææ
æææ
æ
æ
ææ
æ
ææææ
æ
ææææææææææææææ
ææææææææææææ
æ
æ
æææææææææ
æ
æ
æ
æ
æ
ææ
ææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææææ
æ
æ
æ
ææ
æ
æ
æ
æ
ææ
æ
ææ
ææ
æ
æ
ææ
æ
æ
æææææ
ææææ
æ
æ
æææ
ææææ æææææææææ
æææ
æ
æææ
æ
ææææ
æ
æææææææææææ
æææ
ææææææææææææ
æ
æ
æææææææææ
æ
æ state 1
æ state 2
æ state 3
(b)
0 10 20 30 40 50
0.0
0.2
0.4
0.6
0.8
1.0
time HpsL
population
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
ææ
æ
æ
æææ
æ
æ
æ
ææ
æ
æ
æææ
æ
æ
æ
ææ
æææ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æææ
æ
æ
æ æ
æ
ææ
æ
æ
æææ
æ
æ
æ
ææææ
æ
æ
ææ
æ æ
ææ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
ææ
æ
æ
æ
æææ
æ
æ
ææææææææææææ
æææ
æ
ææææ
ææææ
æ
æææææ
ææææææææææææææææ
ææææææææææææææææææææææææææææææææææææææææ
ææææææææ
ç
ç
çç
ç
ç
ç
ççç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
çç
ç
çç
ç
ç
ççç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
ç
çç
ç
ç
ç
çç
ç
ç
ç
ç
ç
çç
ç
çç
çç
ç
ç
ç
ç
ç
ç ç
ç
çç
çç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
çç
ç
çç
ç
ç
ç
çç
ç
ç
ç
çç
ç
ç
ç
çç
ç
ççççç çç
çççççç
ç
ç ç
ç
ç
ç
ç
ççç
ççççççççççççççççççç
çççç
ççç
çç
ç
ççç
ççççç
ç
ç
çççççççç ççç
çç
çççççççç
ç
ç
ççç
ç
æ state 1
æ state 3
ç state 1
ç state 3
FSSH GFSH
(c)
0 10 20 30 40 50
0
5
10
15
20
time HpsL
ð of forbidden transitions
æ
æ
æ
æ
æ
æ
ææææ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
ææ
æ
æ
æ
æ
æææ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
ææ
æ
æææ
æ
æ
æ
æ
ææ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
ææ
ææ
æ
æ
æ
ææ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
ææ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
ææ
ææ
æ
æ
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
ç
ç
ç
ççç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç ç
çççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççççç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
çç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
ç
çç
ç
ç
ç
ç
ç
ç
ç
ç
çç
çç
ç
ç
ç
ç
ç
ç
ç
ç
æ 1®2
æ 1®3
æ 3®2
ç 1®2
ç 1®3
ç 3®2
FSSH GFSH
(d)
0 10 20 30 40 50
0.00
0.05
0.10
0.15
0.20
time HpsL
ð of transitions
FIGURE 2.2: State populations at 500 K, from a 300-trajectory ensemble,
using (a) FSSH and (b) GFSH. Solid horizontal lines are Boltzmann pop-
ulations at thermal equilibrium given by Eq. (2.18). The arrows in (a)
identify when populations deviate from Boltzmann. The corresponding
number of forbidden transitions, while state 1 or 3 is occupied, is shown
in (c). The corresponding number of implemented transitions is shown
in (d). Only transitions into states of higher energy are shown in (d). The
data points are presented with a 0.5 ps interval.
2.5. Results and Discussion 17
K, which gives k
B
T/2 about 3 times smaller than the largest energy gap. The differ-
ence in the number of classically forbidden transitions, between FSSH and GFSH, in-
creases as temperature increases. At thermal equilibrium, kinetic energy is sampled
from a Maxwell-Boltzmann energy distribution (for a one-dimensional particle), which
is monotonically-decreasing; therefore, the probability of .005 a.u. < E < .01 a.u., in-
creases as k
B
T/2 increases. This effect becomes less important towards detailed balance,
however, since the number of implemented and forbidden transitions converge toward
similar orders of magnitude.
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ
æ
æ
æ FSSH
æ GFSH
(a)
0.0005 0.0010 0.0015 0.0020
0.5
0.6
0.7
0.8
0.9
1.0
temperature
-1
HK
-1
L
population
æ æ æ
æ
æ
æ
æ
æ
æ
æ
æ æ æ
æ
æ
æ
æ
æ
æ
æ
(b)
0.0005 0.0010 0.0015 0.0020
0.00
0.05
0.10
0.15
temperature
-1
HK
-1
L
population
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
(c)
0.0005 0.0010 0.0015 0.0020
0.00
0.05
0.10
0.15
0.20
0.25
temperature
-1
HK
-1
L
population
æ
æ æ æ
æ æ
æ
æ
æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ
æ
æ
æ æ æ
æ æ
æ
æ
æ
æ
æ state 1
æ state 2
æ state 3
(d)
0.0005 0.0010 0.0015 0.0020
0
2000
4000
6000
8000
temperature
-1
HK
-1
L
N
GFSH
FIGURE 2.3: Equilibrated population of states (a) 1, (b) 2, and (c) 3, at
various temperatures, where each data point is an average of the last 20
ps of a 300-trajectory ensemble. Dashed black lines in (a) through (c)
are Boltzmann populations given by Eq. (2.18). Figure (d) shows the
required-number of GFSH data points to match the statistical accuracy of
FSSH, where N
FSSH
= 6000.
In Fig. 2.3, each state population was averaged over the last 20 ps, to neglect any
dependence on initial conditions.
2
300 trajectories were generated for each tempera-
ture. Data were recorded every 1 ps, and N = 6000 data points were used to obtain the
averages, i.e., 20 data points per each of the 300 trajectories. The variance in the mean
(m) is s
2
m
= s
2
/N, where s
2
is the variance in the sample.[94] Due to the large sample
size, these errors are insignificant. Nonetheless, GFSH shows less variance and better
convergence than FSSH.
Figs. 2.3a through 2.3c show GFSH agreeing more with Boltzmann overall. This is
particularly true for low temperatures, where paths to states 2 and 3 are more restricted
by energy conservation. Fig. 2.3d shows the number of data points, N
GFSH
, required
2
Initial conditions of the linear chain were sampled from probability distributions using the Hamilto-
nian of a quantized one-dimensional harmonic oscillator.[93] It is worth mentioning, however, that initial
conditions generally have no impact on the properties of equilibrium discussed in this study.
18 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
by GFSH to match the statistical accuracy of FSSH achieved with N
FSSH
= 6000 points.
N
GFSH
was chosen based on the criterion,s
2
GFSH
/N
GFSH
= s
2
FSSH
/N
FSSH
. For states 1 and
3 involved in superexchange, the statistical error in GFSH is consistently smaller than in
FSSH. The errors of state 2 are similar; a consequence of the fact that state 2 is unaffected
by classically forbidden transitions. Overall, GFSH attains the same statistical accuracy
as FSSH with fewer trajectories.
Although the discrepancy is small, surface hopping does not exactly match Boltz-
mann. States 2 and 3 are consistently overpopulated, while state 1, is underpopu-
lated. This effect was previously studied for a two-state system, where the ratio of
attempted upward and downward transitions, k
ab
/k
ba
, was found to be greater than
1, and the fraction of accepted upward transitions was less than the Boltzmann factor,
exp(D/k
B
T), whereD = e
b
e
a
> 0. [26] Detailed balance is defined as the product
of k
ab
/k
ba
and the fraction of upward transitions; therefore, unless the amount overesti-
mated by k
ab
/k
ba
exactly cancels the amount underestimated by exp(D/k
B
T), results
would not yield Boltzmann. Analytic expressions argue that the correct equilibrium
behavior is attained in specific regions: small adiabatic splitting and/or strong nonadi-
abatic coupling.[26] Numerical results show that when the (magnitude of) nonadiabatic
coupling d
ab
was increased from 6.5 Å to 104 Å, the ratio of the excited and ground
state populations more closely resembled exp(D/k
B
T).[25, 26] It is expected that the
same holds true for GFSH. Even outside these limits, FSSH (and GFSH) are likely to
introduce relatively small deviations, while still reproducing the expected exponential
decay of the excited state populations as the temperature of the system is reduced, as
shown in Figs. 2.3b and 2.3c.
2.5.2 Equilibration Rate
Detailed balance is a static assessment of the two surface hopping methods. In this
section, their differences are highlighted from a dynamical perspective, by comparing
rate of equilibration. The temperature was initially set to 1700 K and as before, the quan-
tum subsystem was fully prepared in the ground state. After equilibrium, at approxi-
mately 20 ps, the temperature was abruptly changed to 600 K. A total of 600 trajectories
were used, each 50 ps in length, and the sampling interval was set to 0.05 ps.
The least-squares method was used to fit the data to an exponential function of the
form,
A+ B exp(kt) , (2.19)
where A is the final state population, A+ B is the initial population, and k is the equili-
bration rate. The constants A and B were fixed to the populations given by Eq. (2.18);
thus, the only variable in the fit is k, in order to draw a comparison from the rate of equi-
libration only, as opposed to also determining whether or not the subsystem reaches the
Boltzmann populations, which has been studied in section 2.5.1. Results are shown in
Fig. 2.4. The x-axis is offset to start at the time at which temperature was changed to
600 K.
Not surprisingly, GFSH equilibrates faster than FSSH. This again is due to the unsur-
passable energy barriers encountered by FSSH. GFSH is more free to transition, espe-
cially between states 1 and 3. Fig. 2.4 shows GFSH more consistent with the Boltzmann
populations, while FSSH experiences multiple “oscillations," similar to those observed
in Fig. 2.2a. We emphasize that increasing the number of trajectories does not eliminate
2.6. Conclusions 19
æ æ æ
æ
æ æ æ
æ æ æ æ æ
æ
æ æ
æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ
æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
k= .56 ±.01 ps
-1
k= .36 ±.01 ps
-1
k= 2.0 ±.05 ps
-1
æ state 1
æ state 2
æ state 3
(a)
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
time HpsL
population
æ æ æ
æ
æ
æ
æ
æ
æ æ
æ
æ
æ æ æ æ æ
æ æ æ
æ æ
æ æ æ æ æ æ æ æ æ
æ æ
æ æ
æ æ æ æ
æ æ æ æ æ
æ
æ æ æ æ
æ
æ
æ æ
æ æ æ æ æ
æ æ æ
æ æ æ æ
æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ
æ
æ æ æ æ æ æ
æ æ æ
æ æ
æ æ æ æ æ æ æ æ
æ æ
æ æ æ
æ æ æ
æ æ æ æ æ
æ
æ
æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ
æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ
æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ
æ æ æ
æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ
æ
æ
æ æ æ æ æ
æ æ æ æ æ
æ
æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ
æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ
æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ
æ
æ æ
æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ
æ æ æ
æ
æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ
æ æ æ
æ
æ æ
æ æ æ
æ
æ æ
æ æ æ æ
æ æ æ
æ æ
æ æ æ æ æ æ æ æ æ
æ æ
æ æ
æ æ æ æ
æ æ æ æ æ
æ
æ æ æ æ
æ
æ æ æ
æ æ æ
æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ
æ
æ æ æ æ æ æ
æ æ
æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ
æ
æ æ
æ æ
æ æ æ æ æ
æ æ æ æ æ æ æ
æ
æ æ
æ
æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ
æ
æ æ
æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ
æ
æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ
æ æ
æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ
æ
æ æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ
æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ
æ
æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ æ æ æ
æ æ æ æ
æ
æ æ
æ æ æ æ
æ æ æ
æ æ æ æ æ æ æ æ æ
æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
æ æ æ æ æ æ æ æ æ
æ æ
k= 1.3 ±.04 ps
-1
k= .97 ±.03 ps
-1
k= 2.4 ±.06 ps
-1
(b)
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
time HpsL
population
FIGURE 2.4: State populations at 600 K, from a 600-trajectory ensemble,
using (a) FSSH and (b) GFSH. Solid horizontal lines are the Boltzmann
populations. The quantum-classical system was prepared, in equilib-
rium, at 1700 K. Data was fit to Eq.(2.19), shown by dashed black lines.
this inconsistency. The fluctuation shown in Fig. 2.4b, however, is likely to diminish
with a larger number of trajectories. The difference in equilibration between both meth-
ods continues to diverge as temperature is reduced.
Equilibration obtained by GFSH is not only faster, but also more accurate than FSSH.
Irrespective of the model, Boltzmann must be reproduced. GFSH agrees more with
these populations overall, indicating that the path taken to reach thermal equilibrium
is better explained by GFSH. The significance of this can be illustrated in the context of
charge mobility. For a one-dimensional lattice with static disorders and only nearest-
neighbor electronic coupling, charge transport occurs with many superexchange-like
transitions. GFSH would give a faster rate of population transfer than FSSH, thereby
producing greater charge mobility. These studies are currently underway.
2.6 Conclusions
We showed that GFSH attains the Boltzmann populations at thermal equilibrium.
The calculations were performed in the adiabatic basis with a numerically-chosen Hamil-
tonian.
1
The energies, of the three-level superexchange model, and nonadiabatic cou-
plings were chosen to be independent of the classical coordinate, signifying a model
with weak electron-phonon coupling.
Unlike FSSH, GFSH allows for direct transitions between any two states. It more
accurately attains the correct equilibrium behavior, and with less error, especially for
when the number of implemented transitions is less than the number of classically for-
bidden transitions. In general, GFSH consistently gives better results at temperatures
where k
B
T/2 is less than the largest energy gap. Access to the excited states is more
restricted by FSSH, since larger amounts of dispensable energy is required.
GFSH exhibits a larger number of transitions than FSSH, making population flux
more dynamical. As a result, thermal equilibrium is approached faster with GFSH.
GFSH is particularly useful in describing non-equilibrium, superexchange processes,
which widely occur in physical, chemical, and biological contexts. The distinction be-
tween both methods diminishes as temperature increases.
It would be valuable to study models that further accentuate the differences between
both methods. For instance, one can increase the energy of the intermediate state, V
22
,
20 Chapter 2. Mixed Quantum-Classical Equilibrium in Global Flux Surface Hopping
while making V
11
and V
33
nearly degenerate. In this case, the transitions between states
1 and 3 would be greatly affected by energy conservation within FSSH, which may re-
sult in inaccurate equilibrium behavior.
To recapitulate, the novel GFSH method satisfies detailed balance and leads to ther-
mal equilibrium. GFSH has advantages over FSSH in describing the superexchange
mechanism.[22] It reduces to FSSH in the two-state limit.[22] Available evidence indi-
cates that GFSH can fully replace FSSH, especially for studies at thermal equilibrium.
Investigations are underway to determine the effect of incorporating decoherence. A
common approach that utilizes “a stochastic wave function collapse" is Augmented-
FSSH.[35] It would be beneficial to test this method within the GFSH framework to
determine its degree of applicability and performance. Decoherence is important in ex-
plaining relatively slow transitions between states with significant energy gaps.[34, 39,
40, 95, 96]
21
Chapter 3
Momentum Reversal Following
Classically-Forbidden Electronic
Transitions Improves Detailed
Balance in Surface Hopping
Reprinted with permission from J. Chem. Phys. 144, 211102, (2016). Copyright 2018
American Institute of Physics.
3.1 Introduction
Since the 1990 paper of John C. Tully,[5] surface hopping has become a major field
both in its development and use for the study of nonadiabatic processes from single
molecules all the way to bulk materials.[6, 7] As opposed to the mean-field method,[97,
98] surface hopping propagates the nuclear coordinate on a single surface correspond-
ing to an active electronic state. Stochastic hops depend on population flux, which is
the most important feature of the method. If hops were population-based,[99, 100] a
trajectory could follow a mean-field due to excessive transitions. While this may be
reasonable for deexcitation processes in materials with a high density of states, it is
incorrect for processes such as scattering, [10] photoisomerization, [101–103] and pho-
toionization[104] where motion on a single asymptotic surface is the underlying mecha-
nism. The mean-field method also fails to realize the correct detailed balance at thermal
equilibrium[24, 25] and the hopping mechanism of charge transport. [105]
Surface hopping is a semiclassical theory. Major setbacks include zero-point motion
and nuclear tunneling. Recent progress has been made to correct for overcoherence[32,
34, 35, 37, 75, 106] and second- and higher-order transitions.[22, 107–109] Despite the
progress, most corrections apply to only a small subset of possible physical situations.
Moreover, the fact that surface hopping does not stem from any rigorous theory without
major assumptions[110] renders its domain of validity imprecisely-defined. We study
the treatment of classically-forbidden (or frustrated) hops in the context of system-bath
interactions.
Unlike mean-field, surface hopping approximately attains detailed balance.[25] In
model studies, the quantum subsystem is coupled to a bath and equilibrates to the
22
Chapter 3. Momentum Reversal Following Classically-Forbidden Electronic
Transitions Improves Detailed Balance in Surface Hopping
Boltzmann populations, albeit not rigorously. In fact, it is consistently shown that ex-
cited states are overpopulated, but that the deviation from Boltzmann diminishes in the
case of small adiabatic splitting and/or strong nonadiabatic coupling.[26, 28]
Detailed balance is attributed to the conservation of total quantum-classical energy.
Although population flux may give rise to a target state, a transition cannot take place
unless the classical coordinate has enough kinetic energy to surmount the energy bar-
rier. This is known as a frustrated hop. Within fewest-switches surface hopping[5]
(FSSH), a stochastic parameter, derived from the time-dependent Schrödinger equation,
governs transitions. At thermal equilibrium, detailed balance is attained when (1) the
ratio of attempted upward-to-downward transitions k
ab
/k
ba
is unity, and (2) the fraction
of accepted upward transitions is the Boltzmann factor, exp(D/k
B
T), whereD > 0 is
the energy gap betweenjai andjbi. Frustrated hops are somewhat of a paradox; on one
hand, they violate the self-consistency requirement ensuring that the fraction of trajec-
tories on each surface correspond to state populations, while on the other, they lead to
nearly-correct detailed balance.
The topic of frustrated hops, how to treat them, and their effects, has been discussed
in only a handful of papers.[111–113] A method called fewest-switches with time un-
certainty (FSTU)[112] allows a trajectory that experiences a frustrated hop to hop non-
locally to a region on the upper surface that is energetically accessible so long as the
hopping point is within the time interval obtained by the time-energy Heisenberg un-
certainty relation. Overall, FSTU improved electronic-state distributions when tested
against a series of scattering processes with an electronically-excited atom and a di-
atomic molecule. The treatment of nonlocal frustrated hops however, warranted further
investigation.
A benchmarking of two prescriptions after frustrated hops was employed: leaving
the component of nuclear velocity, along the direction of nonadiabatic coupling, in the
same direction (+) or reversing () it. Findings showed that FSSH (+) and FSTU (+)
are more accurate in predicting the average rotational and vibrational quantum num-
bers of the product diatomic molecule, whereas FSSH () and FSTU () predict more
accurate nonadiabatic transition probabilities and branching ratios.[112, 113] A combi-
nation of these two prescriptions led to therV prescription.[111] In this method, the
components of momentum on the current surface and force on the target surface, along
the direction of nonadiabatic coupling, are computed. A positive dot product leaves
the nuclear velocity unchanged (+), and if negative, an impulse retards the trajectory,
and the () treatment is used. Extensive studies concluded that FSTUrV is the most
successful in the series for trajectory-based surface hopping.
In this chapter, we study FSSH(+) and FSSH(), and find that detailed balance is
more accurately attained with FSSH(). We show that some randomization in the nu-
clear dynamics via reversal drives the system closer to the thermodynamic limit. This
work addresses the concern that surface hopping falls short of detailed balance, and
demonstrates that a simple correction can lead to a major improvement.
3.2. Results and Discussion 23
3.2 Results and Discussion
3.2.1 Recap of Fewest-Switches Surface Hopping
Within FSSH, populations evolve in the adiabatic representation according to
˙ r
ii
=
å
j6=i
b
ij
, (3.1)
where b
ij
=2Re

r

ij
˙
R
SH
d
ij

is the population flux fromjii tojji, R
SH
is the classically-
governed nuclear coordinate, and d
ij
is the nonadiabatic coupling vector. A switch from
surfacejii tojji can take place when
j=k1
å
j=1
g
ij
< x <
j=k
å
j=1
g
ij
, (3.2a)
where
g
ij
= max

Dt b
ij
r
ii
, 0

(3.2b)
is the stochastic parameter and x is a uniform random number between 0 and 1. If
x > å
j
g
ij
, the system remains on the current surface. If a transition takes place, the
nuclear velocity is rescaled along the nonadiabatic coupling vector.[114]
3.2.2 Two-Level Subsystem
We use a variation of a previously-published two-level[25] model with energy gap
D 13 kJ/mol and nonadiabatic coupling d
ab
= 6 Å
1
. The two-level system is
coupled to a single atom in a 20-atom linear chain, each of mass m = 12 amu. The
atom farthest from the quantum subsystem is governed by the Langevin equation with a
Gaussian random force of zero mean and widths =
p
2gmk
B
T/Dt, whereg= 10
14
s
1
is a friction parameter, T is temperature, andDt is the simulation time step. Nearest-
neighbor interaction is given by the attractive Morse potential,
V =
N
å
i=1
V
M
(R
i
R
i+1
) , (3.3a)
where
V
M
(q)= V
0

a
2
R
2
a
3
R
3
+ .58a
4
R
4

. (3.3b)
Here, V
0
= 175 kJ/mol and a= 4 Å
1
.
Simulations were carried out with 300 trajectories, each 50 ps in length, and with
a time step Dt = .01 fs. All dynamical equations were integrated with fourth-order
Runge-Kutta. Populations are averaged over the last 30 ps. A comparison against Boltz-
mann for temperatures ranging from 350 K to 2500 K is shown in Fig. 3.1.
The distinction between reversing and not reversing is small, but noticeable nonethe-
less (Fig. 3.1). Stronger deviations from Boltzmann have been reported whenD is dou-
bled.[25] Here, we use a smaller gap to validate the claim that Boltzmann populations
24
Chapter 3. Momentum Reversal Following Classically-Forbidden Electronic
Transitions Improves Detailed Balance in Surface Hopping
FIGURE 3.1: Equilibrated populations for a two-level quantum subsys-
tem with nuclear velocity reversed and not reversed after a frustrated
hop. (A) State 1 and (B) State 2. Solid line is Boltzmann. (C) Error in State
2 population calculated from surface hopping, relative to Boltzmann.
are more likely realized for small adiabatic splitting.[26, 28] This is true, but what is
most remarkable is that, in spite of already excellent results, reversing the velocity im-
proves detailed balance over nearly the entire range of temperatures, particularly at low
temperature, where relative error is decreased by almost 20 percent (Fig. 3.1(C)).
3.2.3 Three-Level Subsystem
To accentuate deviation from Boltzmann, we use a three-level superexchange model.
Superexchange constitutes a class of systems where two uncoupled low energy states
are connected by an intermediate high-energy state. FSSH lacks superexchange (or
single-step transitions between uncoupled states) because population flux is based on
state-to-state transfer. Recent studies have shown this for calculations in the diabatic
representation.[22] Furthermore, the Auger mechanism, which is a two-particle reac-
tion involving simultaneous electron-hole transitions, is a form of superexchange. FSSH
significantly underestimates the transfer rate in a semiconductor quantum dot where
Auger relaxation is important.[22] The Auger channel of electron relaxation with hole
traps has also been recently studied for a CdSe quantum dot.[115]
In the superexchange model, energies and nonadiabatic couplings are e
1
= 0, e
2

39, e
3
13 kJ/mol and d
12
=6, d
23
= 8, and d
13
= 0 Å
1
, respectively.
1
Clearly,
j2i is the intermediate high-energy state and,j1i andj3i are uncoupled. The model is
challenging for FSSH, because transitions only occur between directly-coupled states,
as seen in b
ij
of Eq. (3.1). Therefore, to occupyj3i, the system must first make the
transitionj1i ! j2i, followed byj2i ! j3i. FSSH is bound to undergo a two-step
process to populatej3i, making detailed balance more difficult to attain than in the two-
level model. We expect to see stronger deviation from Boltzmann, since dispensable
kinetic energy may not overcome the energy barrierD
21
. Trajectories are 100 ps in length
and populations are averaged over the last 30 ps. Results are shown in Fig. 3.2.
3.2. Results and Discussion 25
FIGURE 3.2: Equilibrated populations for a three-level superexchange
model in (A), (B), and (C). Boltzmann is solid line. (D) Error in State 3
population, relative to Boltzmann.
As expected, FSSH deviates from Boltzmann. At low temperature, relative error is
increased from 20 (Fig. 3.1(C)) to 60 (Fig. 3.2(D)) percent. Similar to other works, excited
states are overpopulated (also seen in Fig. 3.1).[26–28] Reversing velocity significantly
alleviates this problem. A couple questions arise: what effect does reversal have on
attempted upward and downward transitions? Are the number of rejected hops from
low- to high-energy surfaces sampling Boltzmann? The detailed balance condition is
N
b
N
a
=

R
ab
(E
a
)

R
ba

E
b
, (3.4)
where N
a
is the equilibrium population ofjai and R
ab
(E
a
) is the transition rate from
jai tojbi with energy E
a
(nuclear energy). The transition rate fromjai tojbi per unit
time is
P
ab
=
2Re

r

ab
˙
R
SH
d
ab

r
aa
. (3.5)
With energy between E
a
and E
a
+ dE
a
, Eq. (3.5) becomes
R
ab
µ
2Re

r

ab
˙
R
SH
d
ab

r
aa
exp(E
a
/k
B
T)
p
E
a
. (3.6)
26
Chapter 3. Momentum Reversal Following Classically-Forbidden Electronic
Transitions Improves Detailed Balance in Surface Hopping
FIGURE 3.3: (A) Normalized energy distribution of the atom coupled
to the quantum subsystem as a function of state occupation. Maxwell-
Boltzmann energy distribution is shown for comparison. Time-evolution
of state population with (B) no reversal and (C) reversal. Horizontal lines
in (B) and (C) show Boltzmann populations. Temperature is 1600 K.
Eq. (3.6) is the product of Eq. (3.5) and the Maxwell-Boltzmann distribution. Due to
energy conservation, E
a
E
b
= e
b
e
a
=D
ba
, the energy gap, wheree
i
is the adiabatic
energy ofjii. This is what leads to detailed balance so long as the ratio of attempted
upward and downward transitions is unity. The bracketh...i in Eq. (3.4) denotes an
average over a conditional probability distribution, which predominantly depends on
the probability of the current state (r
aa
in Eq. (3.6)). The energy of the atom coupled to
the quantum subsystem is shown in Fig. 3.3(A). PDF defines a normalized distribution,
such that the integral over its entire domain is unity. One may conclude that the inter-
play between the quantum subsystem and its classical environment accurately recovers
the Maxwell-Boltzmann distribution. We will show that this is not true without proper
treatment of frustrated hops, but the qualitative description is important nonetheless.
Figs. 3.3(B) and 3.3(C) show dynamical equilibration at 1600 K. Without reversal,j2i
andj3i are overpopulated. Improvement is seen with reversal (Fig. 3.3(C)).
Next, we consider state populations. A distribution of the target state’s population,
r
bb
, given thatjai is occupied, is shown in Fig. 3.4. We consider a two-level model
with gapD
21
and d
12
of the superexchange model. With reversal, the distributions are
symmetric and centered around .5. Not surprisingly,j1i!j2i transitions are more
affected by reversal due to frustrated hops. Without reversal, attempted transitions
3.2. Results and Discussion 27
to the upper surface are accelerated due to large upper-state population. We are in
agreement with Schmidt, Parandekar, and Tully that the ratio of the attempted upward
and downward [transition] rates is actually greater than 1.[26] We qualitatively show this
sincehr
22
i > .5 given thatj1i is occupied, whilehr
11
i < .5 given thatj2i is occupied.
Implementing reversal suppresses the spurious upper-state population shown in Fig.
3.4(A). In Figs. 3.4(B) and 3.4(D), coherences, which also govern electronic transitions
(as seen in Eq. (3.6)) are similarly-distributed for thej1i!j2i andj2i!j1i transitions.
FIGURE 3.4: Normalized distribution of the target state’s population for
the (A)j1i ! j2i and (C)j2i ! j1i transition, along with respective
coherences in (B) and (D). Temperature is 1600 K.
In the superexchange model, Boltzmann predicts N
2
/N
1
= exp(.015/k
B
T) .052
at 1600 K. The area of the histogram in Fig. 3.5(A) is .052 with reversal and .033 without
reversal. Likewise, for thej3i!j2i transition in Fig. 3.5(B), exp(.01/k
B
T) .139
compared to .138 and .079, respectively. Again, we agree that the fraction of accepted hops
is lower than the expected values, exp(D/k
B
T).[26] Reversal induces a quantum back-
reaction, which homogenizes attempted transitions, and makes the classical coordinate
more Maxwellian. The combination inherently recovers a more Boltzmann-like picture
of thermal equilibrium.
It is worth noting that reversal still shows signs of weakness in the low-temperature
regime (Fig. 3.2), as expected due to the inevitable energy barrier that FSSH faces in
superexchange systems. To put things into perspective, 2500 K gives k
B
T almost 2 times
smaller thanD
21
. Surpassing this barrier becomes a challenge, which explains whyj1i
is slightly overpopulated at low temperatures. Global flux surface hopping[22] general-
izes FSSH to undergo the superexchange process (j1i!j3i in a single time step, over-
coming the energy barrier) and improving both scattering[22] and detailed balance.[27]
GFSH combined with velocity reversal is likely most superior, especially in the low-
temperature regime.
28
Chapter 3. Momentum Reversal Following Classically-Forbidden Electronic
Transitions Improves Detailed Balance in Surface Hopping
FIGURE 3.5: Normalized energy distribution given that the (A)j1i!j2i
and (B)j3i!j2i transitions have been attempted. Temperature is 1600
K.
Method Frustrated j1i!j2i j2i!j1i
FSSH(+) 131668 4351 4354
FSSH() 74586 4075 4074
TABLE 3.1: Total number of frustrated and successfully-invoked hops
within 10 ps at 1600 K using 300 trajectories. Data are for the two-level
model withD
21
and d
12
of the superexchange model.
The effect of frustrated hops on the nonadiabatic thermal rate constant in the spin-
boson model was recently studied.[116] Results suggest that incorrect treatment of frus-
trated hops overestimates the rate in the limit of weak friction and diabatic couplings.
Rates were improved with velocities reversed. In the strong frictional regime, ballistic
transport provides more opportunities for successful hops, and as such, reversing ve-
locity had minimal effect. In the large coupling regime, motion is predominantly along
the lower adiabat, and contributions due to frustrated hops were not significant. We
confirm that an excessive number of frustrated hops compromises dynamics (Table 3.1).
The erroneous effect of frustrated hops is likely reduced in the limit of small adiabatic
splitting and/or large nonadiabatic coupling.
3.3 Conclusions
In summary, classically-forbidden electronic transitions affect dynamics within system-
bath interactions. We find that reversing the nuclear velocity after a frustrated hop
leads to better detailed balance in both two- and three-level (superexchange) models.
Our three-level model is a challenging test for detailed balance since hops must un-
dergo a high-energy intermediate. By adding some randomization via reversal, the
ratio of attempted upward-to-downward transitions approaches unity and the number
3.3. Conclusions 29
of frustrated hops is reduced. Our findings are important for electron-phonon relax-
ation, phonon-assisted transport, superexchange phenomena, phonon-assisted Auger
processes, and other types of electron and exciton dynamics. The choice of a semiclas-
sical theory is motivated by a trade-off between feasibility of implementation and rigor.
Treatment of the classical coordinate within trajectory-based surface hopping merits im-
provement, and this work provides insight for studies at thermal equilibrium. Although
reversal improves detailed balance, more work is needed to determine if it has adverse
effects on dynamics.

31
Chapter 4
Numerical Tests of
Coherence-Corrected Surface
Hopping Methods Using a
Donor-Bridge-Acceptor Model
System
4.1 Introduction
Surface hopping[5] is a mixed quantum-classical method that is used for modeling
nonadiabatic excited state processes in molecules and condensed-phase materials.[10–
18] Its underlying approximation is the representation of a nuclear wavepacket as a
swarm of independent and classical nuclear trajectories. The method’s popularity is
leveraged by its efficiency and ease of implementation, but the classical description
of nuclei introduces well-known setbacks including lack of nuclear tunneling, lack of
zero-point energy, and overcoherence. Decoherence is the process by which nuclear
wavepackets on different potential energy surfaces decouple and move independently.
In surface hopping, the electronic wavefunction is integrated coherently along every (in-
dependent) trajectory, resulting in an overcoherence between electronic states.[31] Long-
time dynamics[29] and systems with several regions of nonadiabatic coupling[32] are
susceptible to spurious results without the inclusion of decoherence. From the perspec-
tive of excited state relaxation, decoherence is particularly important when wavepacket
separation is faster than the time needed to stabilize a coherent electronic transition;
this depends on the electronic manifold in which relaxation occurs. For systems with
a quasi-continuum of electronic states, decoherence may not be so important, as a high
density of states facilitates fast electronic transitions.[117, 118] In contrast, well-separated
energy levels and localized states result in much slower relaxation. This effect is seen
for the electron-hole recombination across the band gap in quantum-confined materials
such as quantum-dots and carbon nanotubes.[119, 120] Neglecting decoherence in these
materials leads to underestimating dephasing time scales by several orders of magni-
tude.
There are a number of proposed methods that incorporate decoherence into sur-
face hopping simulations.[33–40] The vast majority can be boiled down to defining a
decay rate indicative of wavepacket separation. Yet even a simple collapse approach
32 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
of the electronic wavefunction has been applied to calculate transition rates in a spin-
boson model, which has proven successful in recovering the correct quadratic-scaling
in diabatic coupling.[121] The decoherence time scale has also been estimated using
a Gaussian wavepacket approximation with width given by the thermal de Broglie
wavelength.[38–40, 122] A more mathematically driven method is Augmented FSSH
(A-FSSH), which involves a moment expansion of the Liouville equation that gives rise
to time-dependent uncertainties in nuclear position and momentum.[35] These uncer-
tainties are propagated along the nuclear trajectory and are variables of the collapse
rate. A-FSSH recovers Marcus theory[123, 124] and improves branching ratios in model
systems.[32, 35] A recent study using a spin-boson model has also found that A-FSSH
improves results compared to exact calculations over a wide range of energetic and
coupling parameters.[125] Here, we would like to benchmark the most popular deco-
herence methods by calculating their nonadiabatic charge transfer rates in a Donor-
bridge-Acceptor (DbA) model system whose standard reaction rates are obtainable us-
ing Marcus theory.
In DbA-type systems, charge transfer from donor to acceptor proceeds through in-
termediate states called bridges.[126] When the energy gap between the donor and
bridge is large relative to the thermal energy, superexchange theory predicts a tunnel-
ing mechanism for charge transfer.[127–129] Experimental works have described elec-
tronic tunneling through DNA hairpins[130, 131] and oligo-p-xylene bridges.[132, 133]
In such studies, the effect of the tunneling energy gap is probed by measuring the rate
of charge transfer from donor to acceptor as a function of the number of bridging units.
The charge transfer rate exponentially decays as a function of distance,[134] and the
distance decay constant can be related to an expression for tunneling through a finite
barrier to estimate an effective tunneling energy gap.[135–137] Not surprisingly, exper-
imental evidence shows that the outcome and efficiency of the tunneling mechanism is
related to the size of the gap. As more DbA systems are predicted, synthesized, and
ultimately used for applications,[56, 138] it is important that the tools of excited state
modeling (e.g., surface hopping) correctly describe the superexchange mechanism for
charge transfer.
In this paper, global flux surface hopping (GFSH)[22, 27] is combined with the fol-
lowing popular decoherence corrections: Truhlar’s decay-of-mixing[139] and Subot-
nik’s augmented surface hopping.[35] Donor-to-acceptor reaction rates are compared to
the rates computed using Marcus theory.[1] This work investigates the surface hopping
description of superexchange for charge transfer and adds to the body of literature[9,
125, 140–142] aimed at assessing the reliability of and improving the surface hopping
method for nonadiabatic molecular dynamics. The paper is organized as follows: Sec-
tion 4.2 reviews the overcoherence problem of surface hopping, Section 4.3 describes
superexchange theory and the superexchange reaction rate given by Marcus theory, Sec-
tion 4.4 describes the tested decoherence methods and simulation details, including the
DbA model used for benchmarking. Finally, results of the simulations and conclusions
are provided in Sections 4.5 and 4.6, respectively.
4.2. Overcoherence Problem of Surface Hopping 33
4.2 Overcoherence Problem of Surface Hopping
In exact dynamics, the total wavefunctionjyi can be represented in terms of coupled
electronic and nuclear wavefunctions through the Born-Oppenheimer expansion,[143]
jyi=
å
i
jc
i
ijf
i
i , (4.1)
wherejf
i
i are adiabatic electronic states determined at fixed nuclear geometries and
jc
i
i are nuclear states. The electronic density matrix (s) is determined by tracing the
combined nuclear-electronic density matrix (jyihyj) over the nuclear degrees of free-
dom (R),[144]
s =
å
i
å
j
Z
dRhc
j
jRihRjc
i
ijf
i
ihf
j
j
=
å
i
å
j
hc
i
jc
j
ijf
i
ihf
j
j . (4.2)
Eq. 4.2 shows that elements of s depend on the overlap of nuclear wavepackets. In the
adiabatic representation, diagonal elements ofs are adiabatic state populations and off-
diagonal elements are coherences. Decoherence is related to the reduction of overlap
between nuclear wavepackets on different surfaces, i.e.,hc
i
jc
j
i! 0 for i6= j.
In Surface Hopping (SH), the initially-prepared nuclear wavepacket is represented
by a swarm of independent and classical trajectories. Each trajectory evolves and stochas-
tically hops between potential energy surfaces according to a hopping algorithm.[5, 22]
The hopping algorithm depends on the adiabatic state coefficients (c
i
) that make up the
state,
jyi=
å
i
c
i
jf
i
i . (4.3)
The electronic density matrix in surface hopping (s
SH
) takes the form,
s
SH
=
å
i
å
j
c
i
c

j
jf
i
ihf
j
j . (4.4)
Eq. 4.4 makes it clear that decoherence must be explicit in the equation of motion of s,
but the electronic Schrödinger equation (with classical nuclear positions treated as pa-
rameters) does not have a decoherent term (see density matrix formulation in Eq. A.6a
below). Thus, in order to recover exact quantum mechanics, surface hopping simula-
tions must be supplemented with a decoherence correction.
A play-by-play of coherence-corrected surface hopping is as follows: A trajectory
enters an interaction region and as a result there is an exchange of quantum amplitude,
c
i
, between the interacting states. In this process, the trajectory stochastically hops be-
tween the surfaces and continues doing so until it is outside of the interaction region.
On account of overcoherence in surface hopping, even when the trajectory is sufficiently
far away from the interaction region, there is still nonzero quantum amplitude associ-
ated to states that were previously involved in the interaction but that are not currently
occupied by the trajectory. Instead, the population of the occupied electronic state (jf
i
i)
should approach unity (c
i
= 1) at a rate known as the decoherence rate. The rate (t
1
ij
)
34 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
at which the occupied electronic state decoheres from all other states (jf
j
i) is the rate at
which the overlap of nuclear wavepackets on different potential energy surfaces decays
to zero (hc
i
jc
j
i
t
1
ij
! 0 in Eq. 4.2).
4.3 Superexchange Theory and Reaction Rates in Donor-Bridge-
Acceptor Systems
Previous studies have benchmarked surface hopping and decoherence methods us-
ing exactly solvable scattering models such as those originally studied by Tully.[5, 22,
32, 35] In this work, we focus on reaction rates using a DbA model that highlights an
electronic tunneling mechanism ubiquitous in many materials.[126] The Hamiltonian of
a DbA model with N bridges is(N+ 2)(N+ 2) and can be written as
H =
0
B
B
B
B
B
B
B
B
B
B
@
E
DD
V
Db
V
bD
E
bb
V
bb
V
bb
E
bb
V
bb
V
bb
...
... V
bb
V
bb
E
bb
V
bb
V
bb
E
bb
V
bA
V
Ab
E
AA
1
C
C
C
C
C
C
C
C
C
C
A
. (4.5)
Here, the diagonal and off diagonal elements are diabatic energies and couplings, re-
spectively. The donor and acceptor states are not explicitly coupled, i.e. V
DA
= V
AD
= 0,
but electron transfer proceeds through intermediates. To convey the main pathways, we
refer readers to Figure 4.1 showing the Liouville space pathways of the density matrix
for a three-level system in the diabatic representation.[108, 109, 128] Fig. 4.1A shows the
superexchange pathway, with electron transfer proceeding through coherences without
populating bridge statej2i. The pathways of Figs. 4.1B and 4.1C that pass through
statej2i’s population are referred to as sequential pathways. Superexchange plays the
dominant role for electron transfer when bridge states are energetically displaced from
the donor and acceptor states (i.e., when bridge states are above the donor and acceptor
states by an energy gap that is greater than k
B
T).
The Marcus theory of electron transfer is the seminal work that explains reaction
rates from a donor to acceptor chemical species.[1] In order to obtain the Marcus reaction
rate during superexchange, the indirect coupling between donor-to-acceptor must be
known. In the two-state limit, the energy gap between the surfaces at the crossing is
2V, where V is diabatic coupling. McConnell derived the energies of the two lowest
symmetric and asymmetric eigenstates corresponding to the donor and acceptor states,
respectively.[129] McConnell’s model assumed an aromatic free radical composed of
two phenyl groups, linked together by a series of N methylene groups. In the limit of
jV
bb
/Dej 1, the first-order energy gap, g, is given by
g=
2V
2
Db
De

V
bb
De

N1
, (4.6)
4.3. Superexchange Theory and Reaction Rates in Donor-Bridge-Acceptor Systems 35
12
23
11 13
21 22
31 32 33
12
23
11 13
33
12 11
22
32 33
12
23
11
22
33
A
B C
FIGURE 4.1: Liouville pathways from the donor to acceptor in the dia-
batic representation. Pathway A is that of superexchange, whereas B and
C are sequential pathways that pass through the population of bridge
statej2i.
where V
Db
= V
bA
andDe is the energy required to remove an electron from the donor
or acceptor orbital and place it in a noninteracting bridge orbital. We will refer toDe as
the tunneling energy gap. By deriving the energy gap between donor and acceptor, the
(N+ 2)(N+ 2) Hamiltonian reduces to an effective 2 2 Hamiltonian, with direct
coupling between donor and acceptor given by g/2. In the presence of a classical bath,
the high-temperature limit reaction rate follows from Marcus theory,
k
ET
=
2p
¯ h
s
1
4plk
B
T
g
2
4
exp

(le
0
)
2
4lk
B
T
!
, (4.7)
where l is the reorganization energy and e
0
is the driving force. Besides the inverted
regime and maximum when e
0
= l, the rate scales as(De)
2N
. The high-temperature
Marcus rate is analogous to taking the classical limit. Similarly, surface hopping as-
sumes classical nuclei and thus the Marcus rate serves as a decent reference solution for
charge transfer in the DbA model.
36 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
4.4 Methods and Simulation Details
We describe GFSH in Section 4.4.1,
1
tested decoherence corrections in Section 4.4.2,
and simulation details in Section 4.4.3.
4.4.1 Global Flux Surface Hopping
Within GFSH,[22, 27] the change of adiabatic state population (Ds
ii
) is calculated at
each time step as
Ds
ii
= s
ii
(t+Dt)s
ii
(t) . (4.8)
All states are classified into one of two groups: one withDs
ii
< 0 (group A) and an-
other withDs
ii
> 0 (group B). Transitions can only occur from group A to group B,
preserving the minimization criterion of Fewest-Switches Surface Hopping (FSSH).[5]
The probability of hopping between the surfaces corresponding to statesjf
i
i andjf
j
i is
given by
g
ij
=
Ds
jj
s
ii
Ds
ii
å
k2A
Ds
kk
. (4.9)
The increase in population of the target statejf
j
i,Ds
jj
, may be attributed to all states
assigned to group A. We reduce this quantity to roughly statejf
i
i’s contribution by
multiplyingDs
jj
with the change in statejf
i
i’s population relative to the total change
in population of group A,Ds
ii
/å
k2A
Ds
kk
. Finally, the expression in Eq. 4.9 is divided
by the probability of the current state,s
ii
, to represent population flux, similar to FSSH.
Scattering results comparing GFSH and exact quantum mechanics are provided in Sup-
porting Information. GFSH combined with the Augmented decoherence method will
be referred to as A-GFSH.
4.4.2 Surface Hopping Dynamics and Decoherence Corrections
In surface hopping, nuclear trajectories evolve piecewise on adiabatic potential en-
ergy surfaces. The electronic density matrix (s) is integrated with the Liouville-von
Neumann equation,[5]
˙ s
i
¯ h
h
E

R
SH

i¯ hd
˙
R
SH
, œ
i
, (4.10)
where E

R
SH

is a diagonal matrix with elements being potential energy surfaces cal-
culated at fixed nuclear geometries (R
SH
),
˙
R
SH
is the velocity of the nuclear trajectory,
and d is the nonadiabatic coupling matrix. Electronic transitions between surfaces oc-
cur according to the GFSH algorithm given by Eq. 4.9. Following a successful hop, the
velocity is rescaled along the direction of nonadiabatic coupling. Transitions that are di-
rected upward in energy are rejected if the dispensable kinetic energy does not exceed
the energy barrier (also known as frustrated hops). We choose to reverse the nuclear
velocity following frustrated hops. In model systems, in which nuclear dynamics are
coupled to an implicit bath, reversing the nuclear velocity after frustrated hops leads to
1
Conventional FSSH is unable to describe the transfer invoking superexchange-like pathways in the
diabatic representation. GFSH was developed to circumvent this deficiency.[22]
4.4. Methods and Simulation Details 37
reaction rates that are in good agreement with Marcus theory[116] and thermal popula-
tions that are in good agreement with the Boltzmann populations.[30]
Truhlar’s Decay-of-Mixing
The electronic density matrix is integrated with coherent (c) and decoherent (d) con-
tributions,[139]
˙ s = ˙ s
coherent
+ ˙ s
decoherent
. (4.11)
The first term is computed with Eq. A.6a. The form of the second term is derived from
several assumptions such as the electronic state populations
kk
for k6= K (where K is the
decoherent state) decays to zero at a rate of 1/t
kK
, i.e.,
˙ s
d
kk
=
s
kk
t
kK
(k6= K) (4.12)
The other conditions used to obtain the time-domain equations for all elements of s
d
are conservation of electronic population,å
k
˙ s
kk
= 0, and conservation of phase angle
(q) in c
k
=jc
k
j exp(iq). Ref. [145] provides a detailed derivation and the explicit form of
s
d
.
The expression for the decoherence rate is given by[146]
t
1
ij

E
i
E
j

¯ h
0
B
@
2mE
0

P
SH

ˆ
d
ij

2
+ C
1
C
A
1
, (4.13)
where P
SH

ˆ
d
ij
is the nuclear momentum in the direction of nonadiabatic coupling, and
C = 1 and E
0
= 0.1 a.u. (atomic units) are empirical parameters chosen based on
numerical testing.[146] Eq. 4.13 is an ad hoc expression that assumes decoherence does
not occur if the momentum in the direction of nonadiabatic coupling is insufficient to
support energy transfer (i.e., t
1
ij

P
SH

ˆ
d
ij

2
/2m when P
SH

ˆ
d
ij
! 0). Additionally,
the decoherence time must be greater than or equal to the shortest electronic time (i.e.,
t
1
ij

E
i
E
j

/¯ h).[95]
Decay-of-Mixing Dephasing-Informed
A drawback of the original decay-of-mixing method (Section 4.4.2) is a decoherence
rate that depends on predefined parameters. Unfortunately, this may limit the method’s
transferability to systems that differ from those used for fitting. Here, we show a plau-
sible improvement of decay-of-mixing that overcomes this problem. On the basis of
Eq. 4.13, the general form of the decoherence rate is
t
1
ij
a

E
i
E
j

, (4.14)
where a is an unknown parameter. Eq. 4.14 eliminates explicit dependence on the
kinetic energy (Eq. 4.13) for simplicity. We propose an ensemble averaged a, a
38 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
ht
1
ij
i/h

E
i
E
j

i, whereh. . .i denotes statistical mechanical averaging. The ensem-
ble averaged decoherence rate,ht
1
ij
i, can be estimated by the decay rate of the pure-
dephasing function,[147]
D
ij
(t)= exp(g
ij
(t)) (4.15a)
g
ij
(t)=
Z
t
0
dt
2
Z
t
2
0
dt
1
R
ij
(t
1
t
2
) (4.15b)
R
ij
(t)=
1
¯ h
2
hdDE
ij
(t)dDE
ij
(0)i (4.15c)
dDE
ij
= E
j
E
i
(hE
j
ihE
i
i), (4.15d)
where g
ij
(t) is approximated up to second order in the cumulant expansion and R
ij
(t) is
the autocorrelation function of the energy gap fluctuation (dDE
ij
). Quantum correlation
functions are generally complex, but in the present case, classical molecular dynamics is
being used to sample the real part of the autocorrelation function that enters the semi-
classical expression for the pure-dephasing function of optical response theory.[148] The
optical response pure-dephasing function has been used in many applications of nona-
diabatic dynamics.[149–152]
We sample potential energy surfaces while the trajectory evolves in the ground
state,
2
and do so for a sufficient amount of time so that thermal equilibrium behav-
ior is realized. Although the use of CPA to computea is not necessary for the relatively
simple DbA model of the present study, we choose to use this method as it is widely
used for dynamics in condensed matter systems.[153] Given adiabatic state energies as
a function of time, the dephasing function between any two adiabatic states and their
decay rate (determined by fitting the dephasing function to a Gaussian
3
) as well as av-
erage gap,h

E
i
E
j

i, are computed to estimate a. An important physical principle
obeyed by this method, which is violated by the out-of-box decay-of-mixing (Eq. 4.13),
is that wavepackets on parallel surfaces do not decohere: D
ij
(t)= 1 because R
ij
(t)= 0
and consequently g
ij
(t) = 0. Further details clarifying this point can be found in Sec-
tion 4.4.2 and Ref. [31].
Subotnik’s Augmented Surface Hopping
Subotnik’s augmented surface hopping is a stochastic collapse approach. Collapse
rate depends on dynamic variables of the system including first-order uncertainties (or
moments) in nuclear position and momentum.[35] The decoherence rate between the
occupied adiabatic state,jf
i
i, and all other states,jf
j
i, is given by
t
1
ij

1
2¯ h

F
jj
F
ii

dR
jj

2
¯ h

F
ij
dR
jj

, (4.16)
2
Here, we make the assumption that the classical path approximation (CPA) is valid. CPA assumes
that ground state trajectories sample nuclear geometries that are seen during excited state dynamics. For
example, CPA is valid for systems in which forces on the ground and excited states are similar.
3
Similar Gaussian behavior can be seen by taking the classical limit of the dephasing function for the
two displaced harmonic oscillators problem of Marcus theory.
4.5. Results and Discussion 39
where F =rVj
R
SH is the force evaluated at the position of the surface hopping (SH)
trajectory anddR= Tr
N

R R
SH

r

is the position moment. The trace is over the nu-
clear (N) degrees of freedom andr is the combined nuclear-electronic density matrix. A
derivation of Eq. 4.16 is available in Supporting Information. Augmented surface hop-
ping stands out as a more rigorous method compared to other approaches because the
decoherence rate outside the interaction region is proportional to the force difference,
F
jj
F
ii
, which is fundamentally correct based on an analysis of the quantum-classical
Liouville equation.[31] Several other works (e.g., Refs. [154] and [155]) have also derived
a decoherence rate that depends on the force difference.
4.4.3 DbA Model Simulations
The DbA model used in our simulations can be rationalized as a linear chain of
diatomic molecules, each representing a specific group: donor, bridge, and acceptor
(Figure 4.2A). The model contains a single bridge state (i.e., N = 1) and diabatic po-
tential energy surfaces are parabolic with respect to the reaction coordinate: E
DD
=
mw
2
x
2
/2+ Mx, E
bb
= mw
2
(x x
b
)
2
/2+De, E
AA
= mw
2
x
2
/2 Mx e
0
. Here,
M =
p
lmw
2
/2, where l is the donor-to-acceptor reorganization energy, m is the re-
duced mass, and w is the angular frequency. Fig. 4.2B shows the adiabatic potential
energy surfaces. The bridge diabat is centered at the transition state configuration be-
tween donor and acceptor at x
b
=e
0
/2M, where e
0
is the driving force. The donor-
bridge and bridge-acceptor diabatic couplings are relatively weak (i.e., V
Db
= V
bA
= V)
in order to simulate nonadiabatic electron transfer between donor and acceptor.
Trajectories started in the donor diabat with initial positions and momenta sampled
from Boltzmann distributions.
4
Each ensemble was made up of 3000 trajectories. Every
trajectory evolved with a Dt = 1.25 a.u. time step for a total of 10
7
time steps. The
population of the acceptor diabat was recorded as a function of time and was computed
by squaring the inner product of the adiabatic state of the occupied surface with the
diabatic state of the acceptor diabat. After averaging over all trajectories, the donor-to-
acceptor reaction rate, k, was obtained by fitting the population of the acceptor diabat
to the function 1 exp(kt). Reaction rates were computed as a function of the driving
force (e
0
), diabatic coupling (V), and tunneling energy (De). Constant parameters of the
model (in a.u.) are m = 1, w = 4.375 10
5
, l = 2.39 10
2
, k
B
T = 9.50 10
4
,
and g = 1.50 10
4
(Langevin friction parameter). These numerical parameters were
borrowed from Ref. [124] and are used to simulate reactions on the nanosecond time
scale.
4.5 Results and Discussion
Figure 4.3 shows donor-to-acceptor reaction rates. The scaling behavior of the reac-
tion rates were determined with a least-squares fit to Eq. 4.7. Table 4.1 shows fitted pa-
rameters of the reorganization energy (l), scaling in diabatic coupling (V), and scaling
in tunneling energy (De). All methods accurately predict the driving force correspond-
ing to the maximum reaction rate (i.e., when e
0
= l), but the magnitude of the rates as
4
For each trajectory, position and momentum were sampled from Gaussian distributions with mean (m)
and standard deviation (s) given byfm,sg=fM/mw
2
,
p
k
B
T/mw
2
g andf0,
p
mk
B
Tg, respectively.
40 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
A
B
1
2
3
Δ
Donor Bridge Acceptor
FIGURE 4.2: (A) Cartoon schematic of the DbA model represented as a
linear chain of diatomic molecules. (B) Adiabatic potential energy sur-
faces of the DbA model with respect to the reaction coordinate. Donor,
bridge, and acceptor diabats are labeled by 1, 2, and 3, respectively. The
tunneling energy gap is labeled byDe.
well as scaling behavior varies among the methods. Generally speaking, we find that
the donor-to-acceptor reaction rate is related to the frequency of nonadiabatic transi-
tions between the two lowest-energy adiabatic surfaces (Fig. 4.2B). In the case of a few
nonadiabatic events, a trajectory may quickly settle in the acceptor diabat (right of the
diabatic crossing in Fig. 4.2B), resulting in a fast reaction rate, k
13
. Conversely, in the case
of many nonadiabatic transitions, the time elapsed before the nuclear coordinate settles
in the acceptor diabat will be longer, resulting in a slower k
13
. The frequency of these
nonadiabatic transitions depends on how the electronic density matrix is propagated,
with or without decoherence.
GFSH without decoherence significantly overestimates the reaction rates across all
tests (Fig. 4.3) and fails to recover the correct scaling in diabatic coupling and tunnel-
ing energy (Table 4.1). In exact quantum dynamics, the coherence between two states
increases as the trajectory enters their interaction region, and likewise decreases (and
eventually decays to zero), as the trajectory leaves their interaction region. Dynamics
are compromised when this relation does not hold, especially if the trajectory exits and
reenters the interaction region; an overcohered electronic density will ultimately give
4.5. Results and Discussion 41
TABLE 4.1: Fitting parameters of the donor-to-acceptor reaction rates of
Fig. 4.3. Data were fit to functions (shown in the column labels) represent-
ing the scaling behavior of the Marcus theory expression of Eq. 4.7 with
respect to driving force (e
0
), coupling (V), and tunneling energy (De).
Method Driving force – a exp
h
(xb)
2
c
i
Coupling – ax
b
Tunneling energy – ax
b
None b= 2.39 10
2
b= 2.00 b=0.97
Decay-of-mixing (C = 1) 2.37 10
2
2.57 1.04
Decay-of-mixing (C = 10) 2.37 10
2
2.92 1.23
DOM Dephasing-Informed 2.35 10
2
2.69 2.18
Augmented (A-GFSH) 2.38 10
2
3.42 1.71
spurious transition probabilities and reaction rates.
Decay-of-mixing is qualitatively correct in that the coherence decays outside the in-
teraction region. However, its decoherence time has adjustable parameters that were
originally chosen based on numerical tests (see Ref. [146]), which may limit the method’s
transferability to new systems.[41] In our case, we performed two sets of simulations by
adjusting the C parameter of the decoherence time (Eq. 4.13) and found that the magni-
tude of the reaction rate is very sensitive to this change; the reaction rate decreased by
approximately three-fold from C = 1 to C = 10 (Fig. 4.3). The C = 10 reaction rates are
in very good agreement with Marcus theory based on their magnitude as well as scaling
in coupling and tunneling energy (Table 4.1), but this choice of C was chosen by mere
trial-and-error. Therefore, the method’s applicability to other systems might be open to
serious challenge without further reparameterization.
On the basis of the results of Fig. 4.3 and Table 4.1, the best performing methods are
decay-of-mixing dephasing-informed and augmented surface hopping. Each method
has notable advantages that make it appealing. In the case of augmented surface hop-
ping, the decoherence rate is relatively rigorous in its formulation and explicitly de-
pends on the force difference between the two decohering potential energy surfaces.
This property proves to be important for recovering Marcus theory with the best quanti-
tative accuracy. Moreover, the decoherence rate does not contain any predefined param-
eters and is presumed to be applicable to various types of systems, although further test-
ing is needed to validate this claim. In the case of decay-of-mixing dephasing-informed,
the decoherence rate is more general than that of the original decay-of-mixing method
since it contains information specific to the system under consideration. As a result,
the magnitude of the method’s reaction rates is in much better agreement to Marcus
theory (e.g., compare C = 1 results to decay-of-mixing dephasing-informed in Fig. 4.3).
Also noteworthy is that reparameterization of the decoherence rate via the dephasing
function obeys the correct limiting behavior in which wavepackets on parallel surfaces
do not decohere. However, a potential downside to the dephasing-informed approach
is the necessity of an accurate and sufficient sampling of the potential energy surfaces
involved during the dynamics to calculate the dephasing function (Eq. 4.15a). For the
case shown here, we were dealing with a one-dimensional model in which these criteria
were easy to adhere to.
42 Chapter 4. Numerical Tests of Coherence-Corrected Surface Hopping Methods
4.6 Summary and Conclusions
Electronic state mixing in a Born-Oppenheimer nuclear-electronic system is con-
trolled by the overlap of nuclear wavepackets on different potential energy surfaces.
Dissimilar forces on potential energy surfaces cause the overlap of nuclear wavepackets
to diminish, leading to decoherence. Due to classical and independent nuclear trajec-
tories in surface hopping, the electronic density matrix is integrated with fictitious co-
herence between states that severely compromise transition probabilities. The evolution
of the electronic density matrix must be corrected such that it continuously decoheres
outside the interaction region. Herein, popular ad hoc and practical first-principles ap-
proaches that incorporate decoherence into the evolution of the electronic density ma-
trix were assessed by comparing their donor-to-acceptor reaction rates to Marcus theory.
The strengths and weaknesses of the decoherence methods were evaluated in the
context of a Donor-bride-Acceptor (DbA) model. Truhlar’s decay-of-mixing decoheres
the electronic density matrix outside the interaction region and qualitatively agrees with
Marcus theory. However, we find that the quality of decay-of-mixing depends on pre-
defined parameters; it was only by trial-and-error that the chosen parameters gave rea-
sonable results. We further introduced a method that alleviates this issue by showing
that the decoherence rate can be parameterized for the system under investigation as
long as an accurate and sufficient sampling of the potential energy surfaces involved
in the dynamics is attainable. The method, which we call decay-of-mixing dephasing-
informed, is in very good agreement with Marcus theory. Finally, the decoherence rate
between electronic states in Subotnik’s augmented surface hopping explicitly depends
on the force difference between their surfaces. This feature proves to be important for
recovering both the magnitude of the reaction rates and the correct scaling in diabatic
coupling and tunneling energy with quantitative accuracy.
On the basis of our benchmark study, the decoherence method used to correct sur-
face hopping simulations must be chosen with great care. Our tests of popular meth-
ods indicate their broadly varying system-dependent performance. Thus, it is recom-
mended that the method of choice be justified for the system under investigation. Fur-
ther testing such as higher dimensional potential energy surfaces and a broader range
of reaction time scales would provide deeper insight into the advantages and disadvan-
tages of the methods discussed in this paper. For example, a follow-up study comparing
decay-of-mixing dephasing-informed and augmented surface hopping would be espe-
cially interesting since they were the most successful methods for the DbA model. In-
formation regarding their versatility to new systems and numerical costs may help to
clarify their realm of applicability.
4.6. Summary and Conclusions 43
0
5
10
Driving force
0
5
k
13
(× 10
7
a.u.)
0
5
1.5 2.5
0
(× 10
2
a.u.)
0
2
Coupling
1.35 1.525
V (× 10
3
a.u.)
None
Tunneling energy
Decay-of-
mixing
DOM
Dephasing-
Informed
9.5 10.5 11.5 12.5
(× 10
2
a.u.)
Augmented
FIGURE 4.3: Donor-to-acceptor reaction rates as a function of driving
force (e
0
), diabatic coupling (V), and tunneling energy gap (De). Reac-
tion rates were computed with surface hopping (red circles) and Marcus
theory (black line). Decay-Of-Mixing (DOM) data include C = 1 (orange
triangles) and C = 10 (red circles) (see Eq. 4.13). Curve fits of the surface
hopping data are shown in light grey.

45
Chapter 5
Photoexcited Nonadiabatic
Dynamics of Solvated Push-Pull
p-Conjugated Oligomers with the
NEXMD Software
Reprinted with permission from J. Chem. Theory Comput. 2018, 14, 8, 3955-3966.
Copyright 2018 American Chemical Society.
5.1 Introduction
Modeling nonadiabatic excited state molecular dynamics is at the core of under-
standing photoinduced processes such as charge and energy transfer,[17, 156–160] pho-
toisomerization,[161, 162] photodissociation,[163, 164] and other types of coupled electron-
vibrational (or vibronic) dynamics.[165–167] In organic conjugated molecules, photoin-
duced relaxation occurs in a dense manifold of vibrational and electronic states due
to strong electronic correlations and electron-phonon couplings.[13, 14, 168] A theoret-
ical understanding of excited state relaxation in these materials not only aids in the
design of functional photoactive materials for technological applications,[169–176] but
also elucidates photochemical and photophysical properties that frequently go unde-
tected in experiments such as conformational changes, multiple reaction pathways, and
electron density distributions.[177, 178] Our ability to gain understanding at the molec-
ular level is largely dictated by the reliability of computational methods. Due to the
large number of degrees of freedom in a solvent, nonadiabatic simulations of realisti-
cally large molecular systems (tens to hundreds of atoms in size) are typically performed
in the gas phase, whereas most experiments are carried out by probing solution-based or
solid-state samples. The use of qualitatively-accurate and numerically-efficient methods
of modeling solvation are necessary for predicting and interpreting structural, optical,
and dynamical properties. Herein, we implement a solvent model in the Nonadiabatic
EXcited-state Molecular Dynamics (NEXMD) software and demonstrate its effects on
the photoinduced nonadiabatic dynamics in a set of organic conjugated push-pull (or
donor-acceptor) molecules.
The solute-solvent system is most accurately modeled by quantum mechanics. How-
ever, this approach is prohibitively expensive for systems with multiple degrees of free-
dom. Therefore, approximations capable of reproducing the effective solvent response
46 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
in large systems are needed. One such method resorts the portion of solvent interact-
ing with the solute to a quantum mechanical description using a combined quantum
mechanical/molecular mechanical (QM/MM) approach.[179–181] At this level, the ab
initio molecular dynamics would still require averaging over many solvent configura-
tions.[182–184] Implicit solvent models reduce computational cost even further by treat-
ing the solvent as a dielectric continuum–an effective average over many solvent config-
urations.[42, 185–188] By embedding the solute in a dielectric cavity,[189] charge density
of the solute induces a surface charge density in the cavity, producing a polarization that
screens Coulombic interactions. Analytical gradients in time-dependent density func-
tional theory (TD-DFT) and time-dependent Hartree Fock (TD-HF) make it feasible to
include solvent effects at the first-order or linear response level.[190–192] In this way,
the solvent responds linearly to the excitation of the solute which is determined by the
spatial distribution of the transition density.[191, 192]
The NEXMD software uses semiempirical Hamiltonians within TD-HF theory, thereby
making it feasible to implement a linear response solvent model.[191, 192] The use
of semiempirical Hamiltonians allows NEXMD to simulate the dynamics of organic
conjugated molecules on the order of hundreds of atoms and for time scales up to
tens of picoseconds. Past developmental versions of NEXMD have been used to sim-
ulate the nonadiabatic dynamics of large systems such as dendrimers,[157, 193–196]
chlorophylls,[197–199] cycloparaphenylenes,[200] and conjugated macrocycles.[159] In-
tegrating an implicit solvent model to the code is both an effort towards retaining the
computational efficiency of NEXMD and incorporating the important solute-solvent
interactions necessary for realistic modeling of molecular systems in contact with po-
larizable environments. Recently, adiabatic dynamics simulations have investigated
solvatochromic shifts in the optical spectra of solvated oligo(para-phenylene vinylene)
(PPVO) derivatives.[201] This work extends dynamics with implicit solvation to the
nonadiabatic realm, where dynamical excited state properties are calculated in order
to study their dependence on donor-acceptor functionalization and solvent polarity.
Herein, we address the importance of solvation on dynamics and introduce a compre-
hensive version of NEXMD capable of efficiently modeling solvated molecules.
5.2 Computational Details
5.2.1 The Nonadiabatic EXcited-state Molecular Dynamics (NEXMD) Frame-
work
The NEXMD software combines the collective electronic oscillator (CEO) method[202,
203] with the semiempirical quantum chemistry (SQM) package from AmberTools.[204]
The CEO approach can be thought of as a generalization of the random phase approx-
imation (RPA) applied to a range of mean field theories such as Hartree-Fock or Kohn-
Sham DFT. NEXMD uses the CEO method to compute electronically excited states with
RPA or configuration interaction with singles (CIS),[205] combined with the diverse set
of semiempirical Hamiltonians that are available in SQM. At this level, the numerical
costs of computing excited states are not substantially more demanding than ground
state calculations.[206, 207] Moreover, semiempirical Hamiltonians such as AM1[208]
provide reasonably accurate ground state geometries and energies, heats of formation,
vertical excitation energies, polarizabilities, and adiabatic excited state potential energy
5.2. Computational Details 47
surfaces (PESs).[209–212] Optical and excited state properties of large systems with
dense manifolds of interacting excited states may be computed as evidenced by suc-
cessful application of this level of theory to systems such as polymers,[168, 202, 213]
dendrimers,[214] light-harvesting complexes,[202] and carbon nanotubes.[215, 216]
Energies and forces as well as nonadiabatic couplings are computed “on the fly”
with nuclei evolving on native excited state PESs. Nonadiabatic transitions between
electronic states are modeled with Tully’s fewest-switches surface hopping (FSSH).[5]
Other practical aspects of calculations that are carried out with NEXMD include but are
not limited to (1) decoherence corrections built on top of FSSH to alleviate inconsisten-
cies due to the classical treatment of nuclei,[31, 41] (2) advanced algorithms for tracking
trivial (unavoided) crossings between noninteracting states,[86] (3) implicit treatment
of solvation including linear response,[191, 192] state-specific,[191, 192] and nonequi-
librium models,[217] and (4) “on-the-fly" limiting to essential excited states that are
needed to sufficiently propagate the electronic Schrödinger equation.[218] The latter
functionality significantly reduces computational time by eliminating the calculation
of unnecessary excited states and nonadiabatic couplings. More detail on the govern-
ing theory that is implemented in developmental versions of NEXMD can be found in
Refs. [14] and [13]. Full description of the NEXMD software such as structure, use, and
benchmarks will be reported elsewhere, along with a public release of the code.
For purposes here, we focus on NEXMD’s functionality of modeling implicit solvation–
specifically, linear response solvation following the theoretical formalism reported in
Ref. [191]. Dynamics are coupled to the conductor-like polarizable continuum model
(CPCM) from the SQM code using a standard tessellation scheme for cavity discretiza-
tion.[219, 220] Solvent response is related to the induced polarization with coefficient
given by f (e)= (e 1) /e, wheree is the dielectric constant of the cavity.[219, 220] The
surface charge distribution at the solute-solvent boundary produces an electrostatic po-
tential which is added to the solute’s Hamiltonian. This accounts for the interaction of
the solute’s electrons and nuclei with the induced polarization charge. In order to prop-
erly simulate nonadiabatic dynamics, solvent effects are also included in the derivative
coupling vectors,
d
ab
=

f
a
(r; R)jr
R
Hjf
b
(r; R)

E
b
E
a
, (5.1)
where H is the electronic Hamiltonian, parameterized by nuclear configuration, and
contains a term for the solute-solvent interaction: H = H
vac
+ H
sol
. The adiabatic eigen-
vectors and eigenvalues of H arejfi and E, respectively.
CPCM approximates PCM and treats the solvent as a conductor which simplifies
numerical integration of Poisson’s equation at the solute-solvent boundary. The coeffi-
cient of polarization f (e) is an artificial function that ensures modeling of a solvent with
a finite dielectric constant as opposed to a perfect conductor with an infinite dielectric
constant. Nevertheless, CPCM has successfully reproduced experimental results.[220]
5.2.2 Studied PPVO Derivatives
Para-phenylene vinylene (PPV) is a prototype conjugated polymer with potential
use in optoelectronic devices due to its rich electronic and optical properties as well
as synthetic flexibility.[221–231] The PPVO derivatives (Figure 5.1) are molecular chro-
mophores with tunable emission energies ranging across the optical spectrum.[232, 233]
48 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
In our simulations, we use the following three oligomers: unsubstitutedfH, Hg, substi-
tution with an acceptorfH, NO
2
g, and substitution with a donor-acceptorfNH
2
, NO
2
g.
These molecules can adapt two distinct asymmetric and symmetric conformations as il-
lustrated in Figure 5.1, parts A and B, respectively.
(B)
(A)
FIGURE 5.1: Chemical structures of PPVO derivatives. (A) and (B)
are two different conformations. fR
1
, R
2
g arefH, Hg,fH, NO
2
g, and
fNH
2
, NO
2
g. Bond lengths defined by b
1
, b
2
, and b
3
will be used to cal-
culate bond length alternation (BLA) between the adjacent carbon-carbon
atoms connecting the aromatic rings. BLA will be discussed in section
5.3.7.
These molecules were chosen because of their varying degrees of polarizability as a
result of chemical substitution.[138] The calculated ground state dipole momentsj~ m
gg
j
vary between 1.5 2.0, 6.8 7.9, and 8.4 9.6 D infH, Hg,fH, NO
2
g, andfNH
2
, NO
2
g,
respectively. The lower and upper bounds of each range correspond toj~ m
gg
j withe= 1
and e = 20, respectively. The dipole moment progressively increases from apolar to
polar molecules. Thus, we expect to see differences in the calculated excited state prop-
erties due to differences in their respective solute-solvent interactions.
5.2.3 Nonadiabatic Dynamics Simulations
The protocol taken to model nonadiabatic dynamics have been reported[13] but for
the sake of completeness, we summarize the steps here as well. A more detailed pro-
cedure is available in section B.1. All molecular geometries were optimized using AM1
in NEXMD and then evolved adiabatically on their respective ground states for several
5.2. Computational Details 49
nanoseconds with a 0.50 fs time step in a Langevin thermostat set to 300 K and fric-
tion parameter of 20 ps
1
. Snapshots were taken throughout the equilibrated ground
state trajectories of both conformers (Figure 5.1) which constitute inputs to excited state
modeling. In our simulations, 80% of initial geometries used for nonadiabatic trajecto-
ries were sampled from the ground state trajectories of the asymmetric conformations of
Figure 5.1A, while the other 20% were sampled from the other ground state trajectories
with the symmetric conformations of Figure 5.1B. These proportions are approximate
Boltzmann populations determined from total energies of the ground state optimized
structures.
1
Single-point calculations at ground state geometries were performed to determine
excited state energies and oscillator strengths. Oscillator strengths were broadened with
a Gaussian-shaped Franck-Condon window at the excitation energies with an empirical
standard deviation of 0.15 eV . A theoretical absorption spectrum of each molecule was
then calculated as an average over all the individual absorption spectra of its sampled
ground state conformations. Emission spectra were also simulated to calculate Stokes
shifts and compare them to experimental data. Similar to the procedure carried out
for absorption, a single trajectory for each molecule evolved adiabatically on the first
excited state S
1
for a few nanoseconds. Following thermal equilibration, geometries
were sampled throughout these trajectories and single-point calculations determined S
1
! S
0
emission energies and oscillator strengths.
Ground state snapshots (i.e. coordinates and velocities) were used as the initial con-
ditions for nonadiabatic trajectories. Each trajectory was prepared in an optically al-
lowed electronic excited state in the vicinity of the chosen photoexcitation energy. The
initial excited state of each trajectory was chosen according to the absorption spectrum
of its sampled ground state geometry such that states with larger oscillator strengths
were more populated than those with lower oscillator strengths. This procedure com-
poses a photoexcited wavepacket sampling the phase space of nuclear configurations
accessible at room temperature. Nonadiabatic trajectories evolved for 1 ps in length
with a 0.10 fs classical time step (nuclei) and a 0.02 fs quantum time step (electrons) in
the same Langevin thermostat as the ground state trajectories. Whenever FSSH signaled
for a hop to occur, the electronic wavefunction was instantaneously collapsed onto the
occupied state following the transition, thereby incorporating the appropriate decoher-
ence correction.[41] Trivial crossings were also accounted for using a method developed
and tested within the NEXMD framework.[86] In order to guarantee statistical conver-
gence, each ensemble was comprised of 635 trajectories. A range of solvents from low
to high polarity were modeled withe=f1, 2, 5, 20g.
5.2.4 Natural Transition Orbitals (NTOs) and Transition Density (TD) Anal-
ysis
Excitonic states of thesep-conjugated oligomers have strong multireference charac-
ter. Due to strong mixing, visualizing these excitations in terms of molecular orbitals
does not provide useful information. A more useful analysis of the excited states was
1
Absolute energy differences between the ground state optimized asymmetric and symmetric confor-
mations of these molecules using the AM1 Hamiltonian ranges from 0.045 to 0.063 eV . For the sake of
simplicity, we used the same proportions of asymmetric (80%) and symmetric (20%) conformations for the
dynamical ensembles of the three PPVO molecules.
50 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
carried out by transforming molecular orbitals to natural transition orbitals (NTOs). By
generating NTOs, the hole and electron wavefunctions were visualized and the ground
to excited state transition characters were more accurately assigned as p! p

, charge
transfer, etc. These holes and electrons come in pairs, where the relative weight of each
pair are excitation amplitudes in the NTO basis. NTOs were calculated in the Gaussian
09[234] software and images showing NTOs were obtained with the Visual Molecular
Dynamics (VMD) software.[235]
Evolution of the excitonic wavefunction was tracked by changes in the spatial lo-
calization of the transition density (TD). Diagonal elements of the single-electron TD
matrices
(r
ge
)
mm
(t)
D
f
e
(t)jc
†
m
c
m
jf
g
(t)
E
(5.2)
represent changes in the electronic density in an atomic orbital (AO) when undergoing
a ground (g) to excited state (e) transition, where c
†
m
and c
m
are Fermi creation and an-
nihilation operators, respectively, and m refers to the AO basis function.[236] Evolution
of a TD matrix contains information regarding charge density fluctuations and spatial
localization of excitations in time.[86, 194, 237, 238] The fraction of TD localized on a
molecular fragment is defined as the sum of all its atomic contributions.[239] For this
study, each molecule was divided into two fragments and the normalized probability
distribution of the TD on each fragment (P
F
) was calculated using
P
F
=
å
fN
F
g
i
jx
ii
j
å
fNg
j

x
jj

(5.3)
where x
nn
is TD on the nth atom,fN
F
g are the atoms associated to the F fragment, and
fNg are all atoms of the molecule.[202] x is a N N reduced TD matrix where diagonal
elementsx
nn
are determined by summing the contributions of TD (r
mm
of Equation 5.2)
from all AO basis functions associated with the nth atom.
In addition to describing the spatial localization of excitations with P
F
, we also mea-
sure exciton localization L
D
. The latter is defined as the inverse participation ratio asso-
ciated to the distribution of populations[202]
L
D
=

å
n
P
2
n
!
1
(5.4a)
P
n
=
jx
nn
j
å
fNg
j

x
jj

. (5.4b)
For a localized excitation, L
D
1, and for a delocalized excitation, L
D
N. Calcu-
lating L
D
is a simple way of determining the localization/delocalization of molecular
excitations as a function of chemical substitution and solvent polarity.
5.3. Results and Discussion 51
5.3 Results and Discussion
5.3.1 Optical Spectra and Initial Excitation
The first step to modeling photoinduced dynamics is photoexcitation into optically
allowed electronic states. The left panel of Figure 5.2 shows the calculated absorption
spectra of the three molecules withe= 1, where contributions of the individual excited
states (S
1
S
9
) are delineated. Three peaks are predicted within the optical window
of interest, the locations of which do not vary by more than 0.20 eV across the three
molecules. The right panels of Figure 5.2 shows the integral spectrum of each molecule
in the four solvent environments. The spectra shift towards lower energy ase increases
due to stabilization of the excited states relative to the ground state caused by the in-
duced polarization (solvatochromic shifts). These shifts are within 0.30 eV . In order to
study nonadiabatic dynamics through multiple excited states, the molecules were pho-
toexcited at the high-energy peak. The laser excitation energy was a Gaussian-shaped
pulse in the energy domain with a mean of 4.30 eV and an empirical linewidth parame-
ter (standard deviation) of 0.15 eV (Figure 5.2).
A calculated absorption spectrum was compared to an experimental spectrum from
the literature in order to support the use of NEXMD’s level of theory for calculating
excited states. The calculated spectrum offH, Hg is red-shifted relative to the experi-
mental spectrum of PPV3[240] by about 0.20 eV , but the Stokes shift (which is approxi-
mately 0.5 eV) is in excellent agreement (Figure B.1). This comparison betweenfH, Hg
and PPV3 is valid since the methoxy groups connected to the central benzene ring do
not contribute to the optical excitation based on NTO analysis (discussed below). While
the absolute excitation energies are slightly red-shifted compared to experiment, rel-
ative energy given by the Stokes shift is described well at the semiempirical level of
theory. Similarly, nonradiative relaxation also depends on energy gaps, and therefore
these results give us confidence in using NEXMD for studying qualitative differences in
the dynamics due to functionalization and solvent polarity.
The molecules of this study have different partial charge transfer character and are
a test bed for investigating the effects of functionalization in different solvent environ-
ments. Transitions are independent of molecular conformations across all ground state
dynamical data, and therefore we only present NTOs of S
1
(Table B.1) and the opti-
cally excited state (Table B.2) of the three molecules in their ground state optimized
geometries with e = 1. ForfH, Hg, the transition to the optically excited state has
p! p

character localized on the aromatic rings. Similar p! p

transitions occur in
fH, NO
2
g, but unlikefH, Hg which is an unsubstituted symmetric molecule,fH, NO
2
g
favors charge transfer towards the NO
2
. Similarly,fNH
2
, NO
2
g also shows significant
charge transfer from NH
2
to NO
2
. The main transitions in all molecules arep! p

, but
NTOs illustrate how charge transfer character can be controlled by chemical substitu-
tion. This is also reflected in the difference of dipole moments in the ground and excited
state

~ m
ee
~ m
gg

which progressively increases from the apolar to polar molecules and
varies from approximately 0.4, 1.1, to 2.4 D across all dynamical simulations withe= 1.
The excited states used to calculate m
ee
are those that contribute most to the absorption
peaks at 4.30 eV , which are S
7
forfH, Hg and S
9
forfH, NO
2
g andfNH
2
, NO
2
g (Figure
5.2).
52 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
† =1 R
1
= H, R
2
= H
Total
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
R
1
= H, R
2
= H
† =1
† =2
† =5
† =20
Intensity (AU)
† =1 R
1
= H, R
2
= NO
2
R
1
= H, R
2
= NO
2
Laser Excitation
1.5 2.0 2.5 3.0 3.5 4.0 4.5
Energy (eV)
† =1 R
1
= NH
2
, R
2
= NO
2
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
R
1
= NH
2
, R
2
= NO
2
FIGURE 5.2: Calculated linear optical absorption spectra in arbitrary
units (AU). Left panels shows the spectra of the three molecules with
e = 1. Both the individual states and the total absorption are shown.
The dotted line shows the emission peak from S
1
. Right panels show
the total absorption of each molecule in four solvent environments with
e =f1, 2, 5, 20g. The dotted lines show emission peaks from S
1
. Each
spectrum is normalized by its maximum absorption/emission intensity.
Also shown in the right panel are the laser excitations at 4.30 eV with
0.15 eV line widths which mimics the initial excitation for nonadiabatic
dynamics into high-energy absorption bands.
5.3.2 Potential Energy Surfaces
Excited state dynamics depend on the topology of potential energy surfaces (PESs).
Figure 5.3 shows solvatochromic shifts when PESs are histogrammed over all time steps
and trajectories. To show relative differences in excited state energies as a function ofe,
the PESs of each molecule were shifted by their mean ground state energies withe= 1.
As a consequence of the electrostatic interaction at the solute-solvent boundary, these
shifts increase in order offH, Hg,fH, NO
2
g, andfNH
2
, NO
2
g. ForfH, Hg,hE
1
i(e)
hE
0
i(e= 1)= 2.79 ate= 1 and 2.29 ate= 20, resulting in an overall shift of 0.50 eV . For
fH, NO
2
g andfNH
2
, NO
2
g, these shifts are 2.56 1.80= 0.76 eV and 2.54 1.59= 0.95
eV , respectively. Increasing solvent polarity stabilizes the energy levels and the degree
of stabilization depends on donor-acceptor groups, where more polar molecules (i.e.
fH, NO
2
g andfNH
2
, NO
2
g) are further stabilized in more polar solvents.
5.3. Results and Discussion 53
(A) R
1
= H, R
2
= H
† = 1
Density
† = 2
† = 5
† = 20
0 1 2 3 4 5 6
Relative Energy (eV)
Mean
Potential
Energies
(B) R
1
= H, R
2
= HNO
2
† = 1
Density
† = 2
† = 5
† = 20
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
0 1 2 3 4 5 6
Relative Energy (eV)
Mean
Potential
Energies
(C) R
1
= NH
2
, R
2
= HNO
2
† = 1
Density
† = 2
† = 5
† = 20
0 1 2 3 4 5 6
Relative Energy (eV)
Mean
Potential
Energies
FIGURE 5.3: Histograms of the PESs over the entire 1 ps ensemble of
trajectories with e =f1, 2, 5, 20g. PESs of each molecule are shifted by
their mean ground state energies with e = 1. The maximum PES shown
for each molecule corresponds to the optically excited state at 4.30 eV . The
bottom panel of each subfigure shows mean PESs averaged over all time
steps and trajectories.
5.3.3 Excited-State Populations and Lifetimes
Figure 5.4 shows the lifetimes of electronic excitations. The initial excitation (black
curve) fully decays in about 500 fs and rapidly transfers population through the inter-
mediate states. The S
1
population was fit to the expression A B exp(kt), where k is
relaxation rate and t is time (Figure 5.4D). Relaxation rates offH, Hg do not strongly
depend on solvent polarity since the molecule is weakly polarizable. In contrast, relax-
ation rates of the polar molecules do depend on solvent polarity.
Relaxation dynamics to S
1
are described by the energy gaps between excited states,
E
ji
= E
j
E
i
. The gaps offH, Hg are larger than those offH, NO
2
g andfNH
2
, NO
2
g
(Figure 5.3); nonadiabatic couplings[241] (Equation 5.1) and larger in the polar molecules,
leading to faster relaxation (Figure 5.4D). In regard to changes due to solvent polarity,
energy gaps can either increase or decrease (Figure B.2).
2
The frequency of small energy
gaps infH, NO
2
g generally decrease withe, thus explaining the monotonic decrease in
2
The number of gaps that fell below a 0.10 eV was recorded (Figure B.3). This was chosen because the
majority of hops occurred below this threshold.
54 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
0.0
0.2
0.4
0.6
0.8
1.0
Population
(A) R
1
= H, R
2
= H
† =1 † =2
0 200 400 600 800
Time (fs)
0.0
0.2
0.4
0.6
0.8
† =5
0 200 400 600 8001000
† =20
S
1
S
2
S
3
S
7
S
m
0.0
0.2
0.4
0.6
0.8
1.0
Population
(B) R
1
= H, R
2
= NO
2
† =1 † =2
0 200 400 600 800
Time (fs)
0.0
0.2
0.4
0.6
0.8
† =5
0 200 400 600 8001000
† =20
S
1
S
2
S
9
S
m
0.0
0.2
0.4
0.6
0.8
1.0
Population
(C) R
1
= NH
2
, R
2
= NO
2
† =1 † =2
0 200 400 600 800
Time (fs)
0.0
0.2
0.4
0.6
0.8
† =5
0 200 400 600 8001000
† =20
S
1
S
2
S
9
S
m
0 5 10 15 20
Dielectric Constant, † (unitless)
0
2
4
6
8
10
S
1
Rate Constant, k (10
−3
fs
−1
)
(D) S
1
Fitting Parameter
R
1
= H, R
2
= H
R
1
= H, R
2
= NO
2
R
1
= NH
2
, R
2
= NO
2
FIGURE 5.4: (A), (B), and (C) are excited state populations offH, Hg,
fH, NO
2
g, andfNH
2
, NO
2
g, respectively. Each subpanel of (A), (B),
and (C) shows excited state populations in one of the solvent environ-
ments with e =f1, 2, 5, 20g. Populations labeled with S
m
are comprised
of all states that are excited by the initial laser excitation at 4.30 eV .
(D) shows relaxation rates to S
1
determined by fitting S
1
populations to
A B exp(kt). Error bars are one standard deviation in the associ-
ated fit.
k (Figure 5.4D). In contrast, the E
32
gap offH, Hg decreases withe, and explains the de-
crease in S
3
’s population in more polar solvents (Figure 5.4A)–the S
3
! S
2
transition be-
comes more easily accessible. The E
98
gap offNH
2
, NO
2
g fluctuates non-monotonically
withe (Figure B.3) and correlates to changes seen in the S
1
population (Figure 5.4D). All
these examples are given to highlight the effect of solvation on relaxation as a result of
changes in energy gaps. The degree to which an excited state is affected by solvation
depends on its polarizability and charge transfer character (discussed in section 5.3.5).
In summary, we find that relaxation to S
1
occurs on a 590 730, 140 350, and 230 370
fs time scale

k
1

infH, Hg,fH, NO
2
g, andfNH
2
, NO
2
g, respectively (Table B.3). The
range of each time scale refers to the effect of the different solvent environments.
5.3.4 Transition Density (TD) Analysis
Evolution of the excitonic wavefunction was qualitatively tracked by changes in spa-
tial localization of the TD. Each molecule was split into two fragments, and the fraction
5.3. Results and Discussion 55
of TD on each fragment was calculated with Equation 5.3. InfH, Hg, the excitation is
split evenly on both fragments throughout the dynamics and is roughly independent of
e (Figure 5.5A). Both properties are expected since the molecule is symmetric and lacks
electron donating and withdrawing groups. InfH, NO
2
g at e = 1, the excitation starts
out evenly distributed and then transfers toward the NO
2
(Figure 5.5B); the exciton be-
comes more localized as trajectories evolve due to the electron withdrawing character
of NO
2
by induction and resonance. InfNH
2
, NO
2
g, the excitation slightly favors the
NH
2
fragment at initial time (Figure 5.5C), in agreement with NTOs of the optically
excited state (Table B.2). The excitation transfers toward the NO
2
and the fraction trans-
ferred decreases ase increases similar to that observed infH, NO
2
g (Figure 5.5B) but to
a lesser extent.
The spatial distribution of the TD is a consequence of the electron withdrawing or
donating character of the functional groups. Both of the polar molecules are function-
alized with NO
2
, but the fraction of TD localized on the fragment containing NO
2
is
larger infNH
2
, NO
2
g (Figure 5.5). The NO
2
group decreases electron density through
an electron withdrawing effect. This effect occurs in both molecules, but in the case of
fNH
2
, NO
2
g, the NH
2
group also increases electron density on the adjacent aromatic
ring through an electron donating effect. As a result, the excitation is more delocalized
infNH
2
, NO
2
g.
In regards to trends observed due to solvent polarity, the TD on each fragment of
both molecules becomes more like the other as e increases, resulting in a more delocal-
ized ground to excited state transition. This effect is a consequence of functionalization
along the major inertial axis, giving rise to relatively large dipole moments. But since
dipole moments are large, an increase ine only slightly affects the distribution of charge
density and the fraction of TD on each fragment (Figure 5.5). This is more so the case
forfNH
2
, NO
2
g than it is forfH, NO
2
g.
A brief discussion of TD as a function of individual excited states and solvent polar-
ities is also available in section B.6.
5.3.5 Excited State Dipole Moments
Figure 5.6 shows histograms of excited state dipole moments. The dipoles offH, Hg
are smaller in magnitude than those offH, NO
2
g andfNH
2
, NO
2
g. The relative direc-
tion also changes from being along the minor intertial axis (Figure 5.6A) in the apolar
molecule (Figure 5.6B) to being nearly parallel along the major inertial axis in the polar
molecules (Figure 5.6C,D) as a result of chemical substitution. Dipole magnitudes in-
crease as e increases and vary in the ranges 1.5 2.1, 8.4 9.6, and 10.6 12.5 D in the
three molecules, respectively. The dipole moments offH, NO
2
g andfNH
2
, NO
2
g also
show small changes in magnitude within the first 200 fs, indicative of charge transfer
character (Figure B.5). A brief discussion on charge transfer states is available in section
B.8.
Dipole moments computed with NEXMD can be validated by considering the iner-
tial axis along which they are oriented. Dipoles vary from approximately 2 D infH, Hg
to 12 D infNH
2
, NO
2
g; a ratio of 6 is in approximate agreement with the length scales
of the major and minor inertial axes for these molecules.
56 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
0 200 400 600 800
0.0
0.2
0.4
0.6
0.8
1.0
Fraction of Transition Density (TD)
(A)
0 200 400 600 800
Time (fs)
(B)
0 200 400 600 800 1000
(C)
† =1
† =2
† =5
† =20
FIGURE 5.5: Fraction of transition density (TD) on specified (boxed)
molecular fragments as a function of time. Asymmetric conformations
of the molecules are shown for visualization, but the TDs of asymmetric
and symmetric conformations are contained in the averaged results.
5.3.6 Exciton Localization
A measure of exciton localization brings together the preceding analyses of TDs and
excited state dipole moments. Figure 5.7 shows a scatter plot of dipole moment ver-
sus exciton localization (Equation 5.4a) across all molecules and solvent polarities. The
dipole moment offH, Hg is inversely related to L
D
(shown in blue); an increase of the
dipole moment along the minor inertial axis (Figure 5.6B) localizes the exciton, but only
to a small extent due to the molecule’s low degree of polarizability. Oncee> 5, L
D
satu-
rates, thus defining a lower bound within this set of molecules. In the case offH, NO
2
g
andfNH
2
, NO
2
g, the excitation becomes more delocalized since functionalization and
dipole moments are aligned along the major inertial axis. ForfH, NO
2
g, L
D
gradually
increases with e, suggesting that the spatial extent of the excitation is more variable as
a function of solvent polarity (shown in green). This is not the case forfNH
2
, NO
2
g,
which experiences an upper bound on L
D
at e > 5 (shown in red); substitution with
donor-acceptor groups attached at the ends of the molecule delocalizes the excitation
so that even a small increase in e maximizes the size of the exciton. L
upper
D
L
lower
D
is
relatively small in these molecules (as it should be for small p-conjugated structures),
but this analysis uncovers qualitative differences among the excitations due to function-
alization and solvent polarity.
5.3. Results and Discussion 57
(B) R
1
= H, R
2
= H
† = 1 † = 1
Density
† = 2 † = 2
† = 5 † = 5
† = 20 † = 20
0 5 10 15
Dipole Moment (Debye)
Mean
Dipole
Moment
0 30 60 90 120 150 180
Direction (Degrees)
Mean
Direction
(C) R
1
= H, R
2
= NO
2
† = 1 † = 1
Density
† = 2 † = 2
† = 5 † = 5
† = 20 † = 20
0 5 10 15
Dipole Moment (Debye)
Mean
Dipole
Moment
0 30 60 90 120 150 180
Direction (Degrees)
Mean
Direction
(D) R
1
= NH
2
, R
2
= NO
2
† = 1 † = 1
Density
† = 2 † = 2
† = 5 † = 5
† = 20 † = 20
0 5 10 15
Dipole Moment (Debye)
Mean
Dipole
Moment
0 30 60 90 120 150 180
Direction (Degrees)
Mean
Direction
FIGURE 5.6: (A) Schematic labeling the relative direction of the perma-
nent excited state dipole moment. The left panels of (B)–(D) show his-
tograms of the magnitude of the dipole moment over all time steps and
trajectories in four different solvent environments with e =f1, 2, 5, 20g.
The right panels of (B)–(D) show directions of the dipole moment, in sim-
ilar format as the left panel. The bottom-most panels show the mean
magnitudes and directions of the dipole moments, respectively.
There is a connection between the results of Figure 5.7 and relaxation time scales of
Figure 5.4D. AlthoughfNH
2
, NO
2
g is more polar thanfH, NO
2
g, its exciton size does
not vary as much with e. Similarly, relaxation time scales offNH
2
, NO
2
g are less de-
pendent on solvent polarity compared tofH, NO
2
g. In other words, the relaxation rate
of the more polar moleculefNH
2
, NO
2
g is less affected by the presence of more polar
solvents since its exciton size is already close to maximum. In the opposite direction,
fH, Hg is the least polar molecule; exciton size and relaxation time scales are changed
only slightly due to the presence of more polar solvents. The simulations presented
here used implicit solvation, and therefore more work is needed to study this effect in
the context of explicit solvation.
5.3.7 Bond Length Alternation (BLA)
Bond length alternation (BLA) between adjacent carbon-carbon atoms,[(b
1
+ b
3
) /2]
b
2
(Figure 5.1A), was calculated to quantify the dependence of chemical substitution
58 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
16 17 18 19 20 21
Exciton Localization, L
D
0
2
4
6
8
10
12
14
Dipole Moment (Debye)
L
D
Lower Bound
L
D
Upper Bound
Increasing †
R
1
= H, R
2
= H
R
1
= H, R
2
= NO
2
R
1
= NH
2
, R
2
= NO
2
FIGURE 5.7: Excited state dipole moment versus exciton localization L
D
.
Each data point was computed as an average over all time steps and tra-
jectories in the nonadiabatic ensembles.
and solvent polarity on structural relaxation. BLAs were calculated between both outer
aromatic rings and the central ring independently (Figure 5.8).
BLA decrease in time, thus verifying that it is greater in the ground state than in the
excited state.[14, 168] As expected, BLA does not vary as a function of solvent polarity
in the apolar molecule, but does in the polar molecules.
The varying dependence of BLA with solvent polarity is a consequence of zwitte-
rions.[242] InfH, NO
2
g, BLA decreases with increasing e on the side containing NO
2
.
A zwitterion offH, NO
2
g shows bond order on the NO
2
side changing from single to
double bond character and vice versa (Figure B.7A). For the same structure, bond char-
acter on the H side is unaffected, in agreement with calculations (middle left panel of
Figure 5.8). In contrast, as a result of a zwitterion that flips bond character throughout
the entire molecule (Figure B.7B), BLA offNH
2
, NO
2
g decreases with increasing e on
both sides of the molecule (bottom panels of Figure 5.8). BLA is further reduced on
the side containing NH
2
, which is likely due to another zwitterion that exclusively flips
bond order on the NH
2
fragment, leaving the NO
2
fragment unchanged (Figure B.7B).
BLA decreases as e increases due to more solvent-stable resonance structures (zwit-
terions), where the contributions of single and double bond character become more
alike. This is not the case forfH, Hg since it lacks functional groups that can change
bond order of the adjacent carbon-carbon atoms. Both observations qualitatively agree
with simple chemical expectations. This analysis not only conveys the dependence of
chemical substitution and solvent polarity on structural relaxation but further confirms
the validity of calculations carried out with NEXMD.
5.4. Conclusions 59
0.02
0.04
0.06
0.08
0.10
0.12
R
1
= H R
2
= H
0.02
0.04
0.06
0.08
0.10
BLA ( )
R
1
= H R
2
= NO
2
0 200 400 600 800
Time (fs)
0.02
0.04
0.06
0.08
0.10
R
1
= NH
2
0 200 400 600 800 1000
R
2
= NO
2
† =1
† =2
† =5
† =20
FIGURE 5.8: Bond length alternation (BLA) as a function of time. Each
row represents a different molecule. The left and right panels show the
BLA on the side of the molecule containing R
1
and R
2
, respectively. Co-
herent oscillations in the BLA are observed within the first 100 fs, which
shows the transfer of optical energy to nuclear vibrational modes and the
ultrafast relaxation on the femtosecond time scale.
5.3.8 Relative Computational Time
The linear response solvent model added very little computational overhead to the
nonadiabatic simulations. CPU times were averaged over all trajectories of each ensem-
ble computed with and without the solvent model at e = 20 and e = 1, respectively
(Figure B.8).
3
The total CPU time increased by slightly less than 10%, which is mainly
due to a 30% increase in the excited state calculation. But compared to the large com-
putational cost of nonadiabatic couplings, this increase is relatively small–excited-state
CPU time scales linearly with the number of excited states, while nonadiabatic cou-
plings scale quadratically. Even the large computational expense of nonadiabatic cou-
plings can be alleviated by reducing the number of computed couplings between states
to a small fraction that are nearby in energy to the occupied state of the system.[218] In
this way, numerical costs of nonadiabatic couplings become practically independent on
the number of excited states.
5.4 Conclusions
Modeling solvation has obvious computational challenges due to the number of de-
grees of freedom in a solvent. Reducing the computational cost needed to model sol-
vation while still capturing the effective solvent response is a desired attribute of sol-
vated atomistic simulations. In this work, we have implemented and demonstrated the
3
CPU times do not vary among simulations withe6= 1.
60 Chapter 5. Nonadiabatic Dynamics of Solvated Push-Pullp-Conjugated Oligomers
use of an implicit solvent model in the Nonadiabatic EXcited-state Molecular Dynamics
(NEXMD) software.
The NEXMD software is an efficient framework for excited state modeling of large
molecular systems. The code calculates electronically excited states at the time depen-
dent Hartree Fock (TD-HF) or configuration interaction with singles (CIS) level com-
bined with a family of optimized semiempirical Hamiltonians. This level of theory,
merged with improved surface hopping methodologies for electronic transitions, makes
it feasible to simulate the nonadiabatic dynamics of molecules with sizes on the order
of hundreds of atoms and for time scales up to tens of picoseconds. This is a signature
property of the code enabling the simulation of large molecular systems where more
elaborate ab initio approaches would be numerically prohibitive. This work introduced
a version of NEXMD based on the semiempirical quantum chemistry (SQM) code from
AmberTools,[204] which contains a diverse set of semiempirical Hamiltonians and a
linear response conductor-like polarizable continuum model (CPCM).
Benchmarks of solvent effects against a test bed of push-pull (donor-acceptor) PPVO
derivatives were meant to both validate the use of this CPCM for future studies seeking
to utilize the code and determine the interplay of functionalization and solvent polarity
on photoinduced nonadiabatic dynamics. Our results showed that solvation can change
dynamic and static properties, where the dependence of solvent polarity is determined
by molecular polarizability due to chemical substitution. In the case of thefH, Hg
oligomer, many of the calculated properties are less dependent on solvent polarity due
to the molecule’s low degree of polarizability. On the other hand, substitution with elec-
tron donating and withdrawing groups infH, NO
2
g andfNH
2
, NO
2
g change excited
state lifetimes, exciton localization, and structural relaxation. For example, we estab-
lished how functionalization and solvent polarity affect spatial confinement of molecu-
lar excitations (Figures 5.5 and 5.7) and evolution of structural parameters such as bond
length alternation (BLA) (Figure 5.8). Of the many properties that were analyzed, a few
followed the expected trends based on chemical and physical intuition such as stabiliza-
tion of energy levels (Figure 5.3) and dependence of excited state dipole moments with
solvent polarity (Figure 5.6). Others however, such as changes in excited state lifetimes
due to solvent-induced fluctuations in energy gaps (Figure 5.4), are evidence of the com-
plex behavior that solvation can have on nonradiative relaxation. These results suggest
that theoretical interpretations may be altered depending on whether or not the effects
of solvation are strong. With an integrated implicit solvent model, the NEXMD software
has further become an appealing framework for realistic modeling of the photoinduced
nonadiabatic dynamics in a diverse set of solvated chemical systems.
This work both demonstrates an important extension to NEXMD’s capabilities and
sets the stage for other advanced features of the code. We have shown that CPCM mod-
eling adds little numerical overhead to the simulations, thus presenting an efficient way
to model solvent effects in extended systems. Our future work will include implemen-
tation of state-specific[191, 192] and non-equilibrium[217] solvent models in the context
of nonadiabatic dynamics. The use of these models is relevant since the dynamics fol-
lowing photoexcitation proceeds through many excited states and are generally out of
equilibrium (i.e. the solute and solvent polarizations are not equilibrated with one an-
other). However, more work is needed to determine how important such effects are on
dynamics as they likely depend on the solute-solvent system of interest. Furthermore,
an important property of the solvent model used for this study is that the coefficient of
5.4. Conclusions 61
induced polarization goes as(e 1) /e; solvent effects asymptotically decrease with in-
creasinge as evidenced by the relaxation time scales of Figure 5.4D. Whether or not high
dielectric constants significantly change dynamics requires further investigation, which
is why extending this work to explicit solvation is important. Indeed there has been an
ongoing effort to introduce a QM/MM solvent model to NEXMD, as implemented in
AmberTools. The goal is to evaluate implicit and explicit solvation on dynamics, and to
assess the validity of each model based on chemical composition of the solute-solvent
system.

63
Chapter 6
NEXMD Modeling of
Photoisomerization Dynamics of
4-Styrylquinoline
Reprinted with permission from J. Phys. Chem. A Just Accepted Manuscript DOI:
10.1021/acs.jpca.8b09103. Copyright 2018 American Chemical Society.
6.1 Introduction
On a molecular scale, optical energy can be converted into mechanical and chemical
energy through isomerization.[162] In many organic molecules, this conversion man-
ifests itself through the trans-to-cis or cis-to-trans conformational change around C=C
bonds[243] and involves pathways mediated by conical intersections.[244] Studying
this process is key to understanding many common yet complex processes including
vision,[245, 246] ion pumping,[247] and organic synthesis.[248] Photoisomerization is
also important for technological applications such as biomimetic materials,[249, 250]
conductive polymers,[251] and optical data storage.[252–254]
Computational modeling of isomerization in organic molecules not only reveals
how this process occurs in nature but also helps identify ways of controlling the dynam-
ics via external stimuli.[255] One such example is solvation, as it has shown to influence
isomerization both in terms of time scales and reaction pathways.[256–264] Molecular
dynamics simulations can be used to determine the effects of solvation by capturing
the changes in molecular geometry and potential energy surfaces (PESs) involved in the
reaction.[265] Other external factors (or tunable stimuli) that can influence isomeriza-
tion include photoexcitation energy[261, 266] and the thermostat.[267] In this paper, we
study the effects of these three external stimuli on the trans-to-cis photoisomerization of
4-styrylquinoline using the Nonadiabatic EXcited-state Molecular Dynamics (NEXMD)
software.
4-styrylquinoline is an organic conjugated molecule that isomerizes upon irradiation
with laser light.[268]. Compared to other commonly studied molecules that undergo
isomerization such as azobenzenes,[45, 103, 161, 269, 270] stilbenes,[103, 271–274] and
ethylenes, [162, 260, 274, 275] 4-styrylquinoline is slightly larger in size; it might be a dif-
ficult task for a first-principles level of theory to compute the excited state dynamics due
to the computational costs. Such numerically prohibitive theories include grid-based
methods or multiconfigurational time-dependent Hartree-Fock.[162] Instead, we run
64Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
our simulations with NEXMD, which uses numerically efficient semiempirical Hamil-
tonian models, and a swarm of classical nuclear trajectories that undergo nonadiabatic
transitions between excited states via surface hopping.[43, 157, 159, 193–200] But the
more compelling reason to study 4-styrylquinoline is its unique isomerization process,
as it has been experimentally shown to proceed from its trans conformation to its final
stable product along one of two different reactions pathways that vary by the number
of photons absorbed. Studying 4-styrylquinoline thus poses the opportunity to com-
plement experimental findings, provide a detailed investigation of some of the external
factors influencing its photoisomerization, and benchmark the capabilities of NEXMD
against a relatively difficult molecule to model from first principles.
6.2 Computational Details
6.2.1 Ground State Sampling
The ground state geometries of t-SQ, c-SQ, DHBP , and BP were optimized with the
AM1 semiempirical Hamiltonian.[208] Born-Oppenheimer molecular dynamics were
then performed on the ground state using AM1. An implicit linear response solvent
model was coupled to the dynamics to simulate the electrostatic effects of solvation.[191]
The induced polarization due to Coulombic interactions at the solute-solvent boundary
increases with the dielectric constant of the solvent e and obeys f (e) = (e 1) /e. The
interaction at the solute-solvent boundary produces an electrostatic potential, which is
added to the solute’s Hamiltonian. The additional effect of the induced polarization
progressively decreases as e increases since f (e) asymptotically approaches one.[43,
276] We limit ourselves to two solvents with low and high polarity–an apolar solvent
of n-hexane (e 2) and a polar solvent of ethanol (e 25). These two solvents were
also used in the associated experimental work (Ref. [268]). Two separate ground state
trajectories for each species that are shown in Figure 6.1 were simulated, differing only
in solvent environment. Ground state trajectories were 1 ns in length with a 0.5 fs time
step. Nuclear dynamics were described by the Langevin equation at a temperature set
to 300 K and friction parameter of 20 ps
1
.[43] Snapshots (i.e., coordinates and veloci-
ties) of each species were recorded throughout these trajectories and were used as initial
conditions for all subsequent NEXMD simulations of the excited states.
6.2.2 Optical Absorption Spectra
Singlepoint calculations were performed at the ground state geometries to deter-
mine vertical excitation energies and oscillator strengths. Excited states were calculated
using the collective electronic oscillator (CEO) method combined with AM1.[202, 203]
Implicit solvation was also included in singlepoint calculations. About 10 excited states
were required to model absorption up to 200 nm. Oscillator strengths were Gaussian-
broadened at the excitation energies with an empirical standard deviation of 0.15 eV .
For each environment, an average over all absorption spectra, one for each snapshot,
formed the convoluted theoretical absorption spectrum (Figure 6.2).
6.2. Computational Details 65
6.2.3 Photoexcitation to S
1
– Adiabatic Dynamics
The photoisomerization process involves the conical intersection between the first
singlet excited state (S
1
) and the ground state (S
0
). Therefore, we begin our analysis by
performing adiabatic dynamics on S
1
in each solvent environment. Following vertical
excitation to S
1
, trajectories were propagated for a maximum of 50 ps with a classical
time step of 0.50 fs using the same Langevin thermostat as the ground state trajectories.
A cartoon schematic of this low-energy (i.e., S
0
! S
1
) simulation is shown in Figure 6.3.
6.2.4 Analysis of NEXMD of Geometries
In order to determine the reaction pathways, we analyzed the geometries along the
adiabatic NEXMD trajectories described in the section 6.2.3. We combined the geome-
tries of several dynamical simulations that were initialized with t-SQ and c-SQ ground
state geometries. Geometries sampled from adiabatic trajectories on S
1
were analyzed
with a Ramachandran diagram[277] off
C=C
andf
CC
(Figure 6.4A). Changes in the po-
tential energy manifests itself in rotation about these two dihedral angles. S
1
adiabatic
dynamics trajectories starting from t-SQ geometries were used to sample geometries
with f
C=C
> 90

. The conical intersection of p-SQ, located at f
C=C
90

(Figure 6.3),
was approached but not passed through (i.e., f
C=C
< 90

). Another set of simulations
starting from c-SQ provided sampled initial geometries with f
C=C
< 90

, including
DHBP . Rotational symmetry around both f
CC
and f
C=C
results in chemically simi-
lar structures. For f
CC
, a C
2
symmetry axis makes the following angles equivalent:
f
CC
=180

, 0

, and 180

. Likewise, structures with f
C=C
=90

and 90

are also
chemically similar. Figure 6.4B labels the different conformers on the Ramachandran
diagram (Figure 6.5).
Molecular orbitals (MOs) were used to describe chemical effects that dictate the re-
action pathways. Since geometries of interest are those that are generated along trajec-
tories from t-SQ to DHBP , passing through both p-SQ and c-SQ, a continuous scan was
performed by changing f
C=C
from 180

(t-SQ) to 0

(c-SQ). This procedure generated
the potential energy surface (PES) of isomerization, and was accomplished by continu-
ously varying the distance between two chosen carbon atoms of SQ (Figure 6.4A). When
the interatomic distance between these two atoms is large, the only plausible conforma-
tion is t-SQ. As interatomic distance decreases, p-SQ, c-SQ, and DHBP become favor-
able.
6.2.5 Isomerization Rate
Nonadiabatic transitions to S
0
are not explicitly simulated in NEXMD as a result
of the inability to describe crossings between a multi-reference excited state (CIS) and
a single-reference ground state (Hartree-Fock).[278] Instead, we apply a simple model
based on energy gaps to qualitatively describe S
1
! S
0
transitions. Since nonadiabatic
coupling scales inversely with energy gap, transitions are very likely to occur at geome-
tries near p-SQ (located nearf
C=C
90

of Figure 6.6). In general, however, trajectories
can transition at finite energy gaps before reaching the conical intersection. We set a
DE = 1.0 eV (Figure 6.3) threshold on the energy gap to qualitatively describe the rate
at which trajectories approach the conical intersection and subsequently transition to
66Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
S
0
.
1
A similar procedure has been applied in works pertaining to photodissociation dy-
namics.[163] Figure 6.7 shows the fraction of trajectories with at least one occurrence
where the energy gap fell below 1.0 eV and thus can be interpreted as the fraction of
trajectories that have reached the conical intersection.
6.2.6 High-Energy Photoexcitation – Nonadiabatic Dynamics
In order to understand the effect of excitation energy on isomerization, we perform
nonadiabatic dynamics from an initial high-energy state, S
n
(n > 1). The excess elec-
tronic energy must first be converted to vibrations during the S
n
! S
1
internal conver-
sion process. On the basis of the absorption spectra of Figure 6.2, the laser excitation
energy was chosen to excite the bright peak at approximately 260 nm. Nonadiabatic dy-
namics used the same 500 snapshots from t-SQ ground state trajectories as those used
for adiabatic dynamics on S
1
. The initial excited state, S
n
, was chosen according to
the absorption spectrum of the corresponding molecular structure, such that excited
states with larger oscillator strengths were more populated than those with lower os-
cillator strengths. This procedure models a photoexcited wave packet sampling the
phase space of different nuclear configurations at room temperature. We find that S
6
is
the most populated state, but due to thermal fluctuations, states in close vicinity to S
6
are also populated due to diabatic state crossings. Trajectories were propagated for 1
ps, with a classical time step of 0.10 fs (nuclei) and quantum time step of 0.02 fs (elec-
trons), using the same Langevin thermostat as the ground state trajectories. Nonadi-
abatic transitions between excited states were modeled with Tully’s fewest switches
surface hopping.[5] Electronic decoherence[41] and transitions at unavoided (trivial)
crossings[86] were treated with established methods within the NEXMD framework.
A cartoon schematic of this high-energy simulation (labeled Nonadiabatic Dynamics) is
shown in Figure 6.3.
6.2.7 Constant Energy Dynamics
In order to assess the effect of the thermostat on nonradiative relaxation from S
m
to
S
1
and subsequent isomerization, we performed the same aforementioned nonadiabatic
simulations from the initial high-energy S
m
state with the thermostat turned off, corre-
sponding to constant energy Newtonian dynamics. For these simulations, we used the
same ground state sampling as preceding simulations, thus making it feasible to isolate
the effects of the bath. For both low- and high-energy simulations (Figure 6.3), the sol-
vent model was turned off (i.e.,e= 1). Upon photoexcitation, it is expected that optical
energy will be converted into nuclear vibrational energy. In a simulation with nuclei
coupled to the Langevin equation, a portion of the vibrational energy is dissipated to
the bath. Since the dissipation is absent in Newtonian simulations, the effect of the bath
on the isomerization can be determined.
1
The S
0
/S
1
conical intersection may not be accurately described with the NEXMD level of theory. Set-
ting a 1.0 eV threshold on the energy gap ensures that dynamics are sufficiently far from the conical inter-
section. Nonadiabatic transitions are likely to occur at much smaller energy gaps, typically less than 0.1
eV .[43]
6.3. Results and Discussion 67
Φ
"#"
Φ
"$"
FIGURE 6.1: Photoexcitation of t-SQ results in a transformation to DHBP ,
with c-SQ being an intermediate in the reaction. Once DHBP is formed, it
is then possible to undergo a final transformation from DHBP to BP with
the addition of oxygen.
6.3 Results and Discussion
The following section is organized as follows. First, we provide an overview of the
conformations of SQ during isomerization (Figure 6.1). We then compute the optical
absorption spectra and compare the theoretical results to the experimental absorption
maxima (Figure 6.2). Finally, we present results of the dynamical simulations in the
presence of different environments. These simulations are carried out at both low and
high photoexcitation energies, corresponding to adiabatic and nonadiabatic dynamics,
respectively (Figure 6.3). As part of our discussion, we describe the reaction pathways
by analyzing SQ conformations and the potential energy surfaces (PESs) involved in
the isomerization process (Figure 6.5). We also measure the isomerization rate, medi-
ated by transitions through the conical intersection between the first singlet excited state
(S
1
) and the ground state (S
0
) (Figure 6.7 and Table 6.1). For the high-energy (nonadi-
abatic) simulations, we compute nonradiative relaxation rates and show that electronic
relaxation to S
1
is primarily coupled to carbon-carbon stretching modes (Figure 6.8).
Computational details of all simulations are provided in section 6.2.
6.3.1 4-styrylquinoline (SQ)
The experimental work of Ref. [268] has described the conformations of SQ during
isomerization. The reaction from the trans conformer (t-SQ) to its final stable product is
shown in Figure 6.1. t-SQ is the most stable in the ground state because of lack of steric
hindrance between the quinoline and phenyl rings. When the central double bond be-
tween the rings (labeled f
C=C
) is rotated by 90

, the quinoline and phenyl rings are
in a nonplanar orientation. This structure is denoted as the perp conformer (p-SQ) (not
shown in Figure 6.1, but will be discussed further). Rotatingf
C=C
by another 90

gener-
ates the cis conformer (c-SQ). This process is met with significant destabilization due to
steric hindrance between the quinoline and phenyl rings. Simultaneous rotation of the
CC bond adjacent to C=C and the phenyl ring (labeled f
CC
) increases the distance
between hydrogen atoms on the rings, which alleviates torsional strain and results in a
structure with nearly perpendicular quinoline and phenyl rings. Rotation about f
CC
,
such that both rings are nearly coplanar, can only be accomplished with a change in hy-
bridization of one carbon atom in each ring, thus forming dihydrobenzophenanthridine
(DHBP). DHBP becomes benzophenanthridine (BP) with the addition of oxygen.
68Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
6.3.2 Optical Absorption Spectra
Optical absorption spectra of the different species, calculated in both n-hexane and
ethanol, are shown in Figure 6.2. The lineshape labeled by total is the sum of all species,
which we directly compare to the experimental absorption maxima of Ref. [268], shown
as dashed lines in Figure 6.2. Each absorption peak of Ref. [268] is assigned to a differ-
ent species. The peaks of interest are the three low-energy peaks, spanning the range
of 500 250 nm. The low- and high-energy peaks are assigned to DHBP and BP , re-
spectively. On the basis of the theoretical results of Figure 6.2, these assignments are
in excellent agreement. The peak between 350 300 nm is experimentally assigned to
t-SQ. Although t-SQ theoretically absorbs within this range, Figure 6.2 shows that c-
SQ and DHBP may also contribute to the absorption; reduction of the 350 300 nm
peak may not be from the decay of t-SQ alone, thus making it difficult to determine the
trans-to-cis isomerization rate from an experimental perspective.
Absorbance
(A) n-hexane
t-SQ
c-SQ
DHBP
BP
total
t-SQ (exp.)
DHBP (exp.)
BP (exp.)
200 250 300 350 400 450 500 550 600
Wavelength (nm)
(B) ethanol
FIGURE 6.2: Optical absorption spectra of t-SQ, c-SQ, DHBP , and BP in
(A) n-hexane and (B) ethanol. Wavelengths of peak maxima from experi-
mental absorption bands are shown with dashed vertical lines. Theoreti-
cal spectra were blueshifted by 0.45 eV .
In order to support the use of the semiempirical level of theory carried out with
6.3. Results and Discussion 69
NEXMD, theoretical and experimental absorption spectra are compared to each other.
The total spectrum was uniformly blueshifted, such that the low-energy peak of DHBP
matched the experimental value at 422 nm. By doing so, the high-energy peak of BP was
shifted to about 267 nm. The experimental wavelength of the corresponding DHBP peak
is reported to be 266 nm. Errors ranging between 0.10 0.20 eV are seen in the peaks
assigned to t-SQ in both n-hexane and ethanol. The ethanol spectra are redshifted rela-
tive to those in n-hexane, which is expected due to stabilization in polar environments.
The main theoretical DHBP peak is redshifted from about 422 to 429 nm ( 0.05 eV
redshift), while experimentally measured maxima are 422 and 443 nm ( 0.14 eV red-
shift), respectively. Despite consistent shifts in the data between theory and experiment,
the semiempirical level of theory used in the present study appears to provide an accu-
rate description of excited states. A further comparison of the absorption spectra using
density functional theory (DFT) is presented in Appendix C (Figures C.1 and C.2).
6.3.3 Photoisomerization Reaction Pathway
The reaction pathway for the photoisomerization process was determined from the
geometries collected from NEXMD trajectories. A schematic of the dynamical simu-
lations that were initialized with low and high photoexcitation energies is shown in
Figure 6.3. In the current subsection, we describe the isomerization pathways in terms
of the low-energy simulation since the process itself occurs between S
1
and S
0
. For this
analysis, conformations of SQ are labeled by the two dihedral angles, f
CC
and f
C=C
,
as they are the degrees of freedom that predominantly dictate the potential energy of
the system (Figure 6.4A). Figure 6.4B represents the different conformers in terms of a
phase diagram, which is more formally known as a Ramachandran diagram.[277]
Figure 6.5 shows the Ramachandran diagram that was made using geometries from
dynamical simulations on S
1
. Geometries were compared to those generated by the
PES scan (Figure C.3). General agreement between the structures allows us to use
scanned geometries for MOs. The predominant transition contributing to the S
1
exci-
tation is p! p

between the highest-occupied molecular orbital (HOMO) and lowest-
unoccupied molecular orbital (LUMO). These MOs are used to describe the reaction of
Figure 6.1 with regards to excitation and relaxation during isomerization and subse-
quent formation of DHBP (Figure C.4).
The following discussion is in reference to Figure 6.5 the MOs presented in Ap-
pendix C (Figure C.4). t-SQ minimizes steric hindrance and is a local minimum in the
ground state. Photoexcitation of t-SQ (I) creates an unstable nodal structure in the or-
bital, resulting in a local energy gap maximum. This nodal structure places strain on the
C=C bond, which is alleviated with a change in f
C=C
. The molecule twists from t-SQ
to p-SQ (II). This process reduces p-p overlap in the C=C bond, which shifts electron
density from the central bond to the adjacent phenyl ring.[279] For p-SQ, the HOMO is
localized on the quinoline ring while the LUMO is localized on the phenyl ring. After
f
C=C
reaches 90

, the molecule transitions to the ground state through the conical in-
tersection (III), and electron density simultaneously moves back to the quinoline ring.
The ground state no longer has a node in the central C=C bond. At this point, there are
two distinct pathways that are energetically accessible. The molecule may twist toward
c-SQ (IV
a
) or directly form DHBP (IV
b
). For the former, the steric hindrance between
the quinoline and phenyl rings requires f
CC
to rotate 90

. A simultaneous rotation of
70Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
TABLE 6.1: Time evolution associated to the fraction of trajectories evolv-
ing on S
1
that encountered an energy gap E
1
E
0
< 1.0 eV . Nuclei evolve
according to Langevin (at 300 K) or energy-conserving Newtonian dy-
namics. Langevin dynamics data were fit to A B exp(t/t). New-
tonian dynamics were only computed with e = 1 and the data were
fit to a double exponential function of the form A B exp(t/t
1
)
C exp(t/t
2
), wheret
1
andt
2
are fast and slow time scales, respectively.
Time scales shown are in ps.
Photoexcitation Energy n-hexane ethanol e= 1 (Langevin) e= 1 (Newtonian)
Low Energy – S
1
18.8 8.97 19.9 2.26, 20.2
High Energy – S
m
(4.5 eV) 16.4 8.90 20.7 0.991, 6.91
f
C=C
generates c-SQ. The second pathway is a direct formation of DHBP through a 90

rotation off
C=C
. Hence, the second pathway is a one photon process.
In the case of IV
a
, c-SQ is a local minimum on the ground state, and therefore another
excitation is required to undergo a transition to DHBP (V). In the excited state, c-SQ is
a local maximum and DHBP is a local minimum; formation of a bond connecting the
quinoline and phenyl rings creates DHBP (VI). The two carbon atoms of the newly
formed bond change hybridization from sp
2
to sp
3
character. The final DHBP product
is stable following subsequent relaxation back to the ground state (VII). The driving
force of the reaction and fundamental conformations are well replicated using NEXMD.
The energy profile of the reaction suggests that there are two pathways from t-SQ to
DHBP that differ by the number of photons absorbed (Figure 6.1).
6.3.4 Trans-to-Cis Photoisomerization Dynamics
Adiabatic dynamics on the S
1
PES were initialized with t-SQ geometries. Figure 6.6A
shows energies of the ground (E
0
) and first excited state (E
1
) as a function off
C=C
taken
from every geometry and at every time step along all trajectories of the ensemble. The
energy gaps E
1
E
0
were also plotted as a function of f
C=C
(Figure 6.6B). The positive
correlation between the energy gap and f
C=C
shows the progression of the molecule
towards p-SQ as the energy gap decreases. Increasing solvent polarity both stabilizes
the energy levels (Figure 6.6A) and decreases the energy gaps in more polar solvents
(Figure 6.6B); E
0
decreases by approximately 0.25 eV from e = 1 to ethanol while E
1
decreases by approximately 0.50 eV . A significant change in the energy gap affects nona-
diabatic transitions, S
1
! S
0
, thus influencing the isomerization rate.
The plots of Figure 6.7 show the trajectory ensemble reaching the conical intersection
and provide a time scale for the trans-to-cis conversion. Table 6.1 shows time scalest of
trans-to-cis from the following stimuli: solvation, photoexcitation energy, and constant
temperature thermostat. Isomerization rate increases with solvent polarity; the relax-
ation rate is approximately twice as fast in the polar solvent than the apolar solvent. In
Ref. [268], the magnitude of the t-SQ absorption peak decreases in time, thus showing
the decay of t-SQ. t-SQ decays faster in ethanol than n-hexane, in agreement with calcu-
lations. It is also interesting that the ratio of the relaxation time scales are in agreement,
i.e., t
nhexane
/t
ethanol
2.1 (theory) and 1.6 (experiment) (see further discussion in
Appendix C, Figure C.5). With regard to photoexcitation energy, we find that it does
6.3. Results and Discussion 71
not affect isomerization rate in different solvents, assuming nuclei are coupled to a ther-
mostat. Isomerization rate does however depend on photoexcitation energy if nuclear
dynamics evolve in vacuum as opposed to a thermostat. However, this dependence is
not so surprising since dynamics in vacuum are energy conserving and therefore excess
photoexcitation energy (i.e. S
m
S
1
) will be transferred to nuclear kinetic energy.
For simulations in vacuum with the thermostat absent (Figure 6.7B, Newtonian), the
isomerization rate took the form of a double exponential function with fast and slow
time scales. We partition the trajectories into two sets, one for each rate, according to
a threshold, which was chosen based on the exponential fit. We find that the differ-
ence in isomerization rate is related to energy redistribution of the molecule, where the
total amount of initial energy present is dependent on the potential energy of the start-
ing configurations. For trajectories that start with f
CC
90

, potential energy is the
highest, while those with f
CC
0

, potential energy is the lowest. The p-orbitals
on the CC bond are the same phase in t-SQ; rotation about this bond destabilizes the
structure. Ramachandran diagrams of each set of trajectories shows the distribution of
f
CC
being narrower for the slow trajectories than the fast trajectories (Figures C.6 and
C.7). Trajectories that begin in an unstable conformation transfer energy between po-
tential and kinetic. Rapid motion of the molecule aroundf
CC
due to this excess energy
eventually leads to a faster transformation from t-SQ to p-SQ through a rotation around
f
C=C
. The distribution off
CC
for a large number of conformations in the slow trajecto-
ries remains roughly planar, which is the most stable conformation. Energy fluctuations
due to the bath eliminates these two distinct time scales (Figure 6.7 and Table 6.1), thus
showing that the bath lessens the dependence of the isomerization process on initial
conditions.
6.3.5 Nonradiative Relaxation Following High-Energy Photoexcitation
Figure 6.8A shows the evolution of the adiabatic state populations for S
1
and S
m
fol-
lowing photoexcitation to the initial high-energy state S
m
, where m corresponds to the
range of states that is an average of 4.5 eV above S
0
. Following high-energy photoexci-
tation, the molecule relaxes nonadiabatically to S
1
. The initial S
m
state persists for about
100 to 150 fs in e = 1 and ethanol, respectively, after which S
1
is primarily populated
for the remainder of the 1 ps dynamics. The molecule remains t-SQ throughout the 1 ps
nonadiabatic dynamics (inset of Figure 6.8A), which demonstrates that isomerization
is roughly independent of photoexcitation energy; relaxation back to S
1
occurs on an
ultrafast time scale (i.e., on the order of 100 fs) whereas isomerization occurs on a much
longer time scale (Table 6.1).
Vibrational dynamics due to the high-energy photoexcitation is apparent from the
bond length alternation (BLA) between the adjacent carbon-carbon atoms connecting
the quinoline and phenyl rings reflecting CC and C=C stretching motions.[43] The
evolution of the average value of BLA is depicted in Figure 6.8B. Coherent oscillations,
with relatively large amplitude, occur in the first 60 fs (inset of Figure 6.8B). These coher-
ent vibronic dynamics are a result of the nonadiabatic transitions between excited states
that are associated to molecular wavefunctions with varying number of nodes.[280] The
amplitude of the oscillations in Figure 6.8B subsequently decreases in time as t-SQ occu-
pies lower-lying excited states. Besides a dampening of the coherent oscillations, BLA
72Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
decreases in time because the molecular wavefunction is more delocalized in lower-
lying states and has fewer nodes. Small values of BLA are generally associated to more
effective p-conjugation and corresponds to increased exciton delocalization. BLA fur-
ther decreases in more polar solvents due to solvent-stable resonance structures, in-
dicative of the increased conjugation; t-SQ becomes polarized since nitrogen is electron
withdrawing and as a result, bond orders of the adjacent carbon-carbon atoms may
change from single-like to double-like in character and vice versa. Finally, further vali-
dation of NEXMD is evidenced by calculating a mass-weighted velocity autocorrelation
function for each nonadiabatic simulation (Figure C.8); spectral peaks are broadened for
simulations carried out in a constant temperature thermostat as opposed to vacuum.
6.4 Conclusions
We used the Nonadiabatic EXcited-state Molecular Dynamics (NEXMD) software
to study the photoisomerization of a conjugated molecule, 4-styrylquinoline (SQ), in
different environments and with different initial conditions. The study confirms that
NEXMD simulations are capable of reproducing experimental observations for the com-
plex photoisomerization process involving conical intersections with the ground state.
NEXMD[13, 14, 43] is a computationally efficient theoretical framework that enables
modeling the nonadiabatic dynamics on realistic length and time scales (hundreds of
atoms and tens of picoseconds). For example, we showed that NEXMD is able to re-
cover an energy profile (Figure 6.5) that shows two different reaction pathways to the
final stable product DHBP (Figure 6.1), involving a single photon or a sequential ab-
sorption of two photons. Indeed, Ref. [268] has reported the possibility of a one photon
reaction from t-SQ to DHBP . This result, in conjunction with agreement in absorption
properties (Figure 6.2 and Appendix C), validates the software’s predictive capability
for our reference molecule, SQ.
We investigated the effect of external stimuli (or tunable parameters) on the photoi-
somerization of SQ. Specifically, solvation, photoexcitation energy, and thermostat were
considered. Using an implicit linear response solvent model (i.e., the conductor-like po-
larizable continuum model), we found that isomerization (specifically, t-SQ! p-SQ) is
twice as fast in ethanol than it is in n-hexane (Figure 6.7A), which is a consequence of
further stabilization of S
1
relative to S
0
(ca. 0.25 eV). In the case of varying photoexcita-
tion energy, dynamics beginning on S
1
versus a higher-lying excited state S
m
(due to a
4.5 eV laser excitation) did not significantly affect isomerization since relaxation to S
1
is
on the order of 100 fs whereas isomerization, which occurs on S
1
, is on the order of tens
of ps (Table 6.1). The dynamics in vacuum shows that trajectories approach the conical
intersection of p-SQ using two distinct pathways with fast and slow time scales (Fig-
ure 6.7B). Our analysis of the geometries from these two subsets shows that rotation of
the phenyl ring destabilizes the molecule, leading to a faster transformation from t-SQ
to p-SQ through rotation of f
C=C
. Our results show that the thermostat modulates the
dynamics in such a way that minimizes the dependence on initial conditions.
Future work will extend NEXMD’s capabilities for modeling complex molecular
processes. For example, the solvation model of this work assumes that solute and sol-
vent polarizations are equilibrated with one another. A natural extension of this work
would be to determine how nonequilibrium[217] or state-specific[191, 192] solvation
6.4. Conclusions 73
affects photoisomerization. Furthermore, since polarization is one of the many interac-
tions that can take place, explicit solvation is likely important for more chemical insight.
There is an ongoing effort to implement a QM/MM solvent model to NEXMD. Other
work of interest would be modeling the coherent control of quantum dynamics with
laser pulses.
74Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
180 90
0
E
TRANS CIS
S
5+
S
4
S
3
S
2
S
1
Ground State
C=C
(°)
FIGURE 6.3: Cartoon schematic of NEXMD simulations of the SQ
molecule. Simulations were performed at low (hn) and high (hn
0
) pho-
toexcitation energies beginning with t-SQ in the ground state. The
schematic only shows t-SQ! c-SQ. The low- and high-energy simula-
tions are modeled as adiabatic and nonadiabatic dynamics, respectively.
While both processes are nonadiabatic in practice, NEXMD cannot explic-
itly simulate nonadiabatic transitions between S
1
and S
0
. As a result, dy-
namics always end on S
1
. Further computational details of the NEXMD
theory as well as the approach taken to qualitatively model S
1
! S
0
tran-
sitions are available in section 6.2.
6.4. Conclusions 75
FIGURE 6.4: (A) Labeling of f
C=C
and f
CC
dihedral angles. The two
carbons atoms colored in purple were used for the PES scan. (B) Differ-
ent conformations of SQ during isomerization that are identified by the
dihedrals.
76Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
FIGURE 6.5: Ramachandran diagram of the low-energy excitation to S
1
in apolar solvent. (A) shows relative energy of the ground state S
0
and
(B) shows the relative energy of the gap E
1
E
0
. Arrows are included to
show photoexcitation, nonradiative relaxation, and movement along the
PESs. I: S
0
! S
1
photoexcitation at t-SQ. II: t-SQ! p-SQ in S
1
. III: S
1
!
S
0
nonadiabatic transition at p-SQ. IV
a
: p-SQ! c-SQ in S
0
. IV
b
: p-SQ!
DHBP in S
0
. V: S
0
! S
1
photoexcitation at c-SQ. VI: c-SQ! DHBP in S
1
.
VII: S
1
! S
0
nonadiabatic transition at DHBP .
6.4. Conclusions 77
FIGURE 6.6: (A) Energies of the ground (E
0
) and first excited state (E
1
) as
a function off
C=C
. (B) Energy gap E
1
E
0
as a function off
C=C
.
78Chapter 6. NEXMD Modeling of Photoisomerization Dynamics of 4-Styrylquinoline
0.0
0.2
0.4
0.6
0.8
1.0
(A)
† =1
n-hexane
ethanol
fit
0 10 20 30 40 50
Time (ps)
0.0
0.2
0.4
0.6
0.8
Frac. of Trajs. w/ E
1
−E
0
<1.0 eV
(B)
S
1
Langevin
S
1
Newtonian
S
m
Newtonian
FIGURE 6.7: Fraction of trajectories evolving on S
1
that encountered
an energy gap E
1
E
0
< 1.0 eV . (A) Simulations were performed in a
Langevin thermostat. (B) Same as part (A) but performed withe= 1.
6.4. Conclusions 79
0.0
0.2
0.4
0.6
0.8
1.0
Population
(A)
S
1
S
m
† =1
n-hexane
ethanol
Newtonian († =1)
0 200 400 600 800 1000
Time (fs)
0.00
0.05
0.10
0.15
BLA ( )
(B)
0 10 20 30 40 50 60
0.06
0.07
0.08
0.09
0.10
0.11
0.12
FIGURE 6.8: (A) Excited state populations as a function of time. The
populations labeled by S
m
are the initial excitations–the combination of
all high-lying excited states with non-zero population at time t = 0 fs.
(B) The bond length alternation (BLA) of adjacent carbon-carbon atoms
connecting the quinoline and benzene ring systems. BLA is defined as
[(b
1
+ b
3
) /2 b
2
], where b
1
and b
3
are CC bonds and b
2
is the C=C
bond. The inset shows coherent oscillations of the BLA within the first 60
fs.

81
Chapter 7
Photoactive Excited States in
Explosive Fe (II) Tetrazine
Complexes: A Time-Dependent
Density Functional Theory Study
Reprinted with permission from J. Phys. Chem. C 2016, 120, 50, 28762-28773. Copy-
right 2018 American Chemical Society.
7.1 Introduction
Coordination complexes have gained worldwide attention due their rich photophys-
ical properties. Fe (II) and Ru (II) polypyridines have been utilized for light-harvesting
in dye-sensitized solar cells[281–292]. Cyclometalated Ir (III) complexes have seen use
in organic light-emitting diodes,[293, 294] light-emitting electrochemical cells,[295–303]
photocatalysis,[293, 297, 304–307] and even biotechnology.[308–311] Needless to say,
coordination complexes have been introduced in a variety of applications. One major
advantage is their synthetic flexibility, which makes tuning their photophysical prop-
erties possible. A new application utilizing coordination complexes is the design of
photoactive explosives.
Current methods used to detonate high explosives rely on mechanical or electri-
cal initiation and are susceptible to accidental initiation from mechanical or electrical
insults. Optical initiation is a promising alternative that offers enhanced safety.[164,
312] Replacing electrical components with fiber optics eliminates the susceptibility to
electrical insult, while replacing sensitive primary explosives with less sensitive pho-
toactive secondary explosives reduces the susceptibility to mechanical insult. In spite
of these advantages, optical initiation systems have been limited by a lack of suitable
photoactive explosives.[313] Sensitive primary explosives such as lead azide have low
laser-initiation thresholds but are too dangerous to employ,[314] while less sensitive sec-
ondary explosives such as pentaerythritol tetranitrate (PETN), 1,3,5-trinitroperhydro-
1,3,5-triazine (RDX), and octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) have laser
initiation thresholds that require too much energy to be of practical use.[315, 316] Pri-
mary explosives that are less sensitive than lead azide have been proposed, but with
initiation thresholds in the NIR at energy densities several orders of magnitude more
than that of lead azide.[317–320] Decreasing the density of PETN can reduce the laser
82
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
initiation threshold, but also renders the material more mechanically sensitive. Doping
PETN with an optically absorbent material such as metal nano-particles can further re-
duce the initiation threshold to a small degree, but does not address the safety concerns.
Recently reported Fe (II) coordination complexes with tetrazine ligands absorb NIR
light.[50, 51] Tetrazines have rich spectroscopic properties both as organic compounds[321]
and as ligands in coordination complexes.[322, 323] The number of CN and NN
bonds in these ligands increases their heat of formation compared to those of conven-
tional carbon-based molecules, thus making them more energetic.[47, 324–326] Their
spectroscopic properties in combination with their low laser-initiation thresholds, me-
chanical sensitivity and explosive performance make them viable laser-initiated sec-
ondary explosives.[51] Recent works outline the syntheses and characterizations of many
high explosive materials[327, 328] and in particular, those of Fe (II) tetrazine complexes
with metal-to-ligand charge transfer (MLCT) character.[50, 51, 329, 330] It is beneficial
to further elucidate structure-property relationships characterizing the low-energy CT
band from a theoretical perspective to enhance control over the initiation threshold.
Molecular structure influences absorption and can be altered to tailor-design ener-
getic materials. A recent theoretical work has studied the effect of molecular structure
on linear and nonlinear optical response in small, conjugated energetic molecules.[53] In
this chapter, time-dependent density functional theory (TD-DFT)[331] is used to study
linear absorption in Fe (II) complexes with different ligand architectures. TD-DFT gener-
ally offers an accurate description of the excited-state electronic structure of large molec-
ular systems.[55, 332] Due to the availability of high power Nd:YAG lasers, the most
practical wavelength for initiation is 1064 nm. By studying how these architectures af-
fect absorption, one can uncover design principles, leading to a class of explosives with
the desired optical response. Most importantly, tuning the low-energy CT band is rele-
vant to many technological applications that utilize the optical properties of octahedral
transition-metal complexes.
7.2 Methods
7.2.1 Computational
Absorption spectra were calculated with various levels of theory and compared to
experimental data. A detailed comparison of these methods are presented and dis-
cussed in section 7.3.1. Overall, the TPSSh density functional[333] and 6-311G basis
set[334] showed the best performance. We have chosen this combination to optimize
ground state geometries and calculate vertical excitation energies for all molecules in
this study. No symmetry constraints were imposed on optimization. Each vertical ex-
citation is represented as a broadened Gaussian with a standard deviation of 0.15 eV .
A total of 70 singlet excited states were requested in the TD-DFT calculation and used
to compute each spectrum. All quantum-chemical calculations were obtained with the
Gaussian 09 software package.[234] The polarizable continuum model (PCM)[335] was
used in the linear response formalism[190, 336, 337] to simulate the dielectric effects of
the acetone (e= 20.493) solvent environment.
The TPSS functional[338] stems from the meta-generalized gradient approximation
(MGGA) and requires finite exchange potential, thereby eliminating erroneous effects
of ground-state one- and two-electron orbital densities, which are ubiquitous due to
7.3. Results and Discussion 83
the hydrogen atom and chemical bond. The TPSS hybrid[333] (TPSSh) incorporates
10% Hartree-Fock exchange and satisfies the same exact constraints as TPSS. TPSSh
surpasses the best available semiempirical methods and is a more uniformly accurate
description of diverse systems and properties.[333]. Past works with the TPSSh func-
tional show that it accurately describes equilibrium geometries in first[339] and second-
row[340] transition-metal complexes, binding energies in transition-metal diatomics[341]
and dimers[342], and ligand dissociation energies of large cationic transition-metal com-
plexes.[343]
Natural transition orbital (NTO) analysis of the dominant electronic transitions within
the low-energy charge transfer (CT) band was performed. NTOs offer the most com-
pact, qualitative representation of the specified transition density expanded in terms of
single-particle transitions.[336, 344] Therefore, NTOs offer a simple way of assigning
the transition character. NTOs were calculated with the TPSSh functional and 6-311G
basis set. Figures showing NTOs were obtained with the Jmol program.[337]
A natural population analysis[345, 346] (NPA) was performed to calculate MLCT
character. The NPA used the configuration interaction singles (CIS) excited and ground
state densities in the natural atomic orbital (NAO) basis using NBO 3.1.[347] The dif-
ference in charge was then used as an MLCT indicator. The spatial diffuseness of the
NAO basis set is optimized for calculating the effective atomic charge and can therefore
be used to calculate local charge shifts that would otherwise require variational contri-
butions from multiple basis functions of variable range such as double zeta, triple zeta,
etc. The B3LYP density functional[348] was used to compute these charges due to the
limited functionality of TPSSh in the Gaussian 09 software package. The B3LYP and 6-
311G combination, however, is in good agreement with UV-Vis and several spectra are
available in section 7.3.1.
7.2.2 Experimental
Atomic coordinates for several of the compounds were obtained from their previ-
ously reported solid-state structures using X-ray diffraction experiments.[51] Absorp-
tion spectra were recorded with an HP 8453 Agilent UV-Vis spectrometer. Materials
were dissolved in solutions of acetone at concentrations of 2 10
4
M. All spectra, both
experimental and theoretical, were normalized by the magnitude of maximum absorp-
tion to convey relative intensity.
7.3 Results and Discussion
The results are organized as follows: First, we systematically choose a model quan-
tum chemistry, which is validated with experimental solid-state structures and absorp-
tion spectra for several compounds (section 7.3.1). Second, we apply the theory to calcu-
late the absorption spectra in a series of compounds with various ligand architectures to
determine structure-property relationships that tune the low-energy CT band (section
7.3.2). Third, we do the same analysis as that done in section 7.3.2, but with oxygenated
analogues of several of the compounds (section 7.3.3). Lastly, we characterize the CT
bands of these compounds by explicitly calculating MLCT character (section 7.3.4).
84
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
FIGURE 7.1: Octahedral geometry and ligands of Fe (II) complexes 1
through 3.
TABLE 7.1: Measured and calculated bond lengths (Å) of complexes 1
through 3. X-ray diffraction measurements are in parentheses.
Complex Fe N
Tz
Fe N
Pyr
N
Tz
N
Tz
1 1.896 (1.907(6)) 1.982 (1.953(6)) 1.339 (1.318(4)), 1.343 (1.318(4))
2 1.899 (1.911(3)) 1.980 (1.980(3)) 1.323 (1.307(3)), 1.371 (1.358(3))
3 1.902 (1.894(5)) 1.974 (1.959(5)) 1.333 (1.311(5)), 1.383 (1.358(5))
7.3.1 Benchmarking Model Quantum Chemistry
In order to utilize an appropriate model quantum chemistry for this study, theo-
retical calculations using different levels of theory were compared to the experimen-
tal data of several recently synthesized compounds. The group of octahedral com-
pounds and their associated ligands are shown in Figure 7.1. Previous TD-DFT rep-
resentative simulations using the B3LYP functional and 6-31G

basis set[349, 350] have
shown very limited predictive accuracy, which warrants a more detailed computational
investigation of this family of molecules.[51] In experiments, perchlorate counterions
were utilized to improve the oxygen balance in the resulting complexes. Ligands 1
through 3 react with [Fe(H
2
O)
6
][ClO
4
]
2
in MeCN to form [(NH
2
TzDMP)
3
Fe][ClO
4
]
2
,
[(TriTzDMP)
3
Fe][ClO
4
]
2
and [(NH
2
TriTzDMP)
3
Fe][ClO
4
]
2
, respectively. Since these coun-
terions do not play a significant role in optical absorption, they were not incorporated
into our simulations.
Single-crystal x-ray crystallography confirmed the geometries of complexes 1, 2, and
3 as distorted octahedrals with three tetrazine ligands. Shown in Table 7.1, the TPSSh
and 6-311G combination calculates bond lengths within 0.03 Å of measured values.
Most notable feature is the effect of the 1,2,4-triazolo[4,3b][1,2,4,5]-tetrazine fused
ring system on the N
Tz
N
Tz
bonds of 2 and 3. As opposed to 1, where N
Tz
N
Tz
are
of similar length, 2 and 3 have N
Tz
N
Tz
bonds of different lengths; the larger of the
two is located farther away from the Fe atom. This is due to the loss of double bond
delocalization imposed by the fused ring system.[51]
Figure 7.2 shows TD-DFT results computed with four different methods. The TPSSh
and 6-311G method (shown in blue) accurately describes many important features of
the absorption spectra. The location of the low-energy CT band matches exceptionally
well with experiment for all three compounds and are recovered to within 0.10 eV . The
CT band shifts to higher wavelengths from 1 to 3 in numerical order, thereby showing
how the triazolo-tetrazine fused ring system tunes the CT band to lower energies. Other
observations that validate the use of the proposed method are the high energy peaks,
which are mainly pp

excitations localized on the ligands. In 1, three peaks ranging
7.3. Results and Discussion 85
FIGURE 7.2: Optical absorption of complexes 1 through 3 computed with
different levels of theory. The indices from left to right refer to functional,
basis set on non-metals atoms, and basis set on the Fe atom. UV-Vis spec-
tra in acetone are shown with dashed lines.
from 300 to 500 nm are recovered in nearly the exact locations. The peaks in 2 and 3 are
also recovered, but are slightly red-shifted compared to experiment.
Several studies of transition-metal complexes have utilized the relativistic effective
core potential (RECP), LANL2, and its associated basis set, LANL2DZ, on the metal
atom.[307, 351–353] The RECP replaces inner core electrons, while leaving explicit treat-
ment of outer electrons.[354, 355] Based on our findings in Figure 7.2, however, LANL2DZ
(shown in green) is slightly inferior to 6-311G on all atoms since the CT band is blue-
shifted compared to experiment for all three compounds. LANL08 is an uncontracted
version of LANL2DZ, giving the basis set more flexibility to adjust to the DFT poten-
tial.[356–358] Performance of LANL08 is shown in Figure D.1. Given these results, we
did not choose a pseudo description of the Fe atom via LANL2DZ or LANL08.
The B3LYP functional with 6-31G

basis set on non-metal atoms and LANL2DZ on
the Fe atom was the method of choice in a previous study (shown in red).[51] It is the
clearly the worst performing method in Figure 7.2, especially at low energies. The low-
energy CT bands are severely blue-shifted from UV-Vis from about 0.50 eV in 1 to 0.70
eV in 3. Also, the relative intensities of the CT bands are noticeably reduced compared
to those observed experimentally and those which are predicted with the TPSSh func-
tional. Results using the B3LYP functional are significantly improved with the 6-311G
basis set on all atoms (shown in cyan). Deviation from UV-Vis is reduced to about 0.20
86
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
FIGURE 7.3: Optical absorption of complexes 1 through 4. The ligands
for each associated octahedral complex are shown. The UV-Vis spectrum
of 1(A) in acetone is shown with a dashed line.
eV in 1 to 0.25 eV in 3. Overall, the B3LYP and 6-311G combination is in good qualita-
tive agreement with UV-Vis spectra. We have chosen this method to calculate natural
populations in section 7.3.4. Again, this is due to the limited functionality of the TPSSh
functional in the Gaussian 09 software package.
The effects of adding polarization functions to the basis sets have been analyzed.
Results are shown in Figure D.2. Polarization on the Fe atom did not affect absorption,
while polarization on non-metal atoms, blue-shifts the spectra by less than couple tenths
of an eV , which is still within reasonable accuracy for DFT. We have chosen not to add
polarization in our final choice of basis set since the spectra for our three test compounds
were not improved. An extension to larger basis sets does not necessarily improve the
results in large conjugated molecules,[359] since optical response originates from mobile
p-electrons that are strongly delocalized. Hence, the contribution of atomic polarization
is minimal.
7.3.2 Optical Absorption of Explosive Compounds
We applied TD-DFT to predict the optical absorption in several compounds. The
first group of ligands and absorption spectra of the associated complexes are shown in
Figure 7.3. In order to explore the influence of chemical structure on absorption, the
substituents bound to position 3 of the tetrazine system were varied: NH
2
(1), OH (2),
H (3), and Cl (4). Compound 1 of Figure 7.1 is labeled as 1(A) in Figure 7.3.
7.3. Results and Discussion 87
FIGURE 7.4: Optical absorption of complexes 5 through 8. The ligands for
each associated octahedral complex are shown. UV-Vis spectra of 5(A)
and 6(A) in acetone are shown with dashed lines.
To reduce carbon content, a pyrazole counterpart was included in the study. Dimethylpyrazole-
and pyrazole-containing ligands are labeled by A and B, respectively. The pyrazole sys-
tems increase oxygen balance and are useful for optical initiation. The low-energy CT
bands of the pyrazole compounds are blue-shifted relative to those of the dimethylpyra-
zole compounds. The peak location of their CT bands varies between 555 nm in 1(B),
555 nm in 3(B), 570 nm in 2(B), and 580 nm in 4(B). In the dimethylpyrazole compounds,
it varies between 590 nm in 1(A), 595 nm in 3(A), 610 nm in 2(A), and 630 nm in 4(A).
The additional oxygen in 2 further increases oxygen balance, which may improve its
explosive performance. Based on Figure 7.3, the CT band can be tuned between approx-
imately 500 and 800 nm within this group of ligand architectures. It would be beneficial
to further shift the CT band towards lower energies since optical initiation with NIR
lasers is the optimal choice for practical purposes.
To determine the effect of molecular size and conjugation of the ligand scaffold on
absorption, a second group of compounds were studied. The ligands and absorption
spectra of the associated complexes are shown in Figure 7.4. These compounds are
composed of more atoms than those shown in Figure 7.3. Compounds 2 and 3 of Figure
7.2 are labeled as 5(A) and 6(A) in Figure 7.4, respectively. The piperidine and pyrrole
systems, bound to position 3 of the tetrazine systems, are labeled as 7 and 8, respectively.
Figure 7.4 also shows the low-energy CT bands of the pyrazole compounds blue-
shifted relative to those of the dimethylpyrazole compounds. The peak location of their
CT bands varies between 555 nm in 7(B), 625 nm in 5(B), 690 nm in 6(B), and 830 nm
in 8(B). In the dimethylpyrazole compounds, it varies between 590 nm in 7(A), 665 nm
88
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
in 5(A), 710 nm in 6(A), and 800 nm in 8(A). Unlike the other compounds, there is an
additional low-energy CT band appearing in 8(A) and 8(B), which will be analyzed in
detail. Based on Figure 7.4, the CT band can be tuned between approximately 500 and
1100 nm within this group of ligand architectures. Overall, these compounds are more
suitable for optical initiation with NIR light.
The triazolo-tetrazine fused ring system shown in 5 and 6 is a relatively large con-
jugated system compared to all other compounds in this data set. This suggests that
conjugation in the ligand scaffold is an important design principle towards pushing the
low-energy CT band into the NIR. The UV-Vis spectrum of 5(A) diminishes slightly past
900 nm, whereas for 6(A), with NH
2
bound to position 5 of the triazole, absorption di-
minishes slightly past 1000 nm. Therefore, functional groups, such as NH
2
, further shift
the CT band towards lower energy. It is worth exploring additional substituents in 6(A)
to increase absorption at 1064 nm. The NTOs of 6(A), shown in Table 7.2, show predom-
inantly partial MLCT and partial intraligand CT. The CT character is from the Fe core
and NH
2
group to the tetrazine system for all electron-hole pairs. The NTOs of 5(A)
show similar excitation character and are available in Table D.1.
TABLE 7.2: NTOs of the photoactive excited states within the low-energy
CT band of 6(A). The percent contribution that each electron-hole pair
carries towards the transition is shown. NTOs displayed amount to at
least 90 percent of the total transition.
l(nm) percent hole electron
708 55
43
707 65
Continued on next page
7.3. Results and Discussion 89
Table 7.2 – Continued from previous page
l(nm) percent hole electron
24
Conformation of the ligand scaffolds in 5 and 6 has been carefully analyzed. There is
a rotational degree of freedom between the tetrazine ring and both, the dimethylpyra-
zole systems of 5(A) and 6(A), and the pyrazole systems of 5(B) and 6(B). It is ener-
getically favorable for the triazolo ring to be further away from the Fe core. The self-
consistent field (SCF) energy difference between this geometry and the alternative ge-
ometry, in which the triazole is pointed towards the Fe core, is on the order of one hun-
dred times k
B
T at room temperature. Therefore, the orientation of the ligand scaffolds
shown in Figure 7.4 are correct to within a high degree of certainty.
Similar to the triazolo-tetrazine fused ring systems of 5 and 6, spectral differences in
7 and 8 also stress the importance of conjugation in the ligand scaffold towards pushing
the low-energy CT band into the NIR. Compounds 7(A) and 7(B), with piperidine sys-
tems bound to positions 3 of the tetrazine systems, have CT bands comparable to those
of Figure 7.3, peaking at 555 and 590 nm, respectively. TD-DFT predicts an additional
low-energy peak in 8(A) and 8(B), however, at 800 and 830 nm, respectively, with ab-
sorption diminishing slightly past 1100 nm. The peaks in 8(A) and 8(B) that are located
at 610 and 575 nm, respectively, have very little MLCT character. These excitations are
predominantly localized on the ligand scaffolds and the associated NTOs are available
in Table D.1. This may imply that the lowest-energy excitations in 8(A) and 8(B) are also
not of MLCT character. We will verify this in a subsequent section. For now, it is clear
that the pyrrole system in 8 increases conjugation, which lowers optical excitation com-
pared to that of the piperidine system in 7. Compounds with the farthest red-shifted CT
bands, and that are most suitable for optical initiation in the NIR, are 5, 6 and 8.
Determining physical descriptors of the ligand scaffold that correlate to the location
of the low-energy CT band is beneficial, as it may aid in the design of materials with a
specific optical response. A recent study of Fe (II) coordination complexes, for example,
shows how ligand field strength can be tuned by ligand design, influencing molecular
and electronic structure.[360] It is generally true that charge distribution of the ground
state geometry affects electronic structure and the nature of electronic transitions. We
calculate the quadrupole moments of the ligand scaffolds shown in Figures 7.3 and 7.4
using[361]
Q
2
2
=
4p
5
å
m
jQ
2m
j
2
, (7.1)
where Q
2m
are elements of the quadrupole moment in spherical coordinates. The az-
imuthal dependence is given by m2f2,1, 0, 1, 2g. Results are shown in Figure 7.5.
The pyrazole ligands, 1(B), 2(B), 3(B), and 4(B), with a single functional group bound
to position 3 of the tetrazine system, are labeled by blue markers. The same ligands
90
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
FIGURE 7.5: Quadrupole moment of the ligand scaffold in its ground
state geometry versus wavelength of the strongest photoactive excited
state within the low-energy CT band. Pyr and DMP stand for pyrazole
and dimethylpyrazole, respectively.
with dimethylpyrazole, 1(A), 2(A), 3(A), and 4(A), are shown in red. The quadrupole
moment of the ligand scaffold correlates to the location of the low-energy CT band in
these compounds. The wavelength shifts from 550 600 nm in the pyrazole compounds
to 600 650 nm in the dimethylpyrazole compounds, with quadrupole moment in-
creasing from 60 75 to 75 90 B, respectively. The same trend is observed with the
triazolo-tetrazine compounds, which are labeled by green markers. The correlation is
more dispersive than that of compounds 1 through 4. The wavelength spans a range
of approximately 630 to 710 nm with quadrupole moment increasing from 110 to 130 B,
respectively. Figure 7.5 shows how the peak of the CT band can be tuned from approxi-
mately 550 to 710 nm between compounds 1 through 6 and that it roughly correlates to
quadrupole moment of the ligand scaffold.
Compounds 7 and 8 do not follow the same general trend as the remaining com-
pounds of Figures 7.3 and 7.4. The pyrazole-containing compound, 7(B), is blue-shifted
compared to its dimeythlpyrazole counterpart, 7(A), but the location of their low-energy
CT bands are similar to those of 1 through 4. The large increase in quadrupole moment
is likely correlated to its molecular size. Compound 8 has the lowest-energy CT band,
exceeding 800 nm. As stated previously, 8 is unique from all other compounds since
TD-DFT predicts an additional low-energy peak, not observed in the other compounds.
It is an outlier and we will show that the excitation character within this CT band is
fundamentally different than all other compounds in this data set.
Tetrazines are readily derivatized with a variety of explosive groups. One such
group is 3,3-dinitroazetidine (DNAZ), which can be attached to position 3 of the tetrazine
system and is henceforth labeled as ligand 10(A). Ligand 10(A) reacts with [Fe(H
2
O)
6
][ClO
4
]
2
in MeCN to form [(DNAZTzDMP)
3
Fe][ClO
4
]
2
. The ligand and absorption spectrum of
7.3. Results and Discussion 91
FIGURE 7.6: Comparison of the optical absorption in complexes 1(A) and
10(A). The ligands of each associated octahedral complex are shown. UV-
Vis spectra of 1(A) and 10(A) in acetone are shown with dashed lines.
the associated complex are shown in Figure 7.6. The TD-DFT calculations are in excel-
lent agreement with the UV-Vis spectrum of 10(A), especially with the location of its
low-energy CT band between 500 and 700 nm. The high energy peak, located at ap-
proximately 350 nm, is also recovered in nearly the exact location. The calculations for
10(A) required 150 excited states to obtain the visible region of the spectrum, as opposed
to only 70 states for the remaining compounds, due to its large density of states. Com-
pound 1(A) is also shown in Figure 7.6 for comparison. The CT bands of 1(A) and 10(A)
are very similar both in position and magnitude, which are attributed to the fact that
the NO
2
substituents of 10(A) are far removed from the tetrazine system. The optical
properties are largely dictated by the excitation character of the Fe core and tetrazine
systems, and substituents that are within close proximity. This is validated by the NTOs
of the low-energy transitions in 10(A), which show no charge distribution on the DNAZ
system. These NTOs are available in Table D.1. Although the CT bands of 1(A) and
10(A) peak in similar locations, i.e., 590 and 605 nm, respectively, oxygenated ligands
are important for explosive applications.
7.3.3 Oxygen-Containing Compounds
To increase oxygen balance and explosive performance, a set of compounds with
oxygen substituents were studied. Ligands and absorption spectra are shown in Figure
7.7. These compounds are labeled with C to denote that they are oxygenated analogues
of their B counterparts. For example, 1(C) is an oxygenated analogue of 1(B), where two
oxygen substituents are bound to positions 2 and 4 of the tetrazine system. Compounds
5(C) and 6(C) are analogues of 5(B) and 6(B), respectively, where a single oxygen sub-
stituent is bound to position 1 of the tetrazine system. Lastly, 11(C) replaces the triazole
92
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
with a tetrazole - a carbon atom is exchanged for a nitrogen - and a single oxygen sub-
stituent is bound to position 1 of the tetrazine system.
FIGURE 7.7: Optical absorption of complexes 1(C), 5(C), 6(C), and 11(C)
with oxygen substituents. The absorption of 1(B), 5(B), and 6(B) without
oxygen substituents are shown for comparison.
There are minor spectral differences between the nonoxygenated and oxygenated
compounds. The peak location of the low-energy CT band in the oxygenated com-
pounds varies between 580 nm in 1(C), 645 nm in 5(C), and 700 nm in 6(C). In the
nonoxygenated compounds, it varies between 555 nm in 1(B), 625 nm in 5(B), and 690
nm in 6(B). Therefore, the CT band is slightly red-shifted in the oxygenated compounds
relative to that of the nonoxygenated compounds. The absorption magnitude is pre-
dicted to be larger in the nonoxygenated compounds however. The peak in 1(B) is over
three times larger than that of 1(C). The same is true for 5(B) and 6(B) relative to 5(C) and
6(C), respectively, but with noticeably smaller differences. This is likely due to molec-
ular size. Unlike 5 and 6, which are relatively large due to the fused ring system, 1 is
smaller and therefore two oxygen substituents constitute a much larger fraction of the
ligand scaffold, greatly affecting its electronegativity and electronic structure. It makes
sense to compare 11(C) to 5(C) since they differ by only one atom. The peak location of
the CT band in 11(C) is at 630 nm, as opposed to 645 nm in 5(C). Also, the absorption
magnitude in 11(C) is predicted to be nearly twice that of 5(C). Similar to the findings
in section 7.3.2, 5(C), 6(C), and 11(C) enhance conjugation due to a fused ring system
and as a result, their CT bands are red-shifted compared to that of 1(C). Although the
syntheses of compounds 5(C) and 6(C) have not yet been reported, these results qual-
itatively show that additional oxygen can increase their oxygen balance, while leaving
their low-energy absorptions more or less unaffected.
Oxygen balance (OB%) provides a measure to which a material can be oxidized.
Generally, optimal explosive performance is achieved as OB% approaches zero.[362]
7.3. Results and Discussion 93
TABLE 7.3: OB% of several compounds with and without oxygen sub-
stituents in their ligand scaffolds.
Complex OB% l (nm)
1(A) 91.7 590
10(A) 74.2 605
1(B) 63.4 555
2(B) 53.5 570
5(B) 66.4 625
6(B) 65.7 690
1(C) 44.8 580
5(C) 57.2 645
6(C) 57.0 700
11(C) 43.2 630
The following equation was used to compute OB%:
OB%=
1600
M

2X+
1
2
Y Z

, (7.2)
where X, Y, and Z, are the number of carbon, hydrogen, and oxygen atoms, respec-
tively, and M is the molecular weight of the compound. Perchlorate counterions were
included in the calculation. Tabulated quantities are shown in Table 7.3. Most notable
differences in OB% are between the base compounds and their oxygenated analogues
such as 1(A)! 10(A), 1(B)! 2(B), 1(B)! 1(C), 5(B)! 5(C), and 6(B)! 6(C). Even a
single oxygen substituent significantly increases the oxygen balance, which has strong
implication on explosive performance. For example, the OB% increases from66.4%
and65.7% in 5(B) and 6(B) to57.2% and57.0% in 5(C) and 6(C), respectively.
Furthermore, the OB% increases from57.2% to43.2% with exchange of the tria-
zole in 5(C) for a tetrazole in 11(C). Also noteworthy is the difference in OB% between
the dimethylpyrazole-containing compound, 1(A), and its pyrazole counterpart, 1(B),
where OB% increases from91.7% to63.4%, respectively. Again, these calculations
support the claim that additional oxygen in the ligand scaffold significantly increases
OB%, while preserving low-energy absorption.
7.3.4 Characterizing MLCT Bands
Transition-metal complexes are generally characterized as having strong MLCT char-
acter. It would be beneficial to compare the excitation character of these energetic com-
pounds to a control group for classification. Two compounds with strong MLCT char-
acter are [Ru(bpy)
3
]
2+
and [Fe(bpy)
3
]
2+
. We quantify the amount of CT from the metal
core to the ligand scaffold by taking the difference of natural charge between the ground
state and strongest photoactive excited states within the low-energy peaks. Results are
shown in Figure 7.8. It is worth mentioning that the B3LYP functional and 6-311G basis
set are used in the following analysis and that the absorption spectra using this method
are usually blue-shifted compared to experiment, as evidenced in Figure 7.2. Neverthe-
less, the main goal is to obtain a qualitative comparison of the MLCT character.
There is a clear distinction between these energetic compounds and [Ru(bpy)
3
]
2+
and [Fe(bpy)
3
]
2+
. Figure 7.8 shows [Ru(bpy)
3
]
2+
having the most MLCT character be-
tween 0.30 and 0.35 electrons. Its MLCT band falls within 400 and 500 nm, peaking
94
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
FIGURE 7.8: MLCT, in units of number of electrons, versus wavelength of
the strongest photoactive excited states within the low-energy CT band.
at about 450 nm. This is in excellent agreement with experiment, where the MLCT
band is reported to peak at 452 3 nm.[363] [Fe(bpy)
3
]
2+
has less MLCT character
than [Ru(bpy)
3
]
2+
, with MLCT between 0.20 and 0.25 electrons. The MLCT band of
[Fe(bpy)
3
]
2+
is slightly red-shifted compared to [Ru(bpy)
3
]
2+
, which also agrees with
experiment.[364] The majority of compounds, which are labeled by red and blue mark-
ers, fall within an MLCT of 0.10 and 0.20 electrons and with low-energy CT bands peak-
ing between 500 and 650 nm. The ligand scaffolds of these energetic compounds have
a higher nitrogen content than the bipyridine ligands of [Ru(bpy)
3
]
2+
and [Fe(bpy)
3
]
2+
.
Therefore, these results suggest that the CT band in complexes with more nitrogen-rich
and electron-poor ligands occur at lower energies and with less MLCT character.
The reason these energetic compounds have less MLCT character than [Ru(bpy)
3
]
2+
and [Fe(bpy)
3
]
2+
is indeed attributed to their high nitrogen content. A higher nitrogen
content of the ligand scaffold increases electronegativity, which means more charge will
be displaced from the metal core to the ligand scaffold in the ground state. Therefore,
during a transition into an excited state, there will be less MLCT. A prime example of
this can be seen between compounds 1(B) of Figure 7.3 and 1(C) of Figure 7.7. The
ligand scaffolds of 1(C) have two additional oxygen substituents not present in 1(B),
making 1(C) more electron-rich. As a result, there will be more MLCT in the excitations
of 1(C) compared to those of 1(B). We calculate approximately 0.14 and 0.21 electrons
in 1(B) and 1(C), respectively. From a visual perspective, the NTOs of 1(C) in Table
7.4 qualitatively show more MLCT character than 1(B). On the other hand, compounds
1(A) and 10(A) have similar MLCT character with 0.13 electrons. Again, the excitation
character in these compounds is similar since the additional substituents in 10(A) are
far removed from the tetrazine system, which withholds most control over the optical
properties.
7.3. Results and Discussion 95
TABLE 7.4: NTOs of photoactive excited states within the low-energy CT
bands of 1(B) and 1(C). The percent contribution that each electron-hole
pair carries towards the transition is shown. NTOs displayed amount to
at least 90 percent of the transition.
complex l(nm) percent hole electron
1(B)
554 63
28
1(C)
580 64
26
Compounds 8(A) and 8(B) have very different excitation character than all other
compounds shown in Figure 7.8. Their lowest-energy CT bands peak at approximately
700 nm using B3LYP and 6-311G, with little to no MLCT character. The NTOs of 8(B),
obtained with TPSSh and 6-311G, are shown in Table 7.5. There is significant CT from
the pyrrole to the tetrazine system for all electron-hole pairs. The CT character on the
Fe atom is minimal. Compound 8 has the lowest-energy CT band and largest amount of
intraligand CT character amongst all other compounds, suggesting that additional con-
jugation in the ligand scaffold via the pyrrole system influences this type of excitation
and is an important design principle that shifts the CT band towards lower energies.
96
Chapter 7. Photoactive Excited States in Explosive Fe (II) Tetrazine Complexes: A
Time-Dependent Density Functional Theory Study
TABLE 7.5: NTOs of photoactive excited states within the low-energy CT
band of 8(B). The percent contribution that each electron-hole pair carries
towards the transition is shown. NTOs displayed amount to at least 90
percent of the transition.
l(nm) percent hole electron
831 90
831 88
11
823 91
7.4 Conclusions
We have utilized TD-DFT to study a class of transition-metal complexes. Past works
support the use of MGGAs to describe bond lengths and dissociation energies. Their
performance towards optically-excited electronic states, however, warrants further in-
vestigation.[365] In this work, we have benchmarked the TPSSh functional against a
series of novel and energetic Fe (II) coordination complexes. TD-DFT is found to be
in excellent agreement with all UV-Vis spectra with a maximum deviation of 0.10 eV .
This gives us confidence in this model quantum chemistry towards predicting ways of
lowering the initiation threshold.
7.4. Conclusions 97
We studied a large set of explosive compounds shown in Figures 7.3 and 7.4. The
pyrazole-containing ligands shift the CT band towards lower energy, but their explosive
performance is potentially weakened by additional carbon and hydrogen. In total, the
CT band can be tuned between 500 and 1100 nm. Compounds with the lowest-energy
CT bands and that are most suitable for optical initiation in the NIR are 5, 6, and 8,
which are due to additional conjugation in the ligand scaffold. This presents itself in
5 and 6 via the 1,2,4-triazolo[4,3b][1,2,4,5]-tetrazine fused ring system, while for 8,
it is the pyrrole system. In order to increase oxygen balance and improve explosive
performance, a set of compounds with oxygen-containing ligands were also studied
and are shown in Figure 7.7. In addition to the fused ring system, electron donating
groups, such as NH
2
, can further shift the CT band towards lower energy, as evidenced
by 6(C) in Figure 7.7.
We characterized the MLCT bands of these compounds by explicitly calculating
effective charge on the metal core in the ground and photoactive excited states. We
found less MLCT character in these compounds than other commonly-studied octahe-
dral transition-metal complexes, such as [Fe(bpy)
3
]
2+
and Ru[(bpy)
3
]
2+
. We attribute
this to the high nitrogen content present in their ligand scaffolds. Compound 8, with
pyrrole bound to position 3 of the tetrazine system, shows little to no MLCT, but rather
mostly intraligand CT character. This suggests that increasing conjugation in the ligand
scaffold both lowers excitation energies and MLCT.
To recapitulate, a theoretical study of the optical absorption in energetic Fe (II) coor-
dination complexes has been accomplished. Vertical excitation energies computed with
the TPSSh density functional and 6-311G basis set match exceptionally well with exper-
iment. This supports the use of this nonempirical MGGA for optical absorption. The
absorption spectra of these compounds strongly depends on the ligand scaffold and can
be controlled by chemical substitution. Altering molecular substituents can push the CT
band into the region of 1064 nm, which is a practical wavelength for optical initiation
due to the availability of high power Nd:YAG lasers. The tetrazine-triazolo fused ring
system shifts absorption to lower energies and is an important design principle. Com-
pound 8 has not yet been synthesized, but is predicted to be within range for optical
initiation. It is worth investigating additional conjugation in the ligand scaffold, such as
fusing two triazolo rings to a central tetrazine. This study not only aids in the ongoing
effort to design explosive materials that are both safer to handle and easier to initiate
with NIR lasers than PETN, but is also important for many light-harvesting applica-
tions, where control over the photophysical properties is desirable.

99
Chapter 8
Cooperative Enhancement of the
Nonlinear Optical Response in
Conjugated Energetic Materials: A
TD-DFT Study
Reprinted with permission from J. Chem. Phys. 146, 114308 (2017). Copyright 2018
American Institute of Physics.
8.1 Introduction
There are has been an increasing effort to design energetic materials that are safe
and easy to handle. While the motivation to do so is obvious, it is rather difficult in
practice, as it requires decoupling the correlation between explosive performance and
sensitivity.[366–373] Current methods used to detonate high explosives are electrical or
mechanical.[374] These methods are not inherently safe, however, because coinciden-
tal electrical or mechanical insults can result in unwarranted detonation. Recent efforts
have focused on photoactive energetic materials with absorption within range of con-
ventional lasers.[50–53, 164, 312, 317–320, 375–378] Introducing a fiber optic cable for di-
rect optical initiation eliminates sensitivity to electrical insults, while replacing primary
explosives with less sensitive secondary explosives reduces sensitivity to mechanical
insults. A class of energetic materials that exhibit the optical, electrical, and mechanical
properties suitable for this application are conjugated energetic molecules (CEMs).
CEMs are secondary explosives with a large number of NN and CN bonds,
which raise their heat of formation.[47, 325–327, 368–371, 379–384] Moreover, their pla-
nar, conjugated structure with electronic delocalization increases density and molecu-
lar stability. Their synthetic versatility has been evidenced in nitrogen-rich heterocy-
cles such as triazole, tetrazole, tetrazine, etc.[382] A recent theoretical work has high-
lighted structure-property relationships, such as adding oxygen to the core framework
to simultaneously increase the two-photon absorption (TPA) cross section and increase
oxygen balance;[53] a measure of the oxygen to fuel ratio that determines explosive
strength.[385] While past works have discussed isolated CEMs, more work is needed to
characterize their associated crystal structures used in practice.
The synthesis and characterization of crystalline CEMs have been recently reported.[386–
391] These materials are generally less sensitive to electrical and mechanical stimuli than
100
Chapter 8. Cooperative Enhancement of the Nonlinear Optical Response in
Conjugated Energetic Materials: A TD-DFT Study
traditional explosives such as pentaerythritol tetranitrate (PETN), 1,3,5-trinitroperhydro-
1,3,5-triazine (RDX), and octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX), while
maintaining strong explosive performance. However, the optical properties of these
materials have not yet been determined, as direct evaluation via optical detonation ex-
periments can often be difficult and costly. It is beneficial to have a qualitative and
quantitative overview of the optical performance a priori, which can be accomplished
using modern computational techniques for predictive modeling. The majority of CEMs
from our last study were optically excited within the range of 500 250 nm (2.5 5.0
eV).[53] However, due to the availability of high power Nd:YAG lasers, the most prac-
tical wavelength for initiation is 1064 nm ( 1.2 eV). One possible avenue for optical
excitation is by degenerate TPA, whereby a material simultaneously absorbs two pho-
tons of the same energy. Increasing the sensitivity of these materials to nonlinear laser
initiation eliminates the need of electrical and mechanical stimuli for detonation. One
such mechanism for increasing TPA intensity is cooperative enhancement.
Cooperative enhancement of TPA is due to the interaction between multiple chro-
mophore units and is observed when the maximum TPA cross section, per chromophore
unit, is considerably larger than that of an individual, non-interacting unit. It is at-
tributed to p-conjugation, where the delocalization of a coherent electronic molecular
wavefunction spans multiple building blocks of the compound, which increases tran-
sition dipole moments. Strong cooperative enhancements have been measured in den-
drimers,[392–396] substituted porphyrins,[397] porphyrin dimers,[398, 399] linear por-
phyrin[400] and squaraine oligomers,[401] multichromophoric compounds,[402] and
self-assembled zinc-porphyrin nanostructures.[403] By utilizing time-dependent den-
sity functional theory (TD-DFT), we calculate a cooperative enhancement in dimers ex-
tracted from crystalline CEMs, thereby providing the first evidence of a strong nonlinear
optical response in energetic materials composed of relatively small molecules for ex-
plosive applications.
8.2 Conjugated Energetic Materials
FIGURE 8.1: Chemical structures of the CEMs.
This section provides readers with a concise overview of recent measurements and
theoretical predictions covering both the safety as well as energetic performance prop-
erties of the materials studied in this paper. It is worth noting that these data are those
of the bulk materials, whereas the focus of this paper is on the optical properties of
8.2. Conjugated Energetic Materials 101
TABLE 8.1: Sensitivity data of materials A through C, PETN, and RDX.
Structure Impact (J) Spark (J) Friction (N)
A 6.0 0.062 109
B 29 0.125 360
C 5.6 0.062 104
PETN 2.5 0.062 92
RDX 6.0 0.062 87
monomers and dimers extracted from their corresponding crystal structures. However,
we find it useful to not only motivate these materials as they are used in practice, but to
also present data from several works in a single location for comparison.
We study molecular materials A through C, which are comprised of the correspond-
ing CEMs shown in Fig. 8.1. Molecule A has a tetrazine-tetrazolo bicyclic framework
with two oxygen substituents and a NH
2
substituent. Molecule B has a triazine-triazolo
bicyclic framework with two NO
2
substituents and a NH
2
substituent. Molecule C has
a single cyclic tetrazine system with two fluorinated chainlike substituents and two
oxygen substituents. The synthesis and characterization of their corresponding crys-
tal structures are provided in Refs. [386] and [387] for A, Refs. [388] and [389] for B, and
Refs. [390] and [391] for C. Table 8.1 shows sensitivity data of these materials, as well
as those of conventional explosive materials. Data for PETN and RDX were taken from
Refs. [386] and [387]. Impact is the minimum energy at which a falling weight causes an
explosive material under total confinement to initiate. Spark, also known as electrostatic
discharge (ESD), is the minimum electrostatic energy discharged from a capacitor to an
explosive material at which initiation occurs. Lastly, friction determines whether or not
an explosive material is susceptible to initiation by a specified frictional force. The de-
tails of the experiments carried out to obtain these results can be found in the references
above.
As shown in Table 8.1, these materials show excellent insensitivities toward destruc-
tive electrical and mechanical stimuli. Material B was found to be relatively insensitive
to impact, spark, and friction. Material A, although more sensitive than B, was found
to be less sensitive than PETN and RDX overall. The insensitivities of C are similar to
those of A. Materials A through C were also found to have good thermal stabilities, with
decomposition occurring above 150

C for A, 232

C for B, and 174

C for C. PETN and
RDX decompose above 164

C[404] and 210

C,[405, 406] respectively.
As a measure of explosive performance, we can compare theoretical predictions of
the detonation pressure, p
D
, and velocity, v
D
, which are defined as the pressure and
velocity of a shock wave front traveling through a detonated explosive. Theoretical
predictions were calculated with the Cheetah thermochemical code.[407] Densities were
measured at 293 K, while heats of formation were calculated with the methods described
in Refs. [408] and [409] for A and B, and Refs. [410] and [411] for C. Densities and heats
of formations were fed into the Cheetah program to obtain p
D
and v
D
. These values
are predicted to be 41.3 GPa and 9.6 km s
1
at 1.93 g cm
3
for A, 32.0 GPa and 8.7 km
s
1
at 1.86 g cm
3
for B, and 40.6 GPa and 8.8 km s
1
at 1.96 g cm
3
for C. For PETN,
these values are 33.2 GPa and 8.3 km s
1
at 1.77 g cm
3
,[404] while for RDX, they are
34.9 GPa and 8.8 km s
1
at 1.80 g cm
3
.[405, 406] Therefore, materials A through C
102
Chapter 8. Cooperative Enhancement of the Nonlinear Optical Response in
Conjugated Energetic Materials: A TD-DFT Study
are predicted to have higher detonation pressures and velocities than those of PETN
and RDX, and with higher densities, which may imply more explosive power per unit
volume. The details of the simulations carried out to obtain these results can be found
in the aforementioned references.
Another factor of explosive performance is the oxygen balance (OB%), which pro-
vides a measure to which a material can be oxidized. On average, optimal explosive per-
formance is achieved as OB% approaches zero.[362] The following equation was used
to compute OB%,
OB%=
1600
M

2X+
1
2
Y Z

, (8.1)
where X, Y, and Z, are the number of carbon, hydrogen, and oxygen atoms, respectively,
and M is the molecular weight of the compound. The OB% of materials A through C are
28.0%,35.4%, and3.8%, respectively. The OB% of C was obtained by combining
each fluorine to a hydrogen to make hydrogen fluoride (HF), which were not included
in the calculation for oxidation. For PETN (C
5
H
8
N
4
O
12
) and RDX (C
3
H
6
N
6
O
6
), these
values are10.1% and22.6%, respectively. Overall, A through C show enhanced
safety properties and promising energetic performance properties compared to conven-
tional explosive materials. By determining their optical properties, we can assess their
viability toward optical initiation.
8.3 Computational Methods
8.3.1 Optical Absorption Spectra
The one-photon absorption (OPA) of the monomers and dimers were calculated.
The ground-state geometry of each monomer was extracted from a single molecule of
its experimental crystal structure. A similar procedure was carried out to obtain the ge-
ometries of the dimers, where a central molecule of the crystal was selected and all its
neighboring molecules were systematically identified. Atomic coordinates were mea-
sured with X-ray crystallography and obtained with the VESTA[412] and Avogadro
programs.[413, 414] Following our previous study,[53] the following model quantum
chemistry was applied. Vertical excitation energies were computed with TD-DFT using
the B3LYP density functional[348] and 6-31G(d
0
) basis set.[349, 350] TD-DFT is the most
practical method for calculating excited states in medium- and large-sized organic and
inorganic molecules.[55, 332] Moreover, this level of theory has produced qualitatively
accurate linear and nonlinear absorption spectra of conjugated organic materials over a
wide range of molecular sizes.[57, 138, 359, 415–418] We validate the use of this model
quantum chemistry, for the molecules of Fig. 8.1 , in the following section 8.3.2.
In order to obtain OPA spectra, A, as a function of energy,W, each vertical excitation
was given a Lorentzian lineshape,
A
e
(W)µ
f
ge
1+ 4

WW
ge
G

2
, (8.2)
where f
ge
is the oscillator strength between the ground and excited state,W
ge
is the en-
ergy of the excited state, andG is the full width at half maximum (FWHM), which was
8.3. Computational Methods 103
set to a homogeneous broadening of 0.15 eV . The complete spectrum, A, is defined as the
sum of all its individual contributions, A
e
, i.e., A=å
e
A
e
. Here, 25 singlet excited states
were requested in the TD-DFT calculation and used to compute each spectrum. Excited
states were computed with the Gaussian 03 software package.[419] Since the most prac-
tical wavelength for initiation is in the NIR (near-infrared), we focus our attention to
low-energy absorption peaks with relatively strong intensities. Therefore, spectra may
be cut off at higher energies.
The two-photon absorption (TPA) of the monomers and dimers were computed us-
ing the extension of adiabatic TD-DFT to nonlinear optical response.[420, 421] TPA spec-
tra were also given Lorentzian lineshapes with FWHM of approximately 0.15 eV . It is
worth noting that a rigorous choice of broadening was not of concern since the objective
was to compare optical response between the monomers and dimers.
In order to determine the nature of the optical excitations in these chromophores,
natural transition orbital (NTO) analysis was performed. NTOs offer a compact, quali-
tative representation of a transition density expanded in terms of single-particle transi-
tions.[336, 344] Therefore, NTOs provide a useful way of assigning transition character.
Figures showing NTOs were obtained with the Jmol program.[337]
8.3.2 Benchmarking Model Quantum Chemistry
To validate the proposed model quantum chemistry, theoretical ground-state ge-
ometries and optical absorption spectra were compared to experimental data. Monomers
were optimized with no symmetry constraints using DFT and compared to those ob-
tained from the experimental crystal structures. Maximum deviation of the bonds lengths
between DFT and experiment was 0.015 Å. Experimental absorption spectra of these
molecules dissolved in acetonitrile were obtained from our previous study,[53] and
which are compared to the new TD-DFT calculations, shown in Fig. 8.2. Our previ-
ous calculations did not account for the effects of solvation on optical properties, but
were rather, carried out in the gas phase. Here, we have calculated the absorption in
solvent, as this makes for a better comparison to the available experimental data. Both
non-equilibrium, linear response[422, 423] (LR) and non-equilibrium, state-specific[424,
425] (SS) solvents were used with a polarizable continuum model (PCM).[188, 190] The
PCM models the solute as a system embedded in a dielectric cavity, with a dielectric
constant e that is characteristic of the solvent. In this case, acetonitrile has a dielectric
constant ofe 37.5. The PCM is the most popular solvent model and has, for example,
been successful in modeling solvated charge transfer states.[426] The theoretical OPA
spectra in Fig. 8.2 were constructed using the method described in section 8.3.1 using
a FWHM of 0.36 eV . This FWHM was chosen to closely match theory to experiment. A
total of 50 singlet excited states were requested in the TD-DFT calculations and used
to compute each spectrum. It is worth noting that we are predominantly concerned
with qualitative models of OPA and TPA lineshapes. Theoretical spectra do not account
for Franck-Condon effects or the changes in zero-point vibrational energies. That being
said, excitation energies of vertical transitions have been found to be blue-shifted by
about 0.20.3 eV relative to those of 00 transitions in rigid molecules.[427] In regard
to error compensation due to choice of density functional, B3LYP with vibronic effects
has proven to be in excellent quantitative agreement to experiment for large organic
molecules.[427] All spectra of Fig. 8.2, experimental and theoretical, were normalized
104
Chapter 8. Cooperative Enhancement of the Nonlinear Optical Response in
Conjugated Energetic Materials: A TD-DFT Study
by maximum absorption to convey relative intensity. Optimized geometries and excited
states were computed with the Gaussian 09 software package.[234]
FIGURE 8.2: Optical absorption of A through C in acetonitrile. Bold lines
are theoretical spectra, determined using TD-DFT, while dashed lines are
experimental. Experimental spectra were recorded with a HP 853 Agilent
UVVis spectrometer. Materials were dissolved in solutions of acetoni-
trile at concentrations on the order of 10
4
M. Extinction coefficients were
averaged over three solutions.
In Fig. 8.2, TD-DFT matches the experimental spectrum of A exceptionally well,
with a maximum deviation of 0.10 eV . The peaks in B and C are also recovered, but
with slightly larger blue-shifts. The relative intensities of all peaks in A through C are
also accurately recovered. Therefore, although TD-DFT is generally more accurate for
larger systems,[55, 332] we expect this level of theory to be valuable for predicting the
linear and nonlinear optical response in these smaller systems as well. It is worth re-
iterating that the geometries used to compute the spectra in section 8.4 were obtained
from experimental crystal structures, as opposed to the optimized structures used in
section 8.3.2.
8.4 Results and Discussion
The OPA spectra of monomer A and its dimers are shown in the bottom panels of
Fig. 8.3. OPA intensities of the dimers were halved in order to eliminate dependence
on the number of chromophore units. Monomer A peaks at 2.76 eV , while dimers 1A
through 5A peak between 2.72 and 2.92 eV . Compared to A, the OPA of 1A, 2A, and
4A are shifted towards lower energy due to the optically-active states at 2.68, 2.64, and
2.60 eV , respectively. The dimers also absorb with larger and more broadened intensities
than the monomer. To account for changes in both peak intensity and broadening, the
8.4. Results and Discussion 105
enhancement factor (EF) is defined as
EF=
1
2
R
A
Dim
(W) dW
R
A
Mon
(W) dW
, (8.3)
where the integral is over the peak of interest, A is absorbance, and the factor of one-half
removes the dependence on the number of chromophore units. EFs of 1A through 5A
are between 0.6 and 1.1. For two non-interacting chromophore units, a 2-fold enhance-
ment from monomer to dimer is expected since OPA depends on the transition dipole
moments between the ground and excited states, which roughly scales linearly with the
number of chromophore units, i.e.,jm
ge
j
2
µ N. The data qualitatively agrees with this
reasoning. Overall, the dimers shift OPA by roughly 0.20 eV compared to that of the
monomer.
The TPA spectra of monomer A and its dimers are shown in the top panels of Fig.
8.3. TPA intensities of the dimers were halved in order to eliminate dependence on the
number of chromophore units. Monomer A peaks in a similar location as its OPA state
at 2.76 eV . TPA of 1A through 5A peak between 2.71 and 2.81 eV . The locations of OPA
and TPA roughly coincide for all dimers, except for dimer 3A, which has a dark OPA,
but bright TPA state at 2.70 eV . This state is red-shifted relative to the lowest-energy
OPA state at 2.78 eV . This shows that TPA can be tuned to lower energies and activate
dark OPA states. Such shifts are more pronounced in larger and more conjugated sys-
tems.[57] Similar to OPA of the dimers, there is also enhanced broadening in the TPA.
The left shoulder in 2A is due to the state at 2.64 eV , which is not optically active in A.
The same is observed in 4A, where the low-energy peak is located at 2.60 eV . Optically-
active low-energy TPA states are important, in practice, for initiation with conventional
NIR lasers.
The most distinctive feature between the optical properties of the monomers and
dimers is a cooperative enhancement in the TPA. Cooperative enhancement occurs when
the TPA of a dimer is more than twice that of the monomer. Following the same def-
inition for enhancement as OPA, EF
TPA
of 1A through 5A are 1.3, 1.5, 2.0, 2.4, and 3.0,
respectively. The relatively large enhancement in 4A, for example, can be attributed
to the low-energy TPA state. Dimer 5A is predicted to have the largest enhancement.
Compared to the other dimers, 5A is the largest in length, with two molecules nearly
head-to-tail, as shown in Fig. 8.3, suggesting that this orientation induces a large en-
hancement in the TPA. We will later justify this within a two-level model.
A similar analysis can be made for the OPA and TPA spectra of materials B and C.
For the sake of brevity, spectra with largest enhancements are shown in Fig. 8.4. OPA of
monomer B peaks at 4.26 eV , while dimer 1B has a broader band that is shifted towards
lower energy and peaks at 3.96 eV . OPA of monomer C and dimer 1C peak in similar
locations around 2.50 and 3.25 eV . The low-energy OPA peak is predicted to have a much
lower intensity than the high-energy OPA peak in both C and 1C. EF
OPA
of 1B and the
high-energy peak in 1C are 1.0 and 1.1, respectively. TPA of 1B is red-shifted relative to
the TPA of B. There is also a TPA state in 1B at 3.13 eV that is not optically active in B.
Again, TPA of C and 1C peak in similar locations around 2.50 eV and 3.25 eV . Unlike
OPA, however, the low-energy TPA peak is predicted to have a larger intensity than the
high-energy TPA peak. EF
TPA
of 1B and the low-energy peak of 1C are estimated to be
3.5 and 5.6, respectively. Similar to A, multiple chromophore units strongly affect TPA
106
Chapter 8. Cooperative Enhancement of the Nonlinear Optical Response in
Conjugated Energetic Materials: A TD-DFT Study
FIGURE 8.3: OPA (bottom panel) and TPA (top panel) spectra of the
monomer and dimers of material A. Results for the monomer are shown
in red, while those of the dimers are shown in blue. Vertical lines in the
bottom panels are vertical excitation energies determined from TD-DFT.
OPA and TPA intensities of the dimers are halved, while the TPA energy
scale is doubled to show total photon energy absorbed. Associated dimer
configurations are shown.
8.4. Results and Discussion 107
FIGURE 8.4: OPA (bottom panel) and TPA (top panel) spectra of the
monomer and dimers of materials B and C. Details of the spectra are the
same as those described in Fig. 8.3.
in these materials. All data are summarized in Table 8.2.
Within this small data set, there are several structure-property relationships. The
first of which is the dependence of conjugation on TPA cross section, s
2
. Monomers A
and B are more conjugated than C due to the tetrazine-tetrazolo and triazine-triazolo
fused ring systems, respectively. The only conjugation in C is from the tetrazine ring.
Therefore, the molecular orbitals of A and B are delocalized to a larger fraction of the
molecule, which increases their transition dipole moments. Table 8.2 shows s
max
2
of A
and B being 2 to 3 times larger than that of C.
A second structure-property relationship is the effect of orientation on cooperative
enhancement. It is beneficial to elucidate the enhancement in a representative dimer
with relatively large s
2
. The head-to-tail orientation of dimer 5A results in an EF
TPA
of 3.0, which can be explained within the two-level model. When the ground and first
couple excited states are well separated from higher-lying excited states, the sum over
states expression for the second hyperpolarizability can be simplified. Furthermore,
when the dipole moment is predominantly along a single molecular axis, only one ten-
sorial component of the second hyperpolarizability is needed to obtains
2
. At maximum
TPA, s
max
2
µ m
2
ge

m
ee
m
gg

2
.[57] For monomer A, m
ge
, m
gg
, and m
ee
are 3.6, 8.1, and 5.6
D, while for 5A, they are 6.5, 17.6, and 14.3 D, respectively. The ratio of s
max
2
at 2.82
eV is s
5A
2
/s
A
2
2.84; a roughly 8% deviation from the full calculation of 3.1, shown
in Table 8.2. The largest contribution is from the transition dipole moment, m
ge
. Fig.
8.5(A) shows how 5A’s orientation both rotatesm
ge
towards the oxygen substituent and
increases its magnitude. The alternation of acceptor-donor substituents between the
oxygen and NH
2
enhances electronic coupling. Moreover, the tetrazine-tetrazolo hete-
rocycles enhances electronic delocalization over both chromophore units. Intermolec-
ular coupling is evidenced by the pp

transition shown in Fig. 8.5(B), where the
excitation is on both chromophore units. A delocalized state due to optical excitation is
common in conjugated systems.[428] Molecular packing in these explosive materials in-
creases the degree of electronic delocalization, which inherently increases the nonlinear
108
Chapter 8. Cooperative Enhancement of the Nonlinear Optical Response in
Conjugated Energetic Materials: A TD-DFT Study
TABLE 8.2: Properties of OPA and TPA spectra in materials A through
C. W
OPA/TPA
are transition energies at maximum OPA and TPA, re-
spectively. EF
OPA/TPA
are enhancement factors, as defined in Eq. 8.3.
EF
max
OPA/TPA
are enhancements factors of peak absorptions. s
max
2
are max-
ima of the halved two-photon cross sections.
Structure W
OPA
(eV) EF
OPA
EF
max
OPA
W
TPA
(eV) EF
TPA
s
max
2
(GM) EF
max
TPA
A 2.76 1.0 1.0 2.79 1.0 10 1.0
1A 2.72 0.8 0.8 2.71 1.3 13 1.3
2A 2.74 0.9 0.7 2.75 1.5 12 1.2
3A 2.92 1.1 0.7 2.72 2.0 14 1.4
4A 2.76 0.9 0.6 2.78 2.4 16 1.6
5A 2.81 1.2 1.1 2.81 3.0 31 3.1
B 4.26 1.0 1.0 4.10 1.0 5.9 1.0
1B 3.96 1.0 0.9 3.96 3.5 20 3.5
C 2.47 1.0 1.0 2.49 1.0 2.2 1.0
1C 2.51 2.51 5.6 12 5.5
C 3.26 1.0 1.0 3.23 1.0 0.5 1.0
1C 3.23 1.1 1.1 3.29 4.0
optical response, potentially making nonlinear laser initiation feasible.
8.5 Conclusions
We have utilized TD-DFT to model the OPA and TPA in energetic molecular ma-
terials that posses both enhanced safety and energetic performance properties com-
pared to conventional energetic materials. The chosen level of theory was validated
against experimental data. The geometries used to calculate the optical response of
the monomers and dimers were extracted from their experimentally-determined crys-
tal structures. OPA scales linearly with the number of chromophore units, while TPA
scales nonlinearly. Monomers A and B are more conjugated than C and have larger
TPA cross sections. The most important feature of the optical response is a coopera-
tive enhancement in the TPA of the dimers. Dimer 5A, for example, showed a 6-fold
enhancement in TPA peak intensity. Several dimers have two-photon active states at
low energy, which is important for optical initiation with conventional NIR lasers. We
predict that materials A and C can be excited via OPA and TPA with widely-available
double frequency (532 nm) and regular (1064 nm) Nd:YAG lasers, respectively. Overall,
material A is predicted to have the optical range, intensity, and enhancement that are
favorable for application.
This study has focused on isolated dimers, but even at this level, TPA is enhanced
within the optical window of interest. A nonlinear increase in the TPA cross section is
expected for these materials in bulk. Moreover, based on the results of Fig. 8.2, theoret-
ical spectra are generally blue-shifted relative to experimental spectra. Therefore, pre-
dicted energies may be overestimated, bringing the true window for optical excitation
closer to the NIR. Due to the nature of cooperative enhancement in the nonlinear optical
8.5. Conclusions 109
FIGURE 8.5: (A) Magnitude and relative direction of m
ge
at 2.82 eV in
monomer A and dimer 5A. (B) NTOs of the vertical excitation in 5A. Per-
centages are NTO eigenvalues.
response, it is worth exploring more extended and conjugated structures for explosive
applications. This work is part of an extended project to predict, design, and assess the
applicability of novel photoactive energetic materials. Future work with these materials
will entail nonadiabatic dynamics to determine photoproducts and relative time scales
of decomposition.

111
Chapter 9
Discovering a Transferable Charge
Assignment Model using Machine
Learning
Reprinted with permission from J. Phys. Chem. Lett. 2018, 9, 16, 4495-4501. Copy-
right 2018 American Chemical Society.
9.1 Introduction
Electrostatic interactions contribute strongly to the forces within and between molecules.
These interactions depend on the charge density fieldr(r), which is computationally de-
manding to compute. Simplified models of the charge density, such as atom-centered
monopoles, are commonly employed. These partial atomic charges result in faster com-
putation as well as provide a qualitative understanding of the underlying chemistry.[63–
65, 429] However, the decomposition of charge density into atomic charges is, by itself,
an ambiguous task. Additional principles are necessary to make the charge assign-
ment task well-defined. Here we show that a machine learning model, trained only on
the dipole moments of small molecules, discovers a charge model that is transferable to
quadrupole predictions and extensible to much larger molecules.
Existing popular charge models have also been designed to reproduce observables
of the electrostatic potential. The Merz-Singh-Kollman (MSK)[67, 430] charge model ex-
actly replicates the dipole moment and approximates the electrostatic potential on many
points surrounding the molecule, resulting in high-quality electrostatic properties exte-
rior to the molecule. However, MSK suffers from basis set sensitivity, particularly for
“buried atoms” located inside large molecules[68, 431, 432]. Charge model 5 (CM5)[68]
is an extension of Hirshfeld analysis,[66] with additional parametrization in order to
approximately reproduce a combination of ab initio and experimental dipoles of 614
gas-phase dipoles. Unlike MSK, Hirshfeld and CM5 are nearly independent of basis
set.[432] This insensitivity allows CM5 to use a single set of model parameters. The
corresponding tradeoff is that its charges do not reproduce electrostatic fields as well as
MSK.
A limitation of these conventional charge models is that they require expensive ab
initio calculation, which can be computationally impractical, especially for large molecules,
long time scales, or systems exhibiting great chemical diversity. Recent advances in
machine learning (ML) have demonstrated great potential to build quantum chemistry
112
Chapter 9. Discovering a Transferable Charge Assignment Model using Machine
Learning
models with ab initio-level accuracy while bypassing ab initio costs.[433] Trained to ref-
erence datasets, ML models can predict energies, forces, and other molecular proper-
ties.[434–449] They have been used to discover materials[450–459] and study dynam-
ical processes such as charge and exciton transfer.[460–463] Most related to this work
are ML models of existing charge models[432, 464–466], which are orders of magnitude
faster than ab initio calculation. Here we show that ML is able to go beyond emulation
and discover a charge model that closely reproduces electrostatic properties by training
directly to the dipole moment.
In this chapter, we use HIP-NN (Hierarchically Interacting Particle Neural Net-
work)[467]—a deep neural network for chemical property prediction—to train our charge
model, called Affordable Charge Assignments (ACA). ACA is effective at predicting
quadrupoles despite being trained only to dipoles, demonstrating the remarkable abil-
ity of ML to infer quantities not given in the training dataset. Furthermore, its predic-
tions are extensible to molecules much larger than those used for training. We validate
ACA by comparing it to other popular charge models, and find that it is similar to CM5.
We then apply ACA to long-time dynamical trajectories of biomolecules, and produce
infrared spectra that agree very well with ab initio calculations.
9.2 HIP-NN
We briefly review HIP-NN’s structure. A more complete description is reported
elsewhere in Ref. [467]. HIP-NN takes a molecular conformation as input. The input
representation consists of the atomic numbers of all atoms and the pairwise distances
between atoms. This representation is simple and ensures that the network predictions
satisfy translational, rotational, and reflection invariances. Figure 9.1 illustrates how
HIP-NN processes molecules using a sequence of on-site and interaction layers. On-site
layers generate information specific to each local atomic environment and interaction
layers allow sharing of information between nearby atomic environments.
H
O
H
On-site Layers
Interaction Layers
Network Input

"
#
,
%
#
,
"
#

"
,
%
,
"

"
&
,
%
&
,
"
&
+

"
(
,
%
(
,
"
(
+

#

&
+

(
+
FIGURE 9.1: Abstract schematic of HIP-NN in the context of dipole pre-
diction, illustrated for a water molecule.
9.3. Predicting Molecular Dipole Moment 113
HIP-NN has previously been successful in modeling energy[467] and pre-existing
charge models.[432] In this work, we extend the model for dipole prediction using
m=
N
atoms
å
i=1
q
i
r
i
, (9.1)
where r
i
and q
i
are the position and charge of atom i. HIP-NN’s learned charge assign-
ment q
i
(the ACA charge) is decomposed as a sum over hierarchical corrections,
q
i
=
N
interactions
å
`=0
q
`
i
. (9.2)
As depicted in Fig. 9.1, each q
`
i
is calculated from the activations (i.e. outputs) of the
`-th set of HIP-NN on-site layers. An equivalent decomposition is m = å
`
m
`
where
m
`
= å
i
q
`
i
r
`
i
is the `-th hierarchical dipole correction. HIP-NN is designed such that
higher-order corrections (i.e. m
`
for larger`) tend to decay rapidly.
Training of HIP-NN proceeds by iterative optimization of the neural network model
parameters using stochastic gradient descent. The goal of training is to maximize the
accuracy of HIP-NN’s dipole predictions (as quantified by the root-mean-square-error)
subject to regularization. The full ACA model was generated by an ensemble of four
networks. More details about HIP-NN and its training process are provided in Ref. [467]
and Appendix E.
9.3 Predicting Molecular Dipole Moment
The HIP-NN training and testing data are drawn from the ANI-1x dataset, which
includes non-equilibrium conformations of molecules with C, H, N, and O atoms.[468]
The ANI-1x dataset was constructed through an active learning procedure[469–471] that
aims to sample chemical space with maximum diversity. Although ANI-1x was origi-
nally designed for potential energy modeling, its chemical diversity also enhances the
transferability of ML predictions for other properties, such as the dipole moment. We
restrict molecule sizes to 30 atoms or less, and randomly select 396k for training and
44k for testing. Dataset calculations were performed with Gaussian 09 using thewB97x
density functional and 6-31G

basis set.[234] This level of theory will be referred to as
the quantum-mechanical (QM) standard.
We benchmark the ACA model according to the accuracy of its dipole and quadrupole
predictions. To demonstrate extensibility, we test on the DrugBank ( 13k structures)
and Tripeptides (2k structures) subsets of the COMP6 benchmark,[468] which contain
non-equilibrium conformations of drug molecules and tripeptides. Figure 9.2 shows
the molecular size distribution of these datasets; the molecules in the extensibility sets
are roughly four times larger on average than those of ANI-1x, which we used to train
ACA.
Figure 9.3 shows 2D histograms comparing ACA predicted dipoles and quadrupoles
to the QM reference, for all three datasets. We measure the root-mean-square-error
(RMSE) and mean-absolute-error (MAE). Left panels of Fig. 9.3 compare Cartesian dipole
components in units of Debye (D). The MAE of 0.078 D for predicting ANI-1x dipoles
is comparable to the error between the QM level of theory and experimental dipole
114
Chapter 9. Discovering a Transferable Charge Assignment Model using Machine
Learning
Count
Extensibility
CHNO
0 10 20 30 40 50 60 70 80
Number of Atoms per Molecule
CNO ANI-1x (Train + Test)
DrugBank
Tripeptides
FIGURE 9.2: Size distributions of molecules in three datasets. Top panel
counts the number of all atoms (C, H, N, O) and bottom panel counts
the number of heavy atoms (C, N, O), per molecule. Each histogram is
normalized by its maximum bin count. Although ACA is only trained to
ANI-1x, its predictions are extensible to the much larger molecules in the
DrugBank and Tripeptides datasets.
measurements[472]. The MAE of 0.3 D for predicting DrugBank and Tripeptides
dipoles demonstrates the strong extensibility of ACA. Right panels of Fig. 9.3 compare
quadrupole Cartesian components in units of Buckingham (B). The agreement with QM
is remarkable (MAE = 0.705 B for the ANI-1x tests) in light of the fact that ACA was
trained only to dipoles. Furthermore, ACA continues to make good quadrupole pre-
dictions for the much larger COMP6 molecules. We conclude that the ACA charges
are physically useful for reproducing electrostatic quantities. Additional material quan-
tifying the distributions depicted in Figs. 9.2 and 9.3, including error as a function of
molecular size, are available in Appendix E.
9.4 Assessing ACA Charges
Next, we compare the dipole-inferred ACA model to some conventional charge
models. This analysis uses a subset of GDB-11, denoted here as GDB-5, which con-
tains up to 5 heavy atoms of types C, N, and O.[473] The dataset contains a total of
517,133 structures, including non-equilibrium conformations. Four charge models were
9.4. Assessing ACA Charges 115
-20.0
-10.0
0.0
10.0
20.0
Dipoles
ANI-1x (Test set)
RMSE = 0.121 D
MAE = 0.078 D
-60.0
-30.0
0.0
30.0
60.0
Quadrupoles
ANI-1x (Test set)
0.951 B
0.705 B
-20.0
-10.0
0.0
10.0
ACA
DrugBank
0.487 D
0.275 D
-60.0
-30.0
0.0
30.0
DrugBank
1.885 B
1.266 B
-20.0 -10.0 0.0 10.0 20.0
QM
-20.0
-10.0
0.0
10.0
Tripeptides
0.465 D
0.309 D
-60.0 -30.0 0.0 30.0 60.0
-60.0
-30.0
0.0
30.0
Tripeptides
1.718 B
1.308 B
Log Count
FIGURE 9.3: 2D histograms showing the correlation between pre-
dicted (ACA) and reference (QM) electrostatic moments using three test
datasets: ANI-1x, DrugBank, and Tripeptides. Left and right panels
show dipole and quadrupole correlations, respectively. The values for
the RMSE and MAE are provided in the lower right corner of each sub-
panel. The color scheme for each histogram is normalized by its maxi-
mum bin count. ACA is surprisingly effective in predicting quadrupoles,
given that it was only trained to ANI-1x dipoles.
included in the reference dataset: Hirshfeld,[66] MSK,[67, 430] CM5,[68] and popula-
tion analysis from natural bond orbitals[345] (NBO). Hirshfeld assigns atomic contribu-
tions to the electron density based on their relative weighting to the proto-density. MSK
charges are constrained to reproduce the dipole moment while attempting to match the
electrostatic potential at many points surrounding the molecule. CM5 is an extension
of Hirshfeld, empirically parametrized to reproduce ab initio and experimental dipoles.
NBO charges are computed as a sum of occupancies from all natural atomic orbitals
on each atom. The NBO model is more popular for capturing features such as bond
character.
Figure 9.4 shows the correlation between each pair of charge models and demon-
strates the inconsistency between different approaches for charge partitioning. The
strongest correspondence is between CM5 and ACA, with a mean-absolute-deviation
of 0.031 e. Other model pairs have mean-absolute-deviations that range from three to
eight times larger—a consequence of differing principles used to design these models.
116
Chapter 9. Discovering a Transferable Charge Assignment Model using Machine
Learning
CM5
ACA
0.043 e
0.031 e
Hirshfeld
0.129 e
0.096 e
NBO
0.198 e
0.167 e
MSK
0.145 e
0.101 e
CM5
0.128 e
0.098 e
0.197 e
0.165 e
0.157 e
0.109 e
Hirshfeld
0.294 e
0.257 e
0.233 e
0.174 e
NBO
0.194 e
0.150 e
Log Count
FIGURE 9.4: 2D histograms showing correlations between all pairs of
charge models. The upper and lower values in each subpanel are root-
mean-square-deviation and mean-absolute-deviation, respectively. The
color scheme for each histogram is normalized by its maximum bin count.
The strong agreement between ACA and CM5 charge assignments was
unexpected.
Conceptually, MSK, CM5, and ACA are similar in that they attempt to partition
charge such that the molecular dipole moment is preserved in the point charge represen-
tation. We note, however, that MSK differs significantly from CM5 and ACA (Fig. 9.4).
MSK is constrained to match the QM dipole exactly for each given input molecular con-
figuration. This constraint alone is under-determined, and so MSK therefore invokes
additional principles for its charge assignment, attempting to fit the far-field electro-
static potential. However, the far-field potential is relatively insensitive to the partial
charge assignments of internal atoms.[68, 431, 432] Because MSK performs its charge
assignments according to global (rather than local) criteria, the assigned charges can de-
viate significantly from the local charge density field. Another related difficulty of MSK
is that it exhibits a noticeable basis set dependence.[431, 432]
CM5 was designed to address such drawbacks.[68] Like CM5, our ACA charge
model is local-by-design, thus averting the problem of artificial long-range effects. Specif-
ically, ACA seeks a local charge assignment model that best reproduces the QM dipoles
9.5. Machine Learned Infrared Spectra with ACA Charges 117
over the whole training dataset. We remark that the ACA dipole predictions do not per-
fectly reproduce the QM dipoles. Allowing for this imperfection may actually be im-
portant; collapsing a charge density field into a relatively small number of monopoles
while simultaneously forcing the molecular dipole to be exact may be incompatible with
locality of the charge model.
As we showed in Fig. 9.4, the CM5 and ACA charges are remarkably consistent; a
result we did not anticipate. CM5 reproduces the molecular dipole well, but not as ac-
curately as ACA (see Appendix E). The reduced accuracy of CM5 dipoles may be due
to the fact that it is fit to a hybrid of ab initio and experimental data. In contrast, ACA
trains to a homogeneous database of QM dipoles. The ML approach has a conceptual
advantage: it is fully automated and requires few design decisions (primarily, the speci-
fication of an error metric for training). As a consequence, the extension of ACA to new
atomic species and to new classes of molecules should be straightforward.
9.5 Machine Learned Infrared Spectra with ACA Charges
A strong practical advantage of ACA is that assignment does not require any new
QM calculations. We highlight this advantage of efficiency by applying ACA to calcu-
late an experimentally-relevant quantity. Inspired by the work of Ref. [448], we use ACA
to calculate dynamic dipoles and subsequently infrared spectra for select molecules.
Ground-state trajectories were generated from the ANI-1x potential[468] and were 100
ps in length with a 0.1 fs time-step—amounting to a total of 10
6
time-steps. Dipoles
were predicted along these trajectories using ACA. Both the molecular dynamics and
dipole prediction were performed using only ML; that is, without any QM calculation.
Spectra were made by Fourier transforming the dipole moment autocorrelation func-
tion. Harmonic spectra were calculated with the Gaussian 09 software. A comparison
of time-domain ML spectra to QM harmonic spectra is shown in Figure 9.5, left panels.
Although time-domain and harmonic spectra are not one-to-one, the comparison is rea-
sonable since spectral features are harmonic to first order. ACA recovers the harmonic
features across all molecules.
To further validate the ACA dipole predictions, QM calculations were performed at
10
3
subsampled time-steps throughout the trajectories. Fig. 9.5, right panels, shows that
the ACA dipole predictions are in excellent agreement with QM; another validation of
ACA’s extensibility. The dipole errors are consistent with those observed in the datasets
of Fig. 9.3. Note that cholesterol and morphine have 74 and 40 atoms, respectively,
whereas our training dataset has no molecules with more than 30 atoms. The quality
of the ML-predicted spectra for cholesterol and morphine is similar to those of smaller
molecules, such as aspirin.
We carried out an additional test with smaller molecules of sizes 6 to 15 atoms,
making it feasible to calculate QM dipoles at all 10
6
time-steps. The resulting infrared
spectra are shown in Appendix E, and are in excellent agreement with our ML-based
approach. For these smaller molecules, ACA yields a factor of greater than 10
4
compu-
tational speed-up. The relative speed-up is even more dramatic for large molecules.
118
Chapter 9. Discovering a Transferable Charge Assignment Model using Machine
Learning
9.6 Conclusions
In summary, the key contribution is the formulation of an electrostatically consistent
charge model called Affordable Charge Assignments (ACA). We construct the ACA
model using a deep neural network that outputs charges. The network is trained to
DFT-computed molecular dipole moments over a diverse set of chemical structures. The
fast and accurate predictive power of the model was evidenced with extensibility tests
(Fig. 9.3) and infrared spectra (Fig. 9.5). Although ACA is only trained directly to the
molecular dipole, we show that it also captures quadrupole moments, demonstrating
transferability.
ACA is compared with four conventional charge models on a dataset containing
over 500k molecules (Fig. 9.4). The rather poor correlation between most model pairs
confirms the ambiguity in charge partitioning. The ACA model correlates well to Charge
Model 5 (CM5). CM5 was designed to combine advantages of the Hirshfeld and MSK
models. It is parameterized to reproduce a combination of ab initio and experimental
dipoles. ACA, like CM5, is a local model that is designed to reproduce dipoles, but
unlike CM5, is built entirely from ab initio data. In addition to fast charge assignments,
a potential advantage of ACA is its applicability to a wide range of chemically diverse
systems, assuming that appropriate training data is available. This work is also a tes-
tament to how physics-informed ML can be used to discover properties (here, charge
assignment) not employed as an explicit target in the training process. We would also
like to note an independent and concurrent study (i.e. Ref. [474]) that took a similar ap-
proach in constructing dipole-driven partial charges. The authors of Ref. [474] confirm
that inferred charges produce interpretable insight into chemical structure.
Future work will focus on improving and utilizing ACA for quantum-chemical pre-
diction. Improvements to extensible dipole prediction may be made by engaging in
dipole-driven active learning. Furthermore, ACA could be trained to higher-order mul-
tipole moments such as quadrupoles—this could be important for systems where the
dipole does not provide enough of a constraint for charge assignments. Currently, ACA
is limited to C, H, N, and O atoms, but this could be overcome when more diverse
datasets are available. Another important drawback of the current model is that charged
systems, such as anionic and cationic species, cannot yet be treated. An application un-
derway is to predict dynamic charges in neutral biomolecular systems to parametrize
force fields for molecular dynamics. We hope that this study not only sheds light on
charge models that best reproduce dynamical data but also helps explain the interesting
correlations among charge models (visible in Fig. 9.4).
9.6. Conclusions 119
Aspirin
(21, 13)
-5
0
5
0.085 D
0.068 D
Caffeine
(24, 14)
QM Harmonic
Time-Domain ML
-5
0
0.163 D
0.122 D
Intensity
Tyrosine
(24, 13)
-5
0
ACA Dipole (D)
0.097 D
0.070 D
Morphine
(40, 21)
-5
0
0.143 D
0.111 D
500 1500 2500 3500
Frequency (cm
−1
)
Cholesterol
(74, 28)
-5 0 5
QM Dipole (D)
-5
0
0.185 D
0.136 D
Log Count
FIGURE 9.5: (Left) Infrared spectra of select molecules, computed with-
out polarization effects due to solvation. The values in parentheses are
the total number of all atoms (C, H, N, O) and of heavy atoms (C, N,
O), respectively. The agreement between QM and ACA-derived spec-
tra is reasonable, given that the harmonic approximation is not exact.
(Right) 2D histograms of predicted (ACA) versus true (QM) dipoles at
10
3
subsampled time-steps throughout the 100 ps trajectories. The up-
per and lower values in each subpanel are RMSE and MAE, respectively.
The color scheme for each histogram is normalized by its maximum bin
count.

121
Chapter 10
Closing Remarks
The interaction of light and matter plays an important role in science, both at the
fundamental level and for applications. In this thesis, we provided an overview of
some theoretical efforts made to advance this field, with special focus on molecular sys-
tems. Our approach was to begin with the development of methods used to model the
evolution of a molecular system as a result of photoexcitation. This task involved de-
scribing how electronic excitations affect nuclear motion and vice versa. Two important
approximations made for computational tractability are the Born-Oppenheimer (BO)
approximation and the classical description of nuclei. Together, they are central to the
surface hopping method in which a nuclear trajectory evolves classically on a single
adiabatic (or BO) potential energy surface (PES) and transitions between PESs depend
on the strength of nonadiabatic coupling. Like any approximate theory, situations arise
in which the theory breaks down. We contributed to areas in which the method can be
improved and even presented numerical tests of insightful benchmarks. The topics cov-
ered were the representation dependence of surface hopping–the unphysical, energy-
forbidden transitions that are encountered when dynamics evolve in the diabatic (as
opposed to adiabatic) representation (Chapter 2 and Appendix A), detailed balance–
the realization of the Boltzmann populations at thermal equilibrium (Chapters 2 and 3),
and decoherence–the decoupling of a mother wavepacket into independent daughter
wavepackets (Chapter 4). These calculations were carried out using exactly solvable
and relatively simple models. Thus, the conclusions drawn using these models and
their relevance to molecular simulations are subject to further investigation.
Meanwhile, we also contributed to developing a software (i.e., NEXMD) to simu-
late photoexcited molecules. We introduced an implicit solvent model to NEXMD and
studied the effects of a polarizable environment on dynamics (Chapter 5). The effects of
solvation on several calculated quantities such as excited state lifetimes, exciton local-
ization, and structural relaxation were found to depend on molecular polarizability due
to chemical substitution. Our results suggest that the effects of polarization are well
captured with implicit solvation, but a study comparing implicit and explicit solva-
tion would provide more insight into the strengths and weaknesses of each model. We
also carried out a case study of the photoinduced isomerization process and compared
theoretical predictions to experimental observation to assess the accuracy of NEXMD
(Chapter 6). Our study of Styrylquinoline suggests that external stimuli such as sol-
vation and the thermostat strongly effect the outcome of the isomerization processes.
NEXMD simulations also predicted an energy profile between the ground and first sin-
glet excited state showing two reaction pathways to the final stable product involving a
122 Chapter 10. Closing Remarks
single photon or a sequential absorption of two photons, in agreement with experimen-
tal observations. Our contributions extend to multiple levels of methods development,
from ensuring surface hopping obeys thermal equilibrium and decoherence to adding
solute-solvent interactions into molecular simulations, thus providing a reliable and
predictive framework for modeling photoexcited molecules.
Besides methods development, we also carried out calculations of photoactive ener-
getic materials for military application. We predicted the optical properties in a set of
experimentally-identified compounds to connect molecular architecture and optical ab-
sorption energy. Lowering the initiation threshold such that excitation energies match
those of economic and readily-available NIR lasers requires theoretical investigation,
as chemical synthesis is costly and time consuming whereas computation is a practical
first-look approach. This part of our work used a more first-principles level of the-
ory (i.e., TD-DFT) than that used for dynamical simulations, as quantitative accuracy
is important for engineering purposes. Furthermore, the diversity of the investigated
compounds, which included transition-metal coordination complexes (Chapter 7) and
dimers extracted from crystalline structures (Chapter 8), required determining the level
of DFT that would reliably predict their optical properties. We compared absorption
spectra from theory to experiment and chose the appropriate model quantum chemistry.
Ultimately, increasing conjugation, such as the fuzed rings of the tetrazine-based ligand
architectures (Chapter 7), is the most influential design principle for lowering optical
excitation energy. We also sought to determine if two-photon absorption can increase
the viability of NIR lasers for systems in which one-photon absorption was not possible.
We identified a material showing an enhanced nonlinear absorption at energies within
the NIR (Chapter 8). Although dynamical studies of these energetic compounds have
not yet been carried out, it would be the next step for their complete understanding,
i.e., photochemical pathways, photoproducts and time scales of decomposition. Yet, the
electronic structure theory used in the dynamics must accurately describe open-shell
systems during photodecomposition. The development of NEXMD to handle these sys-
tems is underway.
A major setback of computational chemistry is the high demand of resources needed
for ab initio calculations. The solution has been to set approximations, but in reality, the
validity of these approximations have limited bandwidth. It is a hard task just knowing
when approximations or what levels of theory are reasonable for a given study. It is not
surprising that a long sought-after goal of theoreticians has been to attain numerically
efficient ab initio calculations. The final chapter of this thesis presented an application
using a novel method to accelerate molecular simulations; highly parametrized statis-
tical models that are trained to learn physical quantities from large reference datasets,
i.e., machine learning. We used a deep neural network (DNN) to predict molecular
dipole moment (Chapter 9). A map was created between the ab initio dipole and the
dipole computed from atom-centered point charges. The DNN was trained to reduce
the error between these two quantities and by doing so, we went beyond emulation and
not only trained a model that predicted dipole but also one that inferred a new charge
model, which we called Affordable Charge Assignments (ACA). ACA accurately pre-
dicted ab initio dipoles on a dataset containing more than 500k structures. The ACA
model was also found to be fast, dynamic, and extensible–the model accurately pre-
dicted the dipoles on datasets containing molecules that are roughly four times larger
(on average) than those used for training. Another encouraging result was that ACA
Chapter 10. Closing Remarks 123
recovered the quadrupole moments of extensible datasets, despite being only trained to
dipoles, thus highlighting the model’s transferability. The strengths of ACA were ev-
idenced by its ability to predict charges that are consistent with electrostatic moments
(i.e., dipoles and quadrupoles) at significantly reduced computational costs. From the
perspective of excited state chemistry, ML would be monumental for nonadiabatic dy-
namics if PESs and nonadiabatic couplings are trainable quantities, as these are major
computational bottlenecks (section B.10). Introducing ML Hamiltonians into NEXMD
may lead to the modeling of very large photoexcited systems (whether it be time scales,
number of atoms, or number of excited states) at ab initio quality. The output of articles
combining ML with nonadiabatic dynamics has already begun.[62, 475]

125
Appendix A
Supporting Information for
Chapter 4
A.1 Global Flux Versus Fewest Switches Surface Hopping in
the Diabatic Representation
FIGURE A.1: Three-level model in the diabatic representation.
FSSH tends to prefer the adiabatic over the diabatic representation.[20] We convey
this point for a certain class of systems and show how a minor correction to the sur-
face hopping algorithm can improve scattering probabilities in the diabatic represen-
tation. First, it is important to recognize why FSSH fails for this class of systems. The
time derivative of state population (which FSSH explicitly uses to compute nonadiabatic
126 Appendix A. Supporting Information for Chapter 4
transition probabilities) is given by,[5]
˙ r
ii
=
2
¯ h
å
j
Im

r

ij
V
ij

2
å
j
Re

r

ij
˙
R d
ij

. (A.1)
The derivative coupling is given by d
ij
=

f
i
jrf
j

wheref is a basis state in the repre-
sentation used to evolve Eq. A.1. In the adiabatic representation, V is diagonal; eigen-
values are PESs. The first term of Eq. A.1 is zero in the adiabatic representation for all
i, j because if i = j, r

ii
V
ii
is real, and if i6= j, V
ij
= 0. Thus, the second term of Eq. A.1
fully describes ˙ r
ii
. Conversely, in the diabatic representation, d
ij
= 0, and the first term
of Eq. A.1 fully describes ˙ r
ii
.
We calculate scattering probabilities using the model shown in Fig. A.1 in the dia-
batic representation.[22] In this model, the two lowest-energy diabatic states (j1i and
j3i) are coupled through a high-energy intermediate (j2i). Diabatic couplings are V
13
=
0, V
12
6= 0, and V
23
6= 0. Assuming trajectories begin onj1i, heading toward the inter-
action region, the pathway toj3i isj1i!j2i!j3i. But sincej2i is of higher energy
thanj3i, conserving energy of the nuclear coordinate prevents thej1i!j2i transition
as long as the hopping algorithm is governed by the first term of Eq. A.1.
FIGURE A.2: Scattering probabilities on diabatj3i as a function of initial
nuclear momentum.
Fig. A.2 shows computed scattering probabilities. Trajectories begin on diabatj1i
heading toward the interaction region. The fraction of trajectories (scattering probabil-
ities) on each surface are computed outside the interaction region. The nuclear kinetic
energy in regime I is not large enough to overcome the V
33
V
11
energy barrier, thus pre-
ventingj1i!j3i population transfer. Surface hopping and exact quantum mechanics
agree in this regime. In regime III, the nuclear kinetic energy is larger than the V
33
V
11
A.2. Scattering Problems: Surface Hopping Versus Exact Quantum Mechanics 127
barrier;j3i becomes populated with FSSH as a result ofj1i!j2i!j3i transitions, also
in agreement with exact results. Finally, the nuclear kinetic energy in regime II is within
the range(V
33
V
22
) < k
2
/2m (V
33
V
11
); scattering probabilities onj3i are under-
estimated in this regime using FSSH sincej1i!j2i is not energetically allowed. In
exact calculations, population does transfer fromj1i toj3i. In summary, FSSH matches
exact results in regimes I and III, but underestimates scattering probabilities in regime
II on account of the diabatic coupling in Eq. A.1. GFSH overcomes the energy conser-
vation problem of regime II (Fig. A.2) because the hopping algorithm (shown in main
text) depends on discrete changes in population between consecutive time steps.
A.2 Scattering Problems: Surface Hopping Versus Exact Quan-
tum Mechanics
A.2.1 Two-Level Models
A-GFSH is tested with the following two-level models: Tully’s extended coupling
with reflection (ECR),[5] a dumbell geometry (DBG),[35] and a modified version of
Tully’s dual-avoided crossing (DAC).[476] In these tests, dynamics evolve in adiabatic
representation. Trajectories start on the lower surface, in the negative region of the re-
action coordinate, in a pure state withjc
1
j
2
= 1. Trajectories continue until they are
well outside of the interaction region and the fraction of trajectories on each surface are
computed to determine reflection and transmission probabilities. The two-level models
are shown in Fig. A.3.
For energies E 0.2 or momenta k . 28.3, trajectories in ECR (Fig. A.3A) reflect
or transmit on the lower surface or reflect on the upper surface. The lower and upper
surfaces are nearly degenerate in the negative region of the reaction coordinate. The
nonadiabatic coupling (d
12
=hf
1
jrVjf
2
i /(E
2
E
1
)) spans over an extended spatial
region. Upon exiting the interaction region, electronic states become decoupled, but
since there is no explicit form of decoherence in surface hopping, the electronic density
matrix for trajectories that reflect back through the interaction region is compromised,
resulting in spurious interference patterns in reflection (Fig. A.4A). A-GFSH corrects
GFSH for this overcoherence. For E > 0.2, trajectories begin to trasmit on the upper
surface; overcoherence is no longer an issue since trajectories do not make a second
pass through the interaction region.
For energies E 0.2 or momenta k. 28.3, trajectories in DBG (Fig. A.3B) are gau-
ranteed to reflect on the lower surface due to the barrier at x10. The barrier is
surpassed at energies E > 0.2. For energies 0.2 < E 0.4 or momenta 28.3. k 40.0,
trajectories either reflect or trasmit on the lower surface. Multiple passes through the
interaction region without decoherence is the source of the inaccurate scattering prob-
abilities shown in Fig. A.4B. Again, A-GFSH corrects for this overcoherence. Finally,
when E> 0.4, overcoherence has less of a negative effect on results as trajectories begin
accessing the upper surface.
The DAC model is famous for its Stueckelberg oscillations and need for a phase
correction. Compared to Tully’s original DAC model,[5] the diabatic coupling in the
model of Fig. A.3C is broadened to enhance overcoherence. For this model, electronic
coefficients were integrated with a phase-corrected Hamiltonian, where the diagonal
elements of the Hamiltonian are changed to H
ii
= p
i
p
j
/m, where j refers to the
128 Appendix A. Supporting Information for Chapter 4
FIGURE A.3: Two-level scattering models. Potential energy surfaces
(bold) and nonadiabatic couplings (dashed) are shown.
residing surface corresponding to adiabatic statejji. Details of the correction can be
found in Ref. [75]. Trajectories reside on the lower surface at energies E 0.05 or
momenta k . 14.1. For E > 0.05, the upper surface becomes accessible. Fig. A.4C
shows phase-corrected A-GFSH matching exact solutions at mid to high energies (i.e.,
k& 15). Both GFSH and A-GFSH do not perform well at low energies (i.e., momenta
within 5. k. 15).
A.2.2 Three-Level Model
The overcoherence problem is accentuated when trajectories encounter many dia-
batic state crossings (or regions with strong nonadiabatic coupling). Model X of Fig. A.5
has three avoided crossings. Trajectories start on the middle surface, in the negative
region of the reaction coordinate, in a pure state withjc
2
j
2
= 1.
A.2. Scattering Problems: Surface Hopping Versus Exact Quantum Mechanics 129
FIGURE A.4: Fraction of trajectories that reflect or transmit on the lower
surfaces of the two-level models.
At energies 0 < E < 0.025 or momenta 0 < k < 10, trajectories reflect on the lower
and middle surfaces (due to the barrier at x = 0) or transmit on the lower surface.
The GFSH reflection on the lower surface is overestimated in Fig. A.6A, while reflection
on the middle surface is underestimated in Fig. A.6B. At energies 0.025 < E 0.035
or momenta 10.0 < k . 11.8, the majority of trajectories transmit on the lower and
middle surfaces, while only a small fraction of trajectories reflect (Figs. A.6A and A.6B).
The well of the upper surface is accessed at energies 0.035 < E 0.06 or momenta
11.8. k. 15.5, which significantly increases the number of crossings a trajectory passes
through before leaving the interaction region. Finally, for energies E 0.06, trajectories
transmit on the upper surface in Fig. A.6D. In this regime, overcoherence does appear
to be a problem. A-GFSH significantly improves results compared to GFSH. Results
are further improved with a phase correction[75], as evidenced by transmission on the
lower surface in Fig. A.6C, over nearly all incoming momenta.
130 Appendix A. Supporting Information for Chapter 4
FIGURE A.5: Model X: three-level scattering model.
A.3 Subotnik’s Augmented Surface Hopping
Here, we summarize the derivation of the augmented surface hopping method. A
detailed derivation is available in Ref. [35]. The method will be formulated in the dia-
batic representation and then in the adiabatic. The reduced electronic density operator
is formally written as
s = Tr
N
(r) , (A.2)
where Tr
N
denotes trace over the nuclear coordinate (N) andr is the combined nuclear-
electronic density matrix. Inserting Eq. A.2 into the Liouville-von Neumann equation,
i¯ h ˙ r= [H,r] , (A.3)
yields
i¯ h ˙ s
ij
=

f
i
jTr
N
[V,r]jf
j

, (A.4)
where

f
i
jTr
N
[T
N
,r]jf
j

= Tr
N

T
N
,r
ij

= 0 since T
N
is the nuclear kinetic energy op-
erator, which is unaffected by the electronic statesjf
i
i. All potential energy terms and
the electronic kinetic energy T
e
are included in V. The diabatic statesjf
i
i are indepen-
dent of nuclear coordinate and time. The derivation starts with expanding V around
the center of the nuclear wavepacket, which is approximated within the surface hop-
ping analog, R
SH
. Expanding V up to second-order yields
V V

R
SH

+
å
i
¶V
¶R
i

R
SH

R
i
R
SH
i

+
å
i,j
¶
2
V
¶R
i
¶R
j

R
SH

R
i
R
SH
i

R
j
R
SH
j

.
(A.5)
A.3. Subotnik’s Augmented Surface Hopping 131
FIGURE A.6: Fraction of trajectories that reflect or transmit in model X.
Inserting Eq. A.5 into Eq. A.4, the density matrix is governed by
˙ s
i
¯ h
h
V

R
SH

,s
i
+
i
¯ h
[F,dR] , (A.6a)
where
dR= Tr
N
n
R R
SH

r
o
(A.6b)
is the position moment and F
ij
=

f
i
jrVj
R
SHjf
j

is the quantum force. Second-
order terms, e.g. dR
i
dR
j
, are neglected in the construction of Eq. A.6a. A derivative
of Eq. A.6b, with respect to time, yields
˙
dR
dP
m

i
¯ h
h
V

R
SH

,dR
i
, (A.7a)
where
dP= Tr
N
n
P P
SH

r
o
(A.7b)
is the momentum moment. Likewise, the derivative ofdP with respect to time is
˙
dP
˙
P
SH
s
i
¯ h
h
V

R
SH

,dP
i
+
1
2
(Fs+sF) . (A.8)
132 Appendix A. Supporting Information for Chapter 4
The density matrix and moments evolve in the adiabatic representation by a simple
extension: V

R
SH

! E

R
SH

i¯ h
˙
R
SH
d, where E is the diagonalized Hamiltonian;
diagonal elements are adiabatic potential energy surfaces and d is the nonadiabatic cou-
pling vector given by d
ij
=

f
i
jrf
j

=

f
i
jrVjf
j

/

E
j
E
i

wherejf
i
i are now adi-
abatic states. Eqs. A.7a and A.8 describe how the position and momentum moments
evolve.
The first-order decoherence rate can be written in terms of position and momentum
moments. The nuclear-electronic wavefunction is given by
y(r, R)=
å
i
x
i
(R)f
i
(r; R) , (A.9)
where adiabatic electronic eigenfunctions, f
i
(r; R) , are parametrized by nuclear coor-
dinates, R , and the coefficients of expansion are nuclear wavefunctions,x
i
(R). Without
loss of generality, we derive the decoherence rate for a two-level system. Assuming
off-diagonal elements of the formjs
12
jµ exp(t/t) , the decay rate is
t
1
=
1
js
12
j
djs
12
j
dt
. (A.10)
In a region of zero nonadiabatic coupling, Eq. A.6a yields
t
1
= Im

(F
11
F
22
)dR
12
¯ hs
12

. (A.11)
Eq. A.11 is computationally unstable due to s
12
in the denominator. The following ap-
proximations are made to obtain a computationally stable decoherence rate:
(i) Nuclear wavefunctions are given a Gaussian ansatz of the form,
x
i
(R)=

1
ps
2
1
4
exp
(

R R
SH

2
2s
2
)
exp

i
¯ h
P
SH

R R
SH

. (A.12)
(ii) Gaussian widths,s
P
k
, are chosen such that the overlap of the nuclear wavefuctions
on different surfaces are maximized, i.e.,¶hx
1
jx
2
i /¶s
2
P
k
= 0.
(iii) A lower bound on the decoherence rate to ensure minimal number of collapsing
events.
These approximations yield
t
1

1
2¯ h
(F
22
F
11
)dR
22
. (A.13)
Eq. A.13 is only valid in regions of zero nonadiabatic coupling and must be modified,
especially since excessive collapsing events in regions of nonadiabatic coupling leads to
spurious results (shown in main text). From Eq. A.6a, it can be shown that
Im

F
12
(dR
22
dR
11
)
¯ hs
12

(A.14)
A.3. Subotnik’s Augmented Surface Hopping 133
contributes to the decoherence rate, where F
ij
µ d
ij
. An empirical modification to en-
sure a minimum number of collapsing events, together with dR
ii
0 whenjii is the
active electronic state leads to
t
1

1
2¯ h
(F
22
F
11
)dR
22

2
¯ h
jF
12
dR
22
j . (A.15)
Eq. A.15 is the collapse rate used in simulations. A step-by-step algorithm is provided
in Ref. [116]

135
Appendix B
Supporting Information for Chapter
5
B.1 Nonadiabatic Modeling Procedure
Nonadiabatic dynamics of each molecule were carried out as follows: Step 1: Opti-
mization The ground state geometry was optimized using the semiempirical AM1[208]
model Hamiltonian within NEXMD.[13, 14, 41, 86, 218, 477]
Step 2: Ground State Sampling A single ground state trajectory evolved adiabatically
for several nanoseconds with a time step of 0.5 fs in an equilibrated Langevin thermostat
with temperature 300 K and frictional parameter 20 ps
1
. A set of nuclear geometries R
and velocities
˙
R were sampled throughout the ground state trajectory and used as the
input geometries and velocities for excited state trajectories. This composes a photoex-
cited wavepacket sampling the phase space of nuclear configurations accessible at room
temperature. In past studies, ensembles of approximately 400 trajectories have provided
statistically converged results to within 5% error.[477] Here, we report 635 trajectories.
Step 3: Optical Spectrum A theoretical optical absorption spectrum from singlepoint
calculations was generated in order to determine the energy at which to excite the sys-
tem. We generated an average spectrum over all geometries (i.e. the sum of all spectra,
divided by the number of geometries). The oscillator strength of each excitation en-
ergy was given a Gaussian lineshape with an empirical standard deviation set to 0.15
eV , thereby approximating a Gaussian-shaped Franck-Condon window. The optical
absorbance of each excitation energy was broadened by this Gaussian lineshape and
weighted by the oscillator strength at that energy,
A
i
e
(W)= f
i
ge
(W
e
)
1
p
2ps
2
exp
"

(W
e
W)
2
2s
2
#
. (B.1)
where A
i
e
defines the contribution to the absorbance of the i-th geometry from excited
statejei. The absorbance over all excitation energies becomes
A
i
(W)=
å
e
A
i
e
(W) . (B.2)
To obtain the combined optical spectrum, the spectra from all geometries were aver-
aged,
A(W)=
1
N
å
i
A
i
(W) , (B.3)
136 Appendix B. Supporting Information for Chapter 5
where N is the number of geometries.
Step 4: Initial Wavepacket The excited states of the optimized ground state struc-
tures were obtained using the collective electronic oscillator (CEO) method[202, 203]
with the configuration interaction with singles (CIS) wavefunction,[205] combined with
AM1.[208] The initial excited state of each trajectory was chosen according to a Gaussian
Franck-Condon window,[194]
c
a
(r; R)= exp
"

(E
laser
W
a
)
2
2s
2
#
, (B.4)
where E
laser
is photon energy,W
a
is the energy of the adiabatic, electronic statejai, and
s is an empirically chosen linewidth broadening parameter (standard deviation) set to
0.15 eV . Oscillator strengths computed for each state were weighted by c
a
(r; R), and
the initial state was chosen by comparing the weighted value with a random number.
Step 5: Excited State Dynamics Nuclei evolved classically on excited state potential
energy surfaces (PESs) in a Langevin thermostat. Excited state forces were computed
“on-the-fly” at each classical time stepDt.[14, 208, 478, 479] The use of native excited
state forces is a more rigorous description of nuclear dynamics than the classical path
approximation (CPA).[480] For the latter, nuclei evolve on a ground state trajectory
while quantum amplitudes evolve in the excited states. Although CPA is appropri-
ate for crystalline systems that have similar ground and excited state forces,[153] it is
discouraged for conjugated molecules, where differential nuclear motion can rule out
the validity of this approximation.[157, 193, 195] Since electrons generally move faster
than nuclei, a smaller time step of integration dt < Dt was required for electronic dy-
namics. Insufficiently small dt can fail to resolve strongly localized nonadiabatic cou-
plings, which can underestimate excited state transition probabilities.[477] Classical and
quantum degrees of freedom evolved with time steps of 0.1 fs and 0.02 fs, respectively.
Quantum coefficients c
a
evolved according to the Schrödinger equation,[5]
i¯ h ˙ c
a
(t)= c
a
(t) E
a
(R) i¯ h
å
b
c
b
(t)
˙
R d
ab
(B.5)
with excited state energies E
a
and analytical nonadiabatic couplings[481, 482]
d
ab
=

f
a
(r; R)jr
R
jf
b
(r; R)

(B.6)
evaluated at each time step dt.[13, 14] A new random seed was used for each trajectory
in order to obtain reliable sampling and avoid nonphysical synchronicity effects of the
Langevin thermostat.[483]
Step 6: Surface Hopping and Decoherence Nonadiabatic evolution of excited state tra-
jectories were computed with Tully’s fewest-switches surface hopping (FSSH).[5, 6, 13,
19, 31] The quantum time step was further reduced by a factor of 40 in the vicinity of
trivial unavoided crossings in order to accurately estimate transition probabilities.[86]
FSSH has been utilized to elucidate nonadiabatic processes in a wide variety of systems
from single molecules to bulk materials.[7, 13, 15, 17, 158, 484–487] However, FSSH fails
to properly decohere the electronic wavefunction.[31] The problem arises from coher-
ently propagating the electronic wavefunction along a classical nuclear trajectory. In
a completely quantum mechanical picture, a wavepacket that approaches an avoided
B.2. Optical Spectra 137
crossing bifurcates, and the daughter wavepacket on one surface eventually moves in-
dependently from the daughter wavepacket on any other surface.[31] Moreover, dy-
namics with several crossings are susceptible to spurious results without the inclusion
of decoherence.[32] Many methods within surface hopping have been designed to de-
cohere the electronic wavefunction.[34, 35, 40, 146, 488] The method used here is the
instantaneous collapse approach, which resets the quantum amplitude of the current
state to unity after an attempted hop. This is based on the notion that wavepackets
immediately separate and move independently in phase space. Instantaneous collapse
is the simplest among all methods and qualitatively improves dynamics at no additional
computational cost.[41]
B.2 Optical Spectra
2.0 2.5 3.0 3.5 4.0
Energy (eV)
Intensity (AU)
Absorption Emission
FIGURE B.1: Comparison between theoretical and experimental absorp-
tion and emission spectra. Theoretical spectra (solid) are those offH, Hg
with e = 2 and experimental (dashed) are those of PPV3 measured in
dioxane (e 2.2) at 293 K.
B.3 Natural Transition Orbitals (NTOs)
138 Appendix B. Supporting Information for Chapter 5
TABLE B.1: Natural transition orbitals (NTOs) of the first excited state S
1
with e = 1. Excitation energy, oscillator strength, and percent contribu-
tion of each NTO are shown.
{R
1
, R
2
} W(eV), f (unitless) percent hole electron
{H, H}
2.93, 0.92 78
{H, NO
2
}
2.89, 1.02 77
{NH
2
, NO
2
}
2.82, 1.17 74
TABLE B.2: Natural transition orbitals (NTOs) of photoactive excited
states in the vicinity of 4.30 eV with e = 1. Excitation energy, oscillator
strength, and percent contribution of each NTO are shown.
{R
1
, R
2
} W(eV), f (unitless) percent hole electron
{H, H}
4.40, 0.54 36
16
13
13
{H, NO
2
}
4.38, 0.63 33
Continued on next page
B.3. Natural Transition Orbitals (NTOs) 139
Table B.2 – Continued from previous page
{R
1
, R
2
} W(eV), f (unitless) percent hole electron
18
15
12
10
{NH
2
, NO
2
}
4.30, 0.45 42
27
13
140 Appendix B. Supporting Information for Chapter 5
B.4 Energy Gaps
(A) R
1
= H, R
2
= H
† = 1
Density
† = 2
† = 5
† = 20
0.0 0.2 0.4 0.6 0.8 1.0
Energy Gap (eV)
Mean
Energy
Gaps
(B) R
1
= H, R
2
= HNO
2
† = 1
Density
† = 2
† = 5
† = 20
E
21
E
32
E
43
E
54
E
65
E
76
E
87
E
98
0.0 0.2 0.4 0.6 0.8 1.0
Energy Gap (eV)
Mean
Energy
Gaps
(C) R
1
= NH
2
, R
2
= HNO
2
† = 1
Density
† = 2
† = 5
† = 20
0.0 0.2 0.4 0.6 0.8 1.0
Energy Gap (eV)
Mean
Energy
Gaps
FIGURE B.2: Histograms of energy gaps over the entire 1 ps ensemble of
trajectories withe =f1, 2, 5, 20g. The maximum PES considered for each
molecule corresponds to the optically excited state at 4.30 eV . The bottom
panel of each subfigure shows mean energy gaps averaged over all time
steps and trajectories.
B.5. Exponential Rate and Time Decay Constants of S
1
141
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(A) R
1
= H, R
2
= H
0.0
0.2
0.4
0.6
0.8
1.0
Fraction of Maximum Small Gaps
(B) R
1
= H, R
2
= NO
2
0 5 10 15 20
Dielectric Constant, † (unitless)
0.0
0.2
0.4
0.6
0.8
1.0
(C) R
1
= NH
2
, R
2
= NO
2
E
21
E
32
E
43
E
54
E
65
E
76
E
87
E
98
FIGURE B.3: Fraction of the maximum number of small energy gaps as
a function dielectric constant. The number of times each gap fell below
a threshold of< 0.1 eV was recorded. For each gap, the total number of
small gaps was divided by that of the dielectric constant with the largest
number of small gaps–maximum y-value shown is 1.0.
B.5 Exponential Rate and Time Decay Constants of S
1
142 Appendix B. Supporting Information for Chapter 5
TABLE B.3: Exponential rate (k) and time decay constant

k
1

of S
1
across all molecules and dielectric constants. The S
1
populations were
fit to a curve of the form A B exp(kt).
{R
1
, R
2
} e k

10
3
fs
1

k
1
(fs)
{H, H} 1 1.71 586
2 1.46 684
5 1.46 682
20 1.38 726
{H, NO
2
} 1 7.33 136
2 4.66 214
5 3.29 304
20 2.87 348
{NH
2
, NO
2
} 1 4.36 230
2 2.73 366
5 3.49 286
20 4.06 246
B.6 Transition Density (TD) Analysis
Each molecule was split into two fragments and the fraction of transition density
(TD) on each fragment was analyzed by considering individual excited states and sol-
vent polarities (Figure B.4). For the sake of brevity, we only discuss the asymmetric con-
formation offH, NO
2
g since it significantly distinguishes TD on each fragment. The
fraction of TD on each fragment was calculated using Equation 5.3. InfH, NO
2
g, the
most populated states throughout the dynamics are S
1
, S
2
, and S
9
, where S
9
is the initial
excitation. We previously saw that at t = 0 fs, TD is split evenly on both fragments
and as time evolves, the excitation becomes more localized on the NO
2
fragment. As
dynamics trickles down into lower-lying excited states, S
n
for n 4, the excitation is
still more localized on the NO
2
fragment. At S
2
and S
3
, however, the excitation becomes
more localized on the H fragment, thereby illustrating the change in exciton localiza-
tion due to the S
m
! S
3
transition. The excitation character of S
2
is similar to that of
S
3
, which is not surprising since they are separated by a relatively small energy gap of
hE
32
i < 0.15 eV (Figure B.2). Finally, the excitation becomes slightly more localized on
the NO
2
fragment in S
1
. As solvent polarity increases frome= 1 toe= 20, we find that
the excitation becomes more delocalized among the fragments, i.e., the contribution of
TD on each fragment converges towards 50%. This is especially true for S
5
through S
9
and S
2
. The most important thing to take from this analysis is that localization of the ex-
citation not only depends on functionalization and solvent polarity, but on the occupied
excited state as well.
B.6. Transition Density (TD) Analysis 143
S
7
† = 1 † = 2 † = 5 † = 20
S
6
S
5
Density
S
4
H
S
3
H
S
2
0.0 0.5
S
1
0.0 0.5
Fraction of TD due to S
0
→ S
n
0.0 0.5 0.0 0.5 1.0
S
9
† = 1 † = 2 † = 5 † = 20
S
8
S
7
S
6
Density
S
5
S
4
H
S
3
NO
2
S
2
0.0 0.5
S
1
0.0 0.5
Fraction of TD due to S
0
→ S
n
0.0 0.5 0.0 0.5 1.0
S
9
† = 1 † = 2 † = 5 † = 20
S
8
S
7
S
6
Density
S
5
S
4
NH
2
S
3
NO
2
S
2
0.0 0.5
S
1
0.0 0.5
Fraction of TD due to S
0
→ S
n
0.0 0.5 0.0 0.5 1.0
FIGURE B.4: Fraction of TD due to ground to excited state transitions.
Data are sampled over all time steps and trajectories. Rows and columns
of each subfigure correspond to different excited states and dielectric con-
stants, respectively.
144 Appendix B. Supporting Information for Chapter 5
B.7 Excited-State Dipole Moments
0 200 400 600 800 1000
Time (fs)
0
2
4
6
8
10
12
14
16
18
Permanent Dipole (Debye)
R
1
= H, R
2
= H
R
1
= H, R
2
= NO
2
R
1
= NH
2
, R
2
= NO
2
† =1
† =2
† =5
† =20
FIGURE B.5: Calculated permanent excited state dipole moment as a
function of time. The solid, dashed, and dotted lines correspond to the
dipole moments offH, Hg,fH, NO
2
g, andfNH
2
, NO
2
g, respectively.
Dipole moments of each molecule were calculated in four different sol-
vent environments withe=f1, 2, 5, 20g.
B.8. Charge Transfer Character 145
B.8 Charge Transfer Character
In addition to visualizing partial charge transfer character from NTOs, we also quan-
tify the degree of charge transfer in different states and in different solvent environments
by calculatingDm =

~ m
ee
~ m
gg

(Figure B.6). ThefH, Hg oligomer has a very smallDm
that does not vary with excited state or solvent polarity (Figure B.6A). InfH, NO
2
g and
fNH
2
, NO
2
g, however, Dm is not only larger overall, but shows a relatively large de-
pendence on the occupied excited state of the system (Figures B.6B and B.6C). For the
sake of brevity, we only discuss the results offNH
2
, NO
2
g since it exhibits the most
charge transfer character in this set of molecules. For example, Dm offNH
2
, NO
2
g is
relatively large in (and near) the optically excited state (S
8
and S
9
) and in the low-lying
excited states (S
1
and S
2
). Charge transfer character of these states is affected by sol-
vation. This is most clearly shown for S
8
and S
9
, where Dm increases as e increases.
This analysis shows that charge transfer character of the excited states varies based on
chemical substitution and solvent polarity.
146 Appendix B. Supporting Information for Chapter 5
S
7
† = 1
(A) R
1
= H, R
2
= H
† = 2 † = 5 † = 20
S
6
S
5
Density
S
4
S
3
S
2
0 1 2 3 4
S
1
0 1 2 3 4
|μ
ee
−μ
gg
| (Debye)
0 1 2 3 4 0 1 2 3 4 5
(B) R
1
= H, R
2
= NO
2
S
9
† = 1 † = 2 † = 5 † = 20
S
8
S
7
S
6
Density
S
5
S
4
S
3
S
2
0 1 2 3 4
S
1
0 1 2 3 4
|~ μ
ee
−~ μ
gg
| (Debye)
0 1 2 3 4 0 1 2 3 4 5
(C) R
1
= NH
2
, R
2
= NO
2
S
9
† = 1 † = 2 † = 5 † = 20
S
8
S
7
S
6
Density
S
5
S
4
S
3
S
2
0 1 2 3 4
S
1
0 1 2 3 4
|~ μ
ee
−~ μ
gg
| (Debye)
0 1 2 3 4 0 1 2 3 4 5
FIGURE B.6: Difference of dipole moments in the ground and excited
state,

~ m
ee
~ m
gg

. Data are sampled over all time steps and trajecto-
ries. Rows and columns of each subfigure correspond to different excited
states and dielectric constants, respectively.
B.9. Solvent-Stable Resonance Structures (Zwitterions) 147
B.9 Solvent-Stable Resonance Structures (Zwitterions)
FIGURE B.7: (A) and (B) are different resonance structures (zwitterions)
offH, NO
2
g andfNH
2
, NO
2
g, respectively.
148 Appendix B. Supporting Information for Chapter 5
B.10 Relative Computational Time
(A) R
1
= H, R
2
= H
Relative CPU Time
(B) R
1
= H, R
2
= NO
2
† =1 † =20
(C) R
1
= NH
2
, R
2
= NO
2
Total
Ground
State
Excited
States
Adiabatic
Forces
Nonadiabatic
Derivatives
FIGURE B.8: Bar graphs showing the relative CPU times of thefH, Hg,
fH, NO
2
g, andfNH
2
, NO
2
g nonadiabatic ensembles computed with
(e= 20) and without (e= 1) the solvent model. Each bar is an average
over all trajectories and the error is one standard deviation.
149
Appendix C
Supporting Information for Chapter
6
C.1 Comparing Absorption Spectra from DFT and Semiempir-
ical Levels of Theory
Vertical excitation energies and oscillator strengths were computed at the optimized
geometries of all stable conformations: t-SQ, c-SQ, DHBP , and BP . Spectra from all
conformations were combined and normalized by maximum absorption intensity (Fig-
ure C.1). The theoretical spectra were blueshifted to match the low-energy DHBP peak
at 422 nm in n-hexane. Both levels of theory predict a peak in close proximity of the
experimental value for t-SQ in n-hexane (approximately 320 nm). However, DFT’s peak
intensity clearly exceeds that of the semiempirical level of theory. This higher intensity
t-SQ peak from DFT agrees better with experiment (Ref. [268]). The experimental BP
peak is predicted by both levels of theory, but while the deviation between experiment
and semiempirics is negligible, DFT shows an 0.36 eV error (247 nm from DFT and 266
nm from experiment). The spectra in ethanol shift towards lower energy as a result of
increasing solvent polarity. For example, these solvatochromic shifts are observed in
the experimental peak absorption of DHBP , which go from 422 nm in n-hexane to 443
nm in ethanol, resulting in a 0.14 eV shift. The theoretical shift using semiempirics is
about 0.05 eV , while for DFT, it is nearly zero. Furthermore, semiempirics accurately
predicts the solvatochromic shifts of t-SQ, while DFT does not. In summary, DFT pre-
dicts a more accurate intensity of the t-SQ peak, but does not perform as well as the
semiempirical level of theory with alignment of the BP peak and solvatochromic shifts
as the environment changes from n-hexane to ethanol.
C.2 Potential Energy Surface Scan
The intrinsic reaction coordinate (IRC) was computed with Gaussian 16 [489] at both
DFT (B3LYP/6-31G

) and semiempirical (CIS/AM1) levels of theory. The QST2 subrou-
tine was used to determine the transition states. The only transition state that was found
via QST2 was the transition state between t-SQ and DHBP . Figure C.2 shows the ground
and excited states along the IRC between t-SQ and DHBP .
The potential energy surface (PES) of isomerization was investigated by generating
structures and computing the total energy using the AM1 Hamiltonian. Initial geome-
tries of the scan were the optimized t-SQ and DHBP conformations. The purple atoms in
150 Appendix C. Supporting Information for Chapter 6
Absorbance
(A) n-hexane
CIS/AM1
B3LYP/6-31G
∗
t-SQ (exp.)
DHBP (exp.)
BP (exp.)
200 250 300 350 400 450 500 550 600
Wavelength (nm)
(B) ethanol
FIGURE C.1: Total absorption spectra using DFT (B3LYP/6-31G

) and
semiempirical (CIS/AM1) levels of theory in (A) n-hexane and (B)
ethanol. Vertical lines are absorption maxima from experimental data.
Total absorption was computed by combining the theoretical spectra of
the following individual conformations: t-SQ, c-SQ, and DHBP . Theoreti-
cal spectra were blueshifted by 0.45 eV (CIS/AM1) and 0.35 eV (B3LYP/6-
31G

).
C.3. Molecular Orbitals of SQ Conformations during Isomerization 151
FIGURE C.2: PES scan from t-SQ to DHBP in three solvents using DFT
and semiempirical levels of theory: (A) B3LYP/6-31G

and (B) CIS/AM1.
The lower and upper curves of each subpanel show the PES of the ground
and first singlet excited state.
Figure C.3A were varied linearly to generate c-SQ. However, merely perturbing the po-
sitions linearly between the initial and final states is not physical since the purple atoms
would approach the C=C bond. Therefore, to reduce the unphysical perturbations, the
scan was guided to ensure that the purple atoms do not approach the C=C bond. Fur-
ther constraints were implemented to prevent the quinoline ring from rotating during
the scan; two atoms of the quinoline ring were held fixed. This constraint should not
affect the energy of the system during the scan because it only dictates the orientation
of the quinoline ring. Although there are slight imperfections in the scanning proce-
dure, relatively strong agreement between scanned results and NEXMD geometries is
evidenced (Figure C.3B).
C.3 Molecular Orbitals of SQ Conformations during Isomer-
ization
Molecular orbitals (MOs) were calculated at the critical conformations of SQ during
isomerization (Figure C.4). General agreement between NEXMD geometries and those
obtained from the PES scan (Figure C.3) allowed us to use scanned geometries for MOs.
MOs were made with GaussView 5.2.[490] A full description of Figure C.4 is provided
in Chapter 6.
C.4 Time-Domain Isomerization Data from Experiment
The authors of Ref. [268] track the magnitude of the absorption peak in the vicinity of
350300 nm, which they assign to t-SQ. Results are shown in Figure C.5. The magnitude
of the peak decreases in time, thus showing the decay of t-SQ during isomerization.
152 Appendix C. Supporting Information for Chapter 6
FIGURE C.3: (A) Diagram that labels the two carbon atoms (in purple)
used for the PES scan of SQ from t-SQ to DHBP . (B) Comparison of ge-
ometries from NEXMD simulations (lighter color conformations) to those
obtained from the PES scan (darker color conformations). The NEXMD
simulation was performed in a Langevin thermostat with e = 1, and dy-
namics evolved adiabatically on S
1
.
Theoretical time scales of Table 6.1 of Chapter 6 are several orders of magnitude
smaller from those shown in Figure C.5 for several reasons: (1) Theoretical results do
not consider quantum yield–likeliness for a photon to be absorbed by t-SQ. (2) Trajec-
tories may undergo a nonadiabatic transition S
1
! S
0
while the molecule remains t-SQ.
In other words, the t-SQ and c-SQ branching ratios are not considered in calculations.
(3) Calculations only capture the dynamics up to p-SQ; c-SQ is assigned at different
(f
C=C
, f
C=C
). (4) The cis-to-trans backreaction is also not considered in the calculations.
All reasons demonstrate that absolute time scales between theory and experiment are
not comparable, but agreement in the relative decay in n-hexane versus ethanol (as men-
tioned in the main text) may be valid.
C.5 NEXMD Simulations in Vacuum
We performed NEXMD simulations in vacuum. Our results show that there are
two subsets of trajectories that approach the conical intersection (located p-SQ) at two
distinct time scales (Table 6.1 of Chapter 6). Ramachandran plots of the geometries from
each subset are shown in Figure C.6.
A notable difference between the data from the fast and slow trajectories is the distri-
bution off
CC
. The distribution is broad in the case of the fast trajectories as opposed to
the slow trajectories. In fact, a significant number of geometries in the slow subset have
a planar orientation (i.e., f
C=C
and f
CC
are approximately180

or 180

). The orien-
tation of the phenyl ring significantly affects the potential energy of t-SQ (Figure C.7).
C.5. NEXMD Simulations in Vacuum 153
Φ C-C
Φ C=C
HOMO LUMO
-180°
-180°

t-SQ

-180°
-90°

p-SQ

-90°
0°

c-SQ

-100°
0°

DHBP

No node
in GS
Node
in ES
Electron
density to
phenyl ring
Electron
density to
C=C bond
FIGURE C.4: HOMO and LUMO of different conformations of SQ during
isomerization.
FIGURE C.5: Experimental decay in the absorbance of t-SQ as a function
of time in n-hexane and ethanol. Curves were fit to A B exp(t/t).
Relaxation time scalest are shown.
154 Appendix C. Supporting Information for Chapter 6
Fast Slow
Φ
"#"
° Φ
"#"
°
Count
Φ
"%"
°
Stable
point
Rotation ofΦ
"#"
destabilizes the
molecule
Fewer
conformations
with nonplanar
orientation of
Φ
"#"
FIGURE C.6: Ramachandran plots of NEXMD simulations in vacuum.
The number of counts are histogrammed as a function of the f
C=C
and
f
CC
dihedral angles. Dynamics evolved adiabatically on S
1
.
Trajectories that begin with less stable conformations have greater potential energy and
greater total energy; the latter is conserved throughout the dynamics. Energy is trans-
ferred between potential and nuclear kinetic energy with rotation of the phenyl ring.
Excess energy eventually allows the molecule to transform to p-SQ by surpassing the
energy barrier present along rotation of f
C=C
, shown in Figure C.7. This process hap-
pens faster for trajectories with the phenyl ring starting in a nonplanar orientation, as
these are the less stable conformations with greater total energy. Interestingly, dynam-
ics in a thermostat gives a single rate (Table 6.1 of Chapter 6), thus showing that the
thermostat minimizes the dependence of isomerization on initial conditions.
C.6 Vibrational Spectra During Nonadiabatic Dynamics
Vibrational spectra during the nonadiabatic dynamics were computed by taking the
Fourier transform of the mass-weighted velocity autocorrelation function:
s(w)µ
Z
¥
¥
dt exp(iwt)
N
å
j
m
j

v
j
(t) v
j
(0)

, (C.1)
where the sum is over N atoms in the molecule.
The peaks of Figure C.8 are broadened when nuclei evolve according to the Langevin
equation as opposed to the energy-conserving Newtonian equation. However, solvent
effects are minimal. For example, spectral differences between e = 1 and n-hexane
(e 2) are nearly indistinguishable. More work is needed to assess whether this is a
drawback of implicit solvation as polarization is not the only solute-solvent interaction.
It is also worth noting that we do not expect the spectra of Figure C.8 to accurately
predict peak magnitudes, as they are derived purely from classical mechanics, which
does not describe the wavefunction overlap between vibrational states.
C.6. Vibrational Spectra During Nonadiabatic Dynamics 155
FIGURE C.7: PES scan off
C=C
for fixedf
CC
.
1000 2000 3000 4000
Frequency (cm
1
)
Intensity
= 1
n-hexane
ethanol
Newtonian ( = 1)
FIGURE C.8: Vibrational spectra from the nonadiabatic simulations in
different environments. Spectra were computed by taking the Fourier
transform of the mass-weighted velocity autocorrelation function.

157
Appendix D
Supporting Information for Chapter
7
D.1 Optical Absorption: Functionals and Basis Sets
FIGURE D.1: Optical absorption of complexes 1 through 3. Each vertical
excitation is represented as a broadened Gaussian with standard devia-
tion of 0.15 eV . Geometry optimization and excited states were computed
with four separate methods. The indices from left to right refer to density
functional, basis set on non-metals atoms, and basis set on the Fe atom.
UV-Vis in acetone are shown with dashed lines. A polarized continuum
was used to model the solvent effects of acetone.
158 Appendix D. Supporting Information for Chapter 7
D.2 Optical Absorption: Polarization Functions
FIGURE D.2: Optical absorption of complexes 1 through 3 using the
TPSSh density functional and various basis sets with and without po-
larization functions (shown in parentheses). Each vertical excitation is
represented as a broadened Gaussian with standard deviation of 0.15 eV .
Geometry optimization and excited states were computed with six sep-
arate methods. The indices from left to right refer to basis set on non-
metals atoms and basis set on the Fe atom. UV-Vis in acetone are shown
with dashed lines. A polarized continuum was used to model the solvent
effects of acetone.
D.3. Natural Transition Orbitals 159
D.3 Natural Transition Orbitals
TABLE D.1: Natural transition orbitals (NTOs) of photoactive excited
states within the charge transfer bands of complexes 5(A), 8(B), and
10(A). The percent contribution that each electron-hole pair carries to-
wards the transition is shown. NTOs displayed amount to at least 90
percent of the transition. Geometry optimization and excited states were
computed with the TPSSh density functional and 6-311G basis set in a
polarized continuum model of acetone.
complex l(nm) percent hole electron
5(A)
666 56
43
665 66
Continued on next page
160 Appendix D. Supporting Information for Chapter 7
Table D.1 – Continued from previous page
complex l(nm) percent hole electron
22
11
8(B)
576 60
34
Continued on next page
D.3. Natural Transition Orbitals 161
Table D.1 – Continued from previous page
complex l(nm) percent hole electron
575 63
24
12
10(A)
607 65
Continued on next page
162 Appendix D. Supporting Information for Chapter 7
Table D.1 – Continued from previous page
complex l(nm) percent hole electron
27
604 57
40
163
Appendix E
Supporting Information for Chapter
9
E.1 HIP-NN Architecture and Training Details
E.1.1 HIP-NN Architecture
The HIP-NN model closely follows the methodology given in Ref. [467]. A key
difference is that linear layers are used to construct partial atomic charge, rather than a
molecular energy, and so no sum over atoms is employed. The network has 2 interaction
blocks, each consisting of 1 interaction layer, followed by 3 on-site layers, and a linear
layer to form hierarchical contribution to charge. Each layer was given a width of 40
neurons. The network architecture contains approximately 60k parameters.
E.1.2 Training Details
Training also closely follows Ref. [467]. The main difference is the cost function,
adapted for dipole regression. The cost function used here consists of dipole RMSE,
total charge RMSE, and L2 regularization (as described in Ref. [467]):
L=
r
1
3
h(m
0
m)
2
i+
q
hQ
0
2
i+L
L2
(E.1)
where the angle bracketsh...i denote a quantity averaged over each training batch of
30 molecules, m
0
and m represent the predicted and QM dipole, respectively, and Q
0
represents the predicted total charge for the molecule (i.e. the total QM charge is set to
zero). The factor of
1
3
is a normalization reflecting the three cartesian degrees of freedom
in the dipole.
Training is then given by the gradient-based optimization and annealing/early-stopping
algorithm in Ref. [467]. A validation set of 1% of the training dataset (approximately
4,385 molecules) was used for the annealing procedure, and the dipole RMSE was used
as the validation criterion for annealing. For training to the ANI-1x dataset used in this
work, the algorithm terminates after roughly 1000 epochs.
E.2 Details of Charge Assignment
The full charge assignments are given by an ensemble prediction using four different
random initializations of HIP-NN, each separately trained to the same data. Figure
164 Appendix E. Supporting Information for Chapter 9
TABLE E.1: Summary of test and extensibility datasets along with statis-
tical measures for dipole and quadrupole prediction.
ANI-1x Drug Bank Tripeptides
Total # molecules 438,481
a
13,379 2,000
Total atoms (CHNO) per molecule, min / mean / max 2 / 14 / 30 8 / 44 / 140 30 / 53 / 70
Heavy atoms (CNO) per molecule, min / mean / max 1 / 7 / 17 3 / 22 / 65 17 / 27/ 37
Dipole MAE, RMSE (D) 0.08
b
, 0.12
b
0.28, 0.49 0.31, 0.47
Quadrupole MAE, RMSE (B) 0.71
b
, 0.95
b
1.27, 1.89 1.31, 1.72
Mean

m
ACA
m
QM

(D) 0.16
b
0.59 0.66
a
This is 10% of the full ANI-1x dataset, which consists of more than 4M molecules.
b
Error metrics computed on held-out test set of 43,849 molecules.
E.1 shows the correlation between charge predictions by the members of the ensemble;
networks agree to approximately 0.01 e (Fig. E.1).The charges produced by the ensemble
are not exactly neutral, and so when predicting the charge on a molecule, excess total
charge is redistributed evenly across atoms. This redistribution constitutes a very small
change, typically 0.001 e or less per atom.
Dipoles for each datapointm are constructed as
m=
N
atoms
å
i=1
r
i
q
i
(E.2)
and traceless quadrupoles are constructed as
Q=
N
atoms
å
i=1

r
i

r
i

1
3
I r
2
i

q
i
(E.3)
where
is the outer product, and I is the unit dyadic (or Kronecker delta).
E.3 Additional Data of Dipole Prediction
This section contains additional data quantifying the performance of ACA. We in-
clude a table summarizing extensibility results (Table E.1), bar charts showing error as
a function of molecule size (Figures E.2, E.3, and E.4), dipole and quadrupole correla-
tions plots comparing ACA to other existing popular charge models (Figures E.5 and
E.6), and infrared spectra computed with ML dynamics + ACA, ML dynamics + QM
dipoles, and harmonic QM (Figure E.7).
E.3. Additional Data of Dipole Prediction 165
Network 0
Network 1
0.012 e
0.008 e
Network 1
0.011 e
0.008 e
Network 2
0.011 e
0.008 e
Network 2
0.012 e
0.008 e
0.012 e
0.009 e
Network 3
0.011 e
0.008 e
Log Count
FIGURE E.1: Pair correlation plots of charge predictions from the four
neural networks constituting our ensemble, labeled as 0, 1, 2, 3. Upper
and lower values in each subpanel are root-mean-square-deviation and
mean-absolute-deviation, respectively. The color scheme for each his-
togram is normalized by its maximum bin count.
166 Appendix E. Supporting Information for Chapter 9
0.0
0.1
0.2
0.3
0.4
0.5
Error (D)
CHNO
RMSE
MAE
0 5 10 15 20 25 30
Number of Atoms per Molecule
0.0
0.1
0.2
0.3
0.4 CNO
FIGURE E.2: Bar charts showing RMSE and MAE in dipole prediction for
each molecule size in the 43,849 test dataset selected from ANI-1x. Top
and bottom panels correspond to total atoms (C, H, N, and O) and heavy
atoms (C, N, and O), respectively.
0.0
0.5
1.0
1.5
2.0
Error (D)
CHNO
RMSE
MAE
0 20 40 60 80 100 120 140
Number of Atoms per Molecule
0.0
0.5
1.0
1.5
CNO
FIGURE E.3: Bar charts showing RMSE and MAE in dipole prediction for
each molecule size for the DrugBank extensibility set. Top and bottom
panels correspond to total atoms (C, H, N, and O) and heavy atoms (C,
N, and O), respectively.
E.3. Additional Data of Dipole Prediction 167
0.0
0.5
1.0
1.5
2.0
Error (D)
CHNO
RMSE
MAE
0 10 20 30 40 50 60 70
Number of Atoms per Molecule
0.0
0.5
1.0
1.5
CNO
FIGURE E.4: Bar charts showing RMSE and MAE in dipole prediction for
each molecule size for the Tripeptide extensibility set. Top and bottom
panels correspond to total atoms (C, H, N, and O) and heavy atoms (C,
N, and O), respectively.
168 Appendix E. Supporting Information for Chapter 9
-10.0
-5.0
0.0
5.0
10.0
Dipoles
ACA
0.040 D
0.028 D
-20.0
-10.0
0.0
10.0
20.0
Quadrupoles
ACA
0.703 B
0.507 B
-10.0
-5.0
0.0
5.0
Hirshfeld
0.540 D
0.397 D
-20.0
-10.0
0.0
10.0
Hirshfeld
1.244 B
0.908 B
-10.0
-5.0
0.0
5.0
Charge Model
MSK
0.000 D
0.000 D
-20.0
-10.0
0.0
10.0
MSK
0.256 B
0.176 B
-10.0
-5.0
0.0
5.0
CM5
0.170 D
0.123 D
-20.0
-10.0
0.0
10.0
CM5
0.715 B
0.512 B
-10.0 -5.0 0.0 5.0 10.0
QM
-10.0
-5.0
0.0
5.0
NBO
0.770 D
0.556 D
-20.0 -10.0 0.0 10.0 20.0
-20.0
-10.0
0.0
10.0
NBO
1.323 B
0.933 B
Log Count
FIGURE E.5: 2D histograms showing correlations between predicted
(Charge Model) and reference (QM) electrostatic moments using five dif-
ferent charge models: ACA, Hirshfeld, MSK, CM5, and NBO. The test
dataset is GDB-5, which contains a total of 517,133 molecules. The up-
per and lower values in each subpanel are RMSE and MAE, respectively.
The color scheme for each histogram is normalized by its maximum bin
count. ACA recovers both the dipole and quadrupole moments of the
test dataset at better accuracy than all other models except for MSK. ACA
recovers quadrupoles, despite being only trained to dipoles. MSK is de-
fined to reproduce QM dipoles exactly.
E.3. Additional Data of Dipole Prediction 169
-20.0
-10.0
0.0
10.0
20.0
Dipoles
ACA
0.129 D
0.083 D
-40.0
-20.0
0.0
20.0
40.0
Quadrupoles
ACA
0.966 B
0.718 B
-20.0
-10.0
0.0
10.0
Charge Model
MSK
0.000 D
0.000 D
-40.0
-20.0
0.0
20.0
MSK
0.283 B
0.200 B
-20.0 -10.0 0.0 10.0 20.0
QM
-20.0
-10.0
0.0
10.0
CM5
0.238 D
0.178 D
-40.0 -20.0 0.0 20.0 40.0
-40.0
-20.0
0.0
20.0
CM5
1.053 B
0.754 B
Log Count
FIGURE E.6: 2D histograms showing correlations between predicted
(Charge Model) and reference (QM) electrostatic moments using three
different charge models: ACA, CM5, and MSK. The test dataset is a dif-
ferent random subset of ANI-1x than that used for training. The dataset
contains a total of 600k molecules. The upper and lower values in each
subpanel are RMSE and MAE, respectively. The color scheme for each
histogram is normalized by its maximum bin count. ACA recovers both
the dipole and quadupole moments of the test dataset at better accuracy
than CM5. ACA recovers quadrupoles, despite being only trained to
dipoles. MSK is defined to reproduce QM dipoles exactly.
170 Appendix E. Supporting Information for Chapter 9
Methanol
(6, 2)
-6
-4
-2
0
2
4
6
0.023 D
0.018 D
Ethanal
(7, 3)
QM Harmonic Modes
Time-Domain QM
Time-Domain ML
-6
-4
-2
0
2
4
0.030 D
0.024 D
Intensity
Acetamide
(9, 4)
-6
-4
-2
0
2
4
ACA Dipole (D)
0.031 D
0.025 D
500 1375 2250 3125 4000
Frequency (cm
−1
)
Dimethylacetamide
(15, 6)
-6 -4 -2 0 2 4 6
QM Dipole (D)
-6
-4
-2
0
2
4
0.041 D
0.032 D
Log Count
FIGURE E.7: Infrared spectra of small molecules calculated using ACA
(red) and QM (black). The values in parentheses are the total number
of all atoms (C, H, N, and O) and heavy atoms (C, N, and O), respec-
tively. For each molecule, both ACA and QM dipoles were predicted at
the same 10
6
time-steps of the ANI-1x molecular dynamics trajectory. Fre-
quencies determined from DFT harmonic mode analysis are also shown
(blue). Right panels are dipole correlation plots of ACA versus QM. Up-
per and lower values in each subpanel are RMSE and MAE, respectively.
The color scheme for each histogram is normalized by its maximum bin
count.
171
Bibliography
1
A. Nitzan, Chemical dynamics in condensed phases: relaxation, transfer and reactions in
condensed molecular systems (Oxford University Press, 2006).
2
T. J. Martinez, M Ben-Nun, and R. Levine, “Multi-electronic-state molecular dynam-
ics: a wave function approach with applications”, J. Phys. Chem. 100, 7884–7895
(1996).
3
M. F. Herman and E. Kluk, “A semiclasical justification for the use of non-spreading
wavepackets in dynamics calculations”, Chem. Phys. 91, 27–34 (1984).
4
A. White, S. Tretiak, and D. Mozyrsky, “Coupled wave-packets for non-adiabatic
molecular dynamics: a generalization of gaussian wave-packet dynamics to multi-
ple potential energy surfaces”, Chem. Sci. 7, 4905–4911 (2016).
5
J. Tully, “Molecular dynamics with electronic transitions”, J. Chem. Phys. 93, 1061–
1071 (1990).
6
J. C. Tully, “Perspective: nonadiabatic dynamics theory”, J. Chem. Phys. 137, 22A301
(2012).
7
M. Barbatti, “Nonadiabatic dynamics with trajectory surface hopping method”, WIREs:
Comput. Mol. Sci. 1, 620–633 (2011).
8
A. W. Jasper, C. Zhu, S. Nangia, and D. G. Truhlar, “Introductory lecture: nonadia-
batic effects in chemical dynamics”, Farad. Discuss. 127, 1–22 (2004).
9
J.-Y. Fang and S. Hammes-Schiffer, “Comparison of surface hopping and mean field
approaches for model proton transfer reactions”, J. Chem. Phys. 110, 11166–11175
(1999).
10
J. C. Tully, “Chemical dynamics at metal surfaces”, Annu. Rev. Phys. Chem. 51, 153–
178 (2000).
11
S. Hammes-Schiffer, “Theoretical perspectives on proton-coupled electron transfer
reactions”, Acc. Chem. Res. 34, 273–281 (2001).
12
S. Hammes-Schiffer and J. C. Tully, “Proton transfer in solution: molecular dynamics
with quantum transitions”, J. Chem. Phys. 101, 4657–4667 (1994).
13
T. Nelson, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, “Nonadiabatic excited-
state molecular dynamics: modeling photophysics in organic conjugated materials”,
Acc. Chem. Res. 47, 1155–1164 (2014).
14
T. Nelson, S. Fernandez-Alberti, V . Chernyak, A. E. Roitberg, and S. Tretiak, “Nona-
diabatic excited-state molecular dynamics modeling of photoinduced dynamics in
conjugated molecules”, J. Phys. Chem. B 115, 5402–5414 (2011).
15
U. Müller and G. Stock, “Surface-hopping modeling of photoinduced relaxation dy-
namics on coupled potential-energy surfaces”, J. Chem. Phys. 107, 6230–6245 (1997).
172 Bibliography
16
R. Long, O. V . Prezhdo, and W. Fang, “Nonadiabatic charge dynamics in novel solar
cell materials”, WIREs: Comput. Mol. Sci. (2017).
17
A. V . Akimov, A. J. Neukirch, and O. V . Prezhdo, “Theoretical insights into photoin-
duced charge transfer and catalysis at oxide interfaces”, Chem. Rev. 113, 4496–4565
(2013).
18
L. J. Wang, O. V . Prezhdo, and D. Beljonne, “Mixed quantum-classical dynamics for
charge transport in organics”, Phys. Chem. Chem. Phys. 17, 12395–12406 (2015).
19
L. J. Wang, A. Akimov, and O. V . Prezhdo, “Recent progress in surface hopping: 2011–
2015”, J. Phys. Chem. Lett. 7, 2100–2112 (2016).
20
J. C. Tully, “Mixed quantum-classical dynamics”, Farad. Discuss. 110, 407–419 (1998).
21
T. Van Voorhis, T. Kowalczyk, B. Kaduk, L.-P . Wang, C.-L. Cheng, and Q. Wu, “The di-
abatic picture of electron transfer, reaction barriers, and molecular dynamics”, Annu.
Rev. Phys. Chem. 61, 149–170 (2010).
22
L. J. Wang, D. Trivedi, and O. V . Prezhdo, “Global flux surface hopping approach for
mixed quantum-classical dynamics”, J. Chem. Theory Comput. 10, 3598–3605 (2014).
23
R. Kapral, “Progress in the theory of mixed quantum-classical dynamics”, Annu. Rev.
Phys. Chem. 57, 129–157 (2006).
24
P . V . Parandekar and J. C. Tully, “Detailed balance in ehrenfest mixed quantum-
classical dynamics”, J. Chem. Theory Comput. 2, 229–235 (2006).
25
P . Parandekar and J. Tully, “Mixed quantum-classical equilibrium”, J. Chem. Phys.
122, 094102 (2005).
26
J. Schmidt, P . V . Parandekar, and J. C. Tully, “Mixed quantum-classical equilibrium:
surface hopping”, J. Chem. Phys. 129, 044104 (2008).
27
A. E. Sifain, L. J. Wang, and O. V . Prezhdo, “Mixed quantum-classical equilibrium in
global flux surface hopping”, J. Chem. Phys. 142, 224102 (2015).
28
M. Sherman and S. Corcelli, “Thermal equilibrium properties of surface hopping with
an implicit langevin bath”, J. Chem. Phys. 142, 024110 (2015).
29
J.-Y. Fang and S. Hammes-Schiffer, “Improvement of the internal consistency in tra-
jectory surface hopping”, J. Phys. Chem. A 103, 9399–9407 (1999).
30
A. E. Sifain, L. Wang, and O. V . Prezhdo, “Communication: proper treatment of clas-
sically forbidden electronic transitions significantly improves detailed balance in sur-
face hopping”, J. Chem. Phys. 144, 211102 (2016).
31
J. E. Subotnik, A. Jain, B. Landry, A. Petit, W. Ouyang, and N. Bellonzi, “Understand-
ing the surface hopping view of electronic transitions and decoherence”, Ann. Rev.
Phys. Chem. 67, 387–417 (2016).
32
J. E. Subotnik, “Fewest-switches surface hopping and decoherence in multiple di-
mensions”, J. Phys. Chem. A 115, 12083–12096 (2011).
33
X. Gao and W. Thiel, “Non-hermitian surface hopping”, Phys. Rev. E 95, 013308
(2017).
34
H. M. Jaeger, S. Fischer, and O. V . Prezhdo, “Decoherence-induced surface hopping”,
J. Chem. Phys. 137, 22A545 (2012).
Bibliography 173
35
J. E. Subotnik and N. Shenvi, “A new approach to decoherence and momentum
rescaling in the surface hopping algorithm”, J. Chem. Phys. 134, 024105 (2011).
36
N. Shenvi, J. E. Subotnik, and W. Yang, “Simultaneous-trajectory surface hopping: a
parameter-free algorithm for implementing decoherence in nonadiabatic dynamics”,
J. Chem. Phys. 134, 144102 (2011).
37
G. Granucci, M. Persico, and A. Zoccante, “Including quantum decoherence in sur-
face hopping”, J. Chem. Phys. 133, 134111 (2010).
38
R. E. Larsen, M. J. Bedard-Hearn, and B. J. Schwartz, “Exploring the role of decoher-
ence in condensed-phase nonadiabatic dynamics: a comparison of different mixed
quantum/classical simulation algorithms for the excited hydrated electron”, J. Phys.
Chem. B 110, 20055–20066 (2006).
39
M. J. Bedard-Hearn, R. E. Larsen, and B. J. Schwartz, “Mean-field dynamics with
stochastic decoherence (mf-sd): a new algorithm for nonadiabatic mixed quantum/classical
molecular-dynamics simulations with nuclear-induced decoherence”, J. Chem. Phys.
123, 234106 (2005).
40
E. R. Bittner and P . J. Rossky, “Quantum decoherence in mixed quantum-classical
systems: nonadiabatic processes”, J. Chem. Phys. 103, 8130–8143 (1995).
41
T. Nelson, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, “Nonadiabatic excited-
state molecular dynamics: treatment of electronic decoherence”, J. Chem. Phys. 138,
224111 (2013).
42
C. J. Cramer and D. G. Truhlar, “Implicit solvation models: equilibria, structure, spec-
tra, and dynamics”, Chem. Rev. 99, 2161–2200 (1999).
43
A. E. Sifain, J. A. Bjorgaard, T. R. Nelson, B. T. Nebgen, A. J. White, B. J. Gifford, D. W.
Gao, O. V . Prezhdo, S. Fernandez-Alberti, A. E. Roitberg, et al., “Photoexcited nonadi-
abatic dynamics of solvated push-pull p-conjugated oligomers with the nexmd soft-
ware”, J. Chem. Theory Comput. (2018).
44
D. H. Waldeck, “Photoisomerization dynamics of stilbenes”, Chem. Rev. 91, 415–436
(1991).
45
H. D. Bandara and S. C. Burdette, “Photoisomerization in different classes of azoben-
zene”, Chem. Soc. Rev. 41, 1809–1825 (2012).
46
A. E. Sifain, B. J. Gifford, D. W. Gao, L. Lystrom, T. R. Nelson, and S. Tretiak, “Nexmd
modeling of photoisomerization dynamics of 4-styrylquinoline”, The Journal of Phys-
ical Chemistry A (2018).
47
P . F. Pagoria, G. S. Lee, A. R. Mitchell, and R. D. Schmidt, “A review of energetic
materials synthesis”, Thermochim. Acta 384, 187–204 (2002).
48
A. Sikder and N. Sikder, “A review of advanced high performance, insensitive and
thermally stable energetic materials emerging for military and space applications”, J.
Hazard. Mater. 112, 1–15 (2004).
49
K. E. Brown, T. W. Myers, C. Snyder, and S. A. Clarke, “Laser initiation of iron-
based photoactive explosives”, in Aip conf. proc. Vol. 1979, 1 (AIP Publishing, 2018),
p. 100008.
174 Bibliography
50
T. W. Myers, D. E. Chavez, S. K. Hanson, R. J. Scharff, B. L. Scott, J. M. Veauthier, and
R. Wu, “Independent control of optical and explosive properties: pyrazole–tetrazine
complexes of first row transition metals”, Inorg. Chem. 54, 8077–8086 (2015).
51
T. W. Myers, J. A. Bjorgaard, K. E. Brown, D. E. Chavez, S. K. Hanson, R. J. Scharff,
S. Tretiak, and J. M. Veauthier, “Energetic chromophores: low-energy laser initiation
in explosive fe (ii) tetrazine complexes”, J. Am. Chem. Soc. 138, 4685–4692 (2016).
52
A. E. Sifain, J. A. Bjorgaard, T. W. Myers, J. M. Veauthier, D. E. Chavez, O. V . Prezhdo,
R. J. Scharff, and S. Tretiak, “Photoactive excited states in explosive fe (ii) tetrazine
complexes: a time-dependent density functional theory study”, J. Phys. Chem. C 120,
28762–28773 (2016).
53
J. A. Bjorgaard, A. E. Sifain, T. Nelson, T. W. Myers, J. M. Veauthier, D. E. Chavez, R. J.
Scharff, and S. Tretiak, “Two-photon absorption in conjugated energetic molecules”,
J. Phys. Chem. A 120, 4455–4464 (2016).
54
V . W. Manner, M. J. Cawkwell, E. M. Kober, T. W. Myers, G. W. Brown, H. Tian,
C. J. Snyder, R. Perriot, and D. N. Preston, “Examining the chemical and structural
properties that influence the sensitivity of energetic nitrate esters”, Chem. Sci. 9, 3649–
3663 (2018).
55
D. Jacquemin, V . Wathelet, E. A. Perpete, and C. Adamo, “Extensive td-dft bench-
mark: singlet-excited states of organic molecules”, J. Chem. Theory Comput. 5, 2420–
2435 (2009).
56
A. E. Sifain, L. F. Tadesse, J. A. Bjorgaard, D. E. Chavez, O. V . Prezhdo, R. J. Scharff,
and S. Tretiak, “Cooperative enhancement of the nonlinear optical response in conju-
gated energetic materials: a td-dft study”, J. Chem. Phys. 146, 114308 (2017).
57
F. Terenziani, C. Katan, E. Badaeva, S. Tretiak, and M. Blanchard-Desce, “Enhanced
two-photon absorption of organic chromophores: theoretical and experimental as-
sessments”, Adv. Mater. 20, 4641–4678 (2008).
58
J. S. Smith, A. E. Roitberg, and O. Isayev, Transforming computational drug discovery
with machine learning and ai, 2018.
59
K. Burke, “Perspective on density functional theory”, J. Chem. Phys. 136, 150901
(2012).
60
B. J. Gifford, S. Kilina, H. Htoon, S. K. Doorn, and S. Tretiak, “Exciton localization
and optical emission in aryl-functionalized carbon nanotubes”, J. Phys. Chem. C 122,
1828–1838 (2017).
61
W. Thiel, “Semiempirical quantum–chemical methods”, Wiley Interdiscip. Rev. Com-
put. Mol. Sci. 4, 145–157 (2014).
62
P . O. Dral, M. Barbatti, and W. Thiel, “Nonadiabatic excited-state dynamics with ma-
chine learning”, J. Phys. Chem. Lett. 9, 5660–5663 (2018).
63
C. J. Cramer and D. G. Truhlar, “A universal approach to solvation modeling”, Acc.
Chem. Res. 41, 760–768 (2008).
64
A. K. Malde, L. Zuo, M. Breeze, M. Stroet, D. Poger, P . C. Nair, C. Oostenbrink, and
A. E. Mark, “An automated force field topology builder (atb) and repository: version
1.0”, J. Chem. Theory Comput. 7, 4026–4037 (2011).
Bibliography 175
65
K. Vanommeslaeghe, E. P . Raman, and A. D. MacKerell Jr, “Automation of the charmm
general force field (cgenff) ii: assignment of bonded parameters and partial atomic
charges”, J. Chem. Inf. Model. 52, 3155–3168 (2012).
66
F. L. Hirshfeld, “Bonded-atom fragments for describing molecular charge densities”,
Theor. Chem. Acc. 44, 129–138 (1977).
67
U. C. Singh and P . A. Kollman, “An approach to computing electrostatic charges for
molecules”, J. Comput. Chem. 5, 129–145 (1984).
68
A. V . Marenich, S. V . Jerome, C. J. Cramer, and D. G. Truhlar, “Charge model 5: an
extension of hirshfeld population analysis for the accurate description of molecular
interactions in gaseous and condensed phases”, J. Chem. Theory Comput. 8, 527–541
(2012).
69
A. E. Sifain, N. Lubbers, B. T. Nebgen, J. S. Smith, A. Y. Lokhov, O. Isayev, A. E.
Roitberg, K. Barros, and S. Tretiak, “Discovering a transferable charge assignment
model using machine learning”, J. Chem. Phys. Lett. 9, 4495–4501 (2018).
70
M. F. Herman, “Dynamics by semiclassical methods”, Ann. Rev. Phys. Chem. 45, 83–
111 (1994).
71
V . N. Gorshkov, S. Tretiak, and D. Mozyrsky, “Semiclassical monte-carlo approach for
modelling non-adiabatic dynamics in extended molecules”, Nat. Commun. 4, 2144
(2013).
72
A. Staib, D. Borgis, and J. T. Hynes, “Proton transfer in hydrogen-bonded acid–base
complexes in polar solvents”, J. Chem. Phys. 102, 2487–2505 (1995).
73
S. Consta and R. Kapral, “Dynamics of proton transfer in mesoscopic clusters”, J.
Chem. Phys. 104, 4581–4590 (1996).
74
S. R. Billeter, S. P . Webb, T. Iordanov, P . K. Agarwal, and S. Hammes-Schiffer, “Hybrid
approach for including electronic and nuclear quantum effects in molecular dynamics
simulations of hydrogen transfer reactions in enzymes”, J. Chem. Phys. 114, 6925–
6936 (2001).
75
N. Shenvi, J. E. Subotnik, and W. Yang, “Phase-corrected surface hopping: correct-
ing the phase evolution of the electronic wavefunction”, J. Chem. Phys. 135, 024101
(2011).
76
M. B. Smith and J. Michl, “Singlet fission”, Chem. Rev. 110, 6891–6936 (2010).
77
T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, “Microscopic theory of singlet
exciton fission. i. general formulation”, J. Chem. Phys. 138, 114102 (2013).
78
D. Beljonne, H Yamagata, J. Brédas, F. Spano, and Y. Olivier, “Charge-transfer exci-
tations steer the davydov splitting and mediate singlet exciton fission in pentacene”,
Phys. Rev. Lett. 110, 226402 (2013).
79
A. V . Akimov and O. V . Prezhdo, “Nonadiabatic dynamics of charge transfer and sin-
glet fission at the pentacene/c
60
interface”, J. Am. Chem. Soc. 136, 1599–1608 (2014).
80
J. Lee, P . Jadhav, P . D. Reusswig, S. R. Yost, N. J. Thompson, D. N. Congreve, E. Hontz,
T. Van Voorhis, and M. A. Baldo, “Singlet exciton fission photovoltaics”, Acc. Chem.
Res. 46, 1300–1311 (2013).
176 Bibliography
81
L. Wang and D. Beljonne, “Flexible surface hopping approach to model the crossover
from hopping to band-like transport in organic crystals”, J. Chem. Phys. Lett. 4, 1888–
1894 (2013).
82
L. Wang and O. V . Prezhdo, “A simple solution to the trivial crossing problem in
surface hopping”, J. Phys. Chem. Lett. 5, 713–719 (2014).
83
G Granucci, M Persico, and A Toniolo, “Direct semiclassical simulation of photo-
chemical processes with semiempirical wave functions”, J. Chem. Phys. 114, 10608–
10615 (2001).
84
E Fabiano, T. Keal, and W Thiel, “Implementation of surface hopping molecular dy-
namics using semiempirical methods”, Chem. Phys. 349, 334–347 (2008).
85
C. Evenhuis and T. J. Martínez, “A scheme to interpolate potential energy surfaces
and derivative coupling vectors without performing a global diabatization”, J. Chem.
Phys. 135, 224110 (2011).
86
S. Fernandez-Alberti, A. E. Roitberg, T. Nelson, and S. Tretiak, “Identification of un-
avoided crossings in nonadiabatic photoexcited dynamics involving multiple elec-
tronic states in polyatomic conjugated molecules”, J. Chem. Phys. 137, 014512 (2012).
87
K. Blum, Density matrix theory and applications (Springer, 1996).
88
G. Hanna and E. Geva, “Vibrational energy relaxation of a hydrogen-bonded complex
dissolved in a polar liquid via the mixed quantum-classical liouville method”, J. Phys.
Chem. B 112, 4048–4058 (2008).
89
P . M. Morse, “Diatomic molecules according to the wave mechanics. ii. vibrational
levels”, Phys. Rev. 34, 57 (1929).
90
J. C. Tully, G. H. Gilmer, and M. Shugard, “Molecular dynamics of surface diffusion.
i. the motion of adatoms and clusters”, J. Chem. Phys. 71, 1630–1642 (1979).
91
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P . Flannery, Numerical recipes: the
art of scientific computing (Cambridge Univ. Press, 1992).
92
E. Hershkovitz, “A fourth-order numerical integrator for stochastic langevin equa-
tions”, J. Chem. Phys. 108, 9253–9258 (1998).
93
L. Wang, A. V . Akimov, L. Chen, and O. V . Prezhdo, “Quantized hamiltonian dynam-
ics captures the low-temperature regime of charge transport in molecular crystals”, J.
Chem. Phys. 139, 174109 (2013).
94
G. Cowan, Statistical data analysis (Oxford University Press, 1998).
95
O. V . Prezhdo, “Mean field approximation for the stochastic schrödinger equation”,
J. Chem. Phys. 111, 8366–8377 (1999).
96
M. D. Hack and D. G. Truhlar, “A natural decay of mixing algorithm for non-born–
oppenheimer trajectories”, J. Chem. Phys. 114, 9305–9314 (2001).
97
X. Li, J. C. Tully, H. B. Schlegel, and M. J. Frisch, “Ab initio ehrenfest dynamics”, J.
Chem. Phys. 123, 084106 (2005).
98
L. J. Wang, D. Beljonne, L. Chen, and Q. Shi, “Mixed quantum-classical simulations
of charge transport in organic materials: numerical benchmark of the su-schrieffer-
heeger model”, J. Chem. Phys. 134, 244116 (2011).
Bibliography 177
99
J. C. Tully and R. K. Preston, “Trajectory surface hopping approach to nonadiabatic
molecular collisions: the reaction of h
+
with d
2
”, J. Chem. Phys. 55, 562–572 (1971).
100
A. V . Akimov, D. Trivedi, L. J. Wang, and O. V . Prezhdo, “Analysis of the trajectory
surface hopping method from the markov state model perspective”, J. Phys. Soc. Jpn
84, 094002 (2015).
101
A. Kazaryan, Z. Lan, L. V . Schäfer, W. Thiel, and M. Filatov, “Surface hopping excited-
state dynamics study of the photoisomerization of a light-driven fluorene molecular
rotary motor”, J. Chem. Theory Comput. 7, 2189–2199 (2011).
102
J Clark, T Nelson, S Tretiak, G Cirmi, and G Lanzani, “Femtosecond torsional relax-
ation”, Nat. Phys. 8, 225–231 (2012).
103
A. J. Neukirch, L. C. Shamberger, E. Abad, B. J. Haycock, H. Wang, J. Ortega, O. V .
Prezhdo, and J. P . Lewis, “Nonadiabatic ensemble simulations of cis-stilbene and cis-
azobenzene photoisomerization”, J. Chem. Theory Comput. 10, 14–23 (2013).
104
S. A. Fischer, D. B. Lingerfelt, J. W. May, and X. Li, “Non-adiabatic molecular dynam-
ics investigation of photoionization state formation and lifetime in mn
2+
-doped zno
quantum dots”, Phys. Chem. Chem. Phys. 16, 17507–17514 (2014).
105
L. J. Wang and D. Beljonne, “Charge transport in organic semiconductors: assessment
of the mean field theory in the hopping regime”, J. Chem. Phys. 139, 064316 (2013).
106
A. J. White, V . N. Gorshkov, S. Tretiak, and D. Mozyrsky, “Non-adiabatic molecu-
lar dynamics by accelerated semiclassical monte carlo”, J. Chem. Phys. 143, 014115
(2015).
107
A. V . Akimov and O. V . Prezhdo, “Second-quantized surface hopping”, Phys. Rev.
Lett. 113, 153003 (2014).
108
L. J. Wang, A. E. Sifain, and O. V . Prezhdo, “Fewest switches surface hopping in liou-
ville space”, J. Phys. Chem. Lett. 6, 3827–3833 (2015).
109
L. J. Wang, A. E. Sifain, and O. V . Prezhdo, “Communication: global flux surface hop-
ping in liouville space”, J. Chem. Phys. 143, 191102 (2015).
110
J. E. Subotnik, W. Ouyang, and B. R. Landry, “Can we derive tully’s surface-hopping
algorithm from the semiclassical quantum liouville equation? almost, but only with
decoherence”, J. Chem. Phys. 139, 214107 (2013).
111
A. W. Jasper and D. G. Truhlar, “Improved treatment of momentum at classically
forbidden electronic transitions in trajectory surface hopping calculations”, Chem.
Phys. Lett. 369, 60–67 (2003).
112
A. W. Jasper, S. N. Stechmann, and D. G. Truhlar, “Fewest-switches with time un-
certainty: a modified trajectory surface-hopping algorithm with better accuracy for
classically forbidden electronic transitions”, J. Chem. Phys. 116, 5424–5431 (2002).
113
A. W. Jasper, M. D. Hack, and D. G. Truhlar, “The treatment of classically forbid-
den electronic transitions in semiclassical trajectory surface hopping calculations”, J.
Chem. Phys. 115, 1804–1816 (2001).
114
J. C. Tully, “Nonadiabatic molecular dynamics”, Int. J. Quant. Chem. 40, 299–309
(1991).
178 Bibliography
115
D. J. Trivedi, L. J. Wang, and O. V . Prezhdo, “Auger-mediated electron relaxation is
robust to deep hole traps: time-domain ab initio study of cdse quantum dots”, Nano
Lett. 15, 2086–2091 (2015).
116
A. Jain and J. E. Subotnik, “Surface hopping, transition state theory, and decoherence.
ii. thermal rate constants and detailed balance”, J. Chem. Phys. 143, 134107 (2015).
117
S. V . Kilina, D. S. Kilin, and O. V . Prezhdo, “Breaking the phonon bottleneck in pbse
and cdse quantum dots: time-domain density functional theory of charge carrier re-
laxation”, ACS Nano 3, 93–99 (2008).
118
B. F. Habenicht, C. F. Craig, and O. V . Prezhdo, “Time-domain ab initio simulation of
electron and hole relaxation dynamics in a single-wall semiconducting carbon nan-
otube”, Phys. Rev. Lett. 96, 187401 (2006).
119
S. V . Kilina, A. J. Neukirch, B. F. Habenicht, D. S. Kilin, and O. V . Prezhdo, “Quan-
tum zeno effect rationalizes the phonon bottleneck in semiconductor quantum dots”,
Phys. Rev. Lett. 110, 180404 (2013).
120
B. F. Habenicht and O. V . Prezhdo, “Nonradiative quenching of fluorescence in a
semiconducting carbon nanotube: a time-domain ab initio study”, Phys. Rev. Lett.
100, 197402 (2008).
121
B. R. Landry and J. E. Subotnik, “Communication: standard surface hopping predicts
incorrect scaling for marcus’ golden-rule rate: the decoherence problem cannot be
ignored”, J. Chem. Phys. 135, 191101 (2011).
122
E. Neria and A. Nitzan, “Semiclassical evaluation of nonadiabatic rates in condensed
phases”, J. Chem. Phys. 99, 1109–1123 (1993).
123
C. A. Schwerdtfeger, A. V . Soudackov, and S. Hammes-Schiffer, “Nonadiabatic dy-
namics of electron transfer in solution: explicit and implicit solvent treatments that
include multiple relaxation time scales”, J. Chem. Phys. 140, 034113 (2014).
124
B. R. Landry and J. E. Subotnik, “How to recover marcus theory with fewest switches
surface hopping: add just a touch of decoherence”, J. Chem. Phys. 137, 22A513 (2012).
125
H.-T. Chen and D. R. Reichman, “On the accuracy of surface hopping dynamics in
condensed phase non-adiabatic problems”, J. Chem. Phys. 144, 094104 (2016).
126
O. S. Wenger, “How donor- bridge- acceptor energetics influence electron tunneling
dynamics and their distance dependences”, Acc. Chem. Res. 44, 25–35 (2010).
127
J. Jortner, M. Bixon, T. Langenbacher, and M. E. Michel-Beyerle, “Charge transfer and
transport in dna”, Proc. Natl. Acad. Sci. USA 95, 12759–12765 (1998).
128
Y. Hu and S. Mukamel, “Superexchange and electron transfer in the photosynthetic
reaction center”, Chem. Phys. Lett. 160, 410–416 (1989).
129
H. M. McConnell, “Intramolecular charge transfer in aromatic free radicals”, J. Chem.
Phys. 35, 508–515 (1961).
130
F. D. Lewis, J. Liu, W. Weigel, W. Rettig, I. V . Kurnikov, and D. N. Beratan, “Donor-
bridge-acceptor energetics determine the distance dependence of electron tunneling
in dna”, Proc. Natl. Acad. Sci. USA 99, 12536–12541 (2002).
Bibliography 179
131
F. D. Lewis, T. Wu, X. Liu, R. L. Letsinger, S. R. Greenfield, S. E. Miller, and M. R.
Wasielewski, “Dynamics of photoinduced charge separation and charge recombina-
tion in synthetic dna hairpins with stilbenedicarboxamide linkers”, J. Am. Chem. Soc.
122, 2889–2902 (2000).
132
D. Hanss and O. S. Wenger, “Tunneling barrier effects on photoinduced charge trans-
fer through covalent rigid rod-like bridges”, Inorg. Chem. 48, 671–680 (2008).
133
D. Hanss and O. S. Wenger, “Electron tunneling through oligo-p-xylene bridges”,
Inorg. Chem. 47, 9081–9084 (2008).
134
G. L. Closs and J. R. Miller, “Intramolecular long-distance electron transfer in organic
molecules”, Science 240, 440–448 (1988).
135
P . P . Edwards, H. B. Gray, M. T. Lodge, and R. J. Williams, “Electron transfer and
electronic conduction through an intervening medium”, Angew. Chem. Int. Ed. 47,
6758–6765 (2008).
136
B. Albinsson, M. P . Eng, K. Pettersson, and M. U. Winters, “Electron and energy
transfer in donor–acceptor systems with conjugated molecular bridges”, Phys. Chem.
Chem. Phys. 9, 5847–5864 (2007).
137
H. B. Gray and J. R. Winkler, “Long-range electron transfer”, Proc. Natl. Acad. Sci.
U.S.A. 102, 3534–3539 (2005).
138
E. A. Badaeva, T. V . Timofeeva, A. Masunov, and S. Tretiak, “Role of donor–acceptor
strengths and separation on the two-photon absorption response of cytotoxic dyes: a
td-dft study”, J. Phys. Chem. A 109, 7276–7284 (2005).
139
D. G. Truhlar, “Decoherence in combined quantum mechanical and classical mechan-
ical methods for dynamics as illustrated for non-born–oppenheimer trajectories”, in
Quantum dynamics of complex molecular systems (Springer, 2007), pp. 227–243.
140
J. Spencer, L. Scalfi, A. Carof, and J. Blumberger, “Confronting surface hopping molec-
ular dynamics with marcus theory for a molecular donor–acceptor system”, Faraday
Discuss. 195, 215–236 (2017).
141
J. Peng, Y. Xie, D. Hu, and Z. Lan, “Performance of trajectory surface hopping method
in the treatment of ultrafast intersystem crossing dynamics”, arXiv preprint arXiv:1811.00254
(2018).
142
A. A. Kananenka, X. Sun, A. Schubert, B. D. Dunietz, and E. Geva, “A comparative
study of different methods for calculating electronic transition rates”, J. Chem. Phys.
148, 102304 (2018).
143
M. Born and K. Huang, Dynamical theory of crystal lattices (Clarendon Press, 1954).
144
O. V . Prezhdo and P . J. Rossky, “Relationship between quantum decoherence times
and solvation dynamics in condensed phase chemical systems”, Phys. Rev. Lett. 81,
5294 (1998).
145
S. C. Cheng, C. Zhu, K. K. Liang, S. H. Lin, and D. G. Truhlar, “Algorithmic decoher-
ence time for decay-of-mixing non–born–oppenheimer dynamics”, J. Chem. Phys.
129, 024112 (2008).
146
C. Zhu, S. Nangia, A. W. Jasper, and D. G. Truhlar, “Coherent switching with decay
of mixing: an improved treatment of electronic coherence for non-born–oppenheimer
trajectories”, J. Chem. Phys. 121, 7658–7670 (2004).
180 Bibliography
147
A. V . Akimov and O. V . Prezhdo, “Persistent electronic coherence despite rapid loss
of electron–nuclear correlation”, J. Phys. Chem. Lett. 4, 3857–3864 (2013).
148
S. Mukamel, Principles of nonlinear optical spectroscopy, Vol. 29 (Oxford University Press
New York, 1995).
149
R. Long, W. Fang, and A. V . Akimov, “Nonradiative electron–hole recombination rate
is greatly reduced by defects in monolayer black phosphorus: ab initio time domain
study”, J. Phys. Chem. Lett. 7, 653–659 (2016).
150
Z. Zhang, R. Long, M. V . Tokina, and O. V . Prezhdo, “Interplay between localized
and free charge carriers can explain hot fluorescence in the ch3nh3pbbr3 perovskite:
time-domain ab initio analysis”, J. Am. Chem. Soc. 139, 17327–17333 (2017).
151
J. He, A. S. Vasenko, R. Long, and O. V . Prezhdo, “Halide composition controls electron–
hole recombination in cesium–lead halide perovskite quantum dots: a time domain
ab initio study”, J. Phys. Chem. Lett. 9, 1872–1879 (2018).
152
Z. Zhang, W.-H. Fang, M. V . Tokina, R. Long, and O. V . Prezhdo, “Rapid decoherence
suppresses charge recombination in multi-layer 2d halide perovskites: time-domain
ab initio analysis”, Nano Lett. 18, 2459–2466 (2018).
153
A. V . Akimov and O. V . Prezhdo, “The pyxaid program for non-adiabatic molecu-
lar dynamics in condensed matter systems”, J. Chem. Theory Comput. 9, 4959–4972
(2013).
154
B. J. Schwartz, E. R. Bittner, O. V . Prezhdo, and P . J. Rossky, “Quantum decoherence
and the isotope effect in condensed phase nonadiabatic molecular dynamics simula-
tions”, J. Chem. Phys. 104, 5942–5955 (1996).
155
O. V . Prezhdo and P . J. Rossky, “Evaluation of quantum transition rates from quantum-
classical molecular dynamics simulations”, J. Chem. Phys. 107, 5863–5878 (1997).
156
L. G. Rego and V . S. Batista, “Quantum dynamics simulations of interfacial electron
transfer in sensitized TiO
2
semiconductors”, J. Am. Chem. Soc. 125, 7989–7997 (2003).
157
S Fernandez-Alberti, V . D. Kleiman, S Tretiak, and A. E. Roitberg, “Unidirectional en-
ergy transfer in conjugated molecules: the crucial role of high-frequency CC bonds”,
J. Phys. Chem. Lett. 1, 2699–2704 (2010).
158
L. Wang, R. Long, and O. V . Prezhdo, “Time-domain ab initio modeling of photoin-
duced dynamics at nanoscale interfaces”, Ann. Rev. Phys. Chem. 66, 549–579 (2015).
159
L. Alfonso Hernandez, T. Nelson, M. F. Gelin, J. M. Lupton, S. Tretiak, and S. Fernandez-
Alberti, “Interference of interchromophoric energy-transfer pathways inp-conjugated
macrocycles”, J. Phys. Chem. Lett. 7, 4936–4944 (2016).
160
P . Goyal and S. Hammes-Schiffer, “Tuning the ultrafast dynamics of photoinduced
proton-coupled electron transfer in energy conversion processes”, ACS Energy Lett.
2, 512–519 (2017).
161
T. Schultz, J. Quenneville, B. Levine, A. Toniolo, T. J. Martínez, S. Lochbrunner, M.
Schmitt, J. P . Shaffer, M. Z. Zgierski, and A. Stolow, “Mechanism and dynamics of
azobenzene photoisomerization”, J. Am. Chem. Soc. 125, 8098–8099 (2003).
162
B. G. Levine and T. J. Martínez, “Isomerization through conical intersections”, Annu.
Rev. Phys. Chem. 58, 613–634 (2007).
Bibliography 181
163
J. C. Vincent, M. Muuronen, K. C. Pearce, L. N. Mohanam, E. Tapavicza, and F. Furche,
“That little extra kick: nonadiabatic effects in acetaldehyde photodissociation”, J. Phys.
Chem. Lett. 7, 4185–4190 (2016).
164
T. Nelson, J. Bjorgaard, M. Greenfield, C. Bolme, K. Brown, S. McGrane, R. J. Scharff,
and S. Tretiak, “Ultrafast photodissociation dynamics of nitromethane”, J. Phys. Chem.
A 120, 519–526 (2016).
165
S. G. Abuabara, L. G. Rego, and V . S. Batista, “Influence of thermal fluctuations on
interfacial electron transfer in functionalized TiO
2
semiconductors”, J. Am. Chem.
Soc. 127, 18234–18242 (2005).
166
A. Torres, R. S. Oliboni, and L. G. Rego, “Vibronic and coherent effects on interfacial
electron transfer dynamics”, J. Phys. Chem. Lett. 6, 4927–4935 (2015).
167
T. R. Nelson, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, “Electronic delocal-
ization, vibrational dynamics and energy transfer in organic chromophores”, J. Phys.
Chem. Lett. (2017).
168
S Tretiak, A Saxena, R. Martin, and A. Bishop, “Conformational dynamics of photoex-
cited conjugated molecules”, Phys. Rev. Lett. 89, 097402 (2002).
169
M Granström, K Petritsch, A. Arias, A Lux, M. Andersson, and R. Friend, “Laminated
fabrication of polymeric photovoltaic diodes”, Nature 395, 257–260 (1998).
170
Y. Cao, I. D. Parker, G. Yu, C. Zhang, and A. J. Heeger, “Improved quantum efficiency
for electroluminescence in semiconducting polymers”, Nature 397, 414–417 (1999).
171
H Sirringhaus, T Kawase, R. Friend, T Shimoda, M Inbasekaran, W Wu, and E. Woo,
“High-resolution inkjet printing of all-polymer transistor circuits”, Science 290, 2123–
2126 (2000).
172
J.-L. Brédas, D. Beljonne, V . Coropceanu, and J. Cornil, “Charge-transfer and energy-
transfer processes in p-conjugated oligomers and polymers: a molecular picture”,
Chem. Rev. 104, 4971–5004 (2004).
173
B. Satishkumar, L. O. Brown, Y. Gao, C.-C. Wang, H.-L. Wang, and S. K. Doorn, “Re-
versible fluorescence quenching in carbon nanotubes for biomolecular sensing”, Nat.
Nanotechnol. 2, 560–564 (2007).
174
J.-L. Brédas, J. E. Norton, J. Cornil, and V . Coropceanu, “Molecular understanding of
organic solar cells: the challenges”, Acc. Chem. Res. 42, 1691–1699 (2009).
175
J.-L. Brédas and R. Silbey, “Excitons surf along conjugated polymer chains”, Science
323, 348–349 (2009).
176
E. Collini and G. D. Scholes, “Coherent intrachain energy migration in a conjugated
polymer at room temperature”, Science 323, 369–373 (2009).
177
J.-L. Brédas, J. Cornil, D. Beljonne, D. A. Dos Santos, and Z. Shuai, “Excited-state
electronic structure of conjugated oligomers and polymers: a quantum-chemical ap-
proach to optical phenomena”, Acc. Chem. Res. 32, 267–276 (1999).
178
F. C. Spano, “Emission from aggregates of oligo-phenylene vinylenes: a recipe for
superradiant h-aggregates”, Chem. Phys. Lett. 331, 7–13 (2000).
182 Bibliography
179
K. A. Kistler and S. Matsika, “Solvatochromic shifts of uracil and cytosine using
a combined multireference configuration interaction/molecular dynamics approach
and the fragment molecular orbital method”, J. Phys. Chem. A 113, 12396–12403
(2009).
180
C. M. Isborn, A. W. Gotz, M. A. Clark, R. C. Walker, and T. J. Martínez, “Electronic ab-
sorption spectra from mm and ab initio qm/mm molecular dynamics: environmental
effects on the absorption spectrum of photoactive yellow protein”, J. Chem. Theory
Comput. 8, 5092–5106 (2012).
181
P . Goyal, C. A. Schwerdtfeger, A. V . Soudackov, and S. Hammes-Schiffer, “Nona-
diabatic dynamics of photoinduced proton-coupled electron transfer in a solvated
phenol–amine complex”, J. Phys. Chem. B 119, 2758–2768 (2015).
182
Y.-l. Lin and J. Gao, “Solvatochromic shifts of the n! p

transition of acetone from
steam vapor to ambient aqueous solution: a combined configuration interaction qm/mm
simulation study incorporating solvent polarization”, J. Chem. Theory Comput. 3,
1484–1493 (2007).
183
U. F. Röhrig, I. Frank, J. Hutter, A. Laio, J. VandeVondele, and U. Rothlisberger, “Qm/mm
car-parrinello molecular dynamics study of the solvent effects on the ground state
and on the first excited singlet state of acetone in water”, ChemPhysChem 4, 1177–
1182 (2003).
184
J. Gao, “Energy components of aqueous solution: insight from hybrid qm/mm simu-
lations using a polarizable solvent model”, J. Comput. Chem. 18, 1061–1071 (1997).
185
G. J. Tawa, R. L. Martin, L. R. Pratt, and T. V . Russo, “Solvation free energy calcu-
lations using a continuum dielectric model for the solvent and gradient-corrected
density functional theory for the solute”, J. Phys. Chem. 100, 1515–1523 (1996).
186
R. L. Martin, P . J. Hay, and L. R. Pratt, “Hydrolysis of ferric ion in water and confor-
mational equilibrium”, J. Phys. Chem. A 102, 3565–3573 (1998).
187
B. Mennucci and R. Cammi, Continuum solvation models in chemical physics: from theory
to applications (Chichester: John Wiley & Sons, 2007).
188
B. Mennucci, “Polarizable continuum model”, WIREs: Comput. Mol. Sci. 2, 386–404
(2012).
189
J. Tomasi, B. Mennucci, and R. Cammi, “Quantum mechanical continuum solvation
models”, Chem. Rev. 105, 2999–3094 (2005).
190
G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi, and V . Barone, “Ge-
ometries and properties of excited states in the gas phase and in solution: theory
and application of a time-dependent density functional theory polarizable contin-
uum model”, J. Chem. Phys. 124, 094107 (2006).
191
J. A. Bjorgaard, V Kuzmenko, K. Velizhanin, and S. Tretiak, “Solvent effects in time-
dependent self-consistent field methods. i. optical response calculations”, J. Chem.
Phys. 142, 044103 (2015).
192
J. A. Bjorgaard, K. A. Velizhanin, and S. Tretiak, “Solvent effects in time-dependent
self-consistent field methods. ii. variational formulations and analytical gradients”, J.
Chem. Phys. 143, 054305 (2015).
Bibliography 183
193
S. Fernandez-Alberti, V . D. Kleiman, S. Tretiak, and A. E. Roitberg, “Nonadiabatic
molecular dynamics simulations of the energy transfer between building blocks in a
phenylene ethynylene dendrimer”, J. Phys. Chem. A 113, 7535–7542 (2009).
194
S Fernandez-Alberti, A. E. Roitberg, V . D. Kleiman, T Nelson, and S Tretiak, “Shishio-
doshi unidirectional energy transfer mechanism in phenylene ethynylene dendrimers”,
J. Chem. Phys. 137, 22A526 (2012).
195
M. A. Soler, A. E. Roitberg, T. Nelson, S. Tretiak, and S. Fernandez-Alberti, “Analysis
of state-specific vibrations coupled to the unidirectional energy transfer in conjugated
dendrimers”, J. Phys. Chem. A 116, 9802–9810 (2012).
196
J. F. Galindo, E. Atas, A. Altan, D. G. Kuroda, S. Fernandez-Alberti, S. Tretiak, A. E.
Roitberg, and V . D. Kleiman, “Dynamics of energy transfer in a conjugated dendrimer
driven by ultrafast localization of excitations”, J. Am. Chem. Soc. 137, 11637–11644
(2015).
197
W. P . Bricker, P . M. Shenai, A. Ghosh, Z. Liu, M. G. M. Enriquez, P . H. Lambrev,
H.-S. Tan, C. S. Lo, S. Tretiak, S. Fernandez-Alberti, et al., “Non-radiative relaxation
of photoexcited chlorophylls: theoretical and experimental study”, Sci. Rep. 5, 13625
(2015).
198
P . M. Shenai, S. Fernandez-Alberti, W. P . Bricker, S. Tretiak, and Y. Zhao, “Internal
conversion and vibrational energy redistribution in chlorophyll a”, J. Phys. Chem. B
120, 49–58 (2015).
199
F. Zheng, S. Fernandez-Alberti, S. Tretiak, and Y. Zhao, “Photoinduced intra-and in-
termolecular energy transfer in chlorophyll a dimer”, J. Phys. Chem. B 121, 5331–5339
(2017).
200
L. Adamska, I. Nayyar, H. Chen, A. K. Swan, N. Oldani, S. Fernandez-Alberti, M. R.
Golder, R. Jasti, S. K. Doorn, and S. Tretiak, “Self-trapping of excitons, violation of
condon approximation, and efficient fluorescence in conjugated cycloparaphenylenes”,
Nano Lett. 14, 6539–6546 (2014).
201
J. A. Bjorgaard, T Nelson, K. Kalinin, V Kuzmenko, K. A. Velizhanin, and S. Tretiak,
“Simulations of fluorescence solvatochromism in substituted ppv oligomers from ex-
cited state molecular dynamics with implicit solvent”, Chem. Phys. Lett. 631, 66–69
(2015).
202
S. Tretiak and S. Mukamel, “Density matrix analysis and simulation of electronic ex-
citations in conjugated and aggregated molecules”, Chem. Rev. 102, 3171–3212 (2002).
203
S. Mukamel, S. Tretiak, T. Wagersreiter, and V . Chernyak, “Electronic coherence and
collective optical excitations of conjugated molecules”, Science 277, 781–787 (1997).
204
D. Case, D. Cerutti, T. Cheatham III, T. Darden, R. Duke, T. Giese, H Gohlke, A. Goetz,
D Greene, N Homeyer, S Izadi, A Kovalenko, T. Lee, S LeGrand, P Li, C Lin, J Liu,
T Luchko, R Luo, D Mermelstein, K. Merz, G Monard, H Nguyen, I Omelyan, A
Onufriev, F Pan, R Qi, D. Roe, A Roitberg, C Sagui, C. Simmerling, W. Botello-Smith,
J Swails, R. Walker, J Wang, R. Wolf, X Wu, L Xiao, D. York, and P . Kollman, Amber
2017, University of California, San Francisco 2017.
205
D. J. Thouless, The quantum mechanics of many-body systems (Courier Corporation,
2013).
184 Bibliography
206
V . Chernyak, M. F. Schulz, S. Mukamel, S. Tretiak, and E. V . Tsiper, “Krylov-space
algorithms for time-dependent hartree–fock and density functional computations”, J.
Chem. Phys. 113, 36–43 (2000).
207
S. Tretiak, C. M. Isborn, A. M. Niklasson, and M. Challacombe, “Representation in-
dependent algorithms for molecular response calculations in time-dependent self-
consistent field theories”, J. Chem. Phys. 130, 054111 (2009).
208
M. J. Dewar, E. G. Zoebisch, E. F. Healy, and J. J. Stewart, “Development and use of
quantum mechanical molecular models. 76. am1: a new general purpose quantum
mechanical molecular model”, J. Am. Chem. Soc. 107, 3902–3909 (1985).
209
S Tretiak, A Saxena, R. Martin, and A. Bishop, “Ceo/semiempirical calculations of
uv–visible spectra in conjugated molecules”, Chem. Phys. Lett. 331, 561–568 (2000).
210
S Tretiak, A Saxena, R. Martin, and A. Bishop, “Collective electronic oscillator/semiempirical
calculations of static nonlinear polarizabilities in conjugated molecules”, J. Chem.
Phys. 115, 699–707 (2001).
211
Z Meng, A Dölle, and W. R. Carper, “Gas phase model of an ionic liquid: semi-
empirical and ab initio bonding and molecular structure”, Comput. Theor. Chem.
585, 119–128 (2002).
212
J. J. Stewart, “Comparison of the accuracy of semiempirical and some dft methods for
predicting heats of formation”, J. Mol. Model. 10, 6–12 (2004).
213
S. Mukamel, A. Takahashi, H. X. Wang, and G. Chen, “Electronic coherence and non-
linear susceptibilities of conjugated polyenes”, Science, 250–250 (1994).
214
S. Tretiak, V . Chernyak, and S. Mukamel, “Localized electronic excitations in pheny-
lacetylene dendrimers”, J. Phys. Chem. B 102, 3310–3315 (1998).
215
S. Kilina, S. Tretiak, S. K. Doorn, Z. Luo, F. Papadimitrakopoulos, A. Piryatinski, A.
Saxena, and A. R. Bishop, “Cross-polarized excitons in carbon nanotubes”, Proc. Natl.
Acad. Sci. USA 105, 6797–6802 (2008).
216
S. Kilina and S. Tretiak, “Excitonic and vibrational properties of single-walled semi-
conducting carbon nanotubes”, Adv. Funct. Mater. 17, 3405–3420 (2007).
217
J. A. Bjorgaard, K. A. Velizhanin, and S. Tretiak, “Nonequilibrium solvent effects in
born-oppenheimer molecular dynamics for ground and excited electronic states”, J.
Chem. Phys. 144, 154104 (2016).
218
T. Nelson, A. Naumov, S. Fernandez-Alberti, and S. Tretiak, “Nonadiabatic excited-
state molecular dynamics: on-the-fly limiting of essential excited states”, Chem. Phys.
(2016).
219
A. Klamt and G. Schüürmann, “Cosmo: a new approach to dielectric screening in
solvents with explicit expressions for the screening energy and its gradient”, J. Chem.
Soc., Perkin Trans. 2, 799–805 (1993).
220
V . Barone and M. Cossi, “Quantum calculation of molecular energies and energy gra-
dients in solution by a conductor solvent model”, J. Phys. Chem. A 102, 1995–2001
(1998).
221
U Rauscher, H Bässler, D. Bradley, and M Hennecke, “Exciton versus band descrip-
tion of the absorption and luminescence spectra in poly(p-phenylenevinylene)”, Phys.
Rev. B 42, 9830 (1990).
Bibliography 185
222
J. Burroughes, D. Bradley, A. Brown, R. Marks, K Mackay, R. Friend, P . Burns, and
A. Holmes, “Light-emitting diodes based on conjugated polymers”, Nature 347, 539–
541 (1990).
223
H. Vestweber, A. Greiner, U. Lemmer, R. F. Mahrt, R. Richert, W. Heitz, and H. Bässler,
“Progress towards processible materials for light-emitting devices using poly(p-phenylphenylenevinylene)”,
Adv. Mat. 4, 661–662 (1992).
224
J Cornil, D Beljonne, Z Shuia, T. Hagler, I Campbell, D. Bradley, J. Brédas, C. Spangler,
and K. Müllen, “Vibronic structure in the optical absorption spectra of phenylene
vinylene oligomers: a joint experimental and theoretical study”, Chem. Phys. Lett.
247, 425–432 (1995).
225
G. C. Bazan, W. J. Oldham, R. J. Lachicotte, S. Tretiak, V . Chernyak, and S. Mukamel,
“Stilbenoid dimers: dissection of a paracyclophane chromophore”, J. Am. Chem. Soc.
120, 9188–9204 (1998).
226
S. Frolov, Z Bao, M Wohlgenannt, and Z. V . Vardeny, “Ultrafast spectroscopy of even-
parity states inp-conjugated polymers”, Phys. Rev. Lett. 85, 2196 (2000).
227
E. Peeters, A. M. Ramos, S. C. Meskers, and R. A. Janssen, “Singlet and triplet excita-
tions of chiral dialkoxy-p-phenylene vinylene oligomers”, J. Chem. Phys. 112, 9445–
9454 (2000).
228
M. Lira-Cantu and F. C. Krebs, “Hybrid solar cells based on meh-ppv and thin film
semiconductor oxides (TiO
2
, Nb
2
O
5
, zno, CeO
2
and CeO
2
–TiO
2
): performance im-
provement during long-time irradiation”, Sol. Energy Mater. Sol. Cells 90, 2076–2086
(2006).
229
C. B. Nielsen, A. Angerhofer, K. A. Abboud, and J. R. Reynolds, “Discrete photopat-
ternable p-conjugated oligomers for electrochromic devices”, J. Am. Chem. Soc. 130,
9734–9746 (2008).
230
S. Ellinger, K. R. Graham, P . Shi, R. T. Farley, T. T. Steckler, R. N. Brookins, P . Taranekar,
J. Mei, L. A. Padilha, T. R. Ensley, et al., “Donor–acceptor–donor-basedp-conjugated
oligomers for nonlinear optics and near-ir emission”, Chem. Mater. 23, 3805–3817
(2011).
231
A. N. Panda, F. Plasser, A. J. Aquino, I. Burghardt, and H. Lischka, “Electronically
excited states in poly(p-phenylenevinylene): vertical excitations and torsional poten-
tials from high-level ab initio calculations”, J. Phys. Chem. A 117, 2181–2189 (2013).
232
Y. I. Park, C.-Y. Kuo, J. S. Martinez, Y.-S. Park, O. Postupna, A. Zhugayevych, S. Kim,
J. Park, S. Tretiak, and H.-L. Wang, “Tailored electronic structure and optical proper-
ties of conjugated systems through aggregates and dipole–dipole interactions”, ACS
Appl. Mater. Int. 5, 4685–4695 (2013).
233
Y. Park, O Postupna, A Zhugayevych, H Shin, Y.-S. Park, B Kim, H.-J. Yen, P Cheruku,
J. Martinez, J. Park, et al., “A new ph sensitive fluorescent and white light emissive
material through controlled intermolecular charge transfer”, Chem. Sci. 6, 789–797
(2015).
186 Bibliography
234
M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman,
G. Scalmani, V . Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X.
Li, H. P . Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada,
M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O.
Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta, F. Ogliaro, M. Bearpark,
J. J. Heyd, E. Brothers, K. N. Kudin, V . N. Staroverov, R. Kobayashi, J. Normand, K.
Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega,
J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V . Bakken, C. Adamo, J. Jaramillo,
R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W.
Ochterski, R. L. Martin, K. Morokuma, V . G. Zakrzewski, G. A. Voth, P . Salvador,
J. J. Dannenberg, S. Dapprich, A. D. Daniels, ï. Farkas, J. B. Foresman, J. V . Ortiz, J.
Cioslowski, and D. J. Fox, Gaussian 09, Revision D. 02 (2009).
235
W. Humphrey, A. Dalke, and K. Schulten, “Vmd: visual molecular dynamics”, J.
Molec. Graphics 14, 33–38 (1996).
236
C. Wu, S. V . Malinin, S. Tretiak, and V . Y. Chernyak, “Exciton scattering and localiza-
tion in branched dendrimeric structures”, Nat. Phys. 2, 631–635 (2006).
237
Y. Li and C. Ullrich, “Time-dependent transition density matrix”, Chem. Phys. 391,
157–163 (2011).
238
T. Nelson, S. Fernandez-Alberti, A. E. Roitberg, and S. Tretiak, “Conformational dis-
order in energy transfer: beyond förster theory”, Phys. Chem. Chem. Phys. 15, 9245–
9256 (2013).
239
M. Baer, Beyond born-oppenheimer: electronic nonadiabatic coupling terms and conical in-
tersections (John Wiley & Sons, 2006).
240
T. M. Cardozo, A. J. Aquino, M. Barbatti, I. Borges Jr, and H. Lischka, “Absorption
and fluorescence spectra of poly(p-phenylenevinylene) (ppv) oligomers: an ab initio
simulation”, J. Phys. Chem. A 119, 1787–1795 (2014).
241
S. Matsika and P . Krause, “Nonadiabatic events and conical intersections”, Ann. Rev.
Phys. Chem. 62, 621–643 (2011).
242
S. R. Marder, C. B. Gorman, F. Meyers, J. W. Perry, G. Bourhill, J.-L. Brédas, and B. M.
Pierce, “A unified description of linear and nonlinear polarization in organic poly-
methine dyes”, Science 265, 632–635 (1994).
243
C. Dugave and L. Demange, “Cis–trans isomerization of organic molecules and biomolecules:
implications and applications”, Chem. Rev. 103, 2475–2532 (2003).
244
L. González, D. Escudero, and L. Serrano-Andrés, “Progress and challenges in the
calculation of electronic excited states”, ChemPhysChem 13, 28–51 (2012).
245
D. Polli, P . Altoè, O. Weingart, K. M. Spillane, C. Manzoni, D. Brida, G. Tomasello,
G. Orlandi, P . Kukura, R. A. Mathies, et al., “Conical intersection dynamics of the
primary photoisomerization event in vision”, Nature 467, 440–443 (2010).
246
A. Mammana, G. T. Carroll, J. Areephong, and B. L. Feringa, “A chiroptical photo-
switchable dna complex”, J. Phys. Chem. B 115, 11581–11587 (2011).
247
H. Kandori, “Ion-pumping microbial rhodopsins”, Front. Mol. Biosci. 2 (2015).
248
K. Tanaka and F Toda, “Solvent-free organic synthesis”, Chem. Rev. 100, 1025–1074
(2000).
Bibliography 187
249
J. Briand, O. Bräm, J. Réhault, J. Léonard, A. Cannizzo, M. Chergui, V . Zanirato, M.
Olivucci, J. Helbing, and S. Haacke, “Coherent ultrafast torsional motion and isomer-
ization of a biomimetic dipolar photoswitch”, Phys. Chem. Chem. Phys. 12, 3178–
3187 (2010).
250
F. Ercole, T. P . Davis, and R. A. Evans, “Photo-responsive systems and biomaterials:
photochromic polymers, light-triggered self-assembly, surface modification, fluores-
cence modulation and beyond”, Polym. Chem. 1, 37–54 (2010).
251
K. Huang and M. Wan, “Self-assembled polyaniline nanostructures with photoiso-
merization function”, Chem. Mater. 14, 3486–3492 (2002).
252
S. Kawata and Y. Kawata, “Three-dimensional optical data storage using photochromic
materials”, Chem. Rev. 100, 1777–1788 (2000).
253
M. Irie, T. Fukaminato, T. Sasaki, N. Tamai, and T. Kawai, “Organic chemistry: a dig-
ital fluorescent molecular photoswitch”, Nature 420, 759–760 (2002).
254
C. C. Corredor, Z.-L. Huang, and K. D. Belfield, “Two-photon 3d optical data stor-
age via fluorescence modulation of an efficient fluorene dye by a photochromic di-
arylethene”, Adv. Mater. 18, 2910–2914 (2006).
255
T. J. Martínez, “Insights for light-driven molecular devices from ab initio multiple
spawning excited-state dynamics of organic and biological chromophores”, Acc. Chem.
Res. 39, 119–126 (2006).
256
S. P . Velsko and G. R. Fleming, “Photochemical isomerization in solution. photo-
physics of diphenyl butadiene”, J. Chem. Phys. 76, 3553–3562 (1982).
257
G Rothenberger, D. Negus, and R. Hochstrasser, “Solvent influence on photoisomer-
ization dynamics”, J. Chem. Phys. 79, 5360–5367 (1983).
258
J. Hicks, M. Vandersall, Z. Babarogic, and K. B. Eisenthal, “The dynamics of barrier
crossings in solution: the effect of a solvent polarity-dependent barrier”, Chem. Phys.
Lett. 116, 18–24 (1985).
259
P . F. Barbara and W. Jarzeba, “Dynamic solvent effects on polar and nonpolar isomer-
izations”, Acc. Chem. Res. 21, 195–199 (1988).
260
R. R. Pappalardo, E. Sanchez Marcos, M. F. Ruiz-Lopez, D. Rinaldi, and J. L. Rivail,
“Solvent effects on molecular geometries and isomerization processes: a study of
push-pull ethylenes in solution”, J. Am. Chem. Soc. 115, 3722–3730 (1993).
261
C.-W. Chang, Y.-C. Lu, T.-T. Wang, and E. W.-G. Diau, “Photoisomerization dynamics
of azobenzene in solution with s1 excitation: a femtosecond fluorescence anisotropy
study”, J. Am. Chem. Soc. 126, 10109–10118 (2004).
262
J. Ding, V . Desikan, X. Han, T. L. Xiao, R. Ding, W. S. Jenks, and D. W. Armstrong,
“Use of chiral ionic liquids as solvents for the enantioselective photoisomerization of
dibenzobicyclo [2.2. 2] octatrienes”, Org. Lett. 7, 335–337 (2005).
263
J. Dokic, M. Gothe, J. Wirth, M. V . Peters, J. Schwarz, S. Hecht, and P . Saalfrank,
“Quantum chemical investigation of thermal cis-to-trans isomerization of azoben-
zene derivatives: substituent effects, solvent effects, and comparison to experimental
data”, J. Phys. Chem. A 113, 6763–6773 (2009).
188 Bibliography
264
S. Mai, P . Marquetand, and L. González, “Intersystem crossing pathways in the non-
canonical nucleobase 2-thiouracil: a time-dependent picture”, J. Chem. Phys. Lett. 7,
1978–1983 (2016).
265
F. Talotta, J.-L. Heully, F. Alary, I. M. Dixon, L. González, and M. Boggio-Pasqua,
“Linkage photoisomerization mechanism in a photochromic ruthenium nitrosyl com-
plex: new insights from an ms-caspt2 study”, J. Chem. Theory Comput. 13, 6120–6130
(2017).
266
J. Bao and P . M. Weber, “Electronic effects on photochemistry: the diverse reaction
dynamics of highly excited stilbenes and azobenzene”, J. Am. Chem. Soc. 133, 4164–
4167 (2011).
267
P . F. Aramendia, R. M. Negri, and E. S. Roman, “Temperature dependence of fluo-
rescence and photoisomerization in symmetric carbocyanines. influence of medium
viscosity and molecular structure”, J. Phys. Chem. 98, 3165–3173 (1994).
268
M. F. Budyka, N. I. Potashova, T. N. Gavrishova, and V . M. Lee, “Solvent-driven adia-
batic trans-to-cis photoisomerization of 4-styrylquinoline”, J. Photochem. Photobiol.
A 203, 100–104 (2009).
269
A. Cembran, F. Bernardi, M. Garavelli, L. Gagliardi, and G. Orlandi, “On the mecha-
nism of the cis–trans isomerization in the lowest electronic states of azobenzene: S
0
,
S
1
, and T
1
”, J. Am. Chem. Soc. 126, 3234–3243 (2004).
270
O. Weingart, Z. Lan, A. Koslowski, and W. Thiel, “Chiral pathways and periodic de-
cay in cis-azobenzene photodynamics”, J. Phys. Chem. Lett. 2, 1506–1509 (2011).
271
J. H. Frederick, Y Fujiwara, J. H. Penn, K. Yoshihara, and H. Petek, “Models for stil-
bene photoisomerization: experimental and theoretical studies of the excited-state
dynamics of 1, 2-diphenylcycloalkenes”, J. Phys. Chem. 95, 2845–2858 (1991).
272
J Saltiel, A. Waller, and D. Sears, “Dynamics of cis-stilbene photoisomerization: the
adiabatic pathway to excited trans-stilbene”, J. Photochem. Photobiol. A 65, 29–40
(1992).
273
D. M. Leitner, B. Levine, J. Quenneville, T. J. Martínez, and P . G. Wolynes, “Quan-
tum energy flow and trans-stilbene photoisomerization: an example of a non-rrkm
reaction”, J. Phys. Chem. A 107, 10706–10716 (2003).
274
J. Quenneville and T. J. Martinez, “Ab initio study of cis–trans photoisomerization in
stilbene and ethylene”, J. Phys. Chem. A 107, 829–837 (2003).
275
M Ben-Nun and T. J. Martinez, “Ab initio molecular dynamics study of cis–trans pho-
toisomerization in ethylene”, Chem. Phys. Lett. 298, 57–65 (1998).
276
A. Masunov, S. Tretiak, J. W. Hong, B. Liu, and G. C. Bazan, “Theoretical study of the
effects of solvent environment on photophysical properties and electronic structure
of paracyclophane chromophores”, J. Chem. Phys. 122, 224505 (2005).
277
G. Ramachandran, C Ramakrishnan, and V Sasisekharan, “Stereochemistry of polypep-
tide chain configurations”, J. Mol. Biol. 7, 95–99 (1963).
278
B. G. Levine, C. Ko, J. Quenneville, and T. J. MartÍnez, “Conical intersections and dou-
ble excitations in time-dependent density functional theory”, Mol. Phys. 104, 1039–
1051 (2006).
Bibliography 189
279
R. Magyar, S Tretiak, Y Gao, H.-L. Wang, and A. Shreve, “A joint theoretical and
experimental study of phenylene–acetylene molecular wires”, Chem. Phys. Lett. 401,
149–156 (2005).
280
T. R. Nelson, D. Ondarse-Alvarez, N. Oldani, B. Rodriguez-Hernandez, L. Alfonso-
Hernandez, J. F. Galindo, V . D. Kleiman, S. Fernandez-Alberti, A. E. Roitberg, and
S. Tretiak, “Coherent exciton-vibrational dynamics and energy transfer in conjugated
organics”, Nat. Commun. 9 (2018).
281
E. Jakubikova and D. N. Bowman, “Fe(ii)-polypyridines as chromophores in dye-
sensitized solar cells: a computational perspective”, Acc. Chem. Res. 48, 1441–1449
(2015).
282
M. K. Nazeeruddin, F. De Angelis, S. Fantacci, A. Selloni, G. Viscardi, P . Liska, S. Ito, B.
Takeru, and M. Grätzel, “Combined experimental and dft-tddft computational study
of photoelectrochemical cell ruthenium sensitizers”, J. Am. Chem. Soc. 127, 16835–
16847 (2005).
283
S. Ardo and G. J. Meyer, “Photodriven heterogeneous charge transfer with transition-
metal compounds anchored to TiO
2
semiconductor surfaces”, Chem. Soc Rev. 38,
115–164 (2009).
284
M. K. Nazeeruddin, P Pechy, and M Grätzel, “Efficient panchromatic sensitization
of nanocrystalline TiO
2
films by a black dye based on a trithiocyanato–ruthenium
complex”, Chem. Commun., 1705–1706 (1997).
285
S. Ferrere, “New photosensitizers based upon [Fe(L)
2
(CN)
2
] and [Fe(L)
3
](L = substi-
tuted 2, 2
0
-bipyridine): yields for the photosensitization of TiO
2
and effects on the
band selectivity”, Chem. Mater. 12, 1083–1089 (2000).
286
S. Ferrere, “New photosensitizers based upon [Fe
II
(L)
2
(CN)
2
] and [Fe
II
L
3
], where l is
substituted 2, 2
0
-Bipyridine”, Inorg. Chim. Acta 329, 79–92 (2002).
287
S. Ferrere and B. A. Gregg, “Photosensitization of TiO
2
by [Fe
II
(2, 2
0
-bipyridine-4,
4
0
-dicarboxylic acid)
2
(CN)
2
]: band selective electron injection from ultra-short-lived
excited states”, J. Am. Chem. Soc. 120, 843–844 (1998).
288
M. Yang, D. W. Thompson, and G. J. Meyer, “Charge-transfer studies of iron cyano
compounds bound to nanocrystalline TiO
2
surfaces”, Inorg. Chem. 41, 1254–1262
(2002).
289
M. Yang, D. W. Thompson, and G. J. Meyer, “Dual pathways for TiO
2
sensitization
by Na
2
[Fe(bpy)(CN)
4
]”, Inorg. Chem. 39, 3738–3739 (2000).
290
H.-L. Xia, S. Ardo, A. A. Narducci Sarjeant, S. Huang, and G. J. Meyer, “Photodriven
spin change of fe (ii) benzimidazole compounds anchored to nanocrystalline TiO
2
thin films”, Langmuir 25, 13641–13652 (2009).
291
J. E. Monat and J. K. McCusker, “Femtosecond excited-state dynamics of an iron (ii)
polypyridyl solar cell sensitizer model”, J. Am. Chem. Soc. 122, 4092–4097 (2000).
292
N. Huse, H. Cho, K. Hong, L. Jamula, F. M. de Groot, T. K. Kim, J. K. McCusker, and
R. W. Schoenlein, “Femtosecond soft x-ray spectroscopy of solvated transition-metal
complexes: deciphering the interplay of electronic and structural dynamics”, J. Phys.
Chem. Lett. 2, 880–884 (2011).
190 Bibliography
293
C. Ulbricht, B. Beyer, C. Friebe, A. Winter, and U. S. Schubert, “Recent developments
in the application of phosphorescent iridium (iii) complex systems”, Adv. Mater. 21,
4418–4441 (2009).
294
R. C. Evans, P . Douglas, and C. J. Winscom, “Coordination complexes exhibiting
room-temperature phosphorescence: evaluation of their suitability as triplet emitters
in organic light emitting diodes”, Coord. Chem. Rev. 250, 2093–2126 (2006).
295
H. J. Bolink, L. Cappelli, E. Coronado, M. Grätzel, E. Ortí, R. D. Costa, P . M. Viru-
ela, and M. K. Nazeeruddin, “Stable single-layer light-emitting electrochemical cell
using 4, 7-diphenyl-1, 10-phenanthroline-bis(2-phenylpyridine)iridium(iii) hexafluo-
rophosphate”, J. Am. Chem. Soc. 128, 14786–14787 (2006).
296
H.-C. Su, F.-C. Fang, T.-Y. Hwu, H.-H. Hsieh, H.-F. Chen, G.-H. Lee, S.-M. Peng, K.-
T. Wong, and C.-C. Wu, “Highly efficient orange and green solid-state light-emitting
electrochemical cells based on cationic ir (iii) complexes with enhanced steric hin-
drance”, Adv. Funct. Mater. 17, 1019–1027 (2007).
297
M. S. Lowry and S. Bernhard, “Synthetically tailored excited states: phosphorescent,
cyclometalated iridium (iii) complexes and their applications”, Chem. Eur. J. 12, 7970–
7977 (2006).
298
S. T. Parker, J. D. Slinker, M. S. Lowry, M. P . Cox, S. Bernhard, and G. G. Malliaras,
“Improved turn-on times of iridium electroluminescent devices by use of ionic liq-
uids”, Chem. Mater. 17, 3187–3190 (2005).
299
H. J. Bolink, L. Cappelli, E. Coronado, A. Parham, and P . Stössel, “Green light-emitting
solid-state electrochemical cell obtained from a homoleptic iridium (iii) complex con-
taining ionically charged ligands”, Chem. Mater. 18, 2778–2780 (2006).
300
M. K. Nazeeruddin, R. Wegh, Z Zhou, C Klein, Q Wang, F De Angelis, S Fantacci, and
M Grätzel, “Efficient green-blue-light-emitting cationic iridium complex for light-
emitting electrochemical cells”, Inorg. Chem. 45, 9245–9250 (2006).
301
J. D. Slinker, J. Rivnay, J. S. Moskowitz, J. B. Parker, S. Bernhard, H. D. Abruña, and
G. G. Malliaras, “Electroluminescent devices from ionic transition metal complexes”,
J. Mater. Chem. 17, 2976–2988 (2007).
302
J. M. Fernández-Hernández, C.-H. Yang, J. I. Beltrán, V . Lemaur, F. Polo, R. Fröhlich, J.
Cornil, and L. De Cola, “Control of the mutual arrangement of cyclometalated ligands
in cationic iridium (iii) complexes. synthesis, spectroscopy, and electroluminescence
of the different isomers”, J. Am. Chem. Soc. 133, 10543–10558 (2011).
303
R. D. Costa, E. Orti, H. J. Bolink, F. Monti, G. Accorsi, and N. Armaroli, “Luminescent
ionic transition-metal complexes for light-emitting electrochemical cells”, Angew. Chem.
Int. Ed. 51, 8178–8211 (2012).
304
M. A. Baldo, D. O’brien, Y You, A Shoustikov, S Sibley, M. Thompson, and S. Forrest,
“Highly efficient phosphorescent emission from organic electroluminescent devices”,
Nature 395, 151–154 (1998).
305
C. Adachi, M. A. Baldo, M. E. Thompson, and S. R. Forrest, “Nearly 100% internal
phosphorescence efficiency in an organic light-emitting device”, J. Appl. Phys. 90,
5048–5051 (2001).
Bibliography 191
306
S. Kwon, K.-R. Wee, J. W. Kim, and S. O. Kang, “The effect of energy level offset
between ir dopant and carbazole hosts on the emission efficiency”, Appl. Phys. Lett.
97, 023309–3 (2010).
307
J. E. Yarnell, C. E. McCusker, A. J. Leeds, J. M. Breaux, and F. N. Castellano, “Exposing
the excited-state equilibrium in an ir (iii) bichromophore: a combined time resolved
spectroscopy and computational study”, Eur. J. Inorg. Chem. 2016, 1808–1818 (2016).
308
F. Shao and J. K. Barton, “Long-range electron and hole transport through dna with
tethered cyclometalated iridium (iii) complexes”, J. Am. Chem. Soc. 129, 14733–14738
(2007).
309
B. Elias, F. Shao, and J. K. Barton, “Charge migration along the dna duplex: hole
versus electron transport”, J. Am. Chem. Soc. 130, 1152–1153 (2008).
310
K. K.-W. Lo, C.-K. Chung, and N. Zhu, “Synthesis, photophysical and electrochemical
properties, and biological labeling studies of cyclometalated iridium (iii) bis(pyridylbenzaldehyde)
complexes: novel luminescent cross-linkers for biomolecules”, Chem. Eur. J. 9, 475–
483 (2003).
311
J.-M. Hsieh, M.-L. Ho, P .-W. Wu, P .-T. Chou, T.-T. Tsai, and Y. Chi, “Iridium-complex
modified cdse/zns quantum dots; a conceptual design for bifunctionality toward
imaging and photosensitization”, Chem. Commun., 615–617 (2006).
312
M. T. Greenfield, S. D. McGrane, C. A. Bolme, J. A. Bjorgaard, T. R. Nelson, S. Tretiak,
and R. J. Scharff, “Photoactive high explosives: linear and nonlinear photochemistry
of petrin tetrazine chloride”, J. Phys. Chem. A 119, 4846–4855 (2015).
313
S. R. Ahmad and M. Cartwright, Laser ignition of energetic materials (John Wiley &
Sons, 2014).
314
V . M. Lisitsyn, V . Tsipilev, G Damamme, and D Malis, “Effect of the laser radiation
wavelength on the energy threshold of initiation of heavy metal azides”, Combust.,
Explos., Shock Waves 47, 591–600 (2011).
315
A. M. Renlund, P . L. Stanton, and W. M. Trott, “Laser initiation of seconday explo-
sives”, Proc. 9th Inter. Det. Symp., 1118–1127 (1989).
316
D. L. Paisley, “Prompt detonation of secondary explosives by laser”, Proc. 9th Inter.
Det. Symp., 1110–1117 (1989).
317
M. Joas, T. M. Klapötke, and N. Szimhardt, “Photosensitive metal (ii) perchlorates
with 1, 2-bis[5-(1-methylhydrazinyl)tetrazol-1-yl]ethane as ligand: synthesis, charac-
terization and laser ignition”, Eur. J. Inorg. Chem. 2014, 493–498 (2014).
318
N. Fischer, M. Joas, T. M. Klapötke, and J. Stierstorfer, “Transition metal complexes of
3-amino-1-nitroguanidine as laser ignitible primary explosives: structures and prop-
erties”, Inorg. Chem. 52, 13791–13802 (2013).
319
J. Evers, I. Gospodinov, M. Joas, T. M. Klapötke, and J. Stierstorfer, “Cocrystallization
of photosensitive energetic copper (ii) perchlorate complexes with the nitrogen-rich
ligand 1, 2-di (1 h-tetrazol-5-yl)ethane”, Inorg. Chem. 53, 11749–11756 (2014).
320
D. Fischer, T. M. Klapötke, D. G. Piercey, and J. Stierstorfer, “Copper salts of halo
tetrazoles: laser-ignitable primary explosives”, J. Energ. Mater. 30, 40–54 (2012).
321
G. Clavier and P . Audebert, “S-tetrazines as building blocks for new functional molecules
and molecular materials”, Chem. Rev. 110, 3299–3314 (2010).
192 Bibliography
322
S. Patra, B. Sarkar, S. Ghumaan, M. P . Patil, S. M. Mobin, R. B. Sunoj, W. Kaim, and
G. K. Lahiri, “Isomeric ruthenium terpyridine complexes [Ru(trpy)(L)Cl]
n+
contain-
ing the unsymmetrically bidentate acceptor l = 3-amino-6-(3, 5-dimethylpyrazol-1-
yl)-1, 2, 4, 5-tetrazine. synthesis, structures, electrochemistry, spectroscopy and dft
calculations”, Dalton Trans., 1188–1194 (2005).
323
A. Nayak, S. Patra, B. Sarkar, S. Ghumaan, V . G. Puranik, W. Kaim, and G. K. Lahiri,
“Tetrazine derived mononuclear Ru
II
(acac)
2
(L) (1), [Ru
II
(bpy)
2
(L)](ClO
4
)
2
(2) and [Ru
II
(bpy)(L)
2
](ClO
4
)
2
(3) (l = 3-amino-6-(3, 5-dimethylpyrazol-1-yl)-1, 2, 4, 5-tetrazine, acac= acetylacet-
onate, bpy= 2, 2
0
-bipyridine): syntheses, structures, spectra and redox properties”,
Polyhedron 24, 333–342 (2005).
324
P . W. Cooper, “Acceleration, formation, and flight of fragments”, Explosives Engi-
neering, Wiley-VCH, New York (1996).
325
D. E. Chavez, M. A. Hiskey, and R. D. Gilardi, “3, 3
0
-azobis(6-amino-1, 2, 4, 5-tetrazine):
a novel high-nitrogen energetic material”, Angew. Chem. 112, 1861–1863 (2000).
326
L. E. Fried, M. R. Manaa, P . F. Pagoria, and R. L. Simpson, “Design and synthesis of
energetic materials”, Annu. Rev. Mater. Res. 31, 291–321 (2001).
327
J. J. Sabatini and K. D. Oyler, “Recent advances in the synthesis of high explosive
materials”, Crystals 6, 5 (2015).
328
C. J. Snyder, P . D. Martin, M. J. Heeg, and C. H. Winter, “Synthesis, structure, and
properties of group 1 metal complexes containing nitrogen-rich hydrotris(tetrazolyl)borate
ligands”, Chem. Eur. J. 19, 3306–3310 (2013).
329
C. R. Benson, A. K. Hui, K. Parimal, B. J. Cook, C.-H. Chen, R. L. Lord, A. H. Flood,
and K. G. Caulton, “Multiplying the electron storage capacity of a bis-tetrazine pincer
ligand”, Dalton Trans. 43, 6513–6524 (2014).
330
M. Ketterle, W. Kaim, and J. Fiedler, “Valence delocalisation in a strongly coupled
(Kc = 10
14
) molecule-bridged cyanodiiron (iii, ii) species”, Chem. Commun., 1701–
1702 (1998).
331
M. Casida, “Recent advances in density-functional methods: part i”, in, Vol. 1, edited
by D. P . Chong (World Scientific: Singapore, 1995) Chap. Time-Dependent Density
Functional Response Theory for Molecules, p. 155.
332
A. Dreuw and M. Head-Gordon, “Single-reference ab initio methods for the calcula-
tion of excited states of large molecules”, Chem. Rev. 105, 4009–4037 (2005).
333
V . N. Staroverov, G. E. Scuseria, J. Tao, and J. P . Perdew, “Comparative assessment of a
new nonempirical density functional: molecules and hydrogen-bonded complexes”,
J. Chem. Phys. 119, 12129–12137 (2003).
334
E. R. Davidson and D. Feller, “Basis set selection for molecular calculations”, Chem.
Rev. 86, 681–696 (1986).
335
B Mennucci, “Wiley interdiscip”, Rev.: Comput. Mol. Sci 2, 386–404 (2012).
336
R. L. Martin, “Natural transition orbitals”, J. Chem. Phys. 118, 4775–4777 (2003).
337
Jmol: an open-source java viewer for chemical structures in 3d. http://www.jmol.org/ (accessed
july 1, 2016).
Bibliography 193
338
J. Tao, J. P . Perdew, V . N. Staroverov, and G. E. Scuseria, “Climbing the density func-
tional ladder: nonempirical meta–generalized gradient approximation designed for
molecules and solids”, Phys. Rev. Lett. 91, 146401 (2003).
339
M. Bühl and H. Kabrede, “Geometries of transition-metal complexes from density-
functional theory”, J. Chem. Theory Comput. 2, 1282–1290 (2006).
340
M. P . Waller, H. Braun, N. Hojdis, and M. Bühl, “Geometries of second-row transition-
metal complexes from density-functional theory”, J. Chem. Theory Comput. 3, 2234–
2242 (2007).
341
K. P . Jensen, “Bioinorganic chemistry modeled with the tpssh density functional”,
Inorg. Chem. 47, 10357–10365 (2008).
342
Y. Zhao and D. G. Truhlar, “Comparative assessment of density functional methods
for 3d transition-metal chemistry”, J. Chem. Phys. 124, 224105 (2006).
343
T. Weymuth, E. P . Couzijn, P . Chen, and M. Reiher, “New benchmark set of transition-
metal coordination reactions for the assessment of density functionals”, J. Chem. The-
ory Comput. 10, 3092–3103 (2014).
344
E. R. Batista and R. L. Martin, “Natural transition orbitals”, in (Wiley Online Library,
2004).
345
A. E. Reed, R. B. Weinstock, and F. Weinhold, “Natural population analysis”, J. Chem.
Phys. 83, 735–746 (1985).
346
A. E. Reed, L. A. Curtiss, and F. Weinhold, “Intermolecular interactions from a natural
bond orbital, donor-acceptor viewpoint”, Chem. Rev. 88, 899–926 (1988).
347
E. Glendening, A. Reed, J. Carpenter, and F Weinhold, NBO, Version 3.1 (1998).
348
A. D. Becke, “Density-functional thermochemistry. iii. the role of exact exchange”, J.
Chem. Phys. 98, 5648–5652 (1993).
349
a. Petersson, A. Bennett, T. G. Tensfeldt, M. A. Al-Laham, W. A. Shirley, and J. Mantzaris,
“A complete basis set model chemistry. i. the total energies of closed-shell atoms and
hydrides of the first-row elements”, J. Chem. Phys. 89, 2193–2218 (1988).
350
G. Petersson and M. A. Al-Laham, “A complete basis set model chemistry. ii. open-
shell systems and the total energies of the first-row atoms”, J. Chem. Phys. 94, 6081–
6090 (1991).
351
E. R. Batista and R. L. Martin, “On the excited states involved in the luminescent
probe [Ru(bpy)
2
dppz]
2+
”, J. Phys. Chem. A 109, 3128–3133 (2005).
352
L. Petit, P . Maldivi, and C. Adamo, “Predictions of optical excitations in transition-
metal complexes with time-dependent density functional theory: influence of basis
sets”, J. Chem. Theory Comput. 1, 953–962 (2005).
353
D. N. Chirdon, C. E. McCusker, F. N. Castellano, and S. Bernhard, “Tracking of tuning
effects in bis-cyclometalated iridium complexes: a combined time resolved infrared
spectroscopy, electrochemical, and computational study”, Inorg. Chem. 52, 8795–8804
(2013).
354
J. A. Pople, P . M. Gill, and B. G. Johnson, “Kohn—sham density-functional theory
within a finite basis set”, Chem. Phys. Lett. 199, 557–560 (1992).
194 Bibliography
355
P . J. Hay and W. R. Wadt, “Ab initio effective core potentials for molecular calcula-
tions. potentials for k to au including the outermost core orbitals”, J. Chem. Phys. 82,
299–310 (1985).
356
L. E. Roy, P . J. Hay, and R. L. Martin, “Revised basis sets for the lanl effective core
potentials”, J. Chem. Theory Comput. 4, 1029–1031 (2008).
357
D. Feller, “The role of databases in support of computational chemistry calculations”,
J. Comput. Chem. 17, 1571–1586 (1996).
358
K. L. Schuchardt, B. T. Didier, T. Elsethagen, L. Sun, V . Gurumoorthi, J. Chase, J. Li,
and T. L. Windus, “Basis set exchange: a community database for computational sci-
ences”, J. Chem. Inf. Model. 47, 1045–1052 (2007).
359
A. Masunov and S. Tretiak, “Prediction of two-photon absorption properties for or-
ganic chromophores using time-dependent density-functional theory”, J. Phys. Chem.
B 108, 899–907 (2004).
360
D. N. Bowman, A. Bondarev, S. Mukherjee, and E. Jakubikova, “Tuning the electronic
structure of fe (ii) polypyridines via donor atom and ligand scaffold modifications: a
computational study”, Inorg. Chem. 54, 8786–8793 (2015).
361
C. Gray and K. Gubbins, Theory of molecular fluids, volume 1: fundamentals (Oxford,
1984).
362
J. P . Agrawal, High energy materials: propellants, explosives and pyrotechnics (John Wiley
& Sons, 2010).
363
K. Kalyanasundaram, “Photophysics, photochemistry and solar energy conversion
with tris(bipyridyl)ruthenium(ii) and its analogues”, Coord. Chem. Rev. 46, 159–244
(1982).
364
P . S. Braterman, J. I. Song, and R. D. Peacock, “Electronic absorption spectra of the
iron (ii) complexes of 2, 2
0
-bipyridine, 2, 2
0
-bipyrimidine, 1, 10-phenanthroline, and 2,
2
0
: 6
0
, 2
00
-terpyridine and their reduction products”, Inorg Chem 31, 555–559 (1992).
365
A. C. Tsipis, “Dft flavor of coordination chemistry”, Coord. Chem. Rev. 272, 1–29
(2014).
366
C. Bian, X. Dong, X. Zhang, Z. Zhou, M. Zhang, and C. Li, “The unique synthesis and
energetic properties of a novel fused heterocycle: 7-nitro-4-oxo-4, 8-dihydro-[1, 2, 4]
triazolo [5, 1d][1, 2, 3, 5] tetrazine 2-oxide and its energetic salts”, J. Mater. Chem. A
3, 3594–3601 (2015).
367
Y. Tang, J. Zhang, L. A. Mitchell, D. A. Parrish, and J. M. Shreeve, “Taming of 3, 4-di
(nitramino) furazan”, J. Am. Chem. Soc. 137, 15984–15987 (2015).
368
P . Yin and J. M. Shreeve, “From n-nitro to n-nitroamino: preparation of high-performance
energetic materials by introducing nitrogen-containing ions”, Angew. Chem. Int. Ed.
54, 14513–14517 (2015).
369
P . Yin and J. M. Shreeve, “From n-nitro to n-nitroamino: preparation of high-performance
energetic materials by introducing nitrogen-containing ions”, Angew. Chem. 127,
14721–14725 (2015).
370
D. E. Chavez, J. C. Bottaro, M. Petrie, and D. A. Parrish, “Synthesis and thermal be-
havior of a fused, tricyclic 1, 2, 3, 4-tetrazine ring system”, Angew. Chem. Int. Ed. 54,
12973–12975 (2015).
Bibliography 195
371
D. E. Chavez, J. C. Bottaro, M. Petrie, and D. A. Parrish, “Synthesis and thermal be-
havior of a fused, tricyclic 1, 2, 3, 4-tetrazine ring system”, Angew. Chem. 127, 13165–
13167 (2015).
372
T. M. Klapötke, M. Leroux, P . C. Schmid, and J. Stierstorfer, “Energetic materials based
on 5, 50-diamino-4, 40-dinitramino-3, 30-bi-1, 2, 4-triazole”, Chem. Asian J. 11, 844–851
(2016).
373
T. M. Klapötke, P . C. Schmid, S. Schnell, and J. Stierstorfer, “3, 6, 7-triamino-[1, 2,
4] triazolo [4, 3-b][1, 2, 4] triazole: a non-toxic, high-performance energetic building
block with excellent stability”, Chem. Eur. J. 21, 9219–9228 (2015).
374
W. Fickett and W. C. Davis, Detonation: theory and experiment (Courier Corporation,
2012).
375
E. Aluker, A. Krechetov, A. Y. Mitrofanov, and D. Nurmukhametov, “Photochemical
and photothermal dissociation of petn during laser initiation”, Russ. J. Phys. Chem.
B 5, 658–660 (2011).
376
M. Joas, T. M. Klapötke, J. Stierstorfer, and N. Szimhardt, “Synthesis and characteri-
zation of various photosensitive copper (ii) complexes with 5-(1-methylhydrazinyl)-
1h-tetrazole as ligand and perchlorate, nitrate, dinitramide, and chloride as anions”,
Chem. Eur. J. 19, 9995–10003 (2013).
377
S. D. McGrane, C. A. Bolme, M. T. Greenfield, D. E. Chavez, S. K. Hanson, and R. J.
Scharff, “Photoactive high explosives: substituents effects on tetrazine photochem-
istry and photophysics”, J. Phys. Chem. A 120, 895–902 (2016).
378
V . V . Chaban, S. Pal, and O. V . Prezhdo, “Laser-induced explosion of nitrated car-
bon nanotubes: non-adiabatic and reactive molecular dynamics simulations”, J. Am.
Chem. Soc. 138, 15927–15934 (2016).
379
D. E. Chavez and M. A. Hiskey, “1, 2, 4, 5-tetrazine based energetic materials”, J.
Energ. Mater. 17, 357–377 (1999).
380
V . Thottempudi, F. Forohor, D. A. Parrish, and J. M. Shreeve, “Tris(triazolo)benzene
and its derivatives: high-density energetic materials”, Angew. Chem. Int. Ed. 51, 9881–
9885 (2012).
381
H. Wei, J. Zhang, and J. M. Shreeve, “Synthesis, characterization, and energetic prop-
erties of 6-amino-tetrazolo [1, 5-b]-1, 2, 4, 5-tetrazine-7-n-oxide: a nitrogen-rich mate-
rial with high density”, Chem. - Asian J. 10, 1130–1132 (2015).
382
P . Yin, Q. Zhang, and J. M. Shreeve, “Dancing with energetic nitrogen atoms: versatile
n-functionalization strategies for n-heterocyclic frameworks in high energy density
materials”, Acc. Chem. Res. 49, 4–16 (2015).
383
M. Schulze, B. Scott, and D. Chavez, “A high density pyrazolo-triazine explosive
(ptx)”, J. Mater. Chem. A 3, 17963–17965 (2015).
384
Y. Li, Y. Shu, B. Wang, S. Zhang, and L. Zhai, “Synthesis, structure and properties
of neutral energetic materials based on n-functionalization of 3, 6-dinitropyrazolo[4,
3-c]pyrazole”, RSC Adv. 6, 84760–84768 (2016).
385
P . W. Cooper, Explosives engineering (Wiley-VCH, New York, 1996).
386
D. E. Chavez, D. A. Parrish, L. Mitchell, and G. H. Imler, “Azido and tetrazolo 1,2,4,5-
tetrazine n-oxides”, Angew. Chem. Int. Ed. (2017).
196 Bibliography
387
D. E. Chavez, D. A. Parrish, L. Mitchell, and G. H. Imler, “Azido and tetrazolo 1,2,4,5-
tetrazine n-oxides”, Angew. Chem. (2017).
388
D. G. Piercey, D. E. Chavez, B. L. Scott, G. H. Imler, and D. A. Parrish, “An ener-
getic triazolo-1, 2, 4-triazine and its n-oxide”, Angew. Chem. Int. Ed. 55, 15315–15318
(2016).
389
D. G. Piercey, D. E. Chavez, B. L. Scott, G. H. Imler, and D. A. Parrish, “An energetic
triazolo-1, 2, 4-triazine and its n-oxide”, Angew. Chem. 128, 15541–15544 (2016).
390
D. E. Chavez, D. A. Parrish, and L. Mitchell, “Energetic trinitro- and dinitro-fluoroethyl
ethers of 1, 2, 4, 5-tetrazines”, Angew. Chem. Int. Ed. 55, 8666–8669 (2016).
391
D. E. Chavez, D. A. Parrish, and L. Mitchell, “Energetic trinitro- and dinitro-fluoroethyl
ethers of 1, 2, 4, 5-tetrazines”, Angew. Chem. 128, 8808–8811 (2016).
392
M. Drobizhev, A. Karotki, Y. Dzenis, A. Rebane, Z. Suo, and C. W. Spangler, “Strong
cooperative enhancement of two-photon absorption in dendrimers”, J. Phys. Chem.
B 107, 7540–7543 (2003).
393
S.-J. Chung, K.-S. Kim, T.-C. Lin, G. S. He, J. Swiatkiewicz, and P . N. Prasad, “Cooper-
ative enhancement of two-photon absorption in multi-branched structures”, J. Phys.
Chem. B 103, 10741–10745 (1999).
394
Y. Wang, G. S. He, P . N. Prasad, and T. Goodson, “Ultrafast dynamics in multibranched
structures with enhanced two-photon absorption”, J. Am. Chem. Soc. 127, 10128–
10129 (2005).
395
M. Drobizhev, A. Karotki, A. Rebane, and C. W. Spangler, “Dendrimer molecules with
record large two-photon absorption cross section”, Opt. Lett. 26, 1081–1083 (2001).
396
O. Varnavski, X. Yan, O. Mongin, M. Blanchard-Desce, and T. Goodson, “Strongly
interacting organic conjugated dendrimers with enhanced two-photon absorption”,
J. Phys. Chem. C 111, 149–162 (2007).
397
M. Drobizhev, A. Karotki, M. Kruk, and A. Rebane, “Resonance enhancement of two-
photon absorption in porphyrins”, Chem. Phys. Lett. 355, 175–182 (2002).
398
M. Drobizhev, Y. Stepanenko, Y. Dzenis, A. Karotki, A. Rebane, P . N. Taylor, and
H. L. Anderson, “Understanding strong two-photon absorption inp-conjugated por-
phyrin dimers via double-resonance enhancement in a three-level model”, J. Am.
Chem. Soc. 126, 15352–15353 (2004).
399
M. Drobizhev, Y. Stepanenko, Y. Dzenis, A. Karotki, A. Rebane, P . N. Taylor, and H. L.
Anderson, “Extremely strong near-ir two-photon absorption in conjugated porphyrin
dimers: quantitative description with three-essential-states model”, J. Phys. Chem. B
109, 7223–7236 (2005).
400
M. Drobizhev, Y. Stepanenko, A. Rebane, C. J. Wilson, T. E. Screen, and H. L. Ander-
son, “Strong cooperative enhancement of two-photon absorption in double-strand
conjugated porphyrin ladder arrays”, J. Am. Chem. Soc. 128, 12432–12433 (2006).
401
H. Ceymann, A. Rosspeintner, M. H. Schreck, C. Mützel, A. Stoy, E. Vauthey, and
C. Lambert, “Cooperative enhancement versus additivity of two-photon-absorption
cross sections in linear and branched squaraine superchromophores”, Phys. Chem.
Chem. Phys. 18, 16404–16413 (2016).
Bibliography 197
402
F. Terenziani, V . Parthasarathy, A. Pla-Quintana, T. Maishal, A.-M. Caminade, J.-P .
Majoral, and M. Blanchard-Desce, “Cooperative two-photon absorption enhancement
by through-space interactions in multichromophoric compounds”, Angew. Chem.
Int. Ed. 48, 8691–8694 (2009).
403
A. Mikhaylov, D. V . Kondratuk, A. Cnossen, H. L. Anderson, M. Drobizhev, and
A. Rebane, “Cooperative enhancement of two-photon absorption in self-assembled
zinc-porphyrin nanostructures”, J. Phys. Chem. C 120, 11663–11670 (2016).
404
T. W. Myers, C. J. Snyder, D. E. Chavez, R. J. Scharff, and J. M. Veauthier, “Synthesis
and electrochemical behavior of electron-rich s-tetrazine and triazolo-tetrazine nitrate
esters”, Chem. Eur. J. 22, 10590–10596 (2016).
405
C. He and J. M. Shreeve, “Energetic materials with promising properties: synthe-
sis and characterization of 4, 40-bis (5-nitro-1, 2, 3-2h-triazole) derivatives”, Angew.
Chem. Int. Ed. 54, 6260–6264 (2015).
406
C. He and J. M. Shreeve, “Energetic materials with promising properties: synthe-
sis and characterization of 4, 40-bis (5-nitro-1, 2, 3-2h-triazole) derivatives”, Angew.
Chem. 127, 6358–6362 (2015).
407
L. Fried, W. Howard, S Bastea, K Glaesmann, P . Souers, P . Vitello, and I. Kuo, Cheetah
thermochemical code, Lawrence Livermore National Laboratory (Livermore, CA, USA,
2007).
408
E. F. Byrd and B. M. Rice, “Improved prediction of heats of formation of energetic
materials using quantum mechanical calculations”, J. Phys. Chem. A 110, 1005–1013
(2006).
409
E. F. Byrd and B. M. Rice, “Improved prediction of heats of formation of energetic
materials using quantum mechanical calculations”, J. Phys. Chem. A 113, 5813–5813
(2009).
410
M. A. Kettner, T. M. Klapötke, T. G. Witkowski, and F. von Hundling, “Synthesis,
characterisation and crystal structures of two bi-oxadiazole derivatives featuring the
trifluoromethyl group”, Chem. Eur. J. 21, 4238–4241 (2015).
411
Q. J. Axthammer, B. Krumm, and T. M. Klapötke, “Synthesis of energetic nitrocarba-
mates from polynitro alcohols and their potential as high energetic oxidizers”, J. Org.
Chem. 80, 6329–6335 (2015).
412
K. Momma and F. Izumi, “Vesta 3 for three-dimensional visualization of crystal, vol-
umetric and morphology data”, J. Appl. Cryst. 44, 1272–1276 (2011).
413
Avogadro: an open-source molecular builder and visualization tool. version 1.1.1 (http://avogadro.openmolecules.net/
(Accessed September 1, 2016)).
414
M. D. Hanwell, D. E. Curtis, D. C. Lonie, T. Vandermeersch, E. Zurek, and G. R.
Hutchison, “Avogadro: an advanced semantic chemical editor, visualization, and anal-
ysis platform”, J. Cheminform. 4, 1 (2012).
415
C. Katan, F. Terenziani, O. Mongin, M. H. Werts, L. Porres, T. Pons, J. Mertz, S. Tretiak,
and M. Blanchard-Desce, “Effects of (multi) branching of dipolar chromophores on
photophysical properties and two-photon absorption”, J. Phys. Chem. A 109, 3024–
3037 (2005).
198 Bibliography
416
J. F. Kauffman, J. M. Turner, I. V . Alabugin, B. Breiner, S. V . Kovalenko, E. A. Badaeva,
A. Masunov, and S. Tretiak, “Two-photon excitation of substituted enediynes”, J.
Phys. Chem. A 110, 241–251 (2006).
417
E. Badaeva and S. Tretiak, “Two photon absorption of extended substituted phenyleneviny-
lene oligomers: a tddft study”, Chem. Phys. Lett. 450, 322–328 (2008).
418
C. Katan, M. Charlot, O. Mongin, C. Le Droumaguet, V . Jouikov, F. Terenziani, E.
Badaeva, S. Tretiak, and M. Blanchard-Desce, “Simultaneous control of emission lo-
calization and two-photon absorption efficiency in dissymmetrical chromophores”, J.
Phys. Chem. B 114, 3152–3169 (2010).
419
M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman,
G. Scalmani, V . Barone, B. Mennucci, G. A. Petersson, H. Nakatsuji, M. Caricato, X.
Li, H. P . Hratchian, A. F. Izmaylov, J. Bloino, G. Zheng, J. L. Sonnenberg, M. Hada,
M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O.
Kitao, H. Nakai, T. Vreven, J. A. Montgomery Jr., J. E. Peralta, F. Ogliaro, M. Bearpark,
J. J. Heyd, E. Brothers, K. N. Kudin, V . N. Staroverov, R. Kobayashi, J. Normand, K.
Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, N. Rega,
J. M. Millam, M. Klene, J. E. Knox, J. B. Cross, V . Bakken, C. Adamo, J. Jaramillo,
R. Gomperts, R. E. Stratmann, O. Yazyev, A. J. Austin, R. Cammi, C. Pomelli, J. W.
Ochterski, R. L. Martin, K. Morokuma, V . G. Zakrzewski, G. A. Voth, P . Salvador,
J. J. Dannenberg, S. Dapprich, A. D. Daniels, ï. Farkas, J. B. Foresman, J. V . Ortiz, J.
Cioslowski, and D. J. Fox, Gaussian 03, Revision E. 01 (2004).
420
O. Berman and S. Mukamel, “Quasiparticle density-matrix representation of nonlin-
ear time-dependent density-functional response functions”, Phys. Rev. A 67, 042503
(2003).
421
S Tretiak and V Chernyak, “Resonant nonlinear polarizabilities in the time-dependent
density functional theory”, J. Chem. Phys. 119, 8809–8823 (2003).
422
R. Cammi, B. Mennucci, and J. Tomasi, “Fast evaluation of geometries and proper-
ties of excited molecules in solution: a tamm-dancoff model with application to 4-
dimethylaminobenzonitrile”, J. Phys. Chem. A 104, 5631–5637 (2000).
423
M. Cossi and V . Barone, “Time-dependent density functional theory for molecules in
liquid solutions”, J. Chem. Phys. 115, 4708–4717 (2001).
424
R. Improta, V . Barone, G. Scalmani, and M. J. Frisch, “A state-specific polarizable
continuum model time dependent density functional theory method for excited state
calculations in solution”, J. Chem. Phys. 125, 054103 (2006).
425
R. Improta, G. Scalmani, M. J. Frisch, and V . Barone, “Toward effective and reliable
fluorescence energies in solution by a new state specific polarizable continuum model
time dependent density functional theory approach”, J. Chem. Phys. 127, 074504
(2007).
426
S. Zheng, E. Geva, and B. D. Dunietz, “Solvated charge transfer states of functional-
ized anthracene and tetracyanoethylene dimers: a computational study based on a
range separated hybrid functional and charge constrained self-consistent field with
switching gaussian polarized continuum models”, J. Chem. Theory Comput. 9, 1125–
1131 (2013).
Bibliography 199
427
M. Dierksen and S. Grimme, “Density functional calculations of the vibronic structure
of electronic absorption spectra”, J. Chem. Phys. 120, 3544–3554 (2004).
428
M. I. Ranasinghe, O. P . Varnavski, J. Pawlas, S. I. Hauck, J. Louie, J. F. Hartwig, and
T. Goodson, “Femtosecond excitation energy transport in triarylamine dendrimers”,
J. Am. Chem. Soc. 124, 6520–6521 (2002).
429
M. R. Provorse, T. Peev, C. Xiong, and C. M. Isborn, “Convergence of excitation en-
ergies in mixed quantum and classical solvent: comparison of continuum and point
charge models”, J. Phys. Chem. B 120, 12148–12159 (2016).
430
B. H. Besler, K. M. Merz, and P . A. Kollman, “Atomic charges derived from semiem-
pirical methods”, J. Comput. Chem. 11, 431–439 (1990).
431
E. Sigfridsson and U. Ryde, “Comparison of methods for deriving atomic charges
from the electrostatic potential and moments”, J. Comput. Chem. 19, 377–395 (1998).
432
B. Nebgen, N. Lubbers, J. S. Smith, A. Sifain, A. Lokhov, O. Isayev, A. Roitberg, K.
Barros, and S. Tretiak, “Transferable molecular charge assignment using deep neural
networks”, arXiv preprint arXiv:1803.04395 (2018).
433
J. Behler, “Constructing high-dimensional neural network potentials: a tutorial re-
view”, Int. J. Quantum Chem. 115, 1032–1050 (2015).
434
J. Behler and M. Parrinello, “Generalized neural-network representation of high-dimensional
potential-energy surfaces”, Phys. Rev. Lett. 98, 146401 (2007).
435
M. Rupp, A. Tkatchenko, K.-R. Müller, and O. A. Von Lilienfeld, “Fast and accurate
modeling of molecular atomization energies with machine learning”, Phys. Rev. Lett.
108, 058301 (2012).
436
G. Montavon, M. Rupp, V . Gobre, A. Vazquez-Mayagoitia, K. Hansen, A. Tkatchenko,
K.-R. Müller, and O. A. Von Lilienfeld, “Machine learning of molecular electronic
properties in chemical compound space”, New J. Phys. 15, 095003 (2013).
437
K. Hansen, G. Montavon, F. Biegler, S. Fazli, M. Rupp, M. Scheffler, O. A. Von Lilien-
feld, A. Tkatchenko, and K.-R. Müller, “Assessment and validation of machine learn-
ing methods for predicting molecular atomization energies”, J. Chem. Theory Com-
put. 9, 3404–3419 (2013).
438
O. A. von Lilienfeld, R. Ramakrishnan, M. Rupp, and A. Knoll, “Fourier series of
atomic radial distribution functions: a molecular fingerprint for machine learning
models of quantum chemical properties”, Int. J. Quantum Chem. 115, 1084–1093 (2015).
439
K. Hansen, F. Biegler, R. Ramakrishnan, W. Pronobis, O. A. Von Lilienfeld, K.-R.
Müller, and A. Tkatchenko, “Machine learning predictions of molecular properties:
accurate many-body potentials and nonlocality in chemical space”, J. Phys. Chem.
Lett. 6, 2326–2331 (2015).
440
M. Rupp, R. Ramakrishnan, and O. A. von Lilienfeld, “Machine learning for quan-
tum mechanical properties of atoms in molecules”, J. Phys. Chem. Lett. 6, 3309–3313
(2015).
441
Z. Li, J. R. Kermode, and A. De Vita, “Molecular dynamics with on-the-fly machine
learning of quantum-mechanical forces”, Phys. Rev. Lett. 114, 096405 (2015).
200 Bibliography
442
S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky, K. T. Schütt, and K.-R. Müller,
“Machine learning of accurate energy-conserving molecular force fields”, Sci. Adv. 3,
e1603015 (2017).
443
K. Yao, J. E. Herr, S. N. Brown, and J. Parkhill, “Intrinsic bond energies from a bonds-
in-molecules neural network”, J. Phys. Chem. Lett. 8, 2689–2694 (2017).
444
J. S. Smith, O. Isayev, and A. E. Roitberg, “Ani-1: an extensible neural network po-
tential with dft accuracy at force field computational cost”, Chem. Sci. 8, 3192–3203
(2017).
445
G. Pilania, J. E. Gubernatis, and T. Lookman, “Multi-fidelity machine learning models
for accurate bandgap predictions of solids”, Comput. Mater. Sci. 129, 156–163 (2017).
446
Y. Zhuo, A. Mansouri Tehrani, and J. Brgoch, “Predicting the band gaps of inorganic
solids by machine learning”, J. Phys. Chem. Lett. 9, 1668–1673 (2018).
447
K. T. Schütt, H. E. Sauceda, P .-J. Kindermans, A. Tkatchenko, and K.-R. Müller, “Schnet–
a deep learning architecture for molecules and materials”, J. Chem. Phys. 148, 241722
(2018).
448
K. Yao, J. E. Herr, D. W. Toth, R. Mckintyre, and J. Parkhill, “The tensormol-0.1 model
chemistry: a neural network augmented with long-range physics”, Chem. Sci. 9, 2261–
2269 (2018).
449
A. Grisafi, D. M. Wilkins, G. Csányi, and M. Ceriotti, “Symmetry-adapted machine
learning for tensorial properties of atomistic systems”, Phys. Rev. Lett. 120, 036002
(2018).
450
J. Hachmann, R. Olivares-Amaya, S. Atahan-Evrenk, C. Amador-Bedolla, R. S. Sánchez-
Carrera, A. Gold-Parker, L. Vogt, A. M. Brockway, and A. Aspuru-Guzik, “The har-
vard clean energy project: large-scale computational screening and design of organic
photovoltaics on the world community grid”, J. Phys. Chem. Lett. 2, 2241–2251 (2011).
451
T. Morawietz, V . Sharma, and J. Behler, “A neural network potential-energy surface
for the water dimer based on environment-dependent atomic energies and charges”,
J. Chem. Phys. 136, 064103 (2012).
452
T. Morawietz and J. Behler, “A density-functional theory-based neural network po-
tential for water clusters including van der waals corrections”, J. Phys. Chem. A 117,
7356–7366 (2013).
453
N. Artrith, B. Hiller, and J. Behler, “Neural network potentials for metals and oxides–
first applications to copper clusters at zinc oxide”, Phys. Status Solidi B 250, 1191–
1203 (2013).
454
T. D. Huan, A. Mannodi-Kanakkithodi, and R. Ramprasad, “Accelerated materials
property predictions and design using motif-based fingerprints”, Phys. Rev. B 92,
014106 (2015).
455
S. K. Natarajan, T. Morawietz, and J. Behler, “Representing the potential-energy sur-
face of protonated water clusters by high-dimensional neural network potentials”,
Phys. Chem. Chem. Phys. 17, 8356–8371 (2015).
456
J. P . Janet and H. J. Kulik, “Predicting electronic structure properties of transition
metal complexes with neural networks”, Chem. Sci. 8, 5137–5152 (2017).
Bibliography 201
457
J. P . Janet and H. J. Kulik, “Resolving transition metal chemical space: feature selection
for machine learning and structure–property relationships”, J. Phys. Chem. A 121,
8939–8954 (2017).
458
J. P . Janet, L. Chan, and H. J. Kulik, “Accelerating chemical discovery with machine
learning: simulated evolution of spin crossover complexes with an artificial neural
network”, J. Phys. Chem. Lett. 9, 1064–1071 (2018).
459
Q. Tong, L. Xue, J. Lv, Y. Wang, and Y. Ma, “Accelerating calypso structure predic-
tion by data-driven learning of potential energy surface”, Faraday Discuss., DOI:
10.1039/C8FD00055G (2018).
460
F. Häse, S. Valleau, E. Pyzer-Knapp, and A. Aspuru-Guzik, “Machine learning exciton
dynamics”, Chem. Sci. 7, 5139–5147 (2016).
461
B. Sun, M. Fernandez, and A. S. Barnard, “Machine learning for silver nanoparticle
electron transfer property prediction”, J. Chem. Inf. Model 57, 2413–2423 (2017).
462
F. Brockherde, L. Vogt, L. Li, M. E. Tuckerman, K. Burke, and K.-R. Müller, “Bypassing
the kohn-sham equations with machine learning”, Nat. Commun. 8, 872 (2017).
463
F. Häse, C. Kreisbeck, and A. Aspuru-Guzik, “Machine learning for quantum dynam-
ics: deep learning of excitation energy transfer properties”, Chem. Sci. 8, 8419–8426
(2017).
464
S. Geidl, T. Bouchal, T. Raˇ cek, R. S. Vaˇ reková, V . Hejret, A. Kˇ renek, R. Abagyan, and
J. Koˇ ca, “High-quality and universal empirical atomic charges for chemoinformatics
applications”, J. Cheminform. 7, 59 (2015).
465
T. Bereau, R. A. DiStasio Jr, A. Tkatchenko, and O. A. Von Lilienfeld, “Non-covalent
interactions across organic and biological subsets of chemical space: physics-based
potentials parametrized from machine learning”, J. Chem. Phys. 148, 241706 (2018).
466
P . Bleiziffer, K. Schaller, and S. Riniker, “Machine learning of partial charges derived
from high-quality quantum-mechanical calculations”, J. Chem. Inf. Model 58, 579–
590 (2018).
467
N. Lubbers, J. S. Smith, and K. Barros, “Hierarchical modeling of molecular energies
using a deep neural network”, J. Chem. Phys. 148, 241715 (2018).
468
J. S. Smith, B. Nebgen, N. Lubbers, O. Isayev, and A. E. Roitberg, “Less is more: sam-
pling chemical space with active learning”, J. Chem. Phys. 148, 241733 (2018).
469
D. Reker and G. Schneider, “Active-learning strategies in computer-assisted drug dis-
covery”, Drug Discov. Today 20, 458–465 (2015).
470
M. Gastegger, J. Behler, and P . Marquetand, “Machine learning molecular dynamics
for the simulation of infrared spectra”, Chem. Sci. 8, 6924–6935 (2017).
471
E. V . Podryabinkin and A. V . Shapeev, “Active learning of linearly parametrized in-
teratomic potentials”, Comput. Mater. Sci. 140, 171–180 (2017).
472
A. L. Hickey and C. N. Rowley, “Benchmarking quantum chemical methods for the
calculation of molecular dipole moments and polarizabilities”, J. Phys. Chem. A 118,
3678–3687 (2014).
473
J. S. Smith, O. Isayev, and A. E. Roitberg, “Ani-1, a data set of 20 million calculated
off-equilibrium conformations for organic molecules”, Sci. Data 4, 170193 (2017).
202 Bibliography
474
K. T. Schütt, M. Gastegger, A. Tkatchenko, and K.-R. Müller, “Quantum-chemical in-
sights from interpretable atomistic neural networks”, arXiv preprint arXiv:1806.10349
(2018).
475
W.-K. Chen, X.-Y. Liu, W. Fang, P . O. Dral, and G. Cui, “Deep learning for nonadi-
abatic excited-state dynamics”, J. Phys. Chem. Lett. (2018) 10.1021/acs.jpclett.
8b03026.
476
A. V . Akimov, R. Long, and O. V . Prezhdo, “Coherence penalty functional: a simple
method for adding decoherence in ehrenfest dynamics”, J. Chem. Phys. 140, 194107
(2014).
477
T. Nelson, S. Fernandez-Alberti, V . Chernyak, A. E. Roitberg, and S. Tretiak, “Nonadi-
abatic excited-state molecular dynamics: numerical tests of convergence and param-
eters”, J. Chem. Phys. 136, 054108 (2012).
478
F. Furche, “On the density matrix based approach to time-dependent density func-
tional response theory”, J. Chem. Phys. 114, 5982–5992 (2001).
479
F. Furche and R. Ahlrichs, “Adiabatic time-dependent density functional methods for
excited state properties”, J. Chem. Phys. 117, 7433–7447 (2002).
480
G. D. Billing, “Classical path method in inelastic and reactive scattering”, Int. Rev.
Phys. Chem. 13, 309–335 (1994).
481
V . Chernyak and S. Mukamel, “Density-matrix representation of nonadiabatic cou-
plings in time-dependent density functional (tddft) theories”, J. Chem. Phys. 112,
3572–3579 (2000).
482
M Tommasini, V Chernyak, and S Mukamel, “Electronic density-matrix algorithm for
nonadiabatic couplings in molecular dynamics simulations”, Int. J. Quantum Chem.
85, 225–238 (2001).
483
D. J. Sindhikara, S. Kim, A. F. Voter, and A. E. Roitberg, “Bad seeds sprout perilous dy-
namics: stochastic thermostat induced trajectory synchronization in biomolecules”, J.
Chem. Theory Comput. 5, 1624–1631 (2009).
484
S. Hammes-Schiffer, “Hydrogen tunneling and protein motion in enzyme reactions”,
Acc. Chem. Res. 39, 93–100 (2006).
485
O. V . Prezhdo, “Photoinduced dynamics in semiconductor quantum dots: insights
from time-domain ab initio studies”, Acc. Chem. Res. 42, 2005–2016 (2009).
486
Z. Lan, E. Fabiano, and W. Thiel, “Photoinduced nonadiabatic dynamics of pyrimi-
dine nucleobases: on-the-fly surface-hopping study with semiempirical methods”, J.
Phys. Chem. B 113, 3548–3555 (2009).
487
E. Tapavicza, G. D. Bellchambers, J. C. Vincent, and F. Furche, “Ab initio non-adiabatic
molecular dynamics”, Phys. Chem. Chem. Phys. 15, 18336–18348 (2013).
488
G. Granucci and M. Persico, “Critical appraisal of the fewest switches algorithm for
surface hopping”, J. Chem. Phys. 126, 134114 (2007).
489
M. Frisch, G. Trucks, H. Schlegel, G. Scuseria, M. Robb, J. Cheeseman, G Scalmani,
V Barone, G. Petersson, H Nakatsuji, et al., Gaussian 16, revision a. 03, 2016.
490
R. Dennington, T. Keith, and J. Millam, Gaussview, version 5, 2009.

Abstract (if available)

Abstract Computational modeling of photoexcited molecules provides a fundamental understanding of processes such as photodissociation, photoisomerization, and charge and energy transfer. To model these processes requires the development and utilization of theories in electronic structure and quantum dynamics. One of the most popular methods for modeling nuclear-electronic dynamics is surface hopping. In this dissertation, surface hopping will be presented in different spotlights. We describe the method along with its advantages, disadvantages, and the contributions we make to improve its reliability and applicability. These efforts are carried out with simple Hamiltonians that can be solved exactly, thus allowing for an assessment of this approximate method. While this is an important step in methods development, modeling the large number of degrees of freedom in realistic systems requires efficient software. We present a software that combines surface hopping with numerically efficient methods for calculating ground and excited state potential energy surfaces—a necessary feature for modeling systems made up of hundreds of atoms and processes lasting up to tens of picoseconds. As part of our contribution to this software, we implement and benchmark an implicit solvent model, including investigating its effects during the nonradiative relaxation in organic conjugated molecules. The ultimate goal is to improve and assess the accuracy of these theoretical tools with the intention of progressing real-life applications. Having said that, a portion of this dissertation explores the early stages of an application called photoactive energetic materials—a field seeking to discover mechanically and electrically insensitive materials that undergo detonation through optical initiation. Our work investigates the optical properties in energetic materials, identified through an experimental collaboration, and is aimed at unveiling design principles to enhance control over the initiation threshold. In summary, we provide a comprehensive view of atomistic simulations of photoexcited molecules, starting from the methods used to describe electronic transitions through a manifold of excited states as a result of photoexcitation, followed by the development of software used to model realistic systems, and finally the use of such tools to discover how desired photophysical properties can be attained for practical use. The final part of this dissertation explores a new and booming field of research in the computational sciences called machine learning. Machine learning is a clever way of attaining calculations of ab initio accuracy at a tiny fraction of the computational cost through the use of statistically trained models. We present an application using machine learning and discuss how the use of this method can change the scope of nonadiabatic molecular dynamics.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Adiabatic and non-adiabatic molecular dynamics in nanoscale systems: theory and applications

PDF

Theoretical modeling of nanoscale systems: applications and method development

PDF

Quantum molecular dynamics and machine learning for non-equilibrium processes in quantum materials

PDF

From first principles to machine intelligence: explaining charge carrier behavior in inorganic materials

PDF

Multi-scale quantum dynamics and machine learning for next generation energy applications

PDF

Designing data-effective machine learning pipeline in application to physics and material science

PDF

Out-of-equilibrium dynamics of inhomogeneous quantum systems

PDF

Applications in optical communications: quantum communication systems and optical nonlinear device

PDF

Dynamics of adiabatic quantum search

PDF

Electronically excited and ionized states in condensed phase: theory and applications

PDF

Coulomb interactions and superconductivity in low dimensional materials

PDF

Dynamic topology reconfiguration of Boltzmann machines on quantum annealers

PDF

Photodissociation dynamics of group V hydrides

PDF

Neural network for molecular dynamics simulation and design of 2D materials

PDF

Feedback control of optical cavity systems

PDF

Electronic structure analysis of challenging open-shell systems: from excited states of copper oxide anions to exchange-coupling in binuclear copper complexes

PDF

Quantum and classical steganography in optical systems

PDF

Molecular simulations of water and monovalent ion dynamics in the electroporation of phospholipid bilayers

PDF

Entanglement parity effects in quantum spin chains

PDF

Dynamical error suppression for quantum information

Asset Metadata

Creator Sifain, Andrew Edwar (author)

Core Title Photoexcitation and nonradiative relaxation in molecular systems: methodology, optical properties, and dynamics

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 02/21/2019

Defense Date 11/30/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag computational chemistry,excited state chemistry,machine learning,molecular dynamics,nonadiabatic dynamics,OAI-PMH Harvest,optical properties,quantum chemistry,surface hopping

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Prezhdo, Oleg (committee chair), Haas, Stephan (committee member), Kresin, Vitaly (committee member), Nakano, Aiichiro (committee member), Wittig, Curt (committee member)

Creator Email aesifain@gmail.com,sifain@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-126992

Unique identifier UC11675649

Identifier etd-SifainAndr-7107.pdf (filename),usctheses-c89-126992 (legacy record id)

Legacy Identifier etd-SifainAndr-7107.pdf

Dmrecord 126992

Document Type Dissertation

Format application/pdf (imt)

Rights Sifain, Andrew Edwar

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA