University of Southern California Dissertations and Theses

Quantum mechanical description of electronic and vibrational degrees of freedom in laser-coolable molecules

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Quantum mechanical description of electronic and vibrational degrees of freedom in laser-coolable molecules

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Content QUANTUM MECHANICAL DESCRIPTION OF ELECTRONIC AND VIBRATIONAL DEGREES
OF FREEDOM IN LASER-COOLABLE MOLECULES
by
Paweł Wójcik
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
May 2025
Copyright 2025 Paweł Wójcik

Dedication
To my parents.
ii

Acknowledgements
I would like to thank my advisor Dr. Krylov for giving me many opportunities through the PhD program,
from working with my scientific idols and meeting them in person to having a chance to attend scientific
conferences and even granting me an administrator access to the world class computing resources.
I would like to thank my scientific collaborators who have taught me the nuts and bolts of computational chemistry programs, especially Dr. Maxim Ivanov and Dr. John Stanton. I have benefited greatly
from interactions with the members of the NSF Center for Advanced Molecular Architectures for Quantum Information Science, including Dr. Anastassia Alexandrova, Dr. Eric Hudson, Dr. Wesley Campbell,
Dr. Miguel García-Garibay, and Dr. Justin Caram. I would also like to acknowledge students and postdocs
from this center who worked with me on closely related topics: Dr. Claire Dickerson, Dr. Guo-Zhu Zhu,
Dr. Guanming Lao, Dr. Haowen Zhou, Taras Khvorost, Cecilia Chang, Mia Calvillo, Antonio Macias.
I would like to thank group members for their support and for many positive and productive interactions: Dr. Goran Giudetti, Dr. Sourav Dey, Dr. Sarai Dery Folkestad, Dr. Ronit Sarangi, Dr. Saikiran
Kotaru, Dr. Kaushik Nanda, Dr. Sven Kahler, Dr. Maristella Alessio, Dr. Tirthendu Sen, Dr. Sahil Gulania,
Dr. Wojciech Skomorowski, Dr. Florian Hampe, Dr. Pavel Pokhilko, Dr. Yongbin Kim, Dr. Tingting Zhao,
Kyle Tanovitz, Jia Hao Soh, George Baffour Pipim, Madhubani Mukherjee, Nayanthara Karippara Jayadev,
and Dr. Arnab Chakraborty.
I would like to thank the USC faculty who have been my teachers and some were also my screening,
qualification and thesis commitee memebrs: Dr. Alexander Benderskii, Dr. Rosa Di Felice, Dr. Oleg
iii

Prezhdo, Dr. Brent Melot, Dr. Daniel Lidar, Dr. Hanna Reisler, Dr. Susumu Takahashi, Dr. Andrey Vilesov,
Dr. Curt Wittig. I would also like to thank USC staff and teaching facult who I have worked with Michele
Dea, Claudia Cortez, Magnolia Benitez, Cesar Sul, Joel Sanchez, Thuc Do, Dr. Catherine Skibo, Dr. Jessica
Parr, Alexis Arellanes, and Melinda Ballengee.
I would also like to thank my previous academic advisors who have supported my application to the
PhD program: Dr. Michał Tomza, Dr. Ignacio Franco, and Dr. Maciej Lewenstein. Finally, I would like to
thank my friends, family and my girlfriend for their support.
iv

Table of Contents
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 2: Towards Ultracold Organic Chemistry: Prospects of Laser Cooling Large Organic
Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 3: On the prospects of optical cycling in diatomic cations: Effects of transition metals,
spin-orbit couplings, and multiple bonds . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Theoretical methods and computational details . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.1 Y and Sc series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3.2 Other groups: Ti or Zr with chalcogens, and V or Nb with N or P . . . . . . . . . . 29
3.3.3 FCFs, electronic charge displacement, and Dyson orbitals . . . . . . . . . . . . . . 30
3.3.4 Best candidates for optical cycling . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Chapter 4: Dual Optical Cycling Centers Mounted on an Organic Scaffold: New Insights From
Quantum Chemistry Calculations and Symmetry Analysis . . . . . . . . . . . . . . . . 35
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Chapter 5: Vibronic Coupling Effects in the Photoelectron Spectrum of Ozone: A Coupled-Cluster
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Theoretical models and computational details . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
v

5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Chapter 6: Photoswitching Molecules Functionalized with Optical Cycling Centers Provide a
Novel Platform for Studying Chemical Transformations in Ultracold Molecules . . . . 68
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Theoretical Methods and Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2.1 Computational protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2.2 Wavefunction analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.1 Electronic structure of AB and CaOH . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.3.2 Electronic states of pAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.3.3 OCC as a spectroscopic probe to monitor isomerization . . . . . . . . . . . . . . . 82
6.3.4 Synthetic Efforts: Preliminary results and future work . . . . . . . . . . . . . . . . 85
6.4 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Chapter 7: Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.1 Novel organic ligands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7.2 Functional ligand of charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Towards Ultracold Organic Chemistry: Prospects of Laser Cooling Large Organic Molecules . . 116
A.1 Theoretical Methods and Computational Details . . . . . . . . . . . . . . . . . . . . . . . . 116
A.2 Results of EOM-CC calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.3 Relevant Cartesian geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
On the prospects of optical cycling in diatomic cations: Effects of transition metals, spin-orbit
couplings, and multiple bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.1 FCFs for the 2
2Π1/2 → 1
2Σ1/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.2 Spectra of singly bonded cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.3 FCFs for the 1
2Σ1/2 → 1
2∆3/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.4 FCFs for the 1
2Π1/2 → 1
2∆3/2 transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.5 Spectra of multiply bonded cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
B.6 Excitation energy with number of bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Dual Optical Cycling Centers Mounted on an Organic Scaffold: New Insights From Quantum
Chemistry Calculations and Symmetry Analysis . . . . . . . . . . . . . . . . . . . . . . . . 137
C.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.2 EOM-DEA-CCSD results for triplet and singlet states . . . . . . . . . . . . . . . . . . . . . 137
C.3 TD-DFT versus EOM-DEA-CCSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.4 (1) and (2) functionalized in ortho position . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.5 Spin-orbit couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Appendix D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Vibronic Coupling Effects in the Photoelectron Spectrum of Ozone: A Coupled-Cluster
Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
vi

D.1 Parameters for the KDC Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
D.2 Relevant Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
D.3 PES of the Ozone Cation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Appendix E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Photoswitching Molecules Functionalized with Optical Cycling Centers Provide a Novel
Platform for Studying Chemical Transformations in Ultracold Molecules . . . . . . . . . . 161
E.1 DFT benchmarking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
E.2 Effect of the isomerization on the X→B transition . . . . . . . . . . . . . . . . . . . . . . . 161
E.3 Franck–Condon factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
E.4 Relevant Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
vii

List of Tables
2.1 Excitation energies (in eV) and oscillator strengths (in parenthesis) calculated using
EOM-EA-CCSD. Molecules are arranged from more to less symmetric. . . . . . . . . . . . 10
3.1 Basis sets used in this studya
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Vertical excitation energies (eV) for ScX+ and YX+ cations at the ground-state equilibrium
geometry. Transition dipole moments µ (a.u.) correspond to the transition involving the
ground state. EOM-EA-CCSD without the inclusion of SOC shifts. . . . . . . . . . . . . . 26
3.3 FCFs for the best candidatesa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Properties of four lowest excited statesa of (1) and (2) in the triplet manifold; the lowest
triplet state is 3B1 and 3B3u, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Excitation energies (eV), oscillator strengths (in parentheses), and FCFs for the 0→0
transition for low-lying excited states in the triplet manifold for (3) - (11)a
. The symmetry
of the lowest triplet state for each system is given next to the system number. . . . . . . . 45
4.3 Properties of four lowest excited states of (12) and (13) in the triplet manifolda
; the lowest
triplet state is 1
3A
′
and 13A1, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 Vertical ionization energies with the error estimates, eV. . . . . . . . . . . . . . . . . . . . 60
5.2 The vertical energy gap between the 2A1 and 2B2 states with error estimates, meV. . . . . 60
5.3 Assignment and decomposition of the eigenvectors of the ozone cation. The first column
shows the offset of simulated peaks from the origin in cm−1
. Note that the decomposition
uses the basis of uncoupled states visible in Fig. 5.7, as such the eigenvector decomposition
does not list components along vibronic states of a mixed symmetry, e.g., A1(001) which
might, artificially, appear as a lack of mixing. . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.1 Excitation energies, eV, and oscillator strengths (in parentheses) of the trans-pAB, cis-pAB,
and CaOH calculated with EOM-EA-CCSD/aug-cc-pVDZ at the EOM-EA-CCSD/cc-pVDZ
geometriesa
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
viii

6.2 Changes in excitation energy (eV) for the X → A and X → B transitions upon
isomerization in bare pAB and bpAB with an interacting group at both ends; EOM-EACCSD/aug-cc-pVDZ (pAB) and EOM-DEA-CCSD/aug-cc-pVDZ (bpAB)a,b
. . . . . . . . . . 85
A.1 FCFs for the decay transitions to the ground X2A1 state in CaBz, CaPh, and CaPy. ν1 is
Ca-ring stretching mode, ν2 is Ca-ring bending mode. . . . . . . . . . . . . . . . . . . . . 119
A.2 FCFs for the decay transitions to the ground state in CaCp and iso-CaPy. ν1 is Ca-ring
stretching mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.3 Vertical excitation energies (Eex, eV), oscillator strengths (fl
), Head-Gordon’s indices (nu,
nu,nl), and the number of entangled states (ZHE) in the three isomers of CaBzCa calculated
using EOM-DEA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]. . . . . . . . . . . . . . . 120
A.4 Vertical excitation energies (Eex, eV), oscillator strengths (fl
), Head-Gordon’s indices (nu,
nu,nl), and the number of entangled states (ZHE) in the three isomers of CaBzCa calculated
using EOM-DEA-CCSD/aug-cc-pwCVTZ-PP[Ca]/cc-pVTZ[H,C]. . . . . . . . . . . . . . . . 121
B.1 Y cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
B.2 SOC-corrected ground state equilibrium geometries. . . . . . . . . . . . . . . . . . . . . . . 127
B.3 The SOC-corrected states of the scandium cations. The vertical excitation energies E
(eV) calculated at the internuclear distance r (Å), zero on the energy scale corresponds to
the ground-state energy of the non-relativistic states. The transition dipole moments µ
(a.u.) between one of the ground level components (the state at the top of the table). The
transition dipole moment of the state at the top of the table is its SOC-corrected dipole
moment. Radiative lifetime τ is the inverse of the sum of the Einstein coefficients, where
the sum goes over all states of lower energy, see Eqs. (5) and (6) from the main text. . . . 128
B.4 The SOC-corrected states of yttrium cations. See the caption of Table B.3. . . . . . . . . . 128
B.5 Sc cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.6 Y cations. FCFs are listed for the 1
2∆3/2 → 1
2Σ1/2 transition as for the Y cations 1
2Σ is
the ground state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.7 Ti cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
B.8 Zr cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.9 V cation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.10 Nb cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
B.11 Sc cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.12 Y cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
ix

B.13 Ti cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B.14 Zr cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.15 V cation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.16 Nb cations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
B.17 SOC-corrected ground state equilibrium geometries. . . . . . . . . . . . . . . . . . . . . . . 133
B.18 The SOC-corrected states of titanium cations. See the caption of Table B.3. . . . . . . . . . 133
B.19 The SOC-corrected states of zirconium cations. See the caption of Table B.3. . . . . . . . . 134
B.20 The SOC-corrected states of vanadium and niobium cations. See the caption of Table B.3. 134
C.1 Vertical excitation energies (Eex, eV), oscillator strengths (fl
) and Head-Gordon’s indices
(nu,nl) in molecule (1); EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca]
ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C.2 Vertical excitation energies (Eex, eV), oscillator strengths (fl
) and Head-Gordon’s indices
(nu,nl) in molecule (2); EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca]
ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C.3 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (3); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 138
C.4 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (4); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 138
C.5 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (5); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 138
C.6 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (6); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 138
C.7 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (7); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 139
C.8 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (8); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 139
C.9 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (9); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 139
C.10 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (10); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 139
x

C.11 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (11); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. . . . . . . . . . . . . . . . . . 139
C.12 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (12); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca,Sr] ECP10MDF[Ca] ECP28MDF[Sr]. . . . . . . . 140
C.13 Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (13); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca,Sr] ECP10MDF[Ca] ECP28MDF[Sr]. . . . . . . . 140
C.14 Excitation energies (Eex, eV) and oscillator strengths (fl
) in the triplet manifolds of
molecules (1) and (2) calculated using PBE0-D3/def2-TZVPPD and EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. The label of the lowest triplet
state is listed next to the molecule’s number. . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.15 Comparison of triplet states properties of in molecules (3)-(11) calculated with PBE0-
D3/def2-TZVPPD and EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca]
ECP10MDF[Ca]. Label listed next to the molecule’s number corresponds to the lowest
triplet state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.16 Continuation of Table C.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.17 Frank–Condon factors for the 0 → 0 transition for different excited states for systems (1)
and (2) functionalized with CH3. The oscillator strengths are given in parentheses. . . . . 147
C.18 SOC-corrected states for (1); energies in cm−1
. . . . . . . . . . . . . . . . . . . . . . . . . . 154
C.19 SOC-corrected states for (2); energies in cm−1
. . . . . . . . . . . . . . . . . . . . . . . . . . 155
C.20 SOC-corrected states for (12); energies in cm−1
. . . . . . . . . . . . . . . . . . . . . . . . . 156
C.21 SOC-corrected states for (13); energies in cm−1
. . . . . . . . . . . . . . . . . . . . . . . . . 157
E.1 Screening of selected DFT functionals with def2-TZVPP basis set. Excitation energy (in
eV) and oscillator strengths for the first four excited are reported. State characterization
is omitted for non-physical states. The reference EOM-CC calculation used aug-cc-pVDZ
basis set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
E.2 FCFs for tpAB A→X fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
E.3 FCFs for cpAB A→X fluorescence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
xi

List of Figures
1 Covers of The Journal of Physical Chemistry A issues highlighting publications 4. and 5. . 1
1.1 Progress in laser cooling of molecules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Proposed laser-coolable candidates with aromatic ligands . . . . . . . . . . . . . . . . . . . 8
2.2 A. Natural transition orbitals for the X2Σ
+ → A2Π and X2Σ
+ → B2Σ
+ transitions in
CaOH. B. Optical cycling schemes with indicated FCFs for each vibronic transition. FCFs
were computed using the experimental values of the vibrational frequencies. . . . . . . . 8
2.3 Dyson orbitals (A) and NTOs (B) of CaBz and CaCp plotted with isovalue = 0.03. . . . . . . 10
2.4 A. Illustration of the normal modes of CaBz with the largest displacements. B. Optical
cycling scheme for the A2B1 → X2A1 and C
2A1 → X2A1 transitions in CaBz with
indicated FCFs for each vibronic transition. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Ca-ring stretching modes of CaCp and iso-CaPy. . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 Proposed optical cycling schemes for the A2E1 → X2A1 and B2A1 → X2A1 transitions
in CaCp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Examples of functionalization in CaBz and CaCp (upper panel) and molecules with
multiple cycling centers (lower panel). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.8 Excited states in ortho-, meta-, and para-CaBzCa calculated using EOM-DEA-CCSD/augcc-pwCVTZ-PP[Ca]/cc-pVTZ[C,H]. Insert shows natural orbitals (isovalue=0.02) of the
ground state in para-CaBzCa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
xii

3.1 Electronic design principle of OCCs in diatomic cations investigated in this study.
Electronic configurations of radical-cations ML+ mimic the electronic structure of the
successfully laser-cooled neutral molecules ML. In these neutral species, an alkali-earth
metal forms an ionic bond with a ligand L (which needs one extra electron to form a
closed shell), creating a +1 local charge and a single valence electron localized on the
metal. This pattern results in electronic transitions localized on the metal[11]. The top
row shows the electronic configurations of the elements in each column. In the bottom
row the full and empty dots depict valence electrons and valence holes (electrons needed
to complete the shell). Color highlights elements studied in this work. . . . . . . . . . . . 21
3.2 Dyson orbitals showing the unpaired electron in low-lying electronic states. Representative examples of YCl+(left) and NbN+(right) (isovalue = 0.02; plotted with iQmol[112]).
In the hydrogen-like orbital labels, the principal quantum number n equals 4 for the Sc,
Ti, and V metal cations and 5 for the Y, Zr, and Nb cations. . . . . . . . . . . . . . . . . . . 25
3.3 Scandium and yttrium halide cations with low-lying localized doublet states. Vertical
excitation energies (EOM-EA-CCSD energies without the inclusion of SOC shifts) at the
ground-state minimum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4 YF+ is the most diagonal cation for the 2
2Π → 1
2Σ transition. Numbers on the arrows
indicate branching ratios. The four decay channels generate population leakage out of the
cycling transition which is undesired in an OCC. . . . . . . . . . . . . . . . . . . . . . . . 28
3.5 Multiply bonded cations with low-lying localized doublet states. Vertical excitation
energies (EOM-EA-CCSD energies without the inclusion of SOC shifts) at the groundstate equilibrium geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.6 Three types of optical cycling schemes exemplified by the best candidate cations. In the
group with the 1
2Σ ground state, YCl+ shows the most diagonal transition. In the group
with the 1
2∆ ground state and highly diagonal transitions, NbN+ stands out with the
shortest radiative lifetime. In the third group, the FCFs for the formally dipole-allowed
transition between the ground and the second excited state favor scandium and zirconium
cations. Large decay to the intermediate state requires also the side branch 1
2Π1/2 → 1
2Σ
transition to show quickly saturating FCFs. This second requirement renders the ZrO+
cation as the best candidate in this group. . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Diradicals: Different electronic configurations derived from distributing two electrons in
two (orthogonal) orbitals and their characters. M1 and M2 denote two radical centers,
e.g., the two metal atoms in bi-OCCs. Configuration (i) is the high-spin (Ms=1) triplet state
and configurations (ii)-(v) are low-spin (Ms=0) states. Case 1 corresponds to perfectly
delocalized frontier orbitals (such as bonding and antibonding combinations of atomic
orbitals) and case 2 corresponds to the molecular orbitals perfectly localized on the radical
centers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Structures of bi-OCC molecules considered in this work. . . . . . . . . . . . . . . . . . . . 40
xiii

4.3 Frontier orbitals and their symmetries for the two simplest bi-OCC molecules — CaO-Ph-3-O-Ca with C2v point group (1) and Ca-O-Ph-4-O-Ca with D2h point group
(2). For both molecules two lowest-lying orbitals have s-type character, while other
orbitals have p-like character. Note: Symmetry labels correspond to Q-Chem’s standard
molecular orientation[161]. EOM-DEA-CCSD/cc-pVDZ[H,C,O]/aug-cc-pwCVDZPP[Ca]/ECP10MDF[Ca]; isovalue 0.03. The respective occupations are given in the SI.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4 Natural orbitals for the lowest triplet states of a mixed bi-OCC CaO-Ph-3-OSr (12) and
CaO-Ph-4-OSr (13). EOM-DEA-CCSD in a composite basis set cc-pVDZ[H,C,O]/aug-ccpwCVDZ-PP[Ca,Sr]/ECP10MDF[Ca]/ECP28MDF[Sr]; isovalue 0.03. . . . . . . . . . . . . . 49
5.1 Two-dimensional slice of the potential energy surface of ozone in hyperspherical
coordinates at the value of hyperradiuns ρ=4 a.u. (the lowest contour is at 1 eV below
the dissociation limit). The slice shows three equivalent minima corresponding to C2v
structures separated by large barriers. Reproduced with permission from Ref. 175.
Copyright 2003 American Institute of Physics. . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 Frontier molecular orbitals of ozone; HF/6-31G*. The lowest states of the ozone cation are
2A1 and 2B2 derived by ionization from the HOMO-1 and HOMO-2, respectively. . . . . . 54
5.3 Three normal modes of the neutral ozone CCSDT/ANO1. Asymmetric stretching vibration
of b2 symmetry can couple the two lowest states of the cation (2A1 and 2B2). . . . . . . . 56
5.4 Cuts of potential energy surfaces of the two diabatic states of the ozone cation
(corresponding to the 2A1 and 2B2 states at the equilibrium geometry of the neutral)
shown as function of the two symmetric modes in dimensionless normal coordinates, Q1
and Q2, at Q3=0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Comparison of the experimental (solid black line digitized from Ref. 195) and the simulated
photoelectron spectra shown as electron binding energies. The simulated spectrum is
blue-shifted by 21 meV and the peaks were broadened using γ = 30 meV. Reproduced with
permission from Ref. 195. Copyright 1974 Royal Society of Chemistry. . . . . . . . . . . . 60
5.6 Simulated photoelectron spectrum of ozone shown as electron binding energies. Bottom
axis shows energy scale in eV. Top axis shows energy offset from the origin in cm−1
. Stick
spectrum shows positions and intensities of all simulated states. Blue and orange colors
mark states of A1 and B2 symmetry, respectively. Gray vertical lines with captions on top
indicate positions of features measured by the PFI-ZEKE experiment. [196] The simulated
spectrum was shifted to match the PFI-ZEKE experimental origin by 21 meV. D0 marks
the dissociation threshold of O+
3
. [196] Gray dotted line marks the energy of the minimum
of the conical intersection (CI) as located by our Hamiltonian. The right panel shows the
uncoupled spectrum (the same as in Fig. 5.7 but formatted in a matching manner). . . . . 62
5.7 Simulated Franck–Condon photoelectron spectrum of ozone with vibronic coupling
removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
xiv

5.8 Comparison of the simulated spectrum with the PFI-ZEKE experiment.[196] The simulated
spectrum was blue-shifted by 21 meV = 170 cm−1
. Color of the simulated peaks marks
the symmetry of the vibronic states: A1 are colored blue and B2 are colored orange. See
Table 5.3 for a detailed listing of state assignments. The bottom panel reproduced with
permission from Ref. 196. Copyright 2005 American Institute of Physics. . . . . . . . . . . 64
5.9 Comparison of the simulated spectrum to an earlier simulation by Tarroni and Carter
(assigned lines). [214] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1 Molecules studied in this work. Counterclockwise from the left top: CaOH, azobenzene
(AB), para-CaO-azobenzene (pAB), and bis-(para-CaO)-azobenzene (bpAB). . . . . . . . . 69
6.2 EOM-CC models used in this work. Color illustrates the action of the EOM operator
(only single excitations are shown). An EOM method is defined by specifying the
reference determinant and the type of generalized excitation operator—e.g., spin and
electron-conserving operators are used in EOM-EE, spin-flipping operators are used in in
EOM-SF, and non-particle conserving operators are used in EOM-EA and EOM-DEA. . . . 71
6.3 NTOs of the relevant electronic transitions in the trans-AB (left) and CaOH (right). . . . . 76
6.4 Electronic states of trans-pAB (central panel; EOM-EA-CCSD/aug-cc-pVDZ at the
EOM-EA-CCSD/cc-pVDZ geometry) as a combination of trans-AB states (the column
on the left; EOM-EE-CCSD/aug-cc-pVDZ at the CCSD/cc-pVDZ geometry), and the
OCC states (CaOH; two columns on the right; EOM-EA-CCSD/aug-cc-pVDZ at the
EOM-EA-CCSD/cc-pVDZ geometry). Energies of the respective ground states are set to
zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.5 Trans-pAB at the EOM-EA-CCSD/cc-pVDZ geometry. Comparison of the EOM-EACCSD/aug-cc-pVDZ states (two columns on the left) with the states computed with
EOM-EE-CCSD/aug-cc-pVDZ (three columns on the right). The rightmost column shows
the EOM-EE-CCSD states absent in the EOM-EA-CCSD calculations; some of these extra
states are spin-contaminated (red color); the corresponding values of

S
2

are given in
the parentheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.6 Electronic states of trans-pAB and the character of the underlying wavefunctions. Top:
state energies (relative to the ground state). Bottom: CTNs characterizing the transitions
from the ground state. The states are ordered from left to right to match three groups: first
are the OCC-like states (labeled s, p, and d), second are the states of a mixed character
(mix), and the last are the AB-like states (labeled 2S+1AB). Compare with Fig. 6.5.
EOM-EE-CCSD/aug-cc-pVDZ at the EOM-EA-CCSD/cc-pVDZ ground state’s geometry. . 80
6.7 Particle and hole NTOs for the d(Ca) (doughnut) state of trans-pAB. EOM-EE-CCSD/augcc-pVDZ at the EOM-EA-CCSD/cc-pVDZ geometry of the ground state. . . . . . . . . . . 83
6.8 The azobenzene molecule functionalized with two OCCs, bpAB. The isomerization reduces
the distance between the two Ca atoms from 16.3 Å to 10.8 Å. Geometries optimized with
EOM-SF-CCSD/cc-pVDZ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
xv

6.9 (a) Synthetic routes for asymmetric azobenzene derivatives using Mills reaction (left) and
diazotization method (right). (b) Synthetic routes for symmetric azobenzene derivatives. . 85
7.1 Franck-Condon factor tuning in intermediate-sized molecules. Functionalization of the
phenyl group imporves Franck-Condon factors, impact of functionalization on smaller
groups (illustrated on the example of the vinyl group) has not been studied yet. . . . . . . 89
7.2 A functional group of charge, examples of -NH+
3
and -TeF+
2
. . . . . . . . . . . . . . . . . . 91
A.1 Structures and equilibrium bond lengths of the candidate molecules in different electronic
states (color-coordinated as indicated). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
A.2 Frontier natural orbitals (isovalue = 0.02) of the ground state of ortho-, meta-, and
para-CaBzCa (top to bottom) calculated using EOM-DEA-CCSD/aug-cc-pwCVDZPP[Ca]/cc-pVDZ[H,C]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.1 Vertical excitation energies at the ground-state minimum. EOM-EA-CCSD energies
without the inclusion of SOC shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
B.2 Vertical excitation energies at the ground-state minimum. EOM-EA-CCSD energies
without the inclusion of SOC shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
C.1 Frontier NOs of singlet states of (1), CaO-Ph-3-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O]
aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol. . . 141
C.2 Frontier NOs of triplet states of (1), CaO-Ph-3-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O]
aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol. . . 142
C.3 Frontier NOs of singlet states of (2), CaO-Ph-4-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O]
aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol. . . 142
C.4 Frontier NOs of triplet states of (2), CaO-Ph-4-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O]
aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol. . . 143
C.5 Model Hamiltonians. To clearly visualize the values across many orders of magnitude,
the etries are rescaled using the formula log
1 +

z/cm−1

. States 0-3 belong to block X,
states 4-11 belong to block A, and states 12-19 belong to block B. . . . . . . . . . . . . . . 151
D.1 Contour maps of the two diabatic PESs of the ozone cation (see Fig. 4 in the main text).
Q1 and Q2 are two symmetric normal coordinates. . . . . . . . . . . . . . . . . . . . . . . 160
E.1 Natural transition orbitals (with weights ≥0.25) for the X→B transition in cis-bpAB. . . . 162
xv

Abstract
This dissertation presents a study of molecular candidates for direct laser cooling. The main results of this
work are: 1) extending the set of promising candidates for producing cold and ultracold samples of molecular gases, 2) introducing new resources, intrinsic to polyatomic molecules, to the field of ultracold matter,
3) developing rigorous, quantum-mechanical protocols for description of laser-coolable molecules. These
contributions resulted from simulations of vibrationally resolved electronic spectra, which included also
more subtle (but crucial to the experiments) effects such as the spin-orbit and vibronic interactions. The
systems studied in this work range in size from diatomics to polyatomic, organic and aromatic molecules
decorated with a functional group of a bivalent metal atom or, more generally, an optical cycling center
(OCC).
The chapters of this dissertation are based on the following publications:
1. M. V. Ivanov, F. H. Bangerter, P. Wójcik, et al., “Toward ultracold organic chemistry: Prospects of
laser cooling large organic molecules”, The Journal of Physical Chemistry Letters 11, 6670–6676
(2020). (Chapter 2)
2. P. Wójcik, E. R. Hudson, and A. I. Krylov, “On the prospects of optical cycling in diatomic cations:
effects of transition metals, spin–orbit couplings, and multiple bonds”, Molecular Physics, e2107582
(2022). (Chapter 3)
xvii

3. T. Khvorost, P. Wójcik, C. Chang, et al., “Dual optical cycling centers mounted on an organic scaffold:
New insights from quantum chemistry calculations and symmetry analysis”, The Journal of Physical
Chemistry Letters 15, 5665–5673 (2024). (Chapter 4)
4. P. Wójcik, H. Reisler, P. G. Szalay, et al., “Vibronic Coupling Effects in the Photoelectron Spectrum
of Ozone: A Coupled-Cluster Approach”, The Journal of Physical Chemistry A 128, 10957–10964
(2024). (Chapter 5)
5. P. Wójcik, T. Khvorost, G. Lao, et al., “Photoswitching Molecules Functionalized with Optical Cycling
Centers Provide a Novel Platform for Studying Chemical Transformations in Ultracold Molecules”,
The Journal of Physical Chemistry A, acs.jpca.4c06320 (2024). (Chapter 6)
xviii

Figure 1: Covers of The Journal of Physical Chemistry A issues highlighting publications 4. and 5.
1

Chapter 1
Introduction
Preparation of cold molecules has a long history reaching at least back to the XIX century.[1] From the
perspective of this work, the most relevant in this history is the accomplishment of laser cooling and
trapping of atoms; an achievement recognized with a Nobel prize in 1997.[2, 3, 4] Several subsequent
awards are connected to this field as well, two of them especially stand out. The 2001 distinction celebrated
lowering the temperature of laser cooled atoms to the limit where a new quantum phase of matter was
detected.[5, 6] The 2012 prize, appreciated the fact that trapped and laser cooled atoms form a platform
where these delicate, quantum objects can be controlled individually.[7, 8]
Early progress and achievements in the field of laser cooling were focused on atoms, especially on the
alkali metals. Molecules or atoms from the middle of the Periodic Table were not considered at first due to
their more complex internal structure. A stepping stone in the field came from the work of Di Rosa who
noted that a class of diatomic molecules is promising for laser cooling.[9] In 2010 the group of DeMille has
demonstrated a direct laser cooling of SrF.[10] This monumental achievement has disproved the idea that
the direct laser cooling is only possible for atoms.
In the following years more diatomic molecules were laser cooled and the perspective of extending the
method to polyatomic molecules was formulated.[11, 12] Soon after, three-atomic molecules molecules
were laser cooled: SrOH in 2017, and YbOH and CaOH in 2020.[13, 14, 15] In the year 2020 another record
2

2010 2012 2014 2016 2018 2020 2022 2024
Year
0
2
4
6
8
10
12
SrF
YO
CaF
SrOH
YbF
YbOH
CaOH
BaH
CaOCH3
BaF
CaH
CaD
Laser-cooled molecules
diatomics
triatomics
>3 atoms
Figure 1.1: Progress in laser cooling of molecules.
was broken as a polyatomic molecule consisting of six atoms (calcium monomethodixe CaOCH3) was laser
cooled.[16] New polyatomic molecules are laser cooled each year and the field is advancing towards more
and more complex systems.[17, 18, 19, 20, 21, 22, 23] Figure 1.1 displays the growing number of laser-cooled
molecules over the years.
The experimental advances in the field of laser cooling of molecules were mirrored by an extensive
computational modeling of such systems, with the goal to understand their electronic structure, to predict
likelihood of laser cooling feasibility, and to introduce new guiding principles in the design and optimization of future experiments. This thesis presents results obtained using the tools of computational chemistry
that contribute towards this purpose.
Chapter 2 introduces the key features of molecular structure that make a molecule a good candidate
for laser cooling. The same chapter also installs ideas from organic chemistry to the field of laser cooling of
molecules. Chapter 3 is concerned with the optical cycling in molecular cations; the role of multiple bonds
is highlighted as a resource in the design of optical cycling centers. Chapter 4 considers an organic scaffold
of polyatomic molecules as a hosts of two optical cycling centers, and demonstrates how the freedom in
3

control over relative orientation of the OCCs translates to an ability for chemical enhancement of properties desired for laser cooling. Chapter 5 studies a system exhibiting strong virbonic couplings, which must
be taken into account account for accurate prediction of the laser cooling feasibility. Chapter 6 marries
the ideas of molecular photo-switching and optical cycling and illustrates how chemical complexity can
be viewed as a new resource in the field of laser cooled molecules. Finally, Chapter 7 presents proposals
for future research directions building on the results of this work.
4

Chapter 2
Towards Ultracold Organic Chemistry: Prospects of Laser Cooling Large
Organic Molecules
Ultracold molecules provide exciting opportunities to probe chemical reactivity with reagents prepared in
a specific quantum state, including vibrational, rotational, spin, and hyperfine levels [24, 25, 26, 27]. The
pioneering studies of ultracold chemistry included collisions between atoms, atom/diatomic molecule, and
a pair of diatomic molecules. By providing means to study reaction dynamics and mechanism in quantum
regime, emerging techniques in molecular physics now open access to more complex chemical systems.
Representative examples include a study of the reaction between the conformers of 3-aminophenol and
laser-cooled Ca+,[28] quantum state-controlled synthesis of CaOBa+,[29] production of the cold beams
of YbOH via excited-state chemistry,[30] quantum state-resolved collisions between OH and NO, between
N
+
2
and Rb, and between KRb molecules.[31, 32, 33, 34] Access to cold and ultracold molecules is also a
prerequisite for precision measurements and quantum information storage and simulation.[35, 36]
Until only very recently, laser-cooling was limited to atoms and a handful of diatomic molecules [37,
10, 38, 17, 18, 39, 19, 20, 40]. Recent breakthroughs by Doyle’s group, who laser-cooled a series of a
polyatomic molecules—from CaOH,[15] SrOH[13], and YbOH[14] to much more complex CaOCH3[16]—
inspire an optimism that polyatomic molecules with virtually any degree of complexity can be laser-cooled.
5

The main challenge in laser-cooling of large polyatomic molecules stems from multiple degrees of
freedom and complex electronic structure. Laser cooling is achieved by a continuous scattering of a large
number (>104
) of photons off the target, with each cycle of absorption and emission slowing down its
translational motion. In a simple diatomic molecule with atom-like electronic structure, such as CaF, the
spontaneous emission returns the excited molecule either to the original ground state or first few excited
vibrational levels, thus providing efficient means of laser cooling, with an aid of a few additional repump
lasers. In contrast, emission in a large polyatomic molecule often populates numerous vibrational levels,
thus requiring an unmanageably large number of repump lasers. It has been estimated that in order to
bring at least 10% of molecules to rest by scattering 4,500 photons, at least 99.95% of the emission has to be
recovered by the lasers.[9] That is why viable polyatomic candidates should afford electronic transitions
with minimal structural relaxation, i.e., with diagonal Franck-Condon factors (FCFs)[9, 11, 12], implying
similar equilibrium geometries and normal mode frequencies of the two electronic states involved in the
cycling. The rotational degrees of freedom need to be cooled too, which requires working with carefully
chosen rotationally closed transitions, as has been recently illustrated for CaOCH3[41, 16].
One successful strategy of designing a laser-coolable molecule is to attach an alkaline earth metal (such
as calcium or strontium) to a ligand that withdraws one of the two valence electrons of the metal creating a
strong ionic bond.[42, 43, 12, 44, 41] The second electron remains localized at the metal, both in the ground
and in the excited states, giving rise to highly diagonal FCFs and thereby closed cycling transitions, if an
appropriate number of repump lasers is introduced. The success in exploiting highly localized excitation
for optical cycling in triatomic molecules (CaOH, SrOH, and YbOH) spurred explorations of optical cycling
schemes in more complex polyatomic molecules [11, 12, 45, 46, 41, 47]. In CaOH (and similar molecules),
the localization of the unpaired electron is facilitated by the strongly ionic Ca-O bond, which effectively
blocks electronic communication between Ca+ and the rest of the molecule. Thus, various saturated and
6

unsaturated hydrocarbon groups can follow O atom without significantly deteriorating the FCFs.[12] Recent laser cooling of CaOCH3 proves this concept, suggesting that optical cycling and laser cooling of large
polyatomic molecules can be as efficient as in simpler linear species.[16]
Prospects of optical cycling with polyatomic molecules are particularly promising for building novel
quantum systems [48]. It also paves the way towards (ultra)cold organic chemistry. Because unsaturated
hydrocarbons, and in particular aromatic compounds, are at the heart of chemistry, the ability to laser-cool
molecules containing aromatic groups would enable studying many important reaction mechanisms in
quantum regime. However, considering delocalized nature of the spin/charge distribution in the aromatic
systems,[49, 50] it is unclear whether the FCFs remain diagonal when alkaline earth metal is attached to
an aromatic ligand.
Following the above design principles, one can consider monovalent alkaline earth metal derivatives
of such aromatic molecules as benzene (Bz) and phenol (Ph) as candidates for laser cooling (Figure 2.1).
One can also exploit (hetero)cyclic π-conjugated ligands that become aromatic according to Hückel’s 4n+2
rule, once the electron is transferred from alkaline earth metal to the ligand. Representative examples of
the aromatic ligands include cyclopentadienyl (C5H
−
5
or Cp) and pyrrolide (C4H4N
− or Py) anions (Figure
2.1). By using high-level electronic structure calculations, here we demonstrate that calcium derivatives
with various aromatic ligands (Figure 2.1) appear to be promising candidates for laser cooling, despite their
structural complexity. All electronic structure calculations were performed using the Q-Chem package[51,
52]; the wavefunction analysis was carried out using libwfa library[53]. The computational details are
described in the Appendix A.1.
We begin with the analysis of the electronic structure of CaOH—an alkaline earth metal derivative that
has been recently laser-cooled.[15] One of the two valence s electrons of the metal is transferred to the
ligand, yielding ionically bound Ca+ and OH− ions, with Mulliken’s charges on Ca and O equal to 0.76
and -1.00, respectively. The interaction between two ions results in the orbital hybridization on the metal
7

Ca
Ca N
Ca
N
Ca
Ca O
CaBz
CaPh
CaCp CaPy
iso-CaPy
Figure 2.1: Proposed laser-coolable candidates with aromatic ligands
atom, which minimizes the repulsion between the unpaired electron and a negatively charged ligand. In
the ground X2Σ
+ state, the unpaired electron is localized at the metal and occupies the sσ−pσ hybridized
orbital. The transitions from the X2Σ
+ state to the A2Π and B2Σ
+ states can be described as atom-like
transitions to pπ − dπ and pσ − dσ hybridized orbitals, respectively (Figure 2.2A). Accordingly, upon
excitation the molecule remains linear, with the most significant structural change being the contraction
of the Ca-O bond. Relative to the ground state, Ca-O bond shortens by 0.026 Å in the A2Π state and by
0.024 Å in the B2Σ
+ state.
Ca-O
stretching
Ca-O-H
bending
Ca-O
stretching
Ca-O-H
0.9287 bending 2.03 eV
0.0696 0.0002
0.0015
0.9371 2.34 eV
0.0615 0.0002
0.0011
A
B
Figure 2.2: A. Natural transition orbitals for the X2Σ
+ → A2Π and X2Σ
+ → B2Σ
+ transitions in CaOH.
B. Optical cycling schemes with indicated FCFs for each vibronic transition. FCFs were computed using
the experimental values of the vibrational frequencies.
8

Our previous studies[44, 46] of alkaline earth derivatives showed that the equation-of-motion coupledcluster method with single and double excitations (EOM-CCSD)[54] yields accurate excitation energies
(within 0.1 eV), bond-length changes (within 0.006 Å), and normal mode frequencies. Relative errors in
the FCFs for the 0 → 0 transition calculated within double-harmonic parallel-mode approximation are in
the range of 2-3%.
In CaOH, our calculations yield the FCFs for the A2Π(ν = 0) → X2Σ
+(ν = 0) and B2Σ
+(ν = 0) →
X2Σ
+(ν = 0) transitions equal 0.9287 and 0.9371, respectively, in good agreement with the experimentally derived values. Decays into the stretching Ca-O and bending Ca-O-H modes are also present and
are not negligible (Figure 2.2B). Although there is no structural relaxation along the bending coordinate,
the bending mode is Franck-Condon active because of a noticeable change in the vibrartional frequency
between the ground and excited states, causing the decay into the bending mode; this relaxation channel
may be further enhanced by anharmonicities.
Our calculations show that this electronic structure is preserved when Ca is attached directly to the
ligand, in an in-plane configuration, for all three considered ligands (CaBz, CaPh, and CaPy). In particular,
the pattern in the electronic excitation spectrum is quite similar to that in CaOH (Table 2.1), subject to the
following differences. In contrast to linear CaOH, in which the first excited state is a doubly degenerate
A2Π state, in CaBz, CaPh, and CaPy, the Π state is split into a B1/B2 pair (separated by 0.02-0.05 eV) due
to lower symmetry (C2v).
Figure 2.3A shows Dyson orbitals of four lowest electronic states in CaBz as a representative example of
this series. The unpaired electron is mostly localized at the metal and occupies hybridized orbitals that are
visually similar to those in CaOH and other alkaline-earth metal derivatives. This is especially surprising,
given that the Ca-C bond in CaBz is less ionic than the Ca-O bond in CaOH—Mulliken’s charges on Ca and
C are 0.47 and -0.42, respectively. Perhaps, the requirement of a strongly ionic bond is less stringent than
previously thought and molecules with a covalent bonding may be laser-coolable as well. This observation
9

Table 2.1: Excitation energies (in eV) and oscillator strengths (in parenthesis) calculated using EOM-EACCSD. Molecules are arranged from more to less symmetric.
CaOH A2Π B2Σ
+
2.03 (0.26) 2.34 (0.20)
CaCp A2E1 B2A1
1.92 (0.19) 2.29 (0.16)
A2B1 B2B2 C
2A1
CaBz 1.90 (0.21) 1.95 (0.21) 2.21 (0.17)
CaPh 2.07 (0.24) 2.09 (0.25) 2.34 (0.20)
CaPy 2.04 (0.23) 2.09 (0.23) 2.33 (0.19)
iso-CaPy A2A
′′ B2A
′ C
2A
′
1.92 (0.19) 1.96 (0.18) 2.28 (0.13)
is consistent with a recent demonstration of laser-cooling of BaH, a molecule with a much less ionic bond
than other coolable diatomics (i.e., CaF and SrF).[20]
CaBz
CaCp
CaBz
CaCp
A
B
Figure 2.3: Dyson orbitals (A) and NTOs (B) of CaBz and CaCp plotted with isovalue = 0.03.
Less ionic character of the Ca-C bond leads to a more extended delocalization of the unpaired electron.
As Dyson orbitals in Figure 2.3A show, the unpaired electron distribution spills over the atoms of the ligand
to a slightly larger extent than in CaOH. The excitation remains localized at Ca, with a minor involvement
10

of the directly connected C atom, as revealed by the NTOs in Figure 2.3B. Accordingly, the changes in the
Ca-C bond length are slightly larger than in CaOH: Ca-C bond is elongated by 0.030 Å in the A2B1 state
and by 0.027 Å in the C
2A1 state relative to the ground state. The structural relaxation is limited to the
Ca-C bond: the C-C bond adjacent to Ca-C bond changes by less than 0.004 Å, while the remaining C-C
bonds are completely unaffected by the excitation.
Estimated FCFs in CaBz are not as diagonal as in CaOH and are consistent with the structural relaxation pattern. The FCFs for the A2B1(ν = 0) → X2A1(ν = 0) and C
2A1(ν = 0) → X2A1(ν = 0)
transitions are 0.7595 and 0.8849, respectively. The next largest FCFs are associated with the decays to the
first vibrational level of the Ca-C stretching mode (Figure 2.4A) and are equal to 0.1318 and 0.1055 for the
A2B1 and C
2A1 states, respectively. If both decay channels are addressed by the lasers, a total of 0.8913
and 0.9904 population can be recovered in the A2B1 → X2A1 and C
2A1 → X2A1 cycling transitions,
respectively (Figure 2.4B). More diagonal FCFs for the C
2A1 → X2A1 transition are consistent with our
previous observation of more diagonal FCFs for the corresponding B2Σ
+ → X2Σ
+ transition in the linear
analogues,[44] suggesting that this could be general feature of the alkaline earth metal derivatives.
An additional source of the population decay in the A2B1 → X2A1 transition is due to the Ca-Bz
bending mode because of the frequency change. With an addition of the third laser, one could recover a
total of 0.9552 of the population, which makes this transition viable for optical cycling despite modestly
diagonal FCFs. Structural reorganization in B2B2 is similar to that in A2B1 (Figure A.1), giving rise to
similar FCFs (Table A.1). The other two molecules that we considered, CaPh and CaPy, show similar
patterns in the structural relaxation; their FCFs are given in Table A.1.
We next turn to the molecules with an out-of-plane arrangement of the cycling center (i.e., CaCp and
iso-CaPy), which have been studied both experimentally and theoretically.[55, 56, 57, 58] Experimental
studies have showed that CaCp has a C5v symmetry and features a reach excitation spectrum with extensive vibrational structure due to metal-ring and intra-rings modes, including Jahn-Teller active modes in
11

Ca-Bz
stretching
Ca-Bz
stretching
Ca-Bz
bending
Ca-Bz stretching Ca-Bz bending
0.7595 1.90 eV
0.1318 0.0639
0.0093
0.8849 2.21 eV
0.1055
0.0049
A
B
Figure 2.4: A. Illustration of the normal modes of CaBz with the largest displacements. B. Optical cycling
scheme for the A2B1 → X2A1 and C
2A1 → X2A1 transitions in CaBz with indicated FCFs for each
vibronic transition.
the doubly degenerate A2E1 state.[56] The spectrum of structurally similar iso-CaPy is similar to that of
CaCp, yet it exhibits two closely lying but distinct A and B states due to a lower Cs symmetry. Despite
complex excitation spectrum, the dispersed fluorescence spectra of CaCp and iso-CaPy are quite simple
and feature decays only to the metal-ring stretching mode (Figure 2.5). The small number of decay channels suggests that the structural relaxation in these molecules is small and, therefore, the FCF matrix is
nearly diagonal.
Ca-Cp stretching Ca-Py stretching
Figure 2.5: Ca-ring stretching modes of CaCp and iso-CaPy.
Our calculations confirm that the electronic excitation spectra of CaCp and iso-CaPy are indeed similar
and are consistent with the spectra of other alkaline earth metal derivatives (Table 2.1). In each electronic
12

state, the unpaired electron occupies hybridized atom-like orbitals and electronic transitions are localized
at the metal (see example of CaCp in Figure 2.3). However, the involvement of the ligand is more significant than in CaBz, as the unpaired electron distribution spills over the aromatic moiety (Figure 2.3A).
An increased bonding between the metal and ring translates into a significant structural relaxation in the
excited states: in CaCp the Ca-Cp distance is reduced by 0.070 Å in the A2E1 state and by 0.031 Å in the
B2A1 state, while the intra-ring C-C bond changes are less than 0.003 Å.
Although the FCFs are not as diagonal as they are for the in-plane structure, more than 99% of the population can be recovered with an addition of more repump lasers. In the A2E1 → X2A1 cycling transition,
0.5435 of the emission decays back to the ground level, while 0.4480 of the population is distributed across
three quanta of the Ca-Cp stretching mode (Figure 2.6). Thus, a total of 0.9915 of the population can be
recovered at each cycle with a total of four lasers. In the B2A1 state the magnitudes of the bond length
changes are smaller than in the A2E1 state. In the B2A1 → X2A1 cycling transition a total of 0.9914 of the
population can be recovered with a total of three lasers (Figure 2.6). The proposed optical cycling schemes
in iso-CaPy are quantitatively similar to those in CaCp due to similarities in their electronic structure and
structural relaxation (see Figure A.1 and Table A.2 for details).
Ca-Cp
0.5435 stretching 1.92 eV
0.3286
0.0994
0.0200
Ca-Cp
0.8896 stretching 2.29 eV
0.0965
0.0052
Figure 2.6: Proposed optical cycling schemes for the A2E1 → X2A1 and B2A1 → X2A1 transitions in
CaCp.
As previously shown[44], the electron-withdrawing strength of the ligand impacts the structural and
optical properties of alkaline earth metal derivatives. A judicious choice of the ligand may result in a
13

favorable electronic structure and diagonal FCFs. In this context, an alkaline earth metal connected to
an aromatic ligand offers a convenient molecular framework suitable for further functionalization. For
example, multiple electron-withdrawing ligands of a varied strength can be incorporated into benzene
and cyclopentadienyl moieties (Figure 2.7). A conceptually similar approach has been successfully utilized
in the rational design of aromatic superhalogens[59, 60].
Ca Ca
Ca
F F
F
F F
Ca
Ca
Ca
Ca
para-CaBzCa
CaBzF6 CaCpF6
meta-CaBzCa ortho-CaBzCa
Ca
F
F
F
F F
Figure 2.7: Examples of functionalization in CaBz and CaCp (upper panel) and molecules with multiple
cycling centers (lower panel).
A scope of possible applications in QIS, precision measurements and ultracold chemistry can be expanded by incorporating multiple cycling centers into a single molecule [45, 46, 47]. In this context, benzene offers a versatile framework that can accommodate multiple cycling centers in various arrangements.
For example, CaBzCa hosts two cycling centers and can be realized in three configurations (para, meta, and
ortho) with varied separation length between the two centers (Figure 2.7). Similarly to CaBz, each calcium
atom in CaBzCa hosts unpaired electron giving rise to a diradical character of the electronic wavefunction.
Our EOM-DEA-CCSD calculations show that the electronic structure of para-CaBzCa is similar to that
of CaCCCa— a molecule with two calcium atoms separated by the acetylene linker.[46] The ground state of
para-CaBzCa features two natural orbitals with near-unity occupations, indicating strong diradical character of the wavefunction (Figure 2.8A). The antibonding orbital has a slightly higher occupation number,
reflecting a considerable role of through-bond interactions, as has been also observed in p-benzyne and
CaCCCa.[61, 62, 46] In meta- and ortho-CaBzCa the ordering of the two natural orbitals in the ground
14

state switches back to the expected, with bonding orbital below antibonding (Figure A.2). We therefore
conclude that the nature of the interaction between two unpaired electrons is dominated by through-bond
interactions in para-CaBzCa and switches to through-space in meta- and ortho-CaBzCa, in part due to a
closer proximity of the two centers.
The structure of the electronic spectrum in para-CaBzCa resembles that of CaCCCa (Figure 2.8B).[46]
In particular, the lowest electronic states in para-CaBzCa arise from the excitonic splitting between the
corresponding states in single-centered CaBz. For example, the B1 and B2 states of CaBz are split by
a value of 24-25 meV into the B1g/B2u and B2g/B1u pairs (Table A.4). The value of the excitonic splitting
shows a systematic variation with varied Ca-Ca distance across three isomers, reflecting the varied change
in the through-bond interactions between unpaired electrons (Figure 2.8B).
Figure 2.8: Excited states in ortho-, meta-, and para-CaBzCa calculated using EOM-DEA-CCSD/aug-ccpwCVTZ-PP[Ca]/cc-pVTZ[C,H]. Insert shows natural orbitals (isovalue=0.02) of the ground state in paraCaBzCa.
In conclusion, here we computationally investigated prospects of laser-cooling large molecules in
which an alkaline earth metal is attached to an aromatic ligand. Building upon previous successful frameworks, we designed laser-coolable aromatic molecules by attaching a cycling center to an electron-withdrawing
ligand and propose CaBz, CaPh, and CaPy as viable candidates. We also explored (hetero)cyclic π-conjugated
ligands that become aromatic once the electron is transferred from the cycling center to the ligand and
15

propose CaCp and iso-CaPy. We focused on the extent of unpaired electron delocalization from the cycling
center and how this delocalization translates into the FCFs. We demonstrated that, although the involvement of the ligand in these aromatic molecules is slightly more significant than in triatomic CaOH, the FCFs
do not deteriorate significantly. We predict that with an additional few repump lasers, the cycling transition becomes nearly closed, with a total of 96-99% of the population being recovered. Although the diagonality of the computed FCFs suggests that scattering multiple photons off the proposed organic molecules
is likely, additional experimental and theoretical investigations are required to quantitatively confirm that
scattering of millions of photons is possible using a reasonable number of lasers to achieve efficient laser
cooling. The incorporation of aromatic ligands provides means for further functionalization—for example,
the optical and structural properties of these molecules can be varied by introducing electron-withdrawing
groups. Incorporation of multiple cycling centers further extends the scope of possible applications in QIS,
precision measurements, and ultracold chemistry.
16

Chapter 3
On the prospects of optical cycling in diatomic cations: Effects of
transition metals, spin-orbit couplings, and multiple bonds
3.1 Introduction
Optical cycling center (OCC) is a chromophore moiety that supports electronic transitions closed for electronic and vibrational decay to other channels. It means that the excited state should decay primarily to the
ground electronic state. The transition should also be closed at the vibrational level, such that the excitedstate decay populates primarily the ground vibrational state. In this case the system can be subjected to
repeated absorption–emission cycles without exciting molecular vibrations. By tuning the photon energy
to be slightly below the electronic transition energy, each cycle results in slowing the translational motion
of the system. This is the essence of laser cooling. The lower-energy spectral features, such as rotational
and fine structure, must also be considered, but it is the closure on the vibronic (vibrational and electronic)
level that is the chief prerequisite for optical cycling. The excited electronic state must also be short lived
(lifetime <10-100 µs) to afford a rapid photon scattering rate necessary for effective laser cooling (this condition, however, is not critical for working with cations[63]). These requirements for molecules suitable
17

for laser cooling are well known from earlier studies. [9, 64, 63, 65, 66] Potential applications of lasercoolable species include quantum computing [67, 68] and sensing, as well as precision measurements of
fundamental physical constants. [69, 70, 71, 72]
Charged particles are particularly attractive for experimental applications because they can be controlled by electric fields, motivating intense experimental and theoretical research in this area. Heavy
molecular ions have been investigated as promising platforms for precision measurements. [69, 70, 71,
72] In charged molecular species, an OCC is useful for system preparation and measurement. [73] Much
experimental and theoretical effort has been focused on the SiO+ cation. [74, 75, 76, 77] Prospective advantages in state control motivate an active search for molecular ions with a large ground-state dipole
moment and a splitting of the ground state into opposite parity states in the radio frequency range [78,
79, 80, 81]. A closed optical cycle with a radiative lifetime of approximately 110 µs was predicted for the
AcOH+ cation. [82] Computational studies have shown that because of high electronic transition energies
RaF+ and RaH+ cations are not well suited for laser manipulations. [83] Favorable vibrational branching
ratios were reported for a transition of TlF+ lying in the UV region. [84] In the context of precision measurements, highly charged heavy cations offer additional benefits. In this domain the PaF3+ cation was
identified as a promising candidate for cooling.[85]
The prospects of laser-cooling anions have been discussed. Anions could afford sympathetic cooling
of other negatively charged particles. [86] However, in contrast to neutral and cationic species, which
support many bound excited states, anions rarely have bound excited states. [87, 88, 89] Among promising
exceptions [86, 90] are C−
2
and BN−, which are isoelectronic to SiO+. Molecular anions of alkali metals
were also investigated computationally [91, 92], however, the authors were not able to establish whether
their electronic states were bound or metastable with respect to electron detachment.
The search for optically cyclable molecular cations is ongoing. Earlier, we investigated diatomic ions
of the main group elements with a goal to find transitions suitable for cycling. [63] The results were
18

disappointing—compared to the isoelectronic neutral species with good OCCs, the transitions in the cations
generally featured less favorable properties. We attributed this difference to more delocalized electronic
structure and higher density of states in the cations, which present challenges in the design of cationic
OCCs. [63] Other computational studies have shown that BO+ and SiBr+ are not promising for laser
cooling.[93, 94] Here, we extend our search for cyclable molecular cations to a series of diatomic radicalcations comprising a d-block metal and a p-block ligand that are isoelectronic (in their valence shell) to
the successfully laser-cooled neutral molecules.
To assess the suitability of these species for optical cycling, we compute potential energy curves of
their low-lying electronic states. We then use these curves to compute respective equilibrium geometries
and Franck-Condon factors (FCFs), defined here as the squared overlap between the vibrational states:
qνI νF = |⟨νI |νF ⟩|2
, (3.1)
where |νI ⟩ and |νF ⟩ are the vibrational states of the initial and final electronic states, respectively.
Within the Franck-Condon model, FCFs determine vibrational branching of the electronic transition.
The most desired property for optical cycling is a diagonal matrix of the FCF, implying no change in the
vibrational state during an electronic transition. Because repumping schemes can mitigate the effect of
leaking population, cumulative FCFs — the sum of the first few FCFs with fixed νI—are also valuable in
evaluating quality of an OCC. [95]
In diatomic molecules, FCFs are determined by the change in the position of the minimum and the
curvature of the potential energy curves of the ground and excited states. The position of the minimum (i.e.,
equilibrium bond length) depends on the bonding pattern, determined by the shape of occupied molecular
orbitals. Hence, the change in electron density upon excitation provides a visual cue whether the bonding
pattern changes and whether or not one may expect diagonal FCF—i.e., large changes in the inter-atomic
19

charge distribution are expected to give rise to a significant bond length change and, consequently, nondiagonal FCFs.
The low-lying electronic transitions in doublet electronic states, such as radical-cations studied here,
are transitions of the unpaired electron. Hence, we use Dyson orbitals to visualizing the states of the
unpaired electron in the ground and excited states[96]:
ϕ
d
(1) = √
N
Z
Ψ
N (1, . . . , n)ΨN−1
(2, . . . , n) d2 . . . dn , (3.2)
where ΨN and ΨN−1
are the solution of the electronic Schrödinger equation for a cation (one unpaired
electron) and a dication (closed-shell), respectively. The pair of states with Dyson orbitals localized outside
of the bond area are likely to have diagonal FCFs. The idea of rational engineering of OCCs by seeking
molecules with localized atomic-like transitions was first described in a break-through paper by Isaev and
Berger[11], who also proposed several promising candidates. Later, neutral molecules featuring such electronic structure have been successfully laser-cooled[12, 13, 16]. Following this strategy, more polyatomic
molecules were proposed as prospective candidates for laser cooling[44, 46, 97, 98, 99].
In this study we follow the same design paradigm as before[63]—seeking molecular cations with a single unpaired electron localized on a metal atom so that electronic transitions resemble atomic transitions
on the metal. The atomic-like transitions localized on the metal do not disturb the bonding pattern in a
molecule, which should lead to diagonal FCFs[11]. Fig. 3.1 illustrates the design principle. We first consider cations of third group elements (yttrium or scandium). These metals lose one of their three valence
electrons by bonding to a 17th group atom, which requires one electron to reach a closed-shell configuration. To make a cation, the second valence electron is removed leading to a molecule with a positive
charge and the desired unpaired electron on the metal. We follow the same strategy to construct other
20

Figure 3.1: Electronic design principle of OCCs in diatomic cations investigated in this study. Electronic
configurations of radical-cations ML+ mimic the electronic structure of the successfully laser-cooled neutral molecules ML. In these neutral species, an alkali-earth metal forms an ionic bond with a ligand L
(which needs one extra electron to form a closed shell), creating a +1 local charge and a single valence
electron localized on the metal. This pattern results in electronic transitions localized on the metal[11].
The top row shows the electronic configurations of the elements in each column. In the bottom row the
full and empty dots depict valence electrons and valence holes (electrons needed to complete the shell).
Color highlights elements studied in this wor

pairs of a metal with n valence electrons and a ligand missing n − 2 electrons to a closed-shell, as shown
in Fig. 3.1.
The structure of the chapter is the following. The next section describes theoretical methods and
computational details. Section 3.3 presents the results for the diatomic cations in which we identified
atomic-like transitions on the metal atom and reports our best candidate molecules for optical cycling.
Our concluding remarks are given in Section 3.4.
3.2 Theoretical methods and computational details
Table 3.1: Basis sets used in this studya
.
atom basis set
Sc, Ti, V aug-cc-pwCVTZ[100]
Y, Zr, Nb aug-cc-pwCVTZ-PP[101]
Y, Zr, Nb aug-cc-pwCVTZ-DK[101]
F, Cl, Br, O, S, N, P aug-cc-pVTZ[102, 103, 104]
a When two basis sets are shown for the same atom, it means that the energy and properties calculations
were computed with a basis with an effective core potential and SOCs were computed with an
all-electron basis.
We solve the electronic structure problem using high-level ab initio methods following the protocols
from our previous studies on laser-coolable species.[44, 63, 46, 98] We use the equation-of-motion coupledcluster method for electron attachment (EOM-EA-CC) to describe the ground and excited electronic states
of doublet radical cations[54, 105]. In these calculations the target states (with +1 charge) are obtained by
attaching an electron to a closed-shell +2 reference state described by the coupled-clusters singles doubles
(CCSD) method. This approach leads to a balanced description of the ground and excited electronic states
and is naturally spin-pure[105]. Table 3.1 lists basis sets used in our calculations[106].
Because the ordering of the electronic states is different in different cations (and sometimes changes at
different level of theory), we do not use spectroscopic notations (i.e., in which the ground state is labeled
22

by ’X’), but instead simply number each state from 1 in each symmetry irrep (i.e., angular momentum
projection group). This convention allows us to use a consistent notation for all molecules.
We include the effect of spin-orbit couplings (SOCs) using the state interaction scheme (sometimes
referred to as a perturbative approach): we first compute a set of low-lying non-relativistic EOM-EACCSD states and use them to evaluate matrix elements of the Breit-Pauli Hamiltonian (we use mean-field
approach to include two-electron contributions)[107, 108]. We then construct and diagonalize the matrix
of the SOC-perturbed Hamiltonian:
U
†
(HEOM
0 + HBP )U = E
SOC, (3.3)
where HEOM
0
is a diagonal matrix composed of EOM-EA energies and HBP is the matrix of the Breit-Pauli
Hamiltonian in the basis of the EOM-EA states. The resulting energies assembled into a diagonal matrix
ESOC are energies of the SOC-perturbed EOM states and matrix U contains the respective eigenstates.
This matrix can be used to obtain properties of the SO-perturbed states, such as permanent and transition
dipole moments:
µ
SOC = U
†µ
0U (3.4)
where µ
0
is the matrix of dipole moments calculated in the basis of the non-relativistic EOM-EA-CCSD
states. The state-interaction treatment of SOCs becomes exact in the limit where all electronic states are
included in the calculation. In the present study, we considered eight lowest electronic states.
We use the computed spin–orbit corrected potential energy curves to determine equilibrium geometries and to compute FCFs. We do not use harmonic approximations[109] but solve the vibrational problem
exactly. Numerical integration of the overlaps between eigen-functions gives the FCFs.
23

Einstein’s AIF coefficient gives the probability of a spontaneous decay from an initial state I to a final
electronic state F:
AIF =
ω
3
IF µ
2
IF
3hc¯
3πε0
, (3.5)
where ωIF is the transition frequency and µIF is the transition dipole moment. The radiative lifetime of
a state I is obtained by summing over all decay channels
τI = 1/X
F
AIF , (3.6)
where the index F goes over all states below state I.
All electronic structure calculations (including energies, state and transition properties) were carried
out using Q-Chem.[52, 110] We computed FCFs with the LEVEL16 program,[111] which solves the quantum
vibrational problem numerically using potential energy curves provided by the user. In these calculations,
the interatomic potential is represented by analytic functions: exponential functions (fit to the provided
potential energy curves) represent asymptotes and the regions in between the sampled points are interpolated with cubic splines.
3.3 Results and discussion
3.3.1 Y and Sc series
Yttrium and scandium cations feature the sought-after atomic-like spectrum, as clearly illustrated by the
Dyson orbitals in Fig. 3.2. In these representative examples of the YCl+ and NbN+ cations, Dyson orbitals
strongly resemble atomic D states. The 1
2Σ and 2
2Π states correspond to the cation with a singly occupied
s- or p-type orbital, respectively, the 1
2∆, 1
2Π, and 2
2Σ correspond to a singly occupied d-type orbital.
Although atomic states originating from a singly occupied d-type orbital have the same electronic angular
24

Figure 3.2: Dyson orbitals showing the unpaired electron in low-lying electronic states. Representative
examples of YCl+(left) and NbN+(right) (isovalue = 0.02; plotted with iQmol[112]). In the hydrogen-like
orbital labels, the principal quantum number n equals 4 for the Sc, Ti, and V metal cations and 5 for the Y,
Zr, and Nb cations.
25

Table 3.2: Vertical excitation energies (eV) for ScX+ and YX+ cations at the ground-state equilibrium
geometry. Transition dipole moments µ (a.u.) correspond to the transition involving the ground state.
EOM-EA-CCSD without the inclusion of SOC shifts.
ScF+ ScCl+ ScBr+
State Eex µ Eex µ Eex µ
1
2∆ 0.000 — 0.000 — 0.000 —
1
2Σ 0.502 0.000 0.143 0.000 0.032 0.000
1
2Π 0.781 0.141 0.431 0.138 0.348 0.144
2
2Σ 2.058 0.000 1.615 0.000 1.471 0.000
2
2Π 4.897 0.578 4.458 0.555 4.216 0.551
re (Å) 1.79 2.21 2.34
YF+ YCl+ YBr+
State Eex µ Eex µ Eex µ
1
2∆ 0.634 0.000 0.620 0.000 0.536 0.000
1
2Σ 0.000 — 0.000 — 0.000 —
1
2Π 1.427 0.923 1.129 0.745 1.056 0.593
2
2Σ 2.160 1.213 1.658 1.125 1.684 0.912
2
2Π 3.829 1.951 3.612 1.906 3.759 1.692
re (Å) 1.90 2.35 2.46
momentum, the diatomic states vary with respect to its projection on the molecular axis, giving rise to
three types of molecular states. Table 3.2 and Fig. 3.3 show non-relativistic EOM-EA-CCSD excitation
energies computed at the ground-state equilibrium geometries for ScF+, ScCl+, ScBr+, YF+, YCl+, and
YBr+.
All states consistently appear in the same order except for the two lowest-lying states. The 1
2∆ is
the ground state for scandium cations whereas it lies above the 1
2Σ state for yttrium cations. By inspecting Table 3.2 and Fig. 3.3, we observe a general trend—that the energy gaps increase with the increasing
electron-withdrawing strength of the halogen ligand. This ability to tune up OCC properties by tuning the
ligand’s electron-withdrawing strength was discussed in previous computational studies[44, 99].
We first assess whether the 2
2Π → 1
2Σ transition (analog of the atomic 2P → 2S transition) is suitable
for optical cycling. Just as in the case of neutral molecules, this transition is characterized by a highly diagonal FCFs (see Table B.1). The large excitation energy (≈ 4 eV) and a strong transition dipole moment give
rise to a short radiative lifetime of the excited state. Unfortunately, these advantageous properties come
26

ScF+ ScCl+ ScBr+ YF+ YCl+ YBr+
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
E (eV)
2
2Π
2
2Σ
1
2Π
1
2Σ
1
2∆
Figure 3.3: Scandium and yttrium halide cations with low-lying localized doublet states. Vertical excitation
energies (EOM-EA-CCSD energies without the inclusion of SOC shifts) at the ground-state minimum.
together with undesired ones. First, there are intermediate states, which can drain the populations out of
the OCC. Second, the excitation energy of this transition is in an undesirable UV range. This transition
is even less attractive in the case of the scandium cations, where the 1
2Σ state is an excited state. Our
calculations indicate that the YF+ cation, the best candidate in this group shown in Fig. 3.4, is unlikely to
form a good OCC.
The simplest way to avoid electronic branching decays is to use the transition between the ground
and the first excited electronic states. Unfortunately the transition between 1
2Σ and 1
2∆ states is dipole
forbidden. However, the SOCs can mix non-relativistic states, giving rise to a non-vanishing transition
dipole moment, which might be sufficiently large to allow for optical cycling.
Tables B.3 and B.4 shows the SOC-corrected states with their energies, transition dipole moments, and
radiative lifetimes. SOCs mix the non-relativistic states and the extent of mixing is determined by the
energy gaps between the states (which varies between the cations) and the magnitude of the Breit-Pauli
Hamiltonian matrix elements (which are larger for cations containing heavier atoms). The energy shifts
27

YF+
0
1
2
3
4
Energy (eV)
1
2Σ
1
2∆
1
2Π
2
2Σ
2
2Π
19.8%
76.3%
3.8%
< 0.1 %
τ = 0.5 ns
Figure 3.4: YF+ is the most diagonal cation for the 2
2Π → 1
2Σ transition. Numbers on the arrows indicate
branching ratios. The four decay channels generate population leakage out of the cycling transition which
is undesired in an OCC.
due to the SOCs do not exceed 0.1 eV and cause splittings of the Π and ∆ states. Crucially, the SOCperturbed states show non-vanishing transition dipole moments between the ∆ and Σ states in the range
of 0.01 a.u.
Consequently, we consider the 1
2∆3/2 → 1
2Σ1/2 transition as the second candidate for an OCC. This
transition is characterized by the excitation energy of about 0.75 eV and a transition dipole moment of
0.01 a.u., yielding the radiative lifetimes longer than hundreds µs. The FCFs saturate slowly, with the most
diagonal YCl+ cation reaching 0.9989 at four states, as shown in Tables B.5 and B.6. Because of the long
radiative lifetime and large vibrational branching, using this transition for optical cycling is challenging.
However, the fact that the inclusion of SOCs can open a way for cycling on a formally forbidden transition
motivates the inspection of similar transitions in other cations.
28

TiO+ TiS+ ZrO+ ZrS+ VN+ NbN+ NbP+
0
1
2
3
4
5
6
7
8
E (eV)
2
2Π
2
2Σ
1
2Π
1
2Σ
1
2∆
Figure 3.5: Multiply bonded cations with low-lying localized doublet states. Vertical excitation energies
(EOM-EA-CCSD energies without the inclusion of SOC shifts) at the ground-state equilibrium geometry.
3.3.2 Other groups: Ti or Zr with chalcogens, and V or Nb with N or P
The spectra of the multiply bonded cations are similar to those of the scandium cations discussed above. All
species feature the 1
2∆ ground state followed by the 1
2Σ and 1
2Π states. Again, we observe an increase in
the energy gaps with an increasing electron withdrawing strength of the ligand. Additionally, a new trend
becomes apparent—the excitation energies increase with the number of bonds in the cation, as shown in
Fig. 3.5. In the case of the transition from yttrium to zirconium and niobium cations, where the order of the
ground and the first excited states changes, taking the energy of 1
2∆ state as the reference point allows
to observe a consistent increase. This new trend is particularly apparent for the series F, O, and N as the
ligands or Cl, S, and P, shown in Figs. B.1 and B.2.
In this set of cations we again inspect the transition between the ground and the first excited states. The
increased energy gaps in Ti and V relative to Sc improve the radiative lifetimes. The heavy atoms Zr and Nb
also have stronger SOCs, which increases transition dipole moment, but an improvement in the radiative
lifetime is achieved only for Nb as in the Zr cations the 1
2∆ and 1
2Σ states become nearly degenerate. The
small improvement in the radiative lifetime of the 1
2Σ → 1
2∆3/2 transition in multiply bonded cations
29

is accompanied by a large improvement in the diagonality of the corresponding FCFs, Sec. B.3, making
this transition very a promising OCC. The best candidate in this group is the NbN+ cation, with FCFs
summarized in Table 3.3 and the radiative lifetime τ= 650 µs.
In search for a shorter radiative lifetime, we focus on the dipole-allowed transition. As the last candidate for an OCC we consider a three-level cycle with the 1
2Π excited state decaying to the ground 1
2∆
and an intermediate 1
2Σ states. The FCFs on the 1
2Π1/2 → 1
2∆3/2 transition single out zirconium and
scandium cations where the sum of the first four FCFs reaches 0.9999; the titanium cations also reach such
saturation by adding the fifth FCF (see Section B.4). In all these cations, the excited state decays significantly (> 10 %) to the intermediate 1
2Σ state. To take advantage of the rapid cycling offered by the short
lifetime of the excited state, this intermediate state must be repumped. Diagonal FCFs, necessary for an
efficient repumping, render the ZrO+ cation to be the most appealing candidate for the three-level OCC.
We do not report details about other screened cations because our exploratory calculations suggest
that they do not feature the localized transitions. Cations of Cr or Mo with a C or Si have very different
electronic structure with significant multi-configurational character and their dications do not converge to
a closed-shell singlet state. We detected similar problems for Ti or Zr with Se cations as well as V and Nb
with As cations. This is not surprising, as transition metals with more than three d-electrons are known
to result in complicated multi-configurational structure and dense manifolds of electronic states.
3.3.3 FCFs, electronic charge displacement, and Dyson orbitals
The transition between the 2
2Π and 1
2Σ states is known for its diagonal FCFs, which are attributed to
the localized character of the transition. The localized transition preserves the charge distribution in the
interatomic area of both states leading to similar bond lengths. We interpret the bond length changes of
other electronic transitions using similar analysis applied to the changes in shapes of the Dyson orbitals.
30

The 1
2Σ → 1
2∆ transition is most promising as it involves the ground and the first excited electronic
states avoiding branching to intermediate levels. The electronic density changes from a spherical distribution of the s-orbital to a clover-shaped shape of the d orbital, see Fig. 3.2. For most of the yttrium and
scandium cations this transition features FCFs indicative of the expected significant vibrational branching
(see Tables B.5 and B.6). For all multiply bonded cations except for ZrS+ this is one of the most diagonal
transitions that we found (Tables B.7–B.10).
In the dipole-allowed 1
2Π → 1
2∆ transition, the d-orbital on a metal atom changes its alignment from
the one that is along the bond to the perpendicular one (Fig. 3.2). This charge displacement leads to the
expected poor FCFs, with a few exceptions of scandium or zirconium cations where the sum the first four
FCFs differs from unity by less than 10−4
(see Sec. B.4).
This analysis suggests that a disruption in the electronic density of the bond area does not necessarily
render the transition to be unusable for optical cycling. One possible reason is that the presence of a
multiple bond is a structurally stabilizing factor.
3.3.4 Best candidates for optical cycling
Table 3.3: FCFs for the best candidatesa
.
YCl+ 1
2∆3/2 → 1
2Σ1/2
qνiνf
sum
q00 0.916 0.9165
q01 0.082 0.9985
q02 0.000 0.9987
q03 0.000 0.9989
ZrO+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.744 0.7444
q01 0.224 0.9687
q02 0.029 0.9978
q03 0.002 0.9999
NbN+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.985 0.9851
q01 0.015 1.0000
q02 0.000 1.0000
q03 0.000 1.0000
Numerical solution of the quantum vibrational problem using spin-orbit perturbed EOM-EA-CCSD
potential energy surfaces.
The variations of the spectra introduced by states derived from a singly occupied d-type orbital led
us to consider four types of OCCs. The molecular analog of the atomic 2P → 2S transition was deemed
not promising (see Section 3.3.1). Among the low-lying states we considered the transitions between the
first excited and the ground states, separately for ground 1
2Σ and for ground 1
2∆ cations, as well as a
31

YCl+ ZrO+ NbN+
0.0
0.5
1.0
1.5
2.0
Energy (eV)
1
2Σ1/2
1
2∆3/2
1
2∆
1
2Σ1/2
1
2Π1/2
1
2∆
1
2Σ1/2
33%
67%
τ = 1.6 ms
τ = 14.5 ns
τ = 650µs
Figure 3.6: Three types of optical cycling schemes exemplified by the best candidate cations. In the group
with the 1
2Σ ground state, YCl+ shows the most diagonal transition. In the group with the 1
2∆ ground
state and highly diagonal transitions, NbN+ stands out with the shortest radiative lifetime. In the third
group, the FCFs for the formally dipole-allowed transition between the ground and the second excited
state favor scandium and zirconium cations. Large decay to the intermediate state requires also the side
branch 1
2Π1/2 → 1
2Σ transition to show quickly saturating FCFs. This second requirement renders the
ZrO+ cation as the best candidate in this group.
three level scheme involving the second excited 1
2Π state. We present the best candidates in each of these
groups in Fig. 3.6; Table 3.3 summarizes the corresponding FCFs.
3.4 Conclusions
By using high-level ab initio calculations, we investigated series of diatomic radical cations designed to
create a localized electronic transition suitable for optical cycling. The results demonstrate that the localized electronic excitation picture exploited in OCCs in neutral molecules [10, 38, 17, 40, 113, 18, 39, 19, 114,
13, 14, 15, 115, 20, 116, 117, 23, 44, 46] to some extent holds for isoelectronic cations formed by transitionmetal atoms. This result applies to molecular cations that either contain a transition metal atom or have
multiple bonds, which is different from the behavior of other cations[63].
The observed trends can help in designing transitions with desired excitation energy. In multiply
bonded cations, we observe persistent diagonal FCFs that are promising for optical cycling, on the 1
2Σ1/2 →
1
2∆3/2 transition. Finally, our analysis of FCFs of selected transitions together with an inspection of the
32

Dyson orbitals of involved states suggests that although the charge displacement often makes the transition
unlikely for optical cycling, it is not the crucial design factor. The most promising candidates in the three
identified optical cycling schemes (discussed in Section 3.3.4) are: 1
2∆3/2 → 1
2Σ in YCl+, 1
2Π1/2 → 1
2∆
with a side branch 1
2Π1/2 → 1
2Σ in ZrO+ and 1
2Σ → 1
2∆ in NbN+.
The most promising optical cycling scheme is the that of the NbN+ cation. The calculated FCFs show
that with lasers addressing the ν
′′ = 0 → ν
′ = 0 and ν
′′ = 1 → ν
′ = 0 branches this molecule should be
able to scatter on average more than 105 photons before population leaks to ν
′′ > 1 states. This number of
scattering events should be sufficient for laser cooling, however, direct laser cooling on a transition with a
radiative lifetime of the order of hundreds microseconds is not attractive.
On the other hand, diatomic cations held in an ion trap can be sympathetically cooled by co-trapped,
laser-cooled atomic ions. [118] Therefore, it is not always necessary that a fast cycling transition be used
for cooling the motion of the molecular ion, but is only needed to enable quantum state preparation and
measurement. [74, 119] This significantly relaxes the constraints on the lifetime of the excited electronic
state of the diatomic cation. In this use case, the optical cycling lifetime need only be significantly faster
than the population redistribution rate from the environment, typically black-body radiation. While the
e.g. black-body redistribution rate is dependent on molecular structure, state lifetimes < 1 ms would
appear to allow for quantum state preparation and measurement in most scenarios.
Further, longer lifetime species with cycling transitions may also be useful for experiments using quantum logic spectroscopy. [120, 121] Here, the cycling transition can be used to optically pump the molecular internal states to greatly improve the rate of the heralded state preparation and measurement used in
quantum logic spectroscopy. The cations presented in this manuscript show properties that make them
promising candidates for use in these applications. The range of described properties allows one to select
the cations that match the best desired experimental requirements for a particular application.
33

Our results can assist engineering of OCCs in other neutral or charged molecules. In particular, a
shortening of the radiative lifetime in the most promising 1
2Σ → 1
2∆ scheme appears possible. The
radiative lifetime of an excited state becomes shorter with an increase in either the excitation energy or
the transition dipole moment. A use of polyatomic ligands of an electron withdrawing strength exceeding
that of atoms appears as the most practical way of raising transition energy. The excitation energy can
be further increased with the growing number of the metal-ligand bonds. Similarly, a formally dipole
forbidden transition can become sufficiently bright in molecules with heavy nuclei due to an increased
SOCs. We expect that the heavy metal analogs of the systems studies in this work (Lu, Hf, Ta cations)
are likely to feature fast OCCs on a transition between its low-lying doublet states and that the design
principle outlined before should help in its fine-tuning.
34

Chapter 4
Dual Optical Cycling Centers Mounted on an Organic Scaffold: New
Insights From Quantum Chemistry Calculations and Symmetry
Analysis
4.1 Introduction
Ultracold molecules are attractive for novel applications[122, 123, 124]. Molecules can be cooled to the
ultracold temperatures by laser cooling, a repeated absorption and emission of photons exploiting the
Doppler effect[125]. When excitation laser is tuned to energy slightly below the electronic transition,
only molecules moving towards the photon source can be excited. The excited molecule then undergoes
spontaneous decay by which it emits recoil momentum in all directions, slowing the average momentum
over time. The key to laser-cooling molecules is the ability to prevent decay to unwanted decay channels,
such as excited vibrational states. The process of scattering many photons using the same closed loop is
called optical cycling. The undesirable population leakage into vibrationally excited states is minimized
when ground and electronically excited states have the same structures, thereby resulting in the diagonal
Franck-Condon factors (FCFs; vibrational overlaps). This goal can be achieved by designing molecules with
localized atomic-like transitions, as was first described by Isaev and Berger[126], who also proposed several promising candidates—all based on a metal-oxygen-ligand (M-O-L) framework with a second-group
35

metals. The essential features of these systems are a strongly ionic M-O bond and a localized unpaired
electron on the metal, giving rise to localized atomic-like transitions. These transitions can be used as the
main photon cycling loop; hence, the name optical cycling center (OCC).
Several molecules featuring such electronic structure have been successfully laser-cooled[127, 128,
129]. Cooling larger molecules could be advantageous for quantum information and sensing applications [130, 131, 132]. Although the initial enthusiasm was curbed by the concerns that the increased complexity is likely to result in more vibrational branching channels, several theoretical studies have shown
that such metal-based OCCs can in fact operate in larger molecules. Following this strategy, more polyatomic molecules were proposed computationally as prospective candidates for laser cooling [133, 134,
135, 136, 137, 138, 139, 140, 141]. The screening of the candidate molecules entails finding a low-lying
bright electronic transition, assessing that it is electronically uncoupled from the rest of the molecule, and
computing FCFs (vibrational overlaps). The electronic transitions and FCFs can be tuned using chemical intuition. As recent studies show, strongly diagonal FCFs are possible in large molecules, leading to
vibrational branching ratios rivaling those of CaOH [137, 138, 139].
It is desirable to further optimize photon cycling in the candidate molecules, such that one can use
fewer repump lasers, introduce additional optical controls, and explore more varied types of molecules.
One way to increase cycling efficiency is to install multiple OCCs on one scaffold, which can increase the
oscillator strength, potentially doubling the photon scatters[134, 133, 142, 143]. Here, we explore this idea
further, by considering a larger set of molecular scaffolds augmented by two OCCs (we refer to them as
bi-OCCs). We analyze the effect of symmetry and use chemical design to control the optical properties of
the prospective bi-OCCs and the respective FCFs (FCFs control vibrational branching ratios in electronic
transitions).
36

13128 | Phys. Chem. Chem. Phys., 2018, 20,13127--13144 This journal is © the Owner Societies 2018
classic papers.7,8 In the context of MMs, the two states of interest
are the lowest singlet and tripletstates.Insystemslikebinuclear
copper complexes, one expects these two states to have covalent
wave functions in which the unpaired electrons are localized on
the two metal centers:
Cs,t(1,2) B [fCu1(1)fCu2(2) ! fCu2(1)fCu1(2)]
" [a(1)b(2) 8 b(1)a(2)] (1)
with little-to-no contributions from ionic configurations
[fCu1(1)fCu1(2) + fCu2(1)fCu2(2)] " [a(1)b(2) # b(1)a(2)]
(2)
In these expressions, fCu1 and fCu2 denote orbitals localized
on the two copper centers, such as copper d-orbitals (perhaps
including small contributions from the nearest ligand atoms).
If the actual MOs hosting the unpaired electrons are delocalized and can be described as (nearly degenerate) bonding and
antibonding combinations of fCu1 and fCu2 (case 1 in Fig. 1), as
is the case in the MMs studied here, the triplet states are
described by configurations (i) and (ii) and have pure covalent
character, as Ct in eqn (1). The character of the lowest singlet
state can vary, depending on the exact weights of configurations (iii) through (v). A purely covalent singlet wave function,
Cs of eqn (1), corresponds to configuration (iv) with l = 1.
A smaller value of l gives rise to the ionic configurations mixed
into the wave function. This happens when the interaction
between the two centers stabilizes the bonding MO relative to
the antibonding one, either due to through-space or throughbond interactions. The ionic configurations can also appear in
the singlet wave functions due to mixing with configuration
(iii), which, in contrast to (ii), has pure ionic character. Since all
configurations, (iii)–(v), can contribute into the singlet state,
the correct description of this state requires an electronic
structure method that treats (iii)–(v) on an equal footing. In
order to describe relative energies of singlet and triplet states,
the method should provide a balanced and unbiased description of all four Ms = 0 configurations from Fig. 1.
From a theoretical perspective, the search for promising
MMs begins with first-principle calculations of the relevant
terms in the phenomenological spin Hamiltonian.2,12,13 Of all
terms in the spin Hamiltonian, the most energetically significant one is the exchange-coupling interaction between
unpaired, spatially separated electrons.14–16 The sign and
magnitude of electronic exchange-coupling between atomic
spin centers determines whether a molecule will behave ferromagnetically or antiferromagnetically upon exposure to an
external magnetic field, and thus, whether the molecule is
suitable for application in a magnetic material. In bimetallic
diradical complexes, like those considered here, the exchangecoupling constant equals the energy difference between the
lowest singlet and triplet states.17 Thus, exchange-coupling
terms of the spin Hamiltonian can be computed by ab initio
methods as singlet–triplet energy gaps. This approach can be
generalized to systems with more unpaired electrons and
multiple polyradical centers.18,19
The main challenge in applying this simple strategy is in the
multiconfigurational character of the low-spin wave functions,
which becomes evident by inspecting Fig. 1. The high-spin
states, such as Ms = 1 triplet (i), are well represented by a single
Slater determinant and, therefore, their energies can be reliably
computed by standard single-reference methods, i.e., coupledcluster theory or DFT. In contrast, wave functions of low-spin
states (ii)–(v) require at least two Slater determinants. Consequently, they are poorly described by single-reference methods.
This imbalance in the description of high-spin and low-spin
states results in large errors in the computed singlet–
triplet gaps.
Several strategies have been employed for describing the
open-shell states and exchange-coupling in MMs. Historically,
the most popular are the broken symmetry (BS) methods20–22
and spin-restricted Kohn–Sham (REKS/ROKS) methods.22–24
Both approaches suffer from imbalance in their treatment—and
sometimes, outright exclusion—of important configurations
depicted in Fig. 1. For example, in BS approach all singlet and
triplet configurations are scrambled. While spin projection
allows one to formally separate singlet and triplet manifolds, it
does not distinguish between open- and closed-shell singlet
states (iii)–(v). Despite belonging to entirely different states
(which may even have different spatial symmetry), these
Fig. 1 Wave functions of diradicals that are eigenfunctions of S2 (only configurations with positive spin projections are shown). Wave function (i)
corresponds to the high-spin Ms = 1 triplet state. Wave functions (ii)–(v) correspond to the low-spin states: Ms = 0 singlets and triplets. Note that all Ms =0
configurations can be formally generated by a spin-flipping excitation of one electron from the high-spin Ms = 1 configuration. The character of the wave
functions (i.e., covalent versus ionic) depends on the nature of molecular orbitals f1 and f2 and the value of l.
Paper PCCP
Published on 22 January 2018. Downloaded by University of Southern California on 16/05/2018 16:05:32.
View Article Online
(i) (ii) (iii) (iv)
Case 1:
Case 2:
1 ⇠ M1 + M2
2 ⇠ M1 M2
1 ⇠ M1
2 ⇠ M2
2
1
Covalent Ionic Covalent (if = 1) Ionic (if = 1)
Covalent Covalent Ionic Ionic
Triplet Singlet Singlet Singlet
13128 | Phys. Chem. Chem. Phys., 2018, 20,13127--13144 This journal is © the Owner Societies 2018
classic papers.7,8 In the context of MMs, the two states of interest
are the lowest singlet and tripletstates.Insystemslikebinuclear
copper complexes, one expects these two states to have covalent
wave functions in which the unpaired electrons are localized on
the two metal centers:
Cs,t(1,2) B [fCu1(1)fCu2(2) ! fCu2(1)fCu1(2)]
" [a(1)b(2) 8 b(1)a(2)] (1)
with little-to-no contributions from ionic configurations
[fCu1(1)fCu1(2) + fCu2(1)fCu2(2)] " [a(1)b(2) # b(1)a(2)]
(2)
In these expressions, fCu1 and fCu2 denote orbitals localized
on the two copper centers, such as copper d-orbitals (perhaps
including small contributions from the nearest ligand atoms).
If the actual MOs hosting the unpaired electrons are delocalized and can be described as (nearly degenerate) bonding and
antibonding combinations of fCu1 and fCu2 (case 1 in Fig. 1), as
is the case in the MMs studied here, the triplet states are
described by configurations (i) and (ii) and have pure covalent
character, as Ct in eqn (1). The character of the lowest singlet
state can vary, depending on the exact weights of configurations (iii) through (v). A purely covalent singlet wave function,
Cs of eqn (1), corresponds to configuration (iv) with l = 1.
A smaller value of l gives rise to the ionic configurations mixed
into the wave function. This happens when the interaction
between the two centers stabilizes the bonding MO relative to
the antibonding one, either due to through-space or throughbond interactions. The ionic configurations can also appear in
the singlet wave functions due to mixing with configuration
(iii), which, in contrast to (ii), has pure ionic character. Since all
configurations, (iii)–(v), can contribute into the singlet state,
the correct description of this state requires an electronic
structure method that treats (iii)–(v) on an equal footing. In
order to describe relative energies of singlet and triplet states,
the method should provide a balanced and unbiased description of all four Ms = 0 configurations from Fig. 1.
From a theoretical perspective, the search for promising
MMs begins with first-principle calculations of the relevant
terms in the phenomenological spin Hamiltonian.2,12,13 Of all
terms in the spin Hamiltonian, the most energetically significant one is the exchange-coupling interaction between
unpaired, spatially separated electrons.14–16 The sign and
magnitude of electronic exchange-coupling between atomic
spin centers determines whether a molecule will behave ferromagnetically or antiferromagnetically upon exposure to an
external magnetic field, and thus, whether the molecule is
suitable for application in a magnetic material. In bimetallic
diradical complexes, like those considered here, the exchangecoupling constant equals the energy difference between the
lowest singlet and triplet states.17 Thus, exchange-coupling
terms of the spin Hamiltonian can be computed by ab initio
methods as singlet–triplet energy gaps. This approach can be
generalized to systems with more unpaired electrons and
multiple polyradical centers.18,19
The main challenge in applying this simple strategy is in the
multiconfigurational character of the low-spin wave functions,
which becomes evident by inspecting Fig. 1. The high-spin
states, such as Ms = 1 triplet (i), are well represented by a single
Slater determinant and, therefore, their energies can be reliably
computed by standard single-reference methods, i.e., coupledcluster theory or DFT. In contrast, wave functions of low-spin
states (ii)–(v) require at least two Slater determinants. Consequently, they are poorly described by single-reference methods.
This imbalance in the description of high-spin and low-spin
states results in large errors in the computed singlet–
triplet gaps.
Several strategies have been employed for describing the
open-shell states and exchange-coupling in MMs. Historically,
the most popular are the broken symmetry (BS) methods20–22
and spin-restricted Kohn–Sham (REKS/ROKS) methods.22–24
Both approaches suffer from imbalance in their treatment—and
sometimes, outright exclusion—of important configurations
depicted in Fig. 1. For example, in BS approach all singlet and
triplet configurations are scrambled. While spin projection
allows one to formally separate singlet and triplet manifolds, it
does not distinguish between open- and closed-shell singlet
states (iii)–(v). Despite belonging to entirely different states
(which may even have different spatial symmetry), these
Fig. 1 Wave functions of diradicals that are eigenfunctions of S2 (only configurations with positive spin projections are shown). Wave function (i)
corresponds to the high-spin Ms = 1 triplet state. Wave functions (ii)–(v) correspond to the low-spin states: Ms = 0 singlets and triplets. Note that all Ms =0
configurations can be formally generated by a spin-flipping excitation of one electron from the high-spin Ms = 1 configuration. The character of the wave
functions (i.e., covalent versus ionic) depends on the nature of molecular orbitals f1 and f2 and the value of l.
Paper PCCP
Published on 22 January 2018. Downloaded by University of Southern California on 16/05/2018 16:05:32.
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(v)
Figure 4.1: Diradicals: Different electronic configurations derived from distributing two electrons in two
(orthogonal) orbitals and their characters. M1 and M2 denote two radical centers, e.g., the two metal
atoms in bi-OCCs. Configuration (i) is the high-spin (Ms=1) triplet state and configurations (ii)-(v) are
low-spin (Ms=0) states. Case 1 corresponds to perfectly delocalized frontier orbitals (such as bonding and
antibonding combinations of atomic orbitals) and case 2 corresponds to the molecular orbitals perfectly
localized on the radical centers.
Although bi-OCC molecules may seem exotic, many features of their electronic structure can be understood by revisiting diradicals. [144, 145, 146, 147, 148] Diradicals are species in which the two frontier molecular orbitals are (nearly)-degenerate and the low-lying electronic states derived by distributing
two electrons in two orbitals. Fig. 4.1 shows resulting electronic configurations. The character of the
wave-functions depends whether the orbitals are delocalized or localized. The triplet states are always
covalent—they correspond to the unpaired electrons localized on the two metal centers whereas the singlets can be either covalent, ionic (or charge-resonance, M+
1 M−
2 ±M−
2 M+
1
), or a mixture of covalent and
ionic contributions. Considering delocalized orbitals (case 1 in Fig. 4.1), configuration (iv) corresponds to
purely covalent diradical configuration with two unpaired electrons when λ=1 and to an equal mixture
of ionic and covalent configurations when λ=0. The extent of diradical character of the wave-function
can be quantified by computing the number of effectively unpaired electrons, nu,nl, using Head-Gordon’s
formula[149]. In the triplet state (configurations (i) and (ii)), the number of unpaired electrons is 2. For
37

configuration (iv) and delocalized orbitals (case 1 in Fig. 4.1), the number of unpaired electrons is related
to λ and can be computed using the following formula[148]:
nu,nl =
32λ
4
(1 + λ2)
4
. (4.1)
Energy gap between the frontier orbitals controls parameter λ and the relative energies of the lowest
singlet and triplet states—lager gaps favor ground singlet states. If the energy separation between ϕ1 and
ϕ2 states is small, the triplet can become the ground state. The gap depends on the through-space and
through-bond interactions. Through-space interaction depends on the overlap of the contributing atomic
orbitals whereas through-bong interaction depends on the molecular scaffold and the relative placements
of the radical centers. This may lead to flipping the ground-state multiplicity in structural isomers—for
example, meta-xylylenes have triplet ground state whereas para-xylylene have singlet ground state.
Previous studies of bi-OCCs considered the following motifs—M1-CC-M2, with M1/M2 = Mg, Ca, Sr
(Ref. 134) and M1 = Yb and M2-Ca/Al (Ref. 142), benzenes functionalized with Ca in ortho-, meta-, and
para- positions[136], as well as Sr and OSr installed on (CH2)n chains and fullerenes (Ref. 143). These
studies revealed the following essential features of bi-OCCs:
• Even in the systems with a short acetylene linker, the frontier orbitals are localized on metal centers
and the low-lying electronic states are largely uncoupled from the scaffold; this can be explained by
the large energy gap between the metal and linker orbitals and the diffuse nature of the former.
• Both through-space and through-bond interactions between the two centers are weak, leading to
nearly or almost degenerate singlet and triplet states.
• Because of the weak interaction between the OCCs, the characters of the singlet and triplet states are
similar—both are covalent (i.e., nu,nl ≥1.8 for the lowest singlet states). Furthermore, the low-lying
38

excited states in each multiplicity follow nearly identical patterns. The energy spacing follows that
of a single OCC very closely, with very small excitonic splitting.
• In bi-OCCs with the short acetylenic linker, the coupling between the centers and the linker and
between the centers themselves is smaller for heavier metals for which the two manifolds become
even more similar. However, in phenoxide with a single OCC, Sr resulted in worse FCFs than Ca.
[150]
In high-symmetry cases, excited states follow excitonic patterns, with one state being dark and the
other carrying double oscillator strength of a single OCC. Depending on the structure, the lowest excited
state can be either bright (desirable for cycling) or dark (undesirable). This can be understood in the
framework of the exciton model[151, 152]—if the transition moments of the two chromophores are parallel
but not aligned, one obtains an H-dimer in which the lowest state is dark. If the two moments are aligned
along the same axis, one obtains a J-dimer in which the lowest state is bright (super-radiant). In between
these two limiting cases, there are structural arrangements in which the transition dipoles of the two
chromophores are oblique[151]. Given plentiful synthetic routes in which the oblique arrangement can be
implemented, understanding this general case is important from the practical point of view. Our results
for concrete bi-OCCs illustrate the range of variation in the intensity patterns corresponding to these
intermediate cases.
The lowest transitions in the metal-based OCCs are of s → p types (the states in which the p-orbital
is aligned along the M-O bond are strongly hybridized with the respective d-orbital). The transition moments are aligned along the direction of the corresponding p-orbitals. Hence, the excitonic pattern can be
controlled by splitting the degeneracy of the p-orbitals and their relative orientation in a molecule. Specifically, in para-benzyne bi-OCC, the lowest transition is dark and in meta- and ortho-benzynes it is bright
(but not super-radiant)[136].
39

OCa
CaO
OCa
CaO
(3)
OCa
CaO
(5)
CaO OCa
(4)
CaO OCa
(6)
OCa
OCa
(7)
OCa
(8)
OCa
CF3
OCa
(9)
OCa
CF3
F3C (10)
CF3
OCa
CaO
CF3
(11)
CF3
OCa
CaO
CaO OCa
(1) (2)
(12) (13)
CaO OCaSr. OCa
CaO
Sr.
(12) (13)
Figure 4.2: Structures of bi-OCC molecules considered in this work.
Here we use M-O-X framework in which the metal centers are even further apart, thus improving the
decoupling relative to the M-X motifs. Fig. 4.2 shows the bi-OCC molecules studied in this work.
4.2 Methods
The type of electronic structure featured by OCCs can be accurately described by the double electron
attachment variant[153] of equation-of-motion coupled-cluster (EOM-DEA-CC) methods[54] in which the
ground and excited target states are described by attaching two electrons to a dicationic reference, as was
done in Refs. 4446. Here we use EOM-DEA with single and double substitutions (EOM-DEA-CCSD) with
the cc-pVDZ basis on H,C,O and aug-cc-pwCVDZ-PP on Ca and Sr with the ECP10MDF and ECP28MDF
pseudopotentials used for the respective metal atoms. To determine relative ordering of the lowest singlet
and triplet states, we carried out EOM-SF-CCSD(fT) calculations[154, 155, 156] with the VTZ basis. To
evaluate the effect of spin–orbit coupling (SOC) on transition energies and state characters, we used stateinteraction approach with SOCs computed using EOM-CC wave-functions[107, 108]. EOM-CC and SOC
calculations were carried out using Q-Chem[110].
For extensive screening involving calculations of FCFs for relatively large molecules it is desirable to
develop a less expensive protocol. Here we take advantage of the fact that the lowest singlet and triplet
manifolds of bi-OCCs are very similar. Hence, one can use the triplet manifold as a proxy for singlet
40

states. Whereas low-spin singlet and triplet states are multi-configurational, the high-spin triplet states
can be well described by a single-reference method (HF, CCSD, DFT), which is exploited, for example, in
the spin-flip approach[154, 155]. Following the same logic, here we compute transitions in the triplet-state
manifold from the lowest high-spin triplet state using density functional theory (DFT) and its extension
to the excited states, TD-DFT. We employ the PBE0-D3/def2-TZVPPD level of theory for geometry optimizations, excitations, frequencies, and FCFs, using Gaussian16 software[157, 158, 159, 160]. The validate
this approach, we benchmarked it against EOM-DEA-CCSD; the results are given in the SI.
4.3 Discussion
We begin by discussing benzenes functionalized by two Ca-O OCCs installed in meta- ((1)) and para- ((2))
positions (see Fig. 4.2). This builds upon previous work where we experimentally realized the single OCC
case, CaOPh [137].
The natural orbitals hosting the unpaired electrons are in-phase and out-of-phase combinations of the
s- and p-orbitals of the to Ca atoms (Fig. 4.3). The ground state is derived by distributing two electrons
in the molecular orbitals derived by linear combinations of the two s-orbitals. The lowest triplet states
are 3B1 in (1) and 3B3u in (2); they appear to be degenerate with the lowest singlet states (1A1 and 1Ag)
at the EOM-DEA-CCSD level of theory. The wave-function analysis yields nu,nl=2 for the singlet states,
confirming that the singlet state has pure diradical character. We note that for the analogous Ca-substituted
benzenes[136], singlet-state nu,nl values were smaller—1.83, 1.92, and 2.00 for ortho-, meta-, and paraisomers, respectively. Hence, linking metals through the oxygen improves the decoupling of the unpaired
electrons and reduces ionic contributions, which should improve the FCFs. This effective decoupling also
makes singlets electronically very similar to the triplets. Hence, we expect that the cycling efficiency
should not be affected by the (spin-forbidden) transitions between the two spin manifolds (more on this
below).
41

z
x a1
a1
b1
b1
b2
a2
ag
y
x
b3u
b2u
b1g
b1u
b2g
(1) (2)
Figure 4.3: Frontier orbitals and their symmetries for the two simplest bi-OCC molecules — Ca-O-Ph3-O-Ca with C2v point group (1) and Ca-O-Ph-4-O-Ca with D2h point group (2). For both molecules
two lowest-lying orbitals have s-type character, while other orbitals have p-like character. Note:
Symmetry labels correspond to Q-Chem’s standard molecular orientation[161]. EOM-DEA-CCSD/ccpVDZ[H,C,O]/aug-cc-pwCVDZ-PP[Ca]/ECP10MDF[Ca]; isovalue 0.03. The respective occupations are
given in the SI.
42

Excited-state properties of (1) and (2) can be rationalized using symmetry analysis. The OCC transitions are derived by promoting electrons to the molecular orbitals composed of linear combinations of
p-orbitals. The degeneracy between the atomic p-orbitals is lifted such that the lowest p-orbitals are inplane, perpendicular to the Ca-O bond, followed by the out-of-plane orbitals, and p-orbitals parallel to
the Ca-O bond (this highest pair of orbitals is not shown in Fig. 4.3). Using point-group symmetry, we
can then determine the number of bright transitions for a given OCC molecule. For example, in (1) (C2v)
combinations of s-orbitals have a1 and b1 symmetries, and combinations of p-orbitals have a1, b1, b2, and
a2 symmetries. One of the in-plane excited states is derived by a1 → a1 and b1 → b1 transitions, yielding
bright transition of A1 symmetry. Transition a1 → b1 (of B1 symmetry) is also optically allowed. Similarly, two out-of-plane transitions have B1 and A2 symmetries, rendering the latter one dark. We remind
that for the transitions from 1A1, the symmetry of the target state is the same as the symmetry of the
transition whereas for the transitions from the open-shell triplet state, the symmetry of the target state is
the product of the symmetry of the transition and the initial triplet state (3B1 in (2)). Following the same
analysis, in (2), (D2h), we anticipate two bright transitions—B1u and B2u. Unfortunately, as anticipated
from the orbital picture, the lowest states are of an H-aggregate type (transition moments are parallel but
not aligned), rendering the lowest excited states dim in (1) and optically forbidden in (2).
Table 4.1 shows the results for (1) and (2) for the lowest transitions in the triplet manifold and the
respective oscillator strength (the results for the singlet manifold are given in the SI). As anticipated from
our analysis, (1) has three bright transitions with symmetries B1, A1, and B2 (corresponding to 3A1,
3B1,
and 3A2 states). The lowest state is optically allowed, but not very bright (fl=0.126). The respective FCF,
3A1(ν = 0) →3B1(ν = 0), is almost twice smaller than that of the fundamental transitions from the 3B1
and 3A2 states (0.5051 versus 0.9176 and 0.9604). Previously, some of us investigated this effect and found
that the diagonality of FCFs is improving proportionally to the distance between the optical centers.[162]
43

Table 4.1: Properties of four lowest excited statesa of (1) and (2) in the triplet manifold; the lowest triplet
state is 3B1 and 3B3u, respectively.
(1)
Ca-O-Ph-3-O-Ca
State Eex, eV (fl
) FCF
1
3A1 2.036 (0.126) 0.5051
1
3B2 2.058 (— ) —
2
3B1 2.085 (0.401) 0.9176
1
3A2 2.097 (0.543) 0.9604
(2)
Ca-O-Ph-4-O-Ca
State Eex, eV (fl
) FCF
1
3B2u 2.044 (— ) —
1
3B1u 2.060 (— ) —
1
3B1g 2.066 (0.518) 0.9663
1
3B2g 2.085 (0.544) 0.9792
a Excitation energies and oscillator strengths:
EOM-DEA-CCSD/cc-pVDZ[H,C,O]/aug-cc-pwCVDZ-PP[Ca]/ECP10MDF[Ca]; FCFs: PBE0-D3/def2-TZVPPD.
(2) has two highly diagonal bright states of symmetries B1g and B2g with FCFs 0.9663 and 0.9792,
respectively. The oscillator strengths of bright transitions in (2) are roughly twice as large as the oscillator
strength of the single-OCC CaOPh molecule (0.518 and 0.544 in bi-OCC versus 0.2075 and 0.2155 in singleOCC) and two subradiant states, following the same pattern as in Ca-X species[46]. Unfortunately, the
lowest transition is dark because of the H-dimer arrangement. In order to obtain superradiant lowest
states in these two model systems, the energies of the two highest molecular orbitals, which are derived
from the hybridized p and d orbitals parallel to the Ca-O bond, have to be brought below the other orbitals
in the p-manifold, which might be possible to achieve by using bulky substituents that destabilize the
other four orbitals. However, given the small size of the benzene scaffold, this route is impractical as
such substituents would deteriorate FCFs. Exploiting the oblique arrangement of the transition dipoles is
promising[151], but to fully explore such motifs larger scaffolds are necessary.
Overall, bi-OCC molecules are good candidates for laser cooling purposes because of good FCFs and
the flexibility they afford to tune excited states. Because the phenyl ring does not allow for complex
functionalization—even a small substituents (such as a CH3 group) interferes with the Ca-centered orbitals, reducing FCFs and increasing orbital mixing (see the SI)—we turn to the larger scaffolds, such as
naphthalene (systems (3)-(11) in Fig. 4.2). Naphthalene and its structural isomer azulene, were previously
considered in its unfunctionalized version for buffer-gas cooling.[163] Below we demonstrate how various
44

Table 4.2: Excitation energies (eV), oscillator strengths (in parentheses), and FCFs for the 0→0 transition
for low-lying excited states in the triplet manifold for (3) - (11)a
. The symmetry of the lowest triplet state
for each system is given next to the system number.
State Eex, eV (fl
) FCF
(3) 1
3Bu
2
3Bu 1.991 (— ) —
1
3Ag 2.005 (0.518) 0.9668
1
3Au 2.015 (— ) —
1
3Bg 2.030 (0.550) 0.9815
(4) 1
3B1
1
3A1 1.987 (0.123) 0.9264
1
3B2 2.013 (— ) —
2
3B1 2.013 (0.400) 0.9646
1
3A2 2.035 (0.550) 0.9796
(5) 1
3Bu
2
3Bu 1.994 (— ) —
1
3Ag 2.014 (0.504) 0.9631
1
3Au 2.020 (— ) —
1
3Bg 2.044 (0.540) 0.9817
(6) 1
3B1
1
3A1 1.979 (0.0001) 0.9409
2
3B1 2.006 (0.503) 0.9649
1
3B2 2.014 (— ) —
1
3A2 2.039 (0.540) 0.9354
(7) 1
3A
′
2
3A
′
1.984 (0.143) 0.8583
1
3A
′′ 2.013 (0.007) 0.9631
3
3A
′
2.015 (0.381) 0.9547
2
3A
′′ 2.042 (0.538) 0.9845
State Eex, eV (fl
) FCF
(8) 1
3A
′
2
3A
′
2.092 (0.077) 0.9145
1
3A
′′ 2.099 (0.013) 0.9603
3
3A
′
2.120 (0.417) 0.9657
2
3A
′′ 2.123 (0.513) 0.9831
(9) 1
3Bu
1
3Au 2.123 (— ) —
2
3Bu 2.130 (— ) —
1
3Bg 2.145 (0.523) 0.9855
1
3Ag 2.149 (0.493) 0.8048
(10) 1
3A
′
2
3A
′
2.074 (0.0002) 0.9619
3
3A
′
2.087 (0.507) 0.9721
1
3A
′′ 2.089 (0.015) 0.9793
2
3A
′′ 2.105 (0.525) 0.9846
(11) 1
3Bu
2
3Bu 2.090 (— ) —
1
3Au 2.100 (— ) —
1
3Ag 2.103 (0.505) 0.951
1
3Bg 2.115 (0.536) 0.970
aExcitation energies and oscillator strengths: EOM-DEA-CCSD cc-pVDZ[H,C,O,F] aug-cc-pwCVDZ-PP[Ca]
ECP10MDF[Ca]; FCFs: PBE0-D3/def2-TZVPPD.
structural modifications affect optical properties of bi-OCCs, illustrating the intensity distribution patterns
corresponding to the oblique transition moments.
Table 4.2 presents the results for naphthalene-based bi-OCCs (see Fig. 4.2). The naphthalene molecule
has three types of symmetry non-equivalent carbon atoms where we can install an OCC, giving rise to
the following five bi-OCC structures: four molecules with OCCs on equivalent carbon sites (systems (3),
(4), (5), and (6)) and one molecule with OCCs on non-equivalent sites (system (7)). In these systems, the
45

lowest singlet and triplet are also degenerate (at the EOM-DEA-CCSD level), and the number of effectively
unpaired electrons in the lowest singlet state is two.
With these arrangements, we observe different patterns of how the combined oscillator strength derived from the s → p transitions is distributed among low-lying manifold of excited states. In systems
with high symmetry, the oscillator strength is concentrated in a smaller number of states whereas in lowsymmetry cases it is distributed among more states so that in non-symmetric cases the number of optically
allowed states is the largest. As for (1) and (2), symmetry analysis helps to understand the pattern of intensity of the transitions in these larger systems. Systems (3) and (5) have the C2h symmetry and two
bright states. (4) and (6) have the C2v symmetry and three bright states whereas (7) has the Cs symmetry
and four bright states.
For C2h systems, both bright states are superradiant (the oscillator strength is on the order of 0.5,
which is twice larger than that in single-OCC molecules), whereas for C2v systems there are two superand one sub-radiant states, with the latter having low oscillator strength (0.123 for (4), and 0.0001 for (6)).
For system (7), both super- and sub- radiant states are bright. Here again the brighter states appear higher
in energy, which is not ideal for optical cycling. We note that the oscillator strength of the lowest state in
(7) is 0.143, to be compared with 0.126 in (1), illustrating that optical properties of these systems can be
improved by identifying structures with better oblique arrangement of the two dipoles.
As was previously shown, electron-withdrawing groups with high Hammett parameters improve FCFs
for single-OCC transitions for benzenes and naphthalenes[137, 164]. We extend this idea to their bi-OCC
counterparts by functionalizing two of the five systems with electron-withdrawing CF3 groups (systems
(8)-(11)). We functionalized systems (3) and (5) because the substituents are positioned at a distance away
from the OCCs such that they do not disrupt the OCC transitions. We then functionalized (3) and (5)
by one or two substituents to produce systems (10), (11) and (8), (9), respectively, to investigate stronger
electron-withdrawing effects.
46

For singly CF3 functionalized systems, (8) and (10), all four excited states are bright due to Cs symmetry point group, similar to system (7). The first excited A′
and A′′ states are subradiant (with oscillator
strengths less than 0.1). For (8), the superradiant transitions (to the third and fourth excited states) have
FCFs of 0.9657 and 0.9831, an improvement over the corresponding ground to excited state transitions of
their unfunctionalized counterpart, (5) (0.9631 and 0.9817). Similarly, (10)’s superradiant transitions have
FCFs of 0.9721 and 0.9846, an improvement over (3)’s (FCFs of 0.9668 and 0.9815).
Double functionalization with CF3 (systems (9) and (11)) recovers the symmetry of the parent species
and, consequently, their numbers of bright excited states. However, the FCFs for doubly functionalized
species deteriorate slightly. This is because functionalization often introduces more vibrational degrees of
freedom that electronic states can couple to, even while improving on the order of this coupling by utilizing
electronic effects. For example, the first two decay channels for the Bu transition of (5) are decays to the
stretching and bending modes with FCFs of 0.0132 and 0.0124, respectively. For the Bu transition of (9) the
two main channels are also stretching and bending, but with FCFs of 0.0041 and 0.0035, respectively. The
same characteristic applies to (11) and (3). This means that we succeed in improving the diagonality of
the transition electronically, however, in this particular case the introduced vibrational degrees of freedom
quench that effect and ultimately result in worsening of FCF. This suggests exploring heteroatomic motifs
in future work—e.g., introducing nitrogen or boron in the scaffold can significantly affect electronic states
(as was observed in organic di- and tri-radicals[165, 166, 167]), but without increasing the number of
degrees of freedom.
Additional flexibility in laser-coolable bi-OCCs is afforded by using different metals,[46, 142] which
makes the two centers distinguishable and independently addressable. Table 4.3 and Figs. 4.4 show the
results for the systems with one Ca and one Sr OCCs (systems (12) and (13) in Fig. 4.2). We note that
the lowest state is reasonably bright in both systems (fl ∼0.3). As we can see from the natural orbitals,
electronic transitions are localized, so that they can be addressed individually. This is different from the
47

Table 4.3: Properties of four lowest excited states of (12) and (13) in the triplet manifolda
; the lowest triplet
state is 1
3A
′
and 13A1, respectively.
(12)
Ca-O-Ph-3-O-Sr
State Eex, eV (fl
) FCF
2
3A
′
1.762 (0.261) 0.8122
1
3A
′′ 1.784 (0.251) 0.9293
3
3A
′
2.045 (0.378) 0.9256
2
3A
′′ 2.066 (0.165) 0.9594
(13)
Ca-O-Ph-4-O-Sr
State Eex, eV (fl
) FCF
1
3B1 1.627 (0.274) 0.9057
1
3B2 1.656 (0.283) 0.9279
2
3A1 1.918 (0.278) 0.9325
2
3B1 2.071 (0.275) 0.9538
a Excitation energies and oscillator strengths:
EOM-DEA-CCSD/cc-pVDZ[H,C,O]/aug-cc-pwCVDZ-PP[Ca,Sr]/ECP10MDF[Ca]/ECP28MDF[Sr]; FCFs:
PBE0-D3/def2-TZVPPD.
bimetallic bi-OCCs linked through the short acetylene bridge[46] where the transitions showed some degree of delocalization, consistent with stronger interactions between the two centers.
Finally, we return to the question of the nature of the ground state. To assess whether using higherlevel correlation treatments can break the degeneracy between the singlet and the triplet, we carried out
EOM-SF-CCSD(fT) calculations for systems (1) and (2). We found that the states remain degenerate within
our resolution (the gap between the states is less than 3 cm−1
, which is below the convergence threshold
for EOM-CC).These results confirm that both the lowest singlet and the lowest triplet have perfect diradical
character. We also carried out calculations of SOCs between the lowest singlet and triplet manifolds for
systems (1), (2), (12), and (13) (see the SI for details). The state-interaction calculations—diagonalization
of the model Hamiltonian including the SOCs between all low-lying states—show that the effect on the
lowest states is small, i.e., the triplet is stabilized by about 0.04 cm−1
in (1) and (2), and by about 0.4 cm−1
in
the Sr-containing bi-OCCs, (12) and (13). The effect of SOC is larger for the excited states—the transition
energy for the lowest triplet decreases by 1 cm−1
in (1) and (2) whereas for (12) and (13) the change
amounts to −50 cm−1
and −30 cm−1
, respectively. Hence, the transitions in the singlet and the triplet
manifolds can be distinguished spectroscopically. State-interaction calculations also provide an estimate
of the probability of inter-system crossings by analyzing the extent of mixing between the triplets and
singlets. The calculated leakage from the lowest bright triplet state to the lowest singlet state is ∼ 0.4%
48

y
x
a’
a’
a’
a’’
a’
a’
(12) (13)
x
z
Sr Ca
Sr Ca
b1
a1
b2
b1
a1
a1
Figure 4.4: Natural orbitals for the lowest triplet states of a mixed bi-OCC CaO-Ph-3-OSr (12) and
CaO-Ph-4-OSr (13). EOM-DEA-CCSD in a composite basis set cc-pVDZ[H,C,O]/aug-cc-pwCVDZPP[Ca,Sr]/ECP10MDF[Ca]/ECP28MDF[Sr]; isovalue 0.03.
and 0% for molecules (1) and (2), respectively. This branching ratio increases to 8.7% and 1.2% for the
Sr-containing molecules, (12) and (13).
In experiment, The bi-OCC molecules can be produced in the cryogenic buffer beam cell with the
methods similar to those for the single-OCC molecules [137, 168, 138]. This can be done by having the
single-OCC molecules react with the meta-stable metal atoms M. As the synthesis of bi-OCC molecules
comprises two steps, i.e., the two metal atoms collide with the ligand one after the other, it is necessary to
predict the the production yield of the bi-OCC molecules, and estimate their signal strength for setting up
the experiment scheme.
Here we start with the reaction rate equations of the single- and bi-OCC molecules:
dNs
dt
= kNLNM, (4.2)
49

dNb
dt
= kNsNM, (4.3)
where k is a reaction rate constant which is roughly the same in the two reactions, and NM, NL, Ns, Nb
are the number density of the meta-stable metal atom, ligand, single- and bi-OCC molecules in the cell,
respectively. By taking an approximation that the number density of ligand is a constant, the solution of
the above two equations are
Ns(t) = kNL
Z t
0
dt1NM(t1), (4.4)
Nb(t) = k
2NL
Z t
0
dt1
Z t1
0
dt2NM(t1)NM(t2) = Ns(t)
2
2NL
. (4.5)
The Eq.4.5 indicates a quadratic relation between the number densities of single- and bi-OCC molecules.
In our current setup[168], NL ≈ 1013cm−3
, Ns ≈ 1010cm−3
, hence the expected number density of biOCC molecules produced in each experimental cycle is Nb ≈ 106
cm−3
. Currently, the sensitivity of our
experimental setup (signal-to-noise ratio < 104
) is not sufficient for observing the fluorescence signal from
bi-OCC molecules, given that the anticipated signal strength is 10−4
compared to that from single-OCC
molecules. To address this challenge, we propose the excitation of ground state atoms using a continuouswave laser, which can amplify Ns by one to two orders of magnitude [169]. Consequently, this would
elevate Nb by two to four orders of magnitude, and such an enhancement will allow the observation of the
signal from bi-OCC molecules.
4.4 Conclusions
In conclusion, in this contribution we investigated details of the electronic structure of bi-OCCs. We have
shown that in bi-OCCs built upon benzene and naphthalene scaffolds by augmenting them with two OM
groups (M=Ca, Sr), the two optical centers are decoupled from each other and from the organic scaffold,
giving rise to degenerate perfect diradical singlet and triplet ground states. Inclusion of the SOC favors the
50

triplet states, but the effect is small (a few wave-numbers an most). However, SOC changes the energy of
the lowest transitions, such that the cycling transitions in the lowest singlet and lowest triplet manifolds
become distinguishable. We believe that the multiplicity of the ground states in these systems is not
important, as the lowest singlet and triplet states are very similar electronically. The probability of ISC
is relatively small (less than 1% for Ca-only molecules). Overall, we expect that the transitions between
the two manifolds should not pose problems for cycling because FCFs in both manifolds should be nearly
identical.
We observed that in bi-OCCs interaction of two closely-spaced dipoles gives rise to superradiant excited states, which is favorable for laser-cooling applications due to the increased oscillator strength. We
also showed that for the same scaffold, different relative placements of the radical groups yield different
amounts of excited states, consistent with the symmetry properties of the molecules. Furthermore, using
two different metals in one molecule allows for two OCCs to be individually addressable. We posit that
these bi-metallic bi-OCCs offer instrumental flexibility to the experiment because both metal centers can
be used simultaneously for laser cooling.
We believe that this study shows that increasing complexity of the system provides several avenues
for a chemical control over the electronic properties of OCCs. Using chemical design principles, one can
control the amount of the excited states in bi-OCC molecules and the degree of coupling between two
metal centers. Since chemist’s toolbox is essentially endless, many more handles and switches can be
introduced into the system to control its electronic properties.
51

Chapter 5
Vibronic Coupling Effects in the Photoelectron Spectrum of Ozone:
A Coupled-Cluster Approach
5.1 Introduction
Amongst the brotherhood of triatomic molecules, it cannot be denied that water (H2O) is the most important, the most highly studied, and the most well-understood. Beyond H2O, many triatomic molecules
that have an environmental, technological or biological importance have been subjected to many studies
and are understood to various levels of detail. Perhaps the most interesting such case is ozone (O3), which
has a vast number of important properties, a very rich history of study [170], and unlike the relatively
simple water molecule, a profound quantum-mechanical complexity [171]. Structurally, while we think of
ozone as having two distinct kinds of oxygen atoms (and an NMR experiment would reveal that), the full
molecular Hamiltonian does not distinguish between them. Rather, it yields the three equivalent structures
separated by a barrier that lies tantalizingly close to the O3 → O2 + O dissociation threshold (102.4 kJ/mol)
[172], as shown in Fig. 5.1. In reality, the energy levels of ozone all have a near triple (e + a) degeneracy,
albeit with a tunneling splitting so small that it can be ignored, along with a semi-infinite lifetime (despite
opposing opinion [173]). More than a half century ago, this intriguing aspect of the ozone molecule was
first discussed by Berry [174].
52

Figure 5.1: Two-dimensional slice of the potential energy surface of ozone in hyperspherical coordinates
at the value of hyperradiuns ρ=4 a.u. (the lowest contour is at 1 eV below the dissociation limit). The slice
shows three equivalent minima corresponding to C2v structures separated by large barriers. Reproduced
with permission from Ref. 175. Copyright 2003 American Institute of Physics.
Beyond the structural aspects of ozone, other mysteries surround this curious molecule. For example,
the distribution of the eighteen distinct 16O/17O/18O isotopologues in the Earth’s atmosphere differs from
what is expected based on the natural isotopic abundance, a puzzle that has been open for more than three
decades [176, 175].
Among quantum chemists, ozone has a notorious history, owing to its strong diradical character causing significant difficulties in calculations of its ostensibly simple ground-state molecular properties. An
early 1989 study [177, 178] by the Bartlett group and collaborators found that the CCSD+T(CCSD) method
predicted that the molecular equilibrium structure of ozone would have Cs symmetry (that is, the asymmetric stretching harmonic frequency computed by this method is imaginary), a finding that led to a search for
better treatment of non-iterative triple excitations, ultimately leading to the well-known CCSD(T) method
[179, 180, 181]. While CCSD(T) and higher-level coupled-cluster methods available today [182, 183, 184]
do a good job in describing the equilibrium structure and vibrational potential, an elaborate multireference
configuration interaction calculations can describe the entire ground-state surface out to the dissociation
53

LUMO (b1)
HOMO (a2)
val = 4a1 1a2 1b1 3b2
3 A1 2 B1 4 A1 5 A1 3 B1
1 B2 4 B1 6 A1 1 A2
6-31G*
EOMIP:
A1 A2 B1 B2
A1 12.40
A2 13.19
B2 12.48
B1 18.35
HOMO-1 (a1)
HOMO-2 (b2)
Figure 5.2: Frontier molecular orbitals of ozone; HF/6-31G*. The lowest states of the ozone cation are 2A1
and 2B2 derived by ionization from the HOMO-1 and HOMO-2, respectively.
limit [185, 175, 186], facilitating calculations of spectroscopy and chemical reactions of ozone. As such,
the quantum chemical understanding and fidelity for the ground electronic state of O3 is now at a mature
level.
Qualitatively, the challenge posed to electronic structure theory by ozone ultimately arises from its
closely spaced highest-occupied (a2) and lowest-unoccupied (b1) molecular orbitals (HOMO and LUMO,
respectively; see Fig. 5.2), giving rise to the following wavefunction:
X1A1 = C1[core]
6
(3a1)
2
(2b2)
2
(4a1)
2
(5a1)
2
(3b2)
2
(1b1)
2
(4b2)
2
(6a1)
2
(1a2)
2
(2b1)
0 + (5.1)
C2[core]
6
(3a1)
2
(2b2)
2
(4a1)
2
(5a1)
2
(3b2)
2
(1b1)
2
(4b2)
2
(6a1)
2
(1a2)
0
(2b1)
2
. (5.2)
54

The two electron configurations [· · · ]a
2
2
b
0
1
and [· · · ]a
0
2
b
2
1 mix strongly (the coefficients of these two leading configurations are C1 ≈0.9 and C2 ≈0.3, according to the EOM-SF-CCSD (EOM-CCSD with spinflip) calculations[187]), posing the aforementioned challenges with (especially single-reference) quantumchemical methods. A second feature consequence of the symmetry and energetic proximity of these two
orbitals is that the ozone cation (which is isoelectronic to the NO2 radical) has closely lying 2A1 and 2B2
electronic states derived by ionization from the HOMO-1 and HOMO-2, respectively (see Fig. 5.2). Like the
associated states in NO2, these two states are plagued by orbital symmetry breaking [188], a problem that
greatly complicates their quantum-chemical treatment. One of the many accomplishments of the Bartlett
group has been their integral role in the development of equation-of-motion coupled-cluster theory [189,
190, 191] (EOM-CC, also known as linear response coupled-cluster theory [192]). These methods provide a
very efficient and simple way to treat certain classes of electronic structure that are often termed “multireference problems” [54, 105], and are ideally suited to studying reactive intermediates, radicals, diradicals,
and electronically excited states. From a somewhat wider viewpoint, the existence of closely spaced electronic states always carries the potential for (possibly strong) vibronic coupling, a phenomenon that can
play an important role in molecular dynamics and spectroscopy. Indeed, one of the great successes of
EOM-CC methods has been in their ability — if and only if combined with vibronic coupling models — to
enable high-quality simulations of complicated electronic spectra. Such work provides important insights
into the nature of vibronic coupling in molecular systems, as has been exemplified by various application
studies (for example, see Refs. 193 and 194). In the case of ozone cation, the two lowest electronic states
separated by a small gap of ∼0.1 eV are coupled by the asymmetric stretch of b2 symmetry (Fig. 5.3).
The photoelectron spectrum of ozone was reported[195] in 1974. Later, much higher resolution of
the positions of vibronic peaks was obtained with pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE)
spectroscopy [196]. From the theory side, the vibronic photoelectron spectrum of ozone was modeled with
55

Symmetric stretch (a1)
1170 cm-1
Bend (a1)
720 cm-1
Asymmetric stretch (b2)
1130 cm-1
Figure 5.3: Three normal modes of the neutral ozone CCSDT/ANO1. Asymmetric stretching vibration of
b2 symmetry can couple the two lowest states of the cation (2A1 and 2B2).
a vibronic Hamiltonian parameterized using multi-reference configuration interaction method by Tarroni
and Carter[196].
5.2 Theoretical models and computational details
Ozone is a C2v molecule (following Mulliken’s convention [197], the molecule is placed in the yz-plane
and the molecular symmetry axis aligns with the z-direction) with three normal modes: symmetric stretch,
symmetric bend, and asymmetric stretch (Fig. 5.3). The asymmetric stretch is of b2 symmetry. The two
lowest electronic states of the ozone cation, 2A1 and 2B2, are very close in energy—separated by the vertical
gap (at the neutral’s geometry) of mere 0.123 eV (or 990 cm−1
, see Table 5.2)—and can be coupled by the
b2 vibration, giving rise to significant vibronic effects in the photoelectron spectrum. The vertical gap
corresponds to the difference in the energies of electronic states at the same geometry (here, the structure
of the neutral). While vertical energy gap is easy to compute, they are not observable experimentally. The
experimentally accessible value is the adiabatic gap T00—the energy difference between the vibronic levels
of the two state.
We simulate the vibronic states of the ozone cation using the model Hamiltonian of Köppel, Domcke,
and Cederbaum—the KDC Hamiltonian[198, 199, 200]. This is a multi-state and multi-mode Hamiltonian
defined in the basis of diabatic states. For the ozone cation, the basis comprises two quasi-diabatic states
(related to the 2A1 and 2B2 states at the minimum of neutral ozone), coupled by one mode (asymmetric
56

stretch, mode number 3). The model also includes two symmetric modes (modes number 1 and 2). The matrix elements of the vibronic Hamiltonian are expanded around the equilibrium geometry of neutral ozone
using the respective dimensionless normal coordinates, Qi
, and the corresponding harmonic frequencies,
ωi
. The overall form of the vibronic Hamiltonian is thus:
H = H01 +


V
(1) λ3Q3
λ3Q3 V
(2)

 , (5.3a)
where H0 contains kinetic energy operator and a potential energy term for the coupling mode
H0 =
1
2
X
3
i=1
−ωi
∂
2
∂Q2
i
!
+
1
2
ω3Q
2
3
, (5.3b)
and V
(1) and V
(2) describe the two diabatic states expanded over Q1 and Q2
V
(α) = E
(α) +
X
i∈{1,2}
κ
(α)
i Qi +
X
i,j∈{1,2}
κ
(α)
ij QiQj +
X
i,j,k∈{1,2}
κ
(α)
ijkQiQjQk +
X
i,j,k,l∈{1,2}
κ
(α)
ijklQiQjQkQl
.
(5.3c)
Here E(α)
are the vertical ionization energies calculated at the equilibrium geometry of the neutral, κ are
the coefficients of expansion of the potential along the fully symmetric coordinates, and λ is the linear diabatic coupling. Fig. 5.4 shows the resulting diabatic potential energy surfaces (the corresponding contour
plots are shown in the SI). The displacements along the two symmetric normal modes are clearly visible,
suggesting vibrational progressions along the bend and symmetric stretch.
We computed the parameters of the KDC Hamiltonian using ab initio CC and EOM-CC methods. [191,
54, 201, 202, 203, 105] We considered methods in which the CC expansion was truncated to singles and
doubles (CCSD), singles, doubles and triples (CCSDT), as well as singles, doubles, triples, and quadruples
(CCSDTQ).[184] We computed the geometry of the neutral ozone and its harmonic frequencies and normal
coordinates with CCSDT/ANO1 [204, 205]; the Cartesian coordinates are given in the SI.
57

Figure 5.4: Cuts of potential energy surfaces of the two diabatic states of the ozone cation (corresponding to
the 2A1 and 2B2 states at the equilibrium geometry of the neutral) shown as function of the two symmetric
modes in dimensionless normal coordinates, Q1 and Q2, at Q3=0.
At the optimized geometry, ozone has a bond length of 1.270 Å and a bond angle of 116.9◦
. The two
symmetric normal modes have frequencies 1,170 cm−1
and 720 cm−1
, and the asymmetric stretch has a
frequency of 1,130 cm−1
.
We described the states of the cation using EOM-CC for ionization energies (known as EOM-IPCC). [206] We used the definition of quasi-diabatic states by Ichino, Gauss, and Stanton[207] based on
the EOM-IP-CC method (EOMIP-CC-QD). We computed the linear diabatic coupling λ and the expansion
coefficients κ at the equilibrium geometry of the neutral; λ was computed with EOM-IP-CCSD-QD/ANO1.
A grid of 17x17 points stretching from −0.2 to 0.2 units along the dimensionless normal coordinates of
the symmetric modes was used to calculate the EOM-IP-CCSD/ANO1 energies that were fitted into the
4th order polynomial in these coordinate to extract the κ parameters. The computed value of the linear
diabatic coupling constant λ is 1,394 cm−1
. The complete set of KDC parameters is given in the SI.
We computed the vertical ionization energies at the geometry of the neutral ozone using a composite
method. The starting value is the complete basis set (CBS) extrapolation of the EOM-IP-CCSDT/cc-pCVnZ
energies with n = 5, 6. [208] These values were augmented with two corrections: the ∆Q (quadrupoles)
58

correction in the cc-pwCVTZ basis set and the relativistic correction calculated using EOM-IP-CCSD/ccpwCVTZ. [209] The error bars were estimated as follows: for the extrapolated CBS reference values, we
used half of the absolute value of the difference between the best ab initio value and the extrapolated value,
and for the remaining corrections (quadruples correction, ∆Q, and the relativistic correction) we used half
of the absolute value of the correction.
To compare the simulated photoelectron spectrum to the experimental one, we used the cross-section
ratio A1:B2 of 1:1.35.[210] Additionally, the stick spectrum was broadened with the Lorenzian envelopes
normalized to the feature intensities
f(x, xi
, Ii) = Ii
(x − xi)
2 + (γ/2)2
, (5.4)
where xi
is the position of the spectral peak, Ii
is its intensity and γ is the peak’s width.
The spectrum was computed using the xsim module in CFOUR, with the basis of 50 harmonic states in
each mode and 6,000 iterations of the Lanczos procedure, starting from the seed vector with no vibrational
excitations, i.e, |seed⟩ = |0 . . . 0⟩. [211] All electrons were correlated in all CC/EOM-CC calculations. All
calculations were performed using CFOUR. [212, 213]
5.3 Results and Discussion
We begin by discussing the accuracy of the computed electronic states. The vertical ionization energy for
the first excited state, E(1), computed using the composite method described above, is 12.827 eV. Our error
estimate for this value is 30 meV (see Table 5.1 for details). We note that the convergence of the vertical
gap between the two states is much faster and this value is estimated as 123±8 meV (see Table 5.2 for
details).
59

Table 5.1: Vertical ionization energies with the error estimates, eV.
Contribution 2A1
2B2 Error estimate
EOM-CCSDT/CBS 12.872 12.981 0.02
∆Q/pwCVTZ -0.037 -0.021 0.02
Relativistic/CCSD/pwCVTZ -0.008 -0.010 0.005
Final value, eV 12.827 12.950 0.03
Table 5.2: The vertical energy gap between the 2A1 and 2B2 states with error estimates, meV.
Contribution Vertical gap Error estimate
EOM-CCSDT/CBS 108.8 0.9
∆Q/pwCVTZ 16.5 8
Relativistic/CCSD/pwCVTZ -2.2 1.1
Final value, meV 123 8
Final value, cm−1
990 65
Figure 5.5: Comparison of the experimental (solid black line digitized from Ref. 195) and the simulated
photoelectron spectra shown as electron binding energies. The simulated spectrum is blue-shifted by
21 meV and the peaks were broadened using γ = 30 meV. Reproduced with permission from Ref. 195.
Copyright 1974 Royal Society of Chemistry.
60

Figure 5.5 compares the simulated spectrum to the lower-resolution experimental spectrum taken from
Ref. 195. Peak A is known to be a hot band, and does not appear in our (0 K) simulation. [210]The simulation
reproduces well the consecutive increase in the intensities of peaks B, C, D, and E as well as the spacing
between these peaks. The drop in the intensity at peak F is captured by the simulation. Starting from peak
G, the simulation shows discrepancy with the experiment. A sudden drop in the intensity past peak H is
not observed in the simulation. We discuss a likely source of this mismatch below.
Our simulation allows for an additional element of analysis—Figure 5.6 shows a decomposition of the
simulated spectrum from Figure 5.5. All lines that contribute to the spectrum are marked individually
and are color-coded indicating which electronic state’s transition intensity the peak draws from. The total
envelope of the spectrum is also decomposed showing contributions from each state. Our simulation
locates the minimum of the conical intersection (marked as CI) at 3,174 cm−1
above the origin (12.92 eV),
where origin means the lowest energy peak appearing in the spectrum. The adiabatic energy gap between
the two lowest vibrational states in each of the cation states is T00=1,368±65 cm−1
.
We assigned the vibronic peaks by comparing to the Franck–Condon simulation. To this end, the
spectrum is simulated once again, this time, however, setting the linear diabatic constant to zero (λ= 0).
This yields the spectra of the two electronic states at an equivalent level of theory but without vibronic
coupling, i.e., the combined Franck–Condon spectra of the two states. Figure 5.7 shows this spectrum.
The non-coupled spectrum is easy to assign using the labels that mark the symmetry of the electronic
state, A1 or B2, and the vibrational state (ν1ν2ν3), where νi
is the number of quanta in mode i (i = 1 for
the symmetric stretch, i = 2 for the symmetric bend and i = 3 for the asymmetric stretch). The assigned
spectrum shows progressions in the symmetric bend. There is one such progression in each electronic state.
Additionally for each state, there is also another progression in the symmetric bend with one excitation in
the symmetric stretch.
61

12.4 12.6 12.8 13.0 13.2
B C D E F CI D0
0 2000 4000 6000
eV
cm 1
12.4 12.6 12.8 13.0 13.2
B C D E F CI D0
0 2000 4000 6000
eV
cm 1
no coupling
Figure 5.6: Simulated photoelectron spectrum of ozone shown as electron binding energies. Bottom axis
shows energy scale in eV. Top axis shows energy offset from the origin in cm−1
. Stick spectrum shows
positions and intensities of all simulated states. Blue and orange colors mark states of A1 and B2 symmetry, respectively. Gray vertical lines with captions on top indicate positions of features measured by
the PFI-ZEKE experiment. [196] The simulated spectrum was shifted to match the PFI-ZEKE experimental
origin by 21 meV. D0 marks the dissociation threshold of O+
3
. [196] Gray dotted line marks the energy of
the minimum of the conical intersection (CI) as located by our Hamiltonian. The right panel shows the
uncoupled spectrum (the same as in Fig. 5.7 but formatted in a matching manner).
62

Figure 5.7: Simulated Franck–Condon photoelectron spectrum of ozone with vibronic coupling removed.
The decomposition of the spectrum presented on Figure 5.6 provides additional insight. The spectrum
shows that peaks B and C are almost purely of the 2A1 electronic and a1 vibrational character. Starting
from peak D, the contributions from two states are of equal magnitude. It is also clear that the following
peaks are mixtures of many vibronic peaks. At the energy of about 2,000 cm−1
above the origin the
density of vibronic peaks increases significantly. This value can be compared to the minimum of the conical
intersection located at about 1000 cm−1 higher in energy, at 3,174 cm−1
above the origin. Our Hamiltonian
is not suitable for describing the dissociation of the molecule, therefore, we expect a discrepancy with the
experiment as the energy gets closer to the dissociation threshold located at 4,898±3 cm−1
above the
origin. [196]
Figure 5.8 shows the assigned spectrum and Table 5.3 lists the decomposition of all peaks in the region
of up to 3100 cm−1
. We compare the resulting assignment with the high-resolution PFI-ZEKE spectrum
from Ref. 196. The PFI-ZEKE spectrum is the best source of information on the location of the peaks,
63

Figure 5.8: Comparison of the simulated spectrum with the PFI-ZEKE experiment.[196] The simulated
spectrum was blue-shifted by 21 meV = 170 cm−1
. Color of the simulated peaks marks the symmetry of
the vibronic states: A1 are colored blue and B2 are colored orange. See Table 5.3 for a detailed listing of
state assignments. The bottom panel reproduced with permission from Ref. 196. Copyright 2005 American
Institute of Physics.
especially the origin, against which the origin peak of the simulation is aligned. The origin of the simulated spectrum is red-shifted by 170 cm−1
relative to the experimental origin at 101,020.5 cm−1
.[196] This
discrepancy is within our error estimate for the vertical ionization energy (30 meV=240 cm−1
).
The comparison with the high-resolution spectrum (Fig. 5.8) reveals more details. Lines of the PFIZEKE experiment and our simulation match very well. Especially the states of A1 symmetry are well
aligned with the experimental features. States close to the origin are similar to the non-coupled states.
The origin peak is of A1(0,0,0) character, the first two excitations in the symmetric bend, A1(0,1,0) and
A1(0,2,0) are also very similar to the non-coupled states, whereas the higher excitations in this progression
show significant mixing. The same progression with one vibrational quanta in the symmetric stretch is
more interesting. The first of its peaks is very well aligned with experimentally visible feature, which was
previously assigned as the origin of the second electronic state. The second peak in this series, A1(1,1,0), has
very high similarity to its uncoupled version. It is a good candidate to assign the experimental, unassigned
feature above 102,600 cm−1
.
64

Figure 5.9: Comparison of the simulated spectrum to an earlier simulation by Tarroni and Carter (assigned
lines). [214]
States of B2 symmetry, on the other hand, do not align well with the experiment. The first such peak,
close to the 12.64 eV mark in Figure 5.8, is assigned to the A1(0,0,1) state. This is a vibronic state that
gains intensity owing to the coupling to the electronic 2B2 state. This vibronic peak lies in the part of the
spectrum where bands of oxygen (which is an impurity in the gas sample) are visible in the experiment.
The origin of the 2B2 state is very similar to the uncoupled B2(0,0,0) state, but it is located in an empty area
of the experimental spectrum. We use the difference in the energy between this peak and the origin peak
to report the adiabatic energy gap between the two cationic states, which, according to our simulation, is
T00=1,368 cm−1
. This value is reported with an error estimate equal to the error estimate for the vertical
excitation energy gap (65 cm−1
), which is listed Table 5.2. The next peak, slightly above a 12.76 eV mark
in Figure 5.8, corresponds to one excitation of the symmetric bend in the 2B2 state, but as the label on the
figure shows, it is only about 55% similar to the uncoupled state.
Finally, Figure 5.9 compares our spectrum to an earlier accurate simulation by Tarroni and Carter[214].
This earlier work used internally contracted multi reference configuration interaction method in a large
basis set (cc-pV5Z) to thoroughly scan the vicinity of the minima on both potentials. Similarly to our work,
Tarroni and Carter did not account for the potential at distances reaching the dissociation limit. In their
65

work, Tarroni and Carter used a diabatization method based on the adiabatic energies and configuration
interaction coefficients, which avoids calculation of the non adiabatic couplings.
Figure 5.9 includes only the lines from the earlier simulation that were tabulated with an assignment in
the article.[214] The comparison shows that both match well with the experiment. Both simulations also
agree in the assignment of the first progression in the bending mode. The previous simulation, however,
shows smaller spacing between the remaining lines, yielding much higher congestion of the spectrum.
Additionally, the origin of the second state falls at lower energy (as compared to our simulation), which
makes it align well with the peak that was assigned in the PFI-ZEKE experiment also as the origin of the
second state. While an additional simulation of the line intensities would allow for the most complete
comparison to the experimental spectrum, the strong alignment of the simulated peaks of A1 symmetry
with the experimental features leads us to believe that our simulation offers the best assignment of the
photoelectron spectrum of ozone to date.
5.4 Conclusion
In this exploration of the photoelectron spectrum of the ozone molecule, we have deployed cutting-edge
high-order CC models together with a vibronic Hamiltonian approach to spectroscopy beyond the Born–
Oppenheimer approximation [198, 199, 200]. We parameterized the Hamiltonian using EOM-IP-CC calculations and the formalism of quasi-diabatic states of Ichino, Gauss and Stanton. [207] The results of the
simulation are in excellent agreement with the experimental spectrum from the ionization onset to about
2,500 cm−1 higher (the restriction arising from the local parametrization of the Hamiltonian). Analysis
of the spectrum and its simulation underscore the importance of vibronic coupling effects in the ozone
cation. Results of this work offer a state-of-the-art insight into the details of the spectrum and allows for
an assignment of the experimental results. Simulated peaks that gain intensity from the A˜B2 state are
absent in the PFI-ZEKE spectrum, which offers an interesting avenue for further investigations.
66

Table 5.3: Assignment and decomposition of the eigenvectors of the ozone cation. The first column shows
the offset of simulated peaks from the origin in cm−1
. Note that the decomposition uses the basis of
uncoupled states visible in Fig. 5.7, as such the eigenvector decomposition does not list components along
vibronic states of a mixed symmetry, e.g., A1(001) which might, artificially, appear as a lack of mixing.
Offset Symmetry Assignment Eigenvector
0 A1 A1(000) +0.99 A1(000)
618 A1 A1(010) +0.99 A1(010)
915 B2 A1(001)
1076 A1 A1(100) −0.99 A1(100)
1222 A1 A1(020) +0.97 A1(020)
1368 B2 B2(000) +0.97 B2(000)
1498 B2 +0.10 B2(000)
1684 A1 A1(110) +0.96 A1(110)
1796 A1 A1(030) +0.87 A1(030) + 0.16 A1(110) + 0.13 A1(020) − 0.13 A1(040)
1848 A1 −0.31 A1(030)
1921 B2 B2(010) +0.74 B2(010) − 0.18 B2(020) − 0.11 B2(000)
1977 B2 −0.24 B2(010)
2085 B2 +0.52 B2(010)
2132 A1 −0.25 A1(030) − 0.25 A1(040) − 0.13 A1(120) + 0.13 A1(050)
2143 A1
2275 A1 A1(120) +0.86 A1(120) + 0.12 A1(040)
2362 A1 −0.62 A1(040) + 0.40 A1(120) − 0.17 A1(030) + 0.11 A1(050)
2437 A1 −0.48 A1(040) − 0.11 A1(120)
2453 B2 B2(020) +0.54 B2(020) + 0.25 B2(010) − 0.17 B2(030)
2537 B2
2576 B2 +0.27 B2(020)
2671 A1 −0.48 A1(040) − 0.26 A1(050) − 0.19 A1(130) − 0.14 A1(060)
2735 B2 +0.62 B2(020)
2743 A1
2780 B2 +0.22 B2(020)
2862 A1 A1(130) +0.74 A1(130) + 0.13 A1(120) − 0.13 A1(050)
2903 A1 −0.42 A1(130)
2969 A1 −0.25 A1(130) + 0.17 A1(050) + 0.12 A1(040)
2978 B2 −0.24 B2(110) − 0.19 B2(020) + 0.12 B2(010)
3006 B2 −0.20 B2(030) + 0.17 B2(110) − 0.15 B2(020)
3045 A1 A1(050) +0.79 A1(050)
3054 B2 +0.29 B2(110) − 0.25 B2(030) − 0.13 B2(020)
67

Chapter 6
Photoswitching Molecules Functionalized with Optical Cycling Centers
Provide a Novel Platform for Studying Chemical Transformations in
Ultracold Molecules
6.1 Introduction
The CaO- group attached to a molecular scaffold can function as an optical cycling center (OCC). Molecules
functionalized with OCCs can be laser-cooled by scattering thousands of photons with a minimal loss of
the population from the cycling pair of states (preferably, the ground state and the lowest excited state).
Laser-cooled molecules are exploited in quantum technologies as well as in studies of fundamental physical phenomena[35, 215, 216, 36, 67, 70, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226] and ultracold
chemistry [227, 24, 228, 25, 34, 26, 27, 229, 230, 98, 231, 232, 233]. The idea of using OCCs in polyatomic
molecules builds on the success of laser-cooling of diatomic molecules [9, 10, 17, 18, 234, 19, 20, 235, 236, 23,
237, 22, 238]. The electronic structure of laser-cooled molecules (all the way through hyperfine structure),
their applications, and recent experimental advances are thoroughly covered in recent reviews [64, 239,
65, 240, 241, 242, 243].
68

Figure 6.1: Molecules studied in this work. Counterclockwise from the left top: CaOH, azobenzene (AB),
para-CaO-azobenzene (pAB), and bis-(para-CaO)-azobenzene (bpAB).
The CaO- group can function as an OCC because it supplies the molecule with an access to an electronically excited state that spontaneously decays almost exclusively to the molecular (electronic and vibrational) ground state [11, 12, 44, 244, 48, 47, 41, 99]. Other quantum functional groups suitable for laser
cooling, such as SrO- and YbO-, have also been identified [13, 245, 246, 45, 247, 14, 30, 248, 249, 250, 251,
16, 252, 253, 254, 255, 256, 66, 257, 258, 259, 260, 99, 261, 262, 263, 264, 63, 265, 46, 266].
One—thus-far only hypothesized but exciting—possibility is to use quantum functional groups, such
as CaO- OCCs, for precise optical sensing of chemical events. For example, it would be desirable to have
an optical reporter that can detect a chemical change, such as a conformational change or a chemical
reaction, in a part of the molecule distant from the reporter. Because OCCs are electronically isolated
from the rest of the molecule by design, it might appear as a contradiction that an OCC can be used for
such a sensing. However, OCCs rely on transitions with narrow linewidths, which may feature small
but detectable shifts and exhibit other delicate effects such as Fermi resonances arising from accidental
degeneracies of vibrational states [264], all of which may serve as an optical signature of chemical reactions.
As a model reaction to investigate this concept, we have chosen photoisomerization of azobenzene
(AB) [267, 268] (shown in Fig. 6.1). The photoisomerization of AB from the more stable trans-conformation
69

to the less stable cis-conformation is triggered by the ultraviolet light of the 300-400 nm wavelength (3.1-
4.1 eV) [269, 270], which is different from the OCC transition (about 650 nm or 1.9 eV). Thus, it should
be possible to excite the respective transitions independently. In the dark, the cis-isomer has a substantial lifetime, slowly relaxing back the trans-isomer at room temperature. We hypothesize that an OCC
located far from the azo bridge can optically report on photoisomerization—a model chemical event that
is relatively easy to probe in the gas phase.
Furthermore, controlling the position and dipole alignment of OCCs through a chemical or physical
handle (such as photoswitching) brings an interesting new dimension into quantum information science
(QIS). For this reason, we also study azobenzene decorated with two CaO- OCCs located one on each of
the two benzene rings. The possibility of chemical sensing with OCCs and controlling the spatial arrangement of multiple OCCs by means of an “orthogonal” chemical transformation are both of fundamental
importance and of interest to the field of QIS.
In this contribution we investigate the electronic structure of molecules comprising the CaO- and
azobenzene (AB) moieties: para-CaO-azobenzene (pAB) and bis-(para-CaO)-azobenzene (bpAB); the relevant structures are shown in Fig. 6.1. The electronic states of pAB are of three types: the electronic states
similar to these of the scaffold (AB-like), the electronic states localized on the OCC (OCC-like), and the
electronic states of a mixed character.
The quantum-chemical description of such systems is far from trivial, owing to their open-shell character[44, 46, 266]. We use methods from the equation-of-motion coupled-cluster (EOM-CC) family to
describe relevant electronic states of these molecules. EOM-CC provides a robust and versatile approach
to electronically excited and open-shell species [105, 54, 202, 203]. As shown in Fig. 6.2, different variants of EOM-CC provide access to different types of states. Molecules with a single unpaired electron
(CaOH, pAB) are well described by EOM-EA-CC (EOM-CC for electron attachment). EOM-EE-CC (EOMCC for excitation energies) describes excited states of closed shell molecules (AB), but it can also be used for
70

Figure 6.2: EOM-CC models used in this work. Color illustrates the action of the EOM operator (only
single excitations are shown). An EOM method is defined by specifying the reference determinant and the
type of generalized excitation operator—e.g., spin and electron-conserving operators are used in EOM-EE,
spin-flipping operators are used in in EOM-SF, and non-particle conserving operators are used in EOM-EA
and EOM-DEA.
open-shell species such as pAB. EOM-DEA-CC (EOM-CC for double electron attachment) and EOM-SF-CC
(EOM-CC with spin-flipping excitations) are the preferred methods for diradicals (bpAB)[153, 187].
The paper is organized as follows. The next section provides details about the computational approach.
In section 6.3 we review the electronic structure of the prototype systems (AB, Ca-OH) and then characterize the electronic states of the pAB and bpAB. We focus on the three questions: (1) How are the states
of the scaffold impacted by the OCC? (2) How are the states of the OCC impacted by the scaffold? (3) Can
we employ the OCC as a spectroscopic probe of the isomerization reaction? We then discuss preliminary
results of the synthetic efforts and outline future directions.
6.2 Theoretical Methods and Computational Details
The EOM-CC family of methods[54, 202, 203, 105] employs the following ansatz
Ψ = ReT Φ0, (6.1)
71

where Φ0 is the Hartree–Fock reference determinant and T is the cluster operator
T =
X
ia
t
a
i a
†
i +
1
2!2
X
ijab
t
ab
ij a
†
b
†
ij + . . . , (6.2)
where the operators, a
†
, b†
, i, j are the fermionic creation and annihilation operators associated with
molecular orbitals φa, φb, φi
, φj . The choice of reference Φ0 determines the separation between the
occupied and virtual spaces [191]. The amplitudes t are found by solving the coupled-cluster equations
obtained by projecting the similarity transformed Schrödinger equation
exp(−T)H exp(T)Φ0 = ECCΦ0 (6.3)
onto the space spanned by the reference determinant Φ0, all singly {Φ
a
i
}, doubly {Φ
ab
ij }, . . . substituted
determinants. In this work we use EOM-CC models with the expansion truncated after the second term,
EOM-CC with single and double substitutions (EOM-CCSD).
The EOM operator, R, has the form that depends on a particular flavor of EOM (as illustrated in Fig.
6.2). We use EOM-CC methods that are best suited for each type of electronic states studied here
• EOM-EA-CCSD [271, 190, 272]. In this method the Hartree–Fock reference corresponds to a closedshell cation and the EOM operator R has one more creation than annihilation operators, giving rise
to the EOM-EA-CCSD wave-functions of the neutral states. This method is suitable for CaOH and
pAB.
• EOM-EE-CCSD. [189] The EOM-EE excitation operator preserves the number of electrons and generates electronically excited states [54]. For the closed-shell references, the excited states are naturally spin-adapted, forming singlet or triplet excitations. EOM-EE-CCSD can also be applied to
72

open-shell references, although in this case the wavefunctions may be slightly or severely spincontaminated [273, 274, 105], depending on the type of an excitation. This method is suitable for AB
and pAB, with the caveats discussed below.
• EOM-SF-CCSD[154, 187, 275, 276, 155]. The EOM-SF operator preserves the number of electrons,
but flips the spin of one electron. This method uses a high-spin (Ms = 1) triplet state as a reference
and describes the singlet and the low-spin (Ms = 0) components of the triplet states. The description
of the higher-lying states is not accurate due to spin-incompleteness.
• EOM-DEA-CCSD [190, 277, 278, 153]. The EOM-DEA operator increases the number of electrons by
two. The Hartree–Fock reference for this method is a closed-shell dication. This method is suitable
for bpAB; it can describe both the diradical states as well as higher excited states, which was exploited
in previous studies of molecules with two OCCs (bi-OCCs)[46, 98, 266].
Below we provide more details about computational protocols employed.
6.2.1 Computational protocols
We described the electronic states of azobenzene functionalized with a single OCC using the EOM-EACCSD method. We also used this method to optimize the structure of the molecule in its ground electronic
state. We then used this method to compute the low-lying excited states at the optimized geometry. We
note that EOM-EA-CCSD can only accurately describe states dominated by the OCC transitions because
states involving molecular orbitals of the scaffold will appear as 2p1h (two-particle-one-hole) excitations.
To gain access to these states, we calculated excited states with EOM-EE-CCSD starting from an open-shell
doublet reference.
73

We described the azobenzene molecule functionalized with two OCCs with the same methods as in our
previous work on bi-OCCs [46, 98, 266]. We optimized geometry of the ground electronic state with EOMSF-CCSD[275] and computed the excitation energies at the optimized geometry with the EOM-DEA-CCSD
method.
We used Dunning’s cc-pVDZ basis set for geometry optimization and aug-cc-pVDZ for excitation energies [102, 279, 103]. All CC and EOM-CC calculations were carried out with Q-Chem [280, 52]. Orbitals
and structures were visualized with the IQmol and Jmol molecular visualization tools [112, 281].
In the supplemental information (SI), we provide Cartesian coordinates of relevant structures, Franck–
Condon factors for the cycling transitions, as well as results of the calculations with time-dependent density functional theory.
6.2.2 Wavefunction analysis
We analyzed transitions between the electronic states using natural transition orbitals (NTOs)[96], which
provide the most compact representation of the transition. NTOs are defined as the left and right singular
vectors of the one-particle transition density matrix[282, 283, 53]:
γ
f←i
pq = ⟨Ψf |p
†
q|Ψi⟩, (6.4)
where Ψi and Ψf are the two states involved in the transition. γ
f←i
pq is related to one-particle transition
density, ρ
f←i
(rh, rp),
ρ
f←i
(rh, rp) = N
Z
Ψi(rh, r2, . . . , rn)Ψf (rp, r2, . . . , rn) dr2 . . . drn
=
X
pq
γ
f←i
pq φp(rh)φq(rp),
(6.5)
where N is the number of electrons.
74

Visual inspection of the NTOs enables characterization of the electronic states. However insightful,
such analysis can be tedious and somewhat subjective. The wavefunction analysis tools afford direct extraction of information about the character of the wavefunction. We use the TheoDORE package to automate this analysis[284]. In particular, we are interested in the extent of charge transfer between the scaffold and OCCs, which can be used to quantify the extent of electronic coupling between the two moieties.
Charge-transfer numbers (CTNs) is an exciton descriptor, which provides a measure of charge transfer
between different parts of the system. The partition of electrons between fragments A and B is given by
integrating the γ
f←i over restricted regions
ΩAB =
Z
A
drh
Z
B
drp |γ(rh, rp)|
2
. (6.6)
Thus, CTNs can be interpreted as the number of electrons transferred between A and B. In these calculations, the only user-required input is a partition of the molecule into fragments [284]. This analysis
depends on the definition of the region boundaries, similarly to the Mulliken charges.
6.3 Results and Discussion
6.3.1 Electronic structure of AB and CaOH
We begin by reviewing the electronic states of AB and CaOH. Fig. 6.3 shows relevant molecular orbitals
(MOs).
The ground state of AB is a closed-shell singlet. The excited states of AB are derived by promotion of an
electron from the nitrogen (azo) bridge—relevant MOs are π and lone-pair n (see Fig. 6.3). The first excited
singlet state is a dark nπ∗
transition. The second excited singlet state is a bright ππ∗
transition. At the
EOM-EE-CCSD/aug-cc-pVDZ level of theory, the energies of these transitions are 3.055 eV and 4.465 eV,
respectively. In both states, electrons of the nitrogen bridge MOs are promoted to an antibonding orbital,
75

Figure 6.3: NTOs of the relevant electronic transitions in the trans-AB (left) and CaOH (right).
effectively weakening the double bond and lowering the barrier along the cis-trans rotation coordinate.
Below, we refer to the excited states generated by transitions localized on the AB scaffold as the AB-like
states, or the scaffold-like states.
Next, we review the electronic states of CaOH, which can be described as a hydrogen-terminated OCC.
Extensive spectroscopic data is available for this molecule [285, 286, 287, 288, 289, 290, 291, 279]. CaOH
is one of a few polyatomic molecules that have been laser-cooled, paving the way for advanced quantumstate control experiments [292, 15, 115, 293, 294, 295, 296]. CaOH is a doublet radical with one unpaired
electron localized on the metal. The low-lying electronic states of CaOH resemble the states of an alkali
atom (Fig. 6.3). Both the ground and the low-lying excited electronic states can be described as a closedshell core with a single valence electron occupying hydrogen-atom-like orbitals. In the ground state (X
state), the unpaired electron is localized on the Ca atom, resembling the 2S atomic state. The first pair of
excited states (A and B states) are 2P-like states, with degeneracy lifted in non-linear structures. The X→A
transition energy for CaOH is around 2 eV. Higher CaOH states also follow this pattern of a localized
atomic-like transitions of a single unpaired electron. Below we refer to these states as OCC-like states.
This type of states is commonly found in molecules functionalized with the CaO- group [292, 99, 297, 260].
76

6.3.2 Electronic states of pAB
The electronic states of pAB can be classified as either AB-like, OCC-like, or as states of a mixed character.
The key question is the extent to which the AB-like and OCC-like states perturb each other. Previously,
the effect of the electronic states of the scaffold on the OCC-like states was explored by Dickerson and
coworkers [259, 258]. They found that when the energy of the first excited state in the unfunctionalized
scaffold comes close to the energy of the cycling transition, the two electronic states begin to interact,
spoiling the localized character of the OCC transitions. Here we apply similar analysis to electronic states
of functionalized AB.
We begin exploring an interplay between the OCC and AB moieties by inspecting the electronic states
of a singly functionalized molecule, pAB, in which the CaO- group is located at the para position with
respect to the nitrogen bridge (see Fig. 6.1). Placing the CaO- group in the para position minimizes the
steric repulsion, which is favorable for laser cooling [260].
The electronic states of pAB, the molecule comprising the CaOH and AB moieties, can be correlated to
the states of its building blocks. Similar to other molecules functionalized with CaO- groups, the low-lying
states of trans-pAB are clearly identifiable as OCC-like. The first state with a non-OCC features appears
at 3.5 eV as the 6th excited doublet state. This state derives from a π
∗ orbital on the azo bridge with a p/d
admixture on the metal. The transition to this state differs from the S0 → S1 transition of trans-AB as it
does not originate from a promotion of a lone-pair n to the π
∗ orbital; rather it is the s-type orbital of
the OCC which is excited. Interestingly, the next two states recover the OCC-like character and can be
identified as s-like and p/d-like. The 9th excited state located vertically at 4.0 eV is of a mixed character.
Fig. 6.4 summarizes the electronic states of trans-pAB.The states of an OCC-like character are preserved
for excitation energies below the first excited state of the unfunctionalized AB, but some of the OCC-like
states are also present above this threshold. The states of a mixed character appear close and above this
threshold.
77

Figure 6.4: Electronic states of trans-pAB (central panel; EOM-EA-CCSD/aug-cc-pVDZ at the EOMEA-CCSD/cc-pVDZ geometry) as a combination of trans-AB states (the column on the left; EOM-EECCSD/aug-cc-pVDZ at the CCSD/cc-pVDZ geometry), and the OCC states (CaOH; two columns on the
right; EOM-EA-CCSD/aug-cc-pVDZ at the EOM-EA-CCSD/cc-pVDZ geometry). Energies of the respective ground states are set to zero.
78

0
1
2
3
4
Energy (eV)
p out-of-plane
d node-in-plane
higher s
p in-plane
d doughnut
d in-and-out
p/d out-of-plane+ *
d/p in-plane
p/d out-of-plane+ *
p in-plane
d doughnut
d in-and-out
p/d out-of-plane+ *
d/p in-plane
p/d out-of-plane+ *
p out-of-plane
d node-in-plane
higher s
4
(
*
)(2.75)
2
(
*
)(0.83)
4
(
*
)(2.75)
2
( s) (0.76)
4
(
*
) (2.72)
t-pAB (EOM-EA) t-pAB (EOM-EE)
Figure 6.5: Trans-pAB at the EOM-EA-CCSD/cc-pVDZ geometry. Comparison of the EOM-EA-CCSD/augcc-pVDZ states (two columns on the left) with the states computed with EOM-EE-CCSD/aug-cc-pVDZ
(three columns on the right). The rightmost column shows the EOM-EE-CCSD states absent in the EOMEA-CCSD calculations; some of these extra states are spin-contaminated (red color); the corresponding
values of

S
2

are given in the parentheses.
79

2.0
3.0
4.0
Energy (eV)
p p d d d s pd mix mix mix 4AB4AB2AB4AB
0.0
0.2
0.4
0.6
0.8
State characters assignment based on NTO analysis
CTNs
CaO to AB
AB to CaO
AB to AB
CaO to CaO
Figure 6.6: Electronic states of trans-pAB and the character of the underlying wavefunctions. Top: state
energies (relative to the ground state). Bottom: CTNs characterizing the transitions from the ground state.
The states are ordered from left to right to match three groups: first are the OCC-like states (labeled s,
p, and d), second are the states of a mixed character (mix), and the last are the AB-like states (labeled
2S+1AB). Compare with Fig. 6.5. EOM-EE-CCSD/aug-cc-pVDZ at the EOM-EA-CCSD/cc-pVDZ ground
state’s geometry.
80

As explained in the methods section, different flavors of EOM-CCSD are suitable for different types of
states. In the case of pAB, we expect that EOM-EA-CCSD describes well the transitions localized on the
OCC, but yields less accurate results for the transitions involving the scaffold orbitals. To gain a better
handle on these states, we employ EOM-EE-CCSD. We also present wavefunction analysis of the resulting
states.
The method of choice for the treatment of the excited states of AB is EOM-EE, however, using this
method for the pAB molecule is complicated by its open-shell character, which can lead to small or severe
spin-contamination [273, 274, 105]. Generally, states derived by excitations to or from open-shell orbitals
are well-behaved[105] whereas the states derived by excitations from doubly occupied to unoccupied orbitals are poorly described because of the spin-incompleteness of the EOM-EE-CCSD ansatz (this problem
can be remedied by including selected triple excitations[298]).
In trans-pAB, the spin-contamination of the reference Hartree–Fock determinant is small; it is also
rather small for the most EOM low-lying states (∼0.002). The EOM-EE-CCSD method yields the spectrum
of trans-pAB that qualitatively agrees with the EOM-EA-CCSD one. The EOM-EE-CCSD excitation energies are higher by about 0.15 eV. The transition dipole moments agree within 30% with the EOM-EA-CCSD
ones.
The EOM-EE-CCSD calculations also yield a set of states that do not have counterparts among the
EOM-EA-CCSD states. As anticipated, the EOM-EE-CCSD method yields excited states derived by promoting electrons localized on the AB moiety. These states can be classified as AB-like states of π → π
∗
type. One of the new states should again be characterized as a state of a mixed character as it derives from
the π → s(Ca) transition. Some of the AB-like states (especially quartets) show high spin-contamination,
which indicates that their proper description would require inclusion of higher terms in the EOM-CC
expansion (see Fig. 25 and the corresponding discussion in Ref. 105). Fig. 6.5 highlights the differences
between EOM-EA-CCSD and EOM-EE-CCSD results.
81

Having described this manifold of excited states, we conclude this section by presenting the CTN-based
analysis of their corresponding wavefunctions (pAB is partitioned into CaO and the rest). Here, CTNs can
be interpreted as the number of electrons transferred between the OCC and AB moieties—perfectly isolated
localized transitions can be identified by zero CTNs. Fig. 6.6 shows the CTNs and the state energies. We
highlight the following observations:
• Among states classified as OCC-like, the transitions are indeed localized on the CaO- moiety.
• The main outlier is the state visually assigned as d (doughnut). Fig. 6.7 shows NTOs for that state.
Although visually the state appears to be localized on the CaO- group, the CTNs analysis shows that
a significant portion of the electronic density is ascribed to the AB fragment. A similar observation about the analogous B2Σ
+ state of SrOH was made in the previous study, where a significant
contribution to the state was attributed to hydrogen’s s orbital [244].
• The sum of all CTNs gives the norm of the one particle transition density, ∥γ∥. The difference,
1 − ∥γ∥, quantifies the multiply excited character of the state. Spin-contaminated states are all
ranking the highest in this metric, confirming that higher excited configurations are needed for a
proper description of these states.
• CTNs highlight that the states labeled as of a mixed character are formed by promoting the ground
state’s s(Ca) electron to an orbital delocalized over the CaO- and AB moieties, or the other way
around.
6.3.3 OCC as a spectroscopic probe to monitor isomerization
The key feature of azobenzene, which carries over to its functionalized versions, is its ability to assume
two distinct geometries, the cis- and trans-isomers. Well known in photochemistry, this is a new element
in the context of molecules functionalized with OCCs.
82

Figure 6.7: Particle and hole NTOs for the d(Ca) (doughnut) state of trans-pAB. EOM-EE-CCSD/aug-ccpVDZ at the EOM-EA-CCSD/cc-pVDZ geometry of the ground state.
Table 6.1: Excitation energies, eV, and oscillator strengths (in parentheses) of the trans-pAB, cis-pAB, and
CaOH calculated with EOM-EA-CCSD/aug-cc-pVDZ at the EOM-EA-CCSD/cc-pVDZ geometriesa
.
Transition trans-pAB cis-pAB CaOH
X → A 1.964 (0.262) 1.964 (0.260) 1.915 (0.291)
X → B 1.983 (0.272) 1.980 (0.268) 1.915 (0.291)
a EOM convergence threshold 10−10
.
The different structures of the two isomers of pAB make the two systems distinct, just as if the OCC
were attached to two different scaffolds. The impact of scaffold on the properties of an OCC, which is of a
central importance in the context of laser-cooling, have been extensively studied [44, 46, 98, 47, 257, 258,
98, 259, 260, 99, 261, 262, 263, 253, 266]. Yet, the effect of the scaffold isomerization on the OCC states has
not been investigated. Furthermore, what makes pAB special is that cis- and trans-isomers can be interconverted. Such chemical complexity, unique to polyatomic molecules, can bring a useful resource to QIS
applications.
We now consider the other isomer of pAB, cis-pAB, and compare the low-lying excited states of the
two isomers, using the same computational protocol as above. The low-lying electronic states of cis-pAB
are OCC-like. Table 6.1 highlights similarities of the X→A and X→B transitions in trans-pAB, cis-pAB,
and CaOH.
The key result of this calculation is the impact of the isomerization on the excitation energy of the
cycling transition, X → A. Although this value can be affected by a change of the ligand, the computed
excitation energy change due to the isomerization is minute (on the order of 1 meV). Thus, pAB results
83

Figure 6.8: The azobenzene molecule functionalized with two OCCs, bpAB. The isomerization reduces the
distance between the two Ca atoms from 16.3 Å to 10.8 Å. Geometries optimized with EOM-SF-CCSD/ccpVDZ.
show that AB acts, to a large extent, as an inert scaffold, carrying the OCC in space without impacting the
characteristics of the cycling transition. This can be rationalized by the observation that in the gas phase,
the OCC does not experience large variations in its environment between the trans- and cis- isomers.
Hence, one can imagine a scenario in which the inert AB scaffold can move the OCC to a different local
environment.
To create a situation when the environment of an OCC changes more drastically upon isomerization,
we consider the AB scaffold with both phenyl rings functionalized in the para position with the CaOmoieties (bpAB), as shown in Fig. 6.8. In contrast to the isomerization of pAB, where the CaO- group did
not sense large changes to its chemical environment, isomerization of bpAB changes the distance between
the two polar CaO- groups.
The optimized geometries of the two isomers show that the structure of the CaO- moieties is preserved,
meaning that the interactions between the two OCCs are weak and do not induce changes in the overall
bonding pattern. The excitation energy from the ground state to the first exited state is shifted; this is an
expected outcome related to the change in the electron withdrawing strength of the ligand.
The trans-to-cis isomerization reduces the distance between the two Ca atoms by almost one third of its
initial value (see Fig. 6.8). Thus, we expect to observe a response to the isomerization in the OCCs optical
84

Table 6.2: Changes in excitation energy (eV) for the X → A and X → B transitions upon isomerization
in bare pAB and bpAB with an interacting group at both ends; EOM-EA-CCSD/aug-cc-pVDZ (pAB) and
EOM-DEA-CCSD/aug-cc-pVDZ (bpAB)a,b
.
pAB (X → A) bpAB (X → A) pAB (X → B) bpAB (X → B)
cis 1.964 1.968 1.980 2.002
trans 1.964 2.000 1.983 2.006
change ≲ 0.001 0.032 0.003 0.004
a Using geometries optimized as described in the methods section.
b EOM convergence threshold 10−10
.
Figure 6.9: (a) Synthetic routes for asymmetric azobenzene derivatives using Mills reaction (left) and diazotization method (right). (b) Synthetic routes for symmetric azobenzene derivatives.
properties. Table 6.2 shows the change in the excitation energy upon isomerization. In contrast to pAB,
bpAB shows a much more appreciable change in the excitation energy.
We also computed excitation energies for the second transition (X → B). The changes are smaller
than for the X→A transition, likely because the out-of-plane orbitals are aligned at an angle relative to the
aromatic rings, which impacts the through-space interaction (see Fig. S1 in the SI).
6.3.4 Synthetic Efforts: Preliminary results and future work
CaO-azobenzene can be produced by reacting the precursor ligand 4-hydroxyazobenzene (trans form purchased from Sigma Aldrich) with metastable Ca atoms generated by laser ablation in a cryogenic buffer
gas cell, as previously discussed in the context of the production of CaOC6H4 molecules and their functionalized derivatives [99, 261, 264]. However, initial experiments have revealed that the yellow powder
85

of 4-hydroxyazobenzene undergoes a color change to black at ∼ 100 ◦C, well below its melting point of
155 ◦C. This suggests the thermal decomposition of azobenzene [299], making it unlikely to generate sufficient precursor vapors for the reaction. To address this issue, the precursor can be functionalized with
electron-withdrawing groups, such as fluorine atoms, to enhance its thermal stability[299]. Alternatively, a
solid-phase method [260] can be adopted. A mixture of CaH2 and precursor powder (1:1 molar ratio), with
polyethylene glycol or silver powder as a binder, can be thoroughly mixed and ground before being pressed
into a pellet for direct laser ablation. Preliminary results with the Sr-equivalent have shown two peaks
in the excitation spectra. Further experiments of using CaH2 and precursor to produce CaO-azobenzene
will be performed. In addition, the precursor ligand only exists in the trans isomeric form, enabling photoswitching to the cis isomer upon laser illumination. Specifically, the precursor 4-hydroxybenzene has been
reported to undergo photoswitching at ∼ 360 nm (3.44 eV)[269, 270]. A 360 nm UV LED or pulsed Nd:YAG
laser at 355 nm (3.49 eV) can be employed to initiate the photoswitching of CaO-azobenzene. To measure
the branching ratios of the X → A and X → B transitions of these molecules, dispersed laser-induced
fluorescence (DLIF) spectroscopy can be taken either with a pulsed dye laser used in the previous works
[261, 264], or with a continuous-wave (CW) dye laser which is currently under installation in our lab. The
narrow line-width and the broad scanning range of the CW dye laser allow us to significantly suppress
the background noise and enable us to obtain high-resolution spectra of these azobenzene molecules.
With many azobenzene derivatives commercially available, properties of the OCC-photoswitch system
can be tuned using varying functionalizations. Several synthetic routes can access those derivatives that
are not commercially available. A widely used synthetic route involves diazotizing an aromatic amine
using sodium nitrite in the presence of strong acid to afford the resulting diazonium salt, which can then
be coupled with phenol to give azobenzene product (Figure 6.9a). [300, 301] Alternatively, the Mills reaction
can also be used where an aromatic amine is oxidized to a nitroso intermediate (Figure 6.9a). This species
can then be reacted with a separate aromatic amine to yield the azobenzene. For symmetric azobenzenes
86

(bi-OCCs), the desired product can be afforded through reductive coupling of aromatic nitro compounds
using a reducing agent or oxidative coupling of aromatic amines using an oxidizing agent to achieve the
same transformation (Figure 6.9 b) [300, 301].
6.4 Conclusions and outlook
We have presented a comprehensive study of electronically excited states in a model photoswitch molecule
functionalized with one or two OCCs. Using state-of-the art ab initio methods, the calculations reveal that
some states of the functionalized azobenzene preserve their original character while other states aquire
mixed character. The excited-state analysis shows how the atomic-like and scaffold-like character becomes
mixed for excitation energies close and above the first excitation energy of the unfunctionalized scaffold.
We have characterized the new type of states that arises from the mixing. The complete description of
these states is challenging and use of higher order CC methods might be necessary not only to achieve
high accuracy but also for qualitative agreement. Interestingly, we have found OCC-like states even at
energies above the energy of the first excited state of the scaffold.
Following the extensive characterization of the relevant electronic states, we have considered spectroscopic signatures of the scaffold’s isomerization. The calculations show that the OCC can witness and
report on the isomerization process. The azobenzene’s scaffold affects the OCCs excitation energy thus
allowing it to report on its changing environment. Functionalization of the azobenzene’s frame with a second OCC results in a model system in which the first OCC can detect anisotropy of the space through which
the isomerization reaction takes it. Probing this process via OCCs differs from traditional UV-Vis transient
absorption as one can use lower excitation energies and follow much narrower transitions. Furthermore,
the UV-Vis measurements are typically probing ensembles of molecules. OCCs capable of cycling about
100 photons allows for such measurement to be performed on individual, trapped, ultracold molecules. The
measurement requires only access to the lowest excited states and assures little to no changes in the final
87

states of the molecules. Thus, in the regime when the experiment targets an individual, trapped, ultracold
molecule OCC probes offer a great advantage.
The presented model system of OCCs attached to organic scaffolds opens up several venues for future
research. Possibilities for modifications of the organic scaffold are as rich as rich is the chemistry of
photoswitches. In particular, one may explore the stability of the system, range of structural changes
induced by the isomerization, or fine-tuning of the transition properties. The latter can also be studied by
considering different (or multiple) metals in the OCCs. In this work, we did not discuss the mechanism
of the isomerization—rather, we focused on exploring the idea of OCCs as optical reporters. The result
indicates that OCCs can indeed be employed to probe chemical transformations, in particular, for studying
ultracold, trapped molecules. One can even imagine the photoswitch considered in this work as a building
block for producing larger, multi-switch systems, where the distance and interactions between many OCCs
are manipulated by structural realignments.
88

Chapter 7
Future work
This Chapter presents two ideas building on the results from previous chapters. These research directions
have not been studied before and can provide a motivation for future work.
7.1 Novel organic ligands
Chapter 2 discussed the functionalization of organic molecules with metal atoms in order to form lasercoolable molecules. The field of organic chemistry is a resource that has not been fully explored in the
context of laser cooling of molecules.
Figure 7.1: Franck-Condon factor tuning in intermediate-sized molecules. Functionalization of the phenyl
group imporves Franck-Condon factors, impact of functionalization on smaller groups (illustrated on the
example of the vinyl group) has not been studied yet.
89

The experimentally most successful avenue has been the study of derivatives of alcohols. In order to
increase the size of the molecular ligand, while preserving the vibrational closure, significant experimental
effort is currently directed towards study of phenols, which thanks to their rigid structure are expected to
be the next largest laser-cooled molecules.[99, 252] This leap leaves a room for exploration of intermediatesized organic molecules – molecules larger than methoxides, yet not as large as phenoxides. A model of
such molecules can be the ethoxide.[297] Such systems were dismissed as unpromising as their vibrational
structure typically includes low-frequency modes, related to rotations around single bonds, which are
promoting vibrational branching.
In recent years, however, new results have emerged in the study of phenoxides, which call for reconsideration of this point. Namely, the CCI group at UCLA has demonstrated that the matrix of FCFs in an
organic molecule can be turned into a more diagonal one by means of chemical functionalization.[258]
The specific design pattern needed for this desired change lies in an increase of the metal-oxygen bond,
achieved by means of substitution of the benzene ring with fluorine atoms. A similar substitution in smaller
hydrocarbons is possible but has not yet been explored.
Small hydrocarbons with that feature a chain of carbon-carbon bonds can benefit from the fluorination
combined with another type of functionalization. Echoing back to the undesired low-frequency modes, a
simple method of removing such features is by forming double and triple bonds in the molecule. While
triply bonded groups, like acetylene or cyanide have already attracted significant attention in the field, the
doubly bonded chains appeared only in the context of phenoxides. The advantage of considering double
bonds stems not only from increased rigidity of the molecule. Double bonds additionally build the ability
to establish resonance effects in molecular chain. The effect that is of most relevance in this context is
the electron-withdrawing potential of the fluorine atoms substituted along the double bonds, which can in
build up the electron withdrawing potential of the ligand and turn the matrix of FCFs into more diagonal
ones. Figure 7.1 gives a graphical overview of this proposal.
90

Figure 7.2: A functional group of charge, examples of -NH+
3
and -TeF+
2
.
7.2 Functional ligand of charge
The progress in cooling of neutral molecules is not matched by the progress in direct laser cooling of
charged species.[63] Chapter 3 formulated pathways for building new optical cycling centers that would
improve the matrix of FCFs in charged diatomics. The use of multiple bonds established a promising path
towards finding better optical cycling centers. If the charged optical cycling center is found to be a good
scatterer of photons, functionalization of larger ligands would follow just as in case of neutral species. This
effort together with other attempts was driven by the desire to find a charged analog of the metal-oxygen
structure, which is so effective in laser cooling of neutral molecules.
An alternative approach to the pathway outlined in above is rooted in a decoupling of the optical
cycling center and a charge center. As discussed throughout this work, a use of organic scaffolds enables
chemical functionalization. Production of molecules with two optical cycling centers is already broadly
discussed in the field.[45, 98, 266] However, a molecule decorated with a single optical cycling center can
also be decorated with a different type of functional group, namely a functional group of charge.
As of yet, the functional group of charge is not a well established object, but conceptually it should
be a chemical moiety that is capable of hosting an electronic charge and hold on to it strong enough that
during an electronic excitation localized on the optical cycling center the charge should still prefer to be
localized at the same spot. A candidate for a function group of charge would be a moiety that is build of a
protonated cation well known in chemistry, e.g., analogs of the ammonium ion NH+
4
; the functional group
91

of charge would then be the -NH+
3
group attached to a molecule functionalized with an optical cycling
center. A recent work on Ra containing ions suggests that this is a promising direction.[253]
An ammonium-ion-based functional group does not need to be the optimal choice, but a chemical
functionalization offers many alternative candidates. The perfluorinated ions -NF+
3
enhance its electronwithdrawing ability, making it a better host of the positive charge. Another alternative is built by replacing
the central atom by a heavier version, e.g., -PH+
3
, -AsH+
3
, or even a heavier SbH+
3
, or BiH+
3
. The larger,
central atom would allow for a greater separation between the hydrogen atoms increasing probability of a
larger stabilization of the charge, which can be delocalized over larger volume. The heavier central atoms
also still be functionalized with a fluorine forming, -PF+
3
, -AsF+
3
, SrH+
3
, or BiF+
3
.
Another avenue for possible exploration is a change of the central atom to the one from the neighbouring groups. An example of such system could be the sulfonium ion SH+
3
, which would translate to the
group of charge of the form -SH+
2
. A similar reasoning leads to a series of analogs: -SeH+
2
, -TeH+
2
, -PoH+
2
and the versions with substituted hydrogen. Figure 7.2 presents a model use of the functional group of
charge.
92

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115

Appendix A
Towards Ultracold Organic Chemistry: Prospects of Laser Cooling Large
Organic Molecules
A.1 Theoretical Methods and Computational Details
We use equation-of-motion coupled-cluster (EOM-CC) approach, a versatile electronic-structure tool capable of describing a variety of multiconfigurational wave functions within the single-reference formalism[54, 202, 203]. For molecules with the doublet target states (i.e., CaBz, CaCp, CaPy, CaPh), we employ
EOM-EA-CC (EOM-CC for electron-attached states) with single and double substitutions (EOM-EA-CCSD)
in which the cationic reference state is described by CCSD and the excitation operators R are of the 1-
particle and 2-particle-1-hole types.
We treat molecules with diradical character (i.e., three isomers of CaBzCa) by EOM-DEA-CC (EOMCC for double electron attachment) starting from the closed-shell dicationic reference. Here we employ
EOM-DEA-CC with single and double substitutions (EOM-DEA-CCSD) in which the reference state is
described by CCSD and the excitation operators R are of the 2-particle and 3-particle-1-hole types. We
used EOM-DEA-CCSD for energies and properties calculations and CCSD/EOM-SF-CCSD (EOM-CCSD
with spin-flipping operators) for structures’ optimizations[275].
116

Previous studies demonstrated importance of core-valence correlation in alkaline earth metal derivatives[302, 303]. Therefore, we employed the aug-cc-pwCV(D,T)Z-PP basis sets[304] with small-core pseudopotentials[305, 306] to treat Ca and aug-cc-pV(D,T)Z[102] to treat remaining atoms. All geometry optimizations and frequency calculations were performed using aug-cc-pwCVDZ-PP[Ca]/aug-cc-pVDZ[H,C,N,O]
basis set. All subsequent single-point calculations, including Dyson orbital and NTOs calculations, were
performed using a composite basis set aug-cc-pwCVTZ-PP[Ca]/aug-cc-pVTZ[H,C,N,O] basis set.
To visually characterize the spatial distribution of the unpaired electron we plot Dyson orbitals, defined
as the overlap between N and N − 1 electron wavefunctions:[307, 308, 309]:
φ
d
IF (1) = √
N
Z
Φ
N
F
(1, . . . , n)ΦN−1
I
(2, . . . , n)d2 . . . dn (A.1)
where I and F denote the two many-body states (e.g., of the neutral and of the cation).
To quantify the locality of electronic transition we use natural transition orbitals (NTOs). NTOs are
obtained from singular value decomposition of the one-particle transition density matrix and allow concise
description of the exciton wavefunction in terms of a small number of hole and particle orbitals:
Ψexc(rh, re) = X
K
σKψ
h
K(rh)ψ
e
K(re), (A.2)
where σK are singular values, ψ
h
K(rh) are hole orbitals, and ψ
e
K(re) are particle orbitals.
The NTO analysis was carried out using libwfa library[53]. Calculations of FCFs were carried out
within the double-harmonic parallel-mode approximation using ezSpectrum[310]. All electronic structure
calculations were performed using the Q-Chem package[51, 52].
117

1.379
1.380
1.381
1.381
1.415
1.416
1.416
1.417
1.431
1.430
1.434
1.432
2.367
2.326
2.296
Y.XXX
1.429
1.429
1.430
1.429
1.429
1.430
1.429
1.429
1.432
2.242
2.219
2.204
2.207
1.389
1.392
1.394
1.392
1.391
1.390
1.388
1.389
1.435
1.435
1.437
1.436
1.429
1.429
1.432
2.376
2.345
2.306
1.429
1.429
1.428
CaCp
CaPy
iso-CaPy
2.043
2.009
2.009
2.010
1.340
1.343
1.346
1.345
1.416
1.414
1.413
1.413
1.404
1.404
1.404
1.404 1.406
1.406
1.406
1.406
CaPh
2.416
2.389
2.386
2.390
1.425
1.424
1.424
1.423
1.408
1.408
1.408
1.408 1.405
1.405
1.405
1.405
CaBz
Figure A.1: Structures and equilibrium bond lengths of the candidate molecules in different electronic
states (color-coordinated as indicated).
A.2 Results of EOM-CC calculations
118

Table A.1: FCFs for the decay transitions to the ground X2A1 state in CaBz, CaPh, and CaPy. ν1 is Ca-ring
stretching mode, ν2 is Ca-ring bending mode.
Transition CaBz CaPh CaPy
A2B1 B2B2 C
2A1 A2B1 B2B2 C
2A1 A2B1 B2B2 C
2A1
0
0
0
0.7595 0.8341 0.8849 0.8329 0.8521 0.8326 0.7186 0.7692 0.9219
1
0
1
0.1318 0.1135 0.1055 0.1232 0.1070 0.1164 0.1278 0.1153 0.0499
1
0
2
0.0093 0.0064 0.0049 0.0074 0.0067 0.0069 0.0084 0.0065 0.0008
2
0
2
0.0639 0.0296 0.0000 0.0000 0.0000 0.0000 0.0892 0.0729 0.0185
2
0
4
0.0081 0.0016 0.0000 0.0000 0.0000 0.0000 0.0166 0.0104 0.0006
SUM 0.9726 0.9852 0.9953 0.9636 0.9658 0.9559 0.9606 0.9743 0.9917
Table A.2: FCFs for the decay transitions to the ground state in CaCp and iso-CaPy. ν1 is Ca-ring stretching
mode.
Transition CaCp iso-CaPy
A2E1 B2A1 A2A′′ B2A′ C
2A′
0
0
0
0.5435 0.8896 0.5232 0.5515 0.8108
1
0
1
0.3286 0.0965 0.3277 0.3170 0.1507
1
0
2
0.0994 0.0052 0.0895 0.0814 0.0157
1
0
3
0.0200 0.0000 0.0138 0.0122 0.0012
SUM 0.9915 0.9914 0.9542 0.9620 0.9783
119

Table A.3: Vertical excitation energies (Eex, eV), oscillator strengths (fl
), Head-Gordon’s indices (nu, nu,nl),
and the number of entangled states (ZHE) in the three isomers of CaBzCa calculated using EOM-DEACCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C].
ortho-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1A1 0.000 – 1.65 1.83 – 1
3B1 0.000 – 2.04 2.00 –
1
1B1 1.764 0.235 2.05 2.10 4.32 1
3A1 1.703 0.171 2.05 2.13 3.72
2
1A1 1.869 0.162 2.06 2.19 4.13 1
3B2 1.809 0.000 2.04 2.19 3.97
1
1A2 1.875 0.000 2.04 2.23 4.10 1
3A2 1.937 0.436 2.05 2.22 4.27
1
1B2 1.953 0.445 2.05 2.21 4.09 2
3B1 1.938 0.279 2.05 2.22 4.28
3
1A1 2.156 0.211 2.08 2.07 4.37 2
3A1 2.237 0.182 2.06 2.18 3.96
2
1B1 2.178 0.140 2.08 2.01 4.35 3
3B1 2.344 0.139 2.06 2.21 4.12
meta-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1A1 0.000 – 1.75 1.90 – 1
3B1 0.000 – 2.04 2.00 –
1
1B1 1.806 0.248 2.04 2.18 4.23 1
3A1 1.773 0.211 2.04 2.19 3.99
1
1A2 1.898 0.000 2.04 2.23 4.14 1
3B2 1.865 0.000 2.04 2.21 4.10
2
1A1 1.914 0.221 2.05 2.20 4.18 2
3B1 1.931 0.256 2.04 2.23 4.28
1
1B2 1.965 0.446 2.04 2.22 4.17 1
3A2 1.953 0.439 2.04 2.23 4.25
3
1A1 2.264 0.132 2.08 2.02 4.30 2
3A1 2.280 0.229 2.05 2.21 4.05
2
1B1 2.272 0.234 2.06 2.06 4.27 3
3B1 2.369 0.136 2.05 2.22 4.14
para-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1Ag 0.000 – 2.02 2.00 – 1
3B3u 0.000 – 2.04 2.00 –
1
1B1g 1.909 0.000 2.04 2.23 4.22 1
3B2u 1.909 0.000 2.04 2.23 4.22
1
1B2u 1.934 0.433 2.04 2.23 4.27 1
3B1g 1.935 0.433 2.04 2.23 4.27
1
1B2g 1.951 0.000 2.04 2.23 4.22 1
3B1u 1.951 0.000 2.04 2.23 4.22
1
1B1u 1.977 0.434 2.04 2.23 4.27 1
3B2g 1.975 0.432 2.04 2.23 4.26
1
1B3u 2.336 0.400 2.05 2.21 4.18 1
3Ag 2.345 0.426 2.05 2.22 4.16
2
1Ag 2.391 0.000 2.05 2.22 4.14 2
3B3u 2.391 0.000 2.05 2.22 4.13
120

Table A.4: Vertical excitation energies (Eex, eV), oscillator strengths (fl
), Head-Gordon’s indices (nu, nu,nl),
and the number of entangled states (ZHE) in the three isomers of CaBzCa calculated using EOM-DEACCSD/aug-cc-pwCVTZ-PP[Ca]/cc-pVTZ[H,C].
ortho-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1A1 0.000 – 1.68 1.85 – 1
3B1 0.000 – 2.04 2.00 –
1
1B1 1.759 0.224 2.04 2.12 4.29 1
3A1 1.707 0.172 2.04 2.14 3.78
1
1A2 1.868 0.000 2.04 2.22 4.09 1
3B2 1.812 0.000 2.04 2.19 3.99
2
1A1 1.874 0.160 2.05 2.18 4.11 2
3B1 1.927 0.270 2.04 2.21 4.26
1
1B2 1.946 0.424 2.04 2.21 4.08 1
3A2 1.932 0.419 2.04 2.22 4.25
3
1A1 2.116 0.207 2.07 2.08 4.24 2
3A1 2.137 0.160 2.05 2.18 3.98
2
1B1 2.125 0.156 2.06 2.04 4.28 3
3B1 2.240 0.122 2.05 2.20 4.09
meta-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1A1 0.000 – 1.78 1.92 – 1
3B1 0.000 – 2.04 2.00 –
1
1B1 1.806 0.235 2.04 2.18 4.21 1
3A1 1.781 0.206 2.04 2.19 4.02
1
1A2 1.891 0.000 2.04 2.22 4.13 1
3B2 1.864 0.000 2.04 2.21 4.10
2
1A1 1.911 0.219 2.04 2.20 4.18 2
3B1 1.928 0.251 2.04 2.22 4.26
1
1B2 1.955 0.424 2.04 2.22 4.16 1
3A2 1.947 0.420 2.04 2.22 4.23
2
1B1 2.182 0.216 2.05 2.16 4.14 2
3A1 2.171 0.202 2.05 2.21 4.06
3
1A1 2.209 0.142 2.06 2.10 4.14 3
3B1 2.252 0.115 2.05 2.21 4.11
para-CaBzCa
State Eex fl nu nu,nl ZHE State Eex fl nu nu,nl ZHE
1
1Ag 0.000 – 2.03 2.00 – 1
3B3u 0.000 – 2.04 2.00 –
1
1B1g 1.905 0.000 2.04 2.22 4.20 1
3B2u 1.910 0.000 2.04 2.22 4.19
1
1B2u 1.930 0.412 2.04 2.22 4.25 1
3B1g 1.934 0.414 2.04 2.22 4.25
1
1B2g 1.941 0.000 2.04 2.22 4.19 1
3B1u 1.945 0.000 2.04 2.22 4.19
1
1B1u 1.965 0.409 2.04 2.22 4.23 1
3B2g 1.969 0.411 2.04 2.22 4.23
1
1B3u 2.202 0.343 2.05 2.21 4.14 1
3Ag 2.211 0.358 2.05 2.22 4.13
2
1Ag 2.250 0.000 2.05 2.22 4.10 2
3B3u 2.258 0.000 2.05 2.22 4.10
121

Figure A.2: Frontier natural orbitals (isovalue = 0.02) of the ground state of ortho-, meta-, and para-CaBzCa
(top to bottom) calculated using EOM-DEA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C].
122

A.3 Relevant Cartesian geometries
CaBz, EOM-EA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 248.7168800043 hartrees
Final energy is -267.605356162170
Ca 0.0000000000 -0.0000000554 -2.7029342725
C 0.0000000000 0.0000000096 -0.2867878578
C 0.0000000000 -1.1977785026 0.4848803108
H 0.0000000000 -2.1768657482 -0.0234284275
C 0.0000000000 -1.2120378996 1.8933003235
H 0.0000000000 -2.1657761186 2.4365350650
C 0.0000000000 0.0000000000 2.6042854584
H 0.0000000000 0.0000000000 3.7008207736
C 0.0000000000 1.2120378932 1.8933003344
H 0.0000000000 2.1657761081 2.4365350915
C 0.0000000000 1.1977785257 0.4848803091
H 0.0000000000 2.1768657854 -0.0234284087
CaPh, EOM-EA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 324.3318642969 hartrees
Final energy is -342.733306784075
Ca -0.5294437615 -0.6786410532 -2.6947380177
O -0.3139801719 -0.4491434456 -0.6762999225
C -0.1725751769 -0.2953401389 0.6474927637
C -0.2504565933 -1.4057816246 1.5221617644
C -0.1016631843 -1.2410705889 2.9082065218
C 0.1283475968 0.0313867903 3.4602491984
C 0.2072226005 1.1396656132 2.5986240178
C 0.0593788396 0.9822477737 1.2116668203
H -0.4297522948 -2.3990497852 1.0935476934
H -0.1664758225 -2.1183715184 3.5633819083
H 0.2440632198 0.1570631480 4.5420903182
H 0.3861264567 2.1407814460 3.0095637536
H 0.1212527917 1.8478472839 0.5413147803
CaPy, EOM-EA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 205.1079665945 hartrees
Final energy is -245.665697902604
Ca -1.3543828933 -3.0052423495 -0.7565557560
N -0.4397660034 -0.9961468601 -0.3656270151
C 0.0166390864 -0.0804724574 -1.3057194090
C 0.5220808454 1.0588180217 -0.6882710235
C 0.3755862121 0.8480825960 0.7235900390
C -0.2091548651 -0.4053582520 0.8707088335
H -0.0469139291 -0.2983920280 -2.3779576138
H 0.9442821749 1.9343320685 -1.1866493975
H 0.6626126562 1.5291271778 1.5279225395
H -0.4802100844 -0.9218395170 1.7986454032
123

iso-CaPy, EOM-EA-CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 217.8085682669 hartrees
Final energy is -245.681924082277
Ca 0.1570212204 -0.0997132837 -0.9932606852
C -0.3721400511 -1.0849659541 1.3650930334
C 0.9156761455 0.6876463555 1.3767752221
C -0.4105697835 1.1810032427 1.4032959516
C -1.2514125872 0.0236266780 1.3956683795
N 0.9481011359 -0.6897240716 1.3258958458
H 1.8455807318 1.2636306997 1.4073707976
H -0.7241341555 2.2255533360 1.4745350966
H -2.3421587260 -0.0015737891 1.4598580598
H -0.6326866303 -2.1475760133 1.3848899988
ortho-CaBzCa triplet, CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 307.02070664 hartrees
Ca 2.6496646000 0.0000000000 -1.3837142175
Ca -2.6496646000 0.0000000000 -1.3837142175
C 0.7247701000 0.0000000000 0.0132250825
C -0.7247701000 0.0000000000 0.0132250825
C 1.3746917000 0.0000000000 1.2765469825
C -1.3746917000 0.0000000000 1.2765469825
C 0.6981259000 0.0000000000 2.5215574825
C -0.6981259000 0.0000000000 2.5215574825
H 2.4873156000 0.0000000000 1.3400619825
H -2.4873156000 0.0000000000 1.3400619825
H 1.2575273000 0.0000000000 3.4662450825
H -1.2575273000 0.0000000000 3.4662450825
meta-CaBzCa triplet, CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 303.39083053 hartrees
Ca 3.1004558000 0.0000000000 -1.1723865113
Ca -3.1004558000 0.0000000000 -1.1723865112
C 0.0000000000 0.0000000000 -0.2520998113
C 1.2787782000 0.0000000000 0.3860430887
C -1.2787782000 0.0000000000 0.3860430887
C 1.2190734000 0.0000000000 1.8067010887
C -1.2190734000 0.0000000000 1.8067010887
C 0.0000000000 0.0000000000 2.5104417888
H 0.0000000000 0.0000000000 -1.3706052113
H 2.1498288000 0.0000000000 2.3966890887
H -2.1498288000 0.0000000000 2.3966890887
H -0.0000000000 0.0000000000 3.6097054887
para-CaBzCa triplet, CCSD/aug-cc-pwCVDZ-PP[Ca]/cc-pVDZ[H,C]
Nuclear Repulsion Energy = 294.48164368 hartrees
Ca 4.0852992000 0.0000000000 0.0000000000
124

Ca -4.0852992000 -0.0000000000 0.0000000000
C 1.5435587000 0.0000000000 0.0000000000
C 0.7341897000 1.1622605000 0.0000000000
C 0.7341897000 -1.1622605000 0.0000000000
C -0.7341897000 1.1622605000 0.0000000000
C -0.7341897000 -1.1622605000 0.0000000000
C -1.5435587000 -0.0000000000 0.0000000000
H -1.2065336000 2.1592117000 0.0000000000
H -1.2065336000 -2.1592117000 0.0000000000
H 1.2065336000 2.1592117000 0.0000000000
H 1.2065336000 -2.1592117000 0.0000000000
125

Appendix B
On the prospects of optical cycling in diatomic cations: Effects of
transition metals, spin-orbit couplings, and multiple bonds
B.1 FCFs for the 2
2Π1/2 → 1
2Σ1/2 transition
Table B.1: Y cations.
YF+ 2
2Π1/2 → 1
2Σ1/2
qνiνf
sum
q00 0.984 0.9840
q01 0.016 1.0000
q02 0.000 1.0000
q03 0.000 1.0000
YCl+ 2
2Π1/2 → 1
2Σ1/2
qνiνf
sum
q00 0.962 0.9620
q01 0.038 0.9999
q02 0.000 1.0000
q03 0.000 1.0000
YBr+ 2
2Π1/2 → 1
2Σ1/2
qνiνf
sum
q00 0.942 0.9417
q01 0.058 0.9993
q02 0.001 1.0000
q03 0.000 1.0000
126

Table B.2: SOC-corrected ground state equilibrium geometries.
Cation re (Å)
ScF+ 1.79
ScCl+ 2.21
ScBr+ 2.34
YF+ 1.90
YCl+ 2.35
YBr+ 2.46
B.2 Spectra of singly bonded cations
127

Table B.3: The SOC-corrected states of the scandium cations. The vertical excitation energies E (eV) calculated at the internuclear distance r (Å), zero on the energy scale corresponds to the ground-state energy
of the non-relativistic states. The transition dipole moments µ (a.u.) between one of the ground level
components (the state at the top of the table). The transition dipole moment of the state at the top of the
table is its SOC-corrected dipole moment. Radiative lifetime τ is the inverse of the sum of the Einstein
coefficients, where the sum goes over all states of lower energy, see Eqs. (5) and (6) from the main text.
ScF+ ScCl+ ScBr+
State E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ
1
2∆3/2 −0.008 2.801 > 1s −0.008 3.136 > 1s −0.007 3.695 > 1s
1
2∆3/2 −0.008 0.000 > 1s −0.008 0.000 > 1s −0.007 0.000 > 1s
1
2∆5/2 0.008 0.002 > 1s 0.008 0.003 > 1s 0.007 0.004 > 1s
1
2∆5/2 0.008 0.000 > 1s 0.008 0.000 > 1s 0.007 0.000 > 1s
1
2Σ1/2 0.502 0.000 539.9ms 0.142 0.001 > 1s 0.032 0.000 > 1s
1
2Σ1/2 0.502 0.001 539.9ms 0.142 0.000 > 1s 0.032 0.001 > 1s
1
2Π1/2 0.775 0.200 4.3µs 0.423 0.196 21.2µs 0.330 0.201 29.6µs
1
2Π1/2 0.775 0.000 4.3µs 0.423 0.000 21.2µs 0.330 0.000 29.6µs
1
2Π3/2 0.788 0.000 4.3µs 0.439 0.002 20.1µs 0.366 0.004 21.6µs
1
2Π3/2 0.788 0.001 4.3µs 0.439 0.000 20.1µs 0.366 0.000 21.6µs
2
2Σ1/2 2.058 0.000 34.0ns 1.615 0.012 50.9ns 1.470 0.021 55.9ns
2
2Σ1/2 2.058 0.007 34.0ns 1.615 0.000 50.9ns 1.470 0.000 55.9ns
2
2Π1/2 4.886 0.817 0.3ns 4.445 0.785 0.4ns 4.196 0.779 0.4ns
2
2Π1/2 4.886 0.000 0.3ns 4.445 0.000 0.4ns 4.196 0.000 0.4ns
2
2Π3/2 4.908 0.006 0.3ns 4.471 0.013 0.4ns 4.237 0.018 0.4ns
2
2Π3/2 4.908 0.000 0.3ns 4.471 0.000 0.4ns 4.237 0.000 0.4ns
r (Å) 1.79 2.21 2.34
Table B.4: The SOC-corrected states of yttrium cations. See the caption of Table B.3.
YF+ YCl+ YBr+
State E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ
1
2Σ1/2 −0.002 2.001 > 1s −0.000 2.604 > 1s −0.000 3.146 > 1s
1
2Σ1/2 −0.002 0.000 > 1s −0.000 0.000 > 1s −0.000 0.000 > 1s
1
2∆3/2 0.634 0.000 12.1ms 0.589 0.000 1.6ms 0.506 0.000 3.1ms
1
2∆3/2 0.634 0.005 12.1ms 0.589 0.017 1.6ms 0.506 0.015 3.1ms
1
2∆5/2 0.635 0.031 368.9µs 0.649 0.000 > 1s 0.564 0.000 > 1s
1
2∆5/2 0.635 0.006 368.9µs 0.649 0.000 > 1s 0.564 0.000 > 1s
1
2Π1/2 1.401 0.270 218.6ns 1.106 0.490 273.3ns 1.025 0.474 364.1ns
1
2Π1/2 1.401 0.000 218.6ns 1.106 0.063 273.3ns 1.025 0.035 364.1ns
1
2Π3/2 1.450 0.000 214.4ns 1.152 0.000 260.0ns 1.088 0.000 307.0ns
1
2Π3/2 1.450 0.238 214.4ns 1.152 0.477 260.0ns 1.088 0.475 307.0ns
2
2Σ1/2 2.161 0.016 23.7ns 1.659 0.022 27.2ns 1.682 0.841 26.4ns
2
2Σ1/2 2.161 0.000 23.7ns 1.659 0.857 27.2ns 1.682 0.046 26.4ns
2
2Π1/2 3.791 0.835 0.5ns 3.569 1.749 0.5ns 3.710 1.702 0.5ns
2
2Π1/2 3.791 0.000 0.5ns 3.569 0.038 0.5ns 3.710 0.043 0.5ns
2
2Π3/2 3.868 0.012 0.5ns 3.656 0.000 0.5ns 3.810 0.000 0.4ns
2
2Π3/2 3.868 0.763 0.5ns 3.656 1.753 0.5ns 3.810 1.702 0.4ns
r (Å) 1.90 2.35 2.46
128

Table B.5: Sc cations.
ScF+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.851 0.8511
q01 0.129 0.9805
q02 0.017 0.9977
q03 0.002 0.9997
ScCl+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.779 0.7791
q01 0.183 0.9625
q02 0.032 0.9944
q03 0.005 0.9992
ScBr+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.741 0.7413
q01 0.211 0.9521
q02 0.040 0.9925
q03 0.006 0.9989
Table B.6: Y cations. FCFs are listed for the 1
2∆3/2 → 1
2Σ1/2 transition as for the Y cations 1
2Σ is the
ground state.
YF+ 1
2∆3/2 → 1
2Σ1/2
qνiνf
sum
q00 0.775 0.7749
q01 0.203 0.9781
q02 0.021 0.9994
q03 0.001 1.0000
YCl+ 1
2∆3/2 → 1
2Σ1/2
qνiνf
sum
q00 0.916 0.9165
q01 0.082 0.9985
q02 0.000 0.9987
q03 0.000 0.9989
YBr+ 1
2∆3/2 → 1
2Σ1/2
qνiνf
sum
q00 0.659 0.6588
q01 0.281 0.9397
q02 0.054 0.9938
q03 0.006 0.9997
B.3 FCFs for the 1
2Σ1/2 → 1
2∆3/2 transition
Table B.7: Ti cations.
TiO+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 1.000 0.9997
q01 0.000 0.9998
q02 0.000 1.0000
q03 0.000 1.0000
TiS+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.938 0.9377
q01 0.060 0.9973
q02 0.003 0.9999
q03 0.000 1.0000
129

Table B.8: Zr cations.
ZrO+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.984 0.9835
q01 0.016 0.9998
q02 0.000 1.0000
q03 0.000 1.0000
ZrS+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.922 0.9217
q01 0.072 0.9941
q02 0.005 0.9993
q03 0.001 0.9999
Table B.9: V cation.
VN+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.847 0.8474
q01 0.148 0.9958
q02 0.004 0.9999
q03 0.000 1.0000
Table B.10: Nb cations.
NbN+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.985 0.9851
q01 0.015 1.0000
q02 0.000 1.0000
q03 0.000 1.0000
NbP+ 1
2Σ1/2 → 1
2∆3/2
qνiνf
sum
q00 0.991 0.9914
q01 0.009 1.0000
q02 0.000 1.0000
q03 0.000 1.0000
130

Table B.11: Sc cations.
ScF+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.747 0.7471
q01 0.227 0.9737
q02 0.025 0.9987
q03 0.001 1.0000
ScCl+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.874 0.8741
q01 0.120 0.9938
q02 0.006 0.9998
q03 0.000 1.0000
ScBr+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.868 0.8679
q01 0.125 0.9928
q02 0.007 0.9998
q03 0.000 1.0000
Table B.12: Y cations.
YF+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.241 0.2411
q01 0.411 0.6519
q02 0.271 0.9231
q03 0.068 0.9913
YCl+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.657 0.6568
q01 0.321 0.9774
q02 0.022 0.9996
q03 0.000 0.9996
YBr+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.812 0.8121
q01 0.169 0.9814
q02 0.016 0.9978
q03 0.002 0.9996
B.4 FCFs for the 1
2Π1/2 → 1
2∆3/2 transition
Table B.13: Ti cations.
TiO+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.643 0.6429
q01 0.292 0.9345
q02 0.058 0.9928
q03 0.007 0.9995
TiS+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.555 0.5553
q01 0.340 0.8954
q02 0.090 0.9853
q03 0.013 0.9987
131

Table B.14: Zr cations.
ZrO+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.744 0.7444
q01 0.224 0.9687
q02 0.029 0.9978
q03 0.002 0.9999
ZrS+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.761 0.7608
q01 0.212 0.9729
q02 0.025 0.9983
q03 0.002 0.9999
Table B.15: V cation.
VN+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.438 0.4376
q01 0.428 0.8660
q02 0.119 0.9855
q03 0.014 0.9990
Table B.16: Nb cations.
NbN+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.712 0.7117
q01 0.246 0.9575
q02 0.039 0.9961
q03 0.004 0.9998
NbP+ 1
2Π1/2 → 1
2∆3/2
qνiνf
sum
q00 0.587 0.5868
q01 0.312 0.8992
q02 0.083 0.9823
q03 0.015 0.9976
132

Table B.17: SOC-corrected ground state equilibrium geometries.
Cation re (Å)
TiO+ 1.55
TiS+ 2.01
ZrO+ 1.70
ZrS+ 2.15
VN+ 1.49
NbN+ 1.62
NbP+ 2.11
Table B.18: The SOC-corrected states of titanium cations. See the caption of Table B.3.
TiO+ TiS+
State E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ
1
2∆3/2 −0.013 2.279 > 1s −0.013 2.502 > 1s
1
2∆3/2 −0.013 0.000 > 1s −0.013 0.000 > 1s
1
2∆5/2 0.013 0.001 > 1s 0.012 0.003 > 1s
1
2∆5/2 0.013 0.000 > 1s 0.012 0.000 > 1s
1
2Σ1/2 1.529 0.000 49.9ms 0.432 0.000 > 1s
1
2Σ1/2 1.529 0.001 49.9ms 0.432 0.000 > 1s
1
2Π1/2 2.091 0.284 116.2ns 1.000 0.301 720.5ns
1
2Π1/2 2.091 0.000 116.2ns 1.000 0.000 720.5ns
1
2Π3/2 2.114 0.001 116.2ns 1.027 0.006 688.3ns
1
2Π3/2 2.114 0.000 116.2ns 1.027 0.000 688.3ns
2
2Σ1/2 4.217 0.000 3.9ns 2.595 0.008 9.3ns
2
2Σ1/2 4.217 0.004 3.9ns 2.595 0.000 9.3ns
2
2Π1/2 6.512 0.636 0.2ns 5.052 0.648 0.3ns
2
2Π1/2 6.512 0.000 0.2ns 5.052 0.000 0.3ns
2
2Π3/2 6.534 0.003 0.2ns 5.075 0.000 0.3ns
2
2Π3/2 6.534 0.000 0.2ns 5.075 0.009 0.3ns
r (Å) 1.55 2.01
B.5 Spectra of multiply bonded cations
133

Table B.19: The SOC-corrected states of zirconium cations. See the caption of Table B.3.
ZrO+ ZrS+
State E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ
1
2∆3/2 −0.047 2.664 > 1s −0.046 2.686 > 1s
1
2∆3/2 −0.047 0.000 > 1s −0.046 0.000 > 1s
1
2∆5/2 0.046 0.005 > 1s 0.045 0.009 > 1s
1
2∆5/2 0.046 0.000 > 1s 0.045 0.000 > 1s
1
2Σ1/2 0.324 0.000 15.6ms 0.093 0.000 > 1s
1
2Σ1/2 0.324 0.011 15.6ms 0.093 0.004 > 1s
1
2Π1/2 1.978 0.506 14.5ns 1.323 0.433 75.2ns
1
2Π1/2 1.978 0.000 14.5ns 1.323 0.000 75.2ns
1
2Π3/2 2.055 0.008 13.7ns 1.389 0.000 71.7ns
1
2Π3/2 2.055 0.000 13.7ns 1.389 0.001 71.7ns
2
2Σ1/2 3.396 0.000 1.9ns 2.361 0.000 6.9ns
2
2Σ1/2 3.396 0.012 1.9ns 2.361 0.018 6.9ns
2
2Π1/2 4.934 0.919 0.2ns 4.398 0.865 0.3ns
2
2Π1/2 4.934 0.000 0.2ns 4.398 0.000 0.3ns
2
2Π3/2 4.988 0.010 0.2ns 4.468 0.023 0.3ns
2
2Π3/2 4.988 0.000 0.2ns 4.468 0.000 0.3ns
r (Å) 1.70 2.15
Table B.20: The SOC-corrected states of vanadium and niobium cations. See the caption of Table B.3.
VN+ NbN+ NbP+
State E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ E (eV) µ (a.u.) τ
1
2∆3/2 −0.020 2.058 > 1s −0.063 2.224 > 1s −0.061 1.864 > 1s
1
2∆3/2 −0.020 0.000 > 1s −0.063 0.000 > 1s −0.061 0.000 > 1s
1
2∆5/2 0.019 0.001 > 1s 0.062 0.004 > 1s 0.060 0.006 > 1s
1
2∆5/2 0.019 0.000 > 1s 0.062 0.000 > 1s 0.060 0.000 > 1s
1
2Σ1/2 2.485 0.000 31.4ms 1.462 0.000 654.9µs 0.819 0.000 > 1s
1
2Σ1/2 2.485 0.000 31.4ms 1.462 0.006 654.9µs 0.819 0.000 > 1s
1
2Π1/2 3.493 0.358 15.3ns 3.350 0.507 5.5ns 2.373 0.415 23.3ns
1
2Π1/2 3.493 0.000 15.3ns 3.350 0.000 5.5ns 2.373 0.000 23.3ns
1
2Π3/2 3.521 0.001 15.4ns 3.448 0.004 5.4ns 2.443 0.000 23.7ns
1
2Π3/2 3.521 0.000 15.4ns 3.448 0.000 5.4ns 2.443 0.002 23.7ns
2
2Σ1/2 5.788 0.001 1.7ns 5.232 0.007 1.1ns 3.723 0.013 3.4ns
2
2Σ1/2 5.788 0.000 1.7ns 5.232 0.000 1.1ns 3.723 0.000 3.4ns
2
2Π1/2 7.754 0.491 0.2ns 6.614 0.680 0.2ns 5.237 0.541 0.5ns
2
2Π1/2 7.754 0.000 0.2ns 6.614 0.000 0.2ns 5.237 0.000 0.5ns
2
2Π3/2 7.771 0.002 0.2ns 6.666 0.005 0.2ns 5.306 0.009 0.5ns
2
2Π3/2 7.771 0.000 0.2ns 6.666 0.000 0.2ns 5.306 0.000 0.5ns
r (Å) 1.49 1.62 2.11
134

ScF+ TiO+ VN+ YF+ ZrO+ NbN+
0
1
2
3
4
5
6
7
8
E (eV)
2
2Π
2
2Σ
1
2Π
1
2Σ
1
2∆
Figure B.1: Vertical excitation energies at the ground-state minimum. EOM-EA-CCSD energies without
the inclusion of SOC shifts.
B.6 Excitation energy with number of bonds
135

ScCl+ TiS+ YCl+ ZrS+ NbP+
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
E (eV)
2
2Π
2
2Σ
1
2Π
1
2Σ
1
2∆
Figure B.2: Vertical excitation energies at the ground-state minimum. EOM-EA-CCSD energies without
the inclusion of SOC shifts.
136

Appendix C
Dual Optical Cycling Centers Mounted on an Organic Scaffold: New
Insights From Quantum Chemistry Calculations and Symmetry
Analysis
C.1 Computational details
The effective number of unpaired electrons were computed using Head-Gordon’s index
nu,nl =
X
i
n
2
i
(2 − ni)
2
(C.1)
where ni denotes occupation of a natural orbital i and the sum extends over all orbitals. [311] This formula
was shown to give physically meaningful results for various closed and open-shell systems.
C.2 EOM-DEA-CCSD results for triplet and singlet states
Tables C.1-C.11 compare EOM-DEA-CCSD excitation energies in the singlet and triplet manifolds.
Table C.1: Vertical excitation energies (Eex, eV), oscillator strengths (fl
) and Head-Gordon’s indices (nu,nl)
in molecule (1); EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl nu,nl State Eex fl nu,nl
1
3B1 0.000 – 2.00 1
1A1 0.000 – 2.00
1
3A1 2.036 0.126 2.23 1
1B1 2.035 0.126 2.23
1
3B2 2.058 0.000 2.23 1
1A2 2.058 0.000 2.23
2
3B1 2.085 0.401 2.23 2
1A1 2.085 0.399 2.23
1
3A2 2.097 0.543 2.23 1
1B2 2.097 0.543 2.23
137

Table C.2: Vertical excitation energies (Eex, eV), oscillator strengths (fl
) and Head-Gordon’s indices (nu,nl)
in molecule (2); EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl nu,nl State Eex fl nu,nl
1
3B3u 0.000 – 2.00 1
1Ag 0.000 – 2.00
1
3B2u 2.044 0.000 2.23 1
1B1g 2.044 0.000 2.23
1
3B1u 2.060 0.000 2.23 1
1B2g 2.060 0.000 2.23
1
3B1g 2.066 0.518 2.23 1
1B2u 2.066 0.518 2.23
1
3B2g 2.085 0.544 2.23 1
1B1u 2.085 0.544 2.23
Table C.3: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (3); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3Bu 0.000 – 1
1Ag 0.000 –
2
3Bu 1.991 0.000 2
1Ag 1.991 0.000
1
3Ag 2.005 0.518 1
1Bu 2.005 0.518
1
3Au 2.015 0.000 1
1Bg 2.015 0.000
1
3Bg 2.030 0.550 1
1Au 2.030 0.550
Table C.4: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (4); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3B1 0.000 – 1
1A1 0.000 –
1
3A1 1.987 0.123 1
1B1 1.987 0.124
1
3B2 2.013 0.000 1
1A2 2.013 0.000
2
3B1 2.013 0.400 2
1A1 2.013 0.401
1
3A2 2.035 0.550 1
1B2 2.035 0.550
Table C.5: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (5); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3Bu 0.000 – 1
1Ag 0.000 –
2
3Bu 1.994 0.000 2
1Ag 1.994 0.000
1
3Ag 2.014 0.504 1
1Bu 2.014 0.504
1
3Au 2.020 0.000 1
1Bg 2.020 0.000
1
3Bg 2.044 0.540 1
1Au 2.044 0.541
Table C.6: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (6); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3B1 0.000 – 1
1A1 0.000 –
1
3A1 1.979 0.0001 1
1B1 1.979 0.0004
2
3B1 2.006 0.503 2
1A1 2.006 0.504
1
3B2 2.014 0.000 1
1A2 2.014 0.000
1
3A2 2.039 0.540 1
1B2 2.039 0.541
138

Table C.7: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (7); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3A
′
0.000 – 1
1A
′
0.000 –
2
3A
′
1.984 0.143 2
1A
′
1.984 0.144
1
3A
′′ 2.013 0.007 1
1A
′′ 2.013 0.007
3
3A
′
2.015 0.381 3
1A
′
2.015 0.380
2
3A
′′ 2.042 0.538 2
1A
′′ 2.042 0.538
Table C.8: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (8); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3A
′
0.000 – 1
1A
′
0.000 –
2
3A
′
2.092 0.077 2
1A
′
2.093 0.071
1
3A
′′ 2.099 0.013 1
1A
′′ 2.099 0.016
3
3A
′
2.120 0.417 3
1A
′
2.121 0.423
2
3A
′′ 2.123 0.513 2
1A
′′ 2.123 0.510
Table C.9: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (9); EOM-DEA-CCSD ccpVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3Bu 0.000 – 1
1Ag 0.000 –
1
3Au 2.123 0.000 1
1Bg 2.123 0.000
2
3Bu 2.130 0.000 2
1Ag 2.130 0.000
1
3Bg 2.145 0.523 1
1Au 2.145 0.524
1
3Ag 2.149 0.493 1
1Bu 2.149 0.493
Table C.10: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (10); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3A
′
0.000 – 1
1A
′
0.001 –
2
3A
′
2.074 0.0002 2
1A
′
2.075 0.0001
3
3A
′
2.087 0.507 3
1A
′
2.088 0.507
1
3A
′′ 2.089 0.015 1
1A
′′ 2.089 0.024
2
3A
′′ 2.105 0.525 2
1A
′′ 2.105 0.514
Table C.11: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (11); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca].
State Eex fl State Eex fl
1
3Bu 0.000 – 1
1Ag 0.000 –
2
3Bu 2.090 0.000 2
1Ag 2.090 0.000
1
3Au 2.100 0.000 1
1Bg 2.100 0.000
1
3Ag 2.103 0.505 1
1Bu 2.103 0.505
1
3Bg 2.115 0.536 1
1Au 2.115 0.536
139

Table C.12: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (12); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca,Sr] ECP10MDF[Ca] ECP28MDF[Sr].
State Eex fl State Eex fl
1
3A
′
0.000 – 1
1A
′
0.000 –
2
3A
′
1.762 0.261 2
1A
′
1.762 0.259
1
3A
′′ 1.784 0.251 1
1A
′′ 1.784 0.249
3
3A
′
2.045 0.378 3
1A
′
2.044 0.364
4
3A
′
2.066 0.165 4
1A
′
2.065 0.179
Table C.13: Vertical excitation energies (Eex, eV) and oscillator strengths (fl
) in (13); EOM-DEA-CCSD
cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca,Sr] ECP10MDF[Ca] ECP28MDF[Sr].
State Eex fl State Eex fl
1
3A1 0.000 – 1
1A1 0.000 –
1
3B1 1.627 0.274 1
1B1 1.627 0.274
1
3B2 1.656 0.283 1
1B2 1.655 0.284
2
3A1 1.918 0.278 2
1A1 1.917 0.275
2
3B1 2.071 0.275 2
1B1 2.071 0.275
140

Figure C.1: Frontier NOs of singlet states of (1), CaO-Ph-3-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O] augcc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol.
This section presents occupations of the natural orbitals (NOs) that host unpaired electrons. Figures C.1
and C.2 present singlets and triplets of (1); Figures C.3 and C.4 present the same information for (2).
The effective number of unpaired electrons for all considered states of molecules (1) and (2) are given in
Tables C.1 and Table C.2, respectively.
141

Figure C.2: Frontier NOs of triplet states of (1), CaO-Ph-3-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O] augcc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol.
Figure C.3: Frontier NOs of singlet states of (2), CaO-Ph-4-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O] augcc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol.
142

Figure C.4: Frontier NOs of triplet states of (2), CaO-Ph-4-OCa. EOM-DEA-CCSD cc-pVDZ[H,C,O] augcc-pwCVDZ-PP[Ca] ECP10MDF[Ca]; all electrons are active. Isovalue 0.03. Jmol.
143

C.3 TD-DFT versus EOM-DEA-CCSD
Tables C.14, C.15, and C.16 compare the excitation energies and oscillator strengths calculated with the
TD-DFT and EOM-DEA-CCSD methods.
Table C.14: Excitation energies (Eex, eV) and oscillator strengths (fl
) in the triplet manifolds of molecules
(1) and (2) calculated using PBE0-D3/def2-TZVPPD and EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-ccpwCVDZ-PP[Ca] ECP10MDF[Ca]. The label of the lowest triplet state is listed next to the molecule’s
number.
transition symmetry state label EDF T
ex (fl
) EEOM
ex (fl
)
(1) 1
3B1
B1 1
3A1 1.889 (0.110) 2.036 (0.126)
A2 1
3B2 1.933∗
(— ) 2.058 (— )
A1 2
3B1 1.911∗
(0.315) 2.085 (0.401)
B2 1
3A2 1.945 (0.437) 2.097 (0.543)
(2) 1
3B3u
B1g 1
3B2u 1.899 (— ) 2.044 (— )
B2g 1
3B1u 1.918∗
(— ) 2.060 (— )
B2u 1
3B1g 1.915∗
(0.425) 2.066 (0.518)
B1u 1
3B2g 1.936 (0.443) 2.085 (0.544)
∗Order of DFT states is different from that of EOM.
144

Table C.15: Comparison of triplet states properties of in molecules (3)-(11) calculated with PBE0-D3/def2-
TZVPPD and EOM-DEA-CCSD cc-pVDZ[H,C,O] aug-cc-pwCVDZ-PP[Ca] ECP10MDF[Ca]. Label listed
next to the molecule’s number corresponds to the lowest triplet state.
transition symmetry state label EDF T
ex (fl
) EEOM
ex (fl
)
(3) 1
3Bu
Ag 2
3Bu 1.911 (— ) 1.991 (— )
Bu 1
3Ag 1.922 (0.410) 2.005 (0.518)
Bg 1
3Au 1.926 (— ) 2.015 (— )
Au 1
3Bg 1.939 (0.432) 2.030 (0.550)
(4) 1
3B1
B1 1
3A1 1.908 (0.090) 1.987 (0.123)
A2 1
3B2 1.925 (— ) 2.013 (— )
A1 2
3B1 1.929 (0.319) 2.013 (0.400)
B2 1
3A2 1.942 (0.429) 2.035 (0.550)
(5) 1
3Bu
Ag 2
3Bu 1.906 (— ) 1.994 (— )
Bu 1
3Ag 1.920 (0.387) 2.014 (0.504)
Bg 1
3Au 1.925 (— ) 2.020 (— )
Au 1
3Bg 1.943 (0.411) 2.044 (0.540)
(6) 1
3B1
B1 1
3A1 1.894 (0.003) 1.979 (0.0001)
A1 2
3B1 1.920∗
(0.389) 2.006 (0.503)
A2 1
3B2 1.914∗
(— ) 2.014 (— )
B2 1
3A2 1.939 (0.397) 2.039 (0.540)
∗Order of DFT states is different from that of EOM.
145

Table C.16: Continuation of Table C.15
transition symmetry state label EDF T
ex (fl
) EEOM
ex (fl
)
(7) 1
3A
′
A
′
2
3A
′
1.902 (0.132) 1.984 (0.143)
A
′′ 1
3A
′′ 1.924 (0.001) 2.013 (0.007)
A
′
3
3A
′
1.928 (0.274) 2.015 (0.381)
A
′′ 2
3A
′′ 1.946 (0.421) 2.042 (0.538)
(8) 1
3A
′
A
′
2
3A
′
1.919 (0.044) 2.092 (0.077)
A
′′ 1
3A
′′ 1.931 (0.006) 2.099 (0.013)
A
′
3
3A
′
1.937 (0.329) 2.120 (0.417)
A
′′ 2
3A
′′ 1.949 (0.391) 2.123 (0.513)
(9) 1
3Bu
Bg 1
3Au 1.935∗
(— ) 2.123 (— )
Ag 2
3Bu 1.934∗
(— ) 2.130 (— )
Au 1
3Bg 1.951∗
(0.380) 2.145 (0.523)
Bu 1
3Ag 1.948∗
(0.361) 2.149 (0.493)
(10) 1
3A
′
A
′
2
3A
′
1.922 (0.001) 2.074 (0.0002)
A
′
3
3A
′
1.932 (0.397) 2.087 (0.507)
A
′′ 1
3A
′′ 1.931 (0.014) 2.089 (0.015)
A
′′ 2
3A
′′ 1.944 (0.406) 2.105 (0.525)
(11) 1
3Bu
Ag 2
3Bu 1.931 (— ) 2.090 (— )
Bg 1
3Au 1.935 (— ) 2.100 (— )
Bu 1
3Ag 1.941 (0.385) 2.103 (0.505)
Au 1
3Bg 1.946 (0.404) 2.115 (0.536)
∗Order of DFT states is different from that of EOM.
146

C.4 (1) and (2) functionalized in ortho position
Table C.17 shows the results for systems (1) and (2) functionalized with the methyl group. As one can
see, introducing this functional group on a small scaffolds leads to deterioration of FCFs because of its
proximity to the OCCs.
Table C.17: Frank–Condon factors for the 0 → 0 transition for different excited states for systems (1) and
(2) functionalized with CH3. The oscillator strengths are given in parentheses.
System Eex (symmetry) FCF 0 → 0 (f)
CaO OCa
1.8887 eV (A′
) 0.7417 (0.0818)
1.9007 eV (A′′) 0.4737 (0.0219)
1.9280 eV (A′′) 0.8413 (0.3263)
1.9357 eV (A′′) 0.8541 (0.4065)
OCa
CaO
1.8995 eV (A′
) 0.7417 (0.0054)
1.9040 eV (A′′) 0.4715 (0.0378)
1.9150 eV (A′′) 0.8413 (0.4101)
1.9294 eV (A′′) 0.8541 (0.3948)
147

C.5 Spin-orbit couplings
We evaluate the effect of the spin-orbit couplings (SOCs) by using the state interaction approach in which
he model Hamiltonian is constructed in the basis of non-relativistic states. The diagonal elements of the
model Hamiltonian are the EOM-DEA-CCSD energies. The off-diagonal elements of the model Hamiltonian are the matrix elements of the spin-orbit operator of the Breit-Pauli Hamiltonian, HSO
BP . [108] The
SOC-corrected states are the eigenvectors of the model Hamiltonian, summarized in Tabels C.18-C.21.
The matrix of transition dipole moments (TDMs) calculated in the basis of the non-relativistic states is
transformed into the matrix of SOC-corrected TDMs with the unitary matrix that diagonalized the model
Hamiltonian.
The SOC-induced mixing of the singlet and triplet manifolds enables transitions, which are forbidden in
the non-relativistic framework. We estimate the probability of a radiative spin-changing transition using
the Einstein’s A coefficient. The probability for an excited state I to decay to a lower state F is given by
the Einstein’s AIF coefficient
AIF =
ω
3
IF µ
2
IF
3hc¯
3πϵ0
(C.2)
where ωIF is the energy gap between the two states and µIF is the corresponding transition dipole moments. The remaining constants are the reduced Planck constants ¯h, the speed of light c, and the vacuum
permittivity ϵ0. The branching ratio in a decay of an excited state I to any of a lower states Fi
is given by
a ratio
BRFi = P
AIFi
Fj
AIFj
, (C.3)
where the sum extends over all states Fj of energy lower than the initial state I.
The EOM-DEA-CCSD energies were calculated in the basis cc-pVDZ for H, C, and O atoms, aug-ccpwCVDZ-PP for Ca and Sr atoms and for the last two atoms the core electrons were substituted by the
effective core potentials ECP10MDF for Ca and ECP28MDF for Sr.
148

The current Q-Chem implementation of the EOM-DEA-CCSD method does not include calculation of
the SOC. Instead, we calculate the matrix elements of HSO
BP using the EOM-EE-CCSD states starting from
a closed-shell singlet as a reference, and computing both singlet and triplet target states. We carried out
these calculations molecules (1), (2), (12) and (13).
The SOC calculations use an all-electron basis set aug-cc-pwCVDZ-X2C for the metal atoms Ca and
Sr. [304] In this basis set the excitation energies reproduce the degeneracy patterns and states order that was
predicted by previous EOM-DEA-CCSD calculations in molecules (1) and (2), whereas in the Sr-containing
molecules (12) and (13) the degeneracy pattern and states ordering differs.
In the discussion below, we call block X the four lowest states (the lowest singlet state and the three
components of the lowest triplet states) derived from the ground state of the mono-OCC molecules (often
labelled X). Similarly, we call block A the eight states derived from states with an electron promoted to the
in-plane p-like orbital. Block B is the out-of-plane version of block A. See the below example for molecule
(1):
block X




1
1A1, Sz = 0

1
3B1, Sz = −1

1
3B1, Sz = 0

1
3B1, Sz = 1
(C.4)
149

block A




1
1B1, Sz = 0

1
3A1, Sz = −1

1
3A1, Sz = 0

1
3A1, Sz = 1

1
1A2, Sz = 0

1
3B2, Sz = −1

1
3B2, Sz = 0

1
3B2, Sz = 1
(C.5)
block B




2
1A1, Sz = 0

2
3B1, Sz = −1

2
3B1, Sz = 0

2
3B1, Sz = 1

1
1B2, Sz = 0

1
3A2, Sz = −1

1
3A2, Sz = 0

1
3A2, Sz = 1
(C.6)
The model Hamiltonians for the four selected systems are presented in Fig. C.5. Below we discuss in
detail the couplings between the selected blocks. First, we analyze the block X, to highlight the role of
the couplings on the ground-state character. Next, we discuss the couplings to and between the blocks A
150

0 2 4 6 8 1012141618
0
2
4
6
8 10 12 14 16 18
(1)
0
1
2
3
4
0 2 4 6 8 1012141618
0
2
4
6
8 10 12 14 16 18
(2)
0
1
2
3
4
0 2 4 6 8 1012141618
0
2
4
6
8 10 12 14 16 18
(12)
0
1
2
3
4
0 2 4 6 8 1012141618
0
2
4
6
8 10 12 14 16 18
(13)
0
1
2
3
4
Figure C.5: Model Hamiltonians. To clearly visualize the values across many orders of magnitude, the
etries are rescaled using the formula log
1 +

z/cm−1

. States 0-3 belong to block X, states 4-11 belong
to block A, and states 12-19 belong to block B.
and B. We then discuss the overall effect of SOCs on the state ordering and probabilities of inter-system
crossing (intensity borrowing).
Within block X, molecule (1) has the largest matrix elements of HSO
BP of the order of 0.01 cm−1
, those
values increase by about an order of magnitude in the Sr-containing molecule (12), however, in both parasubstituted molecules (2) and (13) these matrix elements vanish. Thus, the calculations predict that the
coupling between these states should be very small.
The diagonal, non-relativistic splitting between the triplet and the singlet states of block X was calculated also using methods that include perturbative triples. Even at this highest level of theory, the splitting
15

was found to be under the applied convergence thresholds (about 0.1 cm−1
). The small diagonal energy
gap makes it possible for even the smallest couplings to generate a strong mixing of the singlet and triplet
character.
An example that highlights this property is a Hamiltonian submatrix of molecule (1) formed by restricting the model Hamiltonian to block X.
H =

















1
1A1, 0

1
3B1, −1

1
3B1, 0

1
3B1, 1

1
1A1, 0

c c

1
3B1, −1

c
∗ Eex

1
3B1, 0

Eex

1
3B1, 1

c
∗ Eex
















(C.7)
where the Eex is the energy gap between the singlet and triplet states, and the number c is the nonzero matrix element of the HSO
BP Hamiltonian. This particular form of the model Hamiltonian reproduces
structure obtained with EOM-EE-CCSD for block X of (1).
Diagonalization of the model Hamiltonian Eq. (C.7) shows that for the ratio |c|/Eex = 0.1, the contribution of the singlet state to one of the triplet components is about 0.14. This coefficient squared equals
0.02, which gives an estimated that about 2% of the transitions from singlets can leak to the triplet manifold. The same estimate increases to 6% for the ratio 0.2 and it becomes 20% at the ratio of 0.5. This model
highlights that the smaller is the singlet–triplet splitting, the stronger is the impact of small SOCs on the
ground state’s properties.
When it comes to the couplings between X and A blocks, the couplings vanish for molecule (2), in
molecule (1) there are a few non-zero couplings of the order of a few wavenumbers, whereas for molecules
(12) and (13), the matrix of couplings is populated by values of the order of tens of wavenumbers, with
the larges one being 70 cm−1
.
152

The increase in the magnitude of the couplings follows the El-Sayed rule. Despite that increase, the
ratio of the coupling strength to the energy gap remains small, as the energy gap between X and the A
blocks is of the order of 104
cm−1
. Therefore, the state mixing due to the X-A couplings is minute.
The couplings of X and B blocks show values close to 20 cm−1
for molecules (1) and (2), and surprising
are slightly lower for the Sr-containing molecules (12) and (13).
Couplings withing A block reach values of about 10 cm−1
in (1) and about ten times as much for the
Sr-containing (12) and (13), whereas they vanish again for (2). Block B shows the same characteristics for
(1) and (2) but the increase in the coupling strength is not observed for (12) and (13) as the largest values
there are also of the order of 10 cm−1
. Finally, the couplings between A and B blocks show a large number
of couplings close to 15 cm−1
for (1) and (2), and in this case the Sr-containing (12) and (13) show an
increase in the coupling strength to about 100 cm−1
.
The increase in the coupling strength in A and B blocks is very relevant as the energy spacing between
these two blocks remains small. The couplings of the magnitude comparable to the energy spacings generate significant mixing in block A and B. In particular, the SOC-corrected lowest state of block A acquires
primarily a triplet character for (12) and (13) and is energetically separated from the other states of the
block A, see Table C.18-C.21.
153

Table C.18: SOC-corrected states for (1); energies in cm−1
.
id EOM state Eex ∆ESOC SOC-corrected eigenvector
|0⟩

1
1A1, 0

0.00 -0.04 0.997 |0⟩ + . . .
|1⟩

1
3B1, −1

0.14 -0.04 0.705 |1⟩ + 0.705 |3⟩ + . . .
|2⟩

1
3B1, 0

0.14 -0.03 1.000 |2⟩
|3⟩

1
3B1, 1

0.14 -0.01 0.707 |3⟩ − 0.707 |1⟩
|4⟩

1
1B1, 0

16416.48 -2.17 0.965 |4⟩ + 0.176 |5⟩ + 0.176 |7⟩ + . . .
|5⟩

1
3A1, −1

16418.09 -1.09 −0.997 |6⟩ + . . .
|6⟩

1
3A1, 0

16418.09 -0.92 −0.706 |7⟩ + 0.706 |5⟩ + . . .
|7⟩

1
3A1, 1

16418.09 0.25 −0.685 |5⟩ − 0.685 |7⟩ + 0.249 |4⟩
|8⟩

1
3B2, −1

16595.53 -2.16 0.704i|10⟩ − 0.704i|8⟩ + . . .
|9⟩

1
3B2, 0

16595.53 -0.43 0.984i|11⟩ + . . .
|10⟩

1
3B2, 1

16595.53 0.01 0.702i|8⟩ + 0.702i|10⟩ + 0.121i|11⟩
|11⟩

1
1A2, 0

16596.34 0.14 −0.997i|9⟩ + . . .
|12⟩

2
1A1, 0

16812.49 -0.73 −0.803 |12⟩ − 0.412 |15⟩ − 0.412 |13⟩ + . . .
|13⟩

2
3B1, −1

16814.10 -1.36 −0.993 |14⟩ − 0.114i|16⟩
|14⟩

2
3B1, 0

16814.10 1.23 −0.575 |15⟩ − 0.575 |13⟩ + 0.574 |12⟩ + . . .
|15⟩

2
3B1, 1

16814.10 2.34 0.703 |13⟩ − 0.703 |15⟩ − 0.103i|11⟩
|16⟩

1
1B2, 0

16914.92 0.80 −0.706i|19⟩ − 0.706i|17⟩ + . . .
|17⟩

1
3A2, −1

16915.73 0.97 −0.706i|17⟩ + 0.706i|19⟩ + . . .
|18⟩

1
3A2, 0

16915.73 1.51 −0.991i|16⟩ + 0.114 |14⟩ + . . .
|19⟩

1
3A2, 1

16915.73 1.74 −0.992i|18⟩ − 0.128 |12⟩
154

Table C.19: SOC-corrected states for (2); energies in cm−1
.
id EOM state Eex ∆ESOC SOC-corrected eigenvector
|0⟩

1
1Ag, 0

0.00 -0.04 1.000 |0⟩
|1⟩

1
3B3u, −1

0.10 -0.04 1.000 |2⟩
|2⟩

1
3B3u, 0

0.10 -0.04 0.707 |1⟩ + 0.707 |3⟩
|3⟩

1
3B3u, 1

0.10 0.00 0.707 |3⟩ − 0.707 |1⟩
|4⟩

1
1B1g, 0

16484.23 -1.92 −0.997i|4⟩ + . . .
|5⟩

1
3B2u, −1

16484.23 -1.91 0.705i|7⟩ − 0.705i|5⟩ + . . .
|6⟩

1
3B2u, 0

16484.23 -0.00 0.760e
−0.16πi |6⟩ − 0.460 |5⟩ − 0.460 |7⟩
|7⟩

1
3B2u, 1

16484.23 -0.00 0.650e
+0.92πi |6⟩ + . . .
|8⟩

1
1B2g, 0

16614.89 -10.81 −0.920 |8⟩ + 0.278i|13⟩ − 0.278i|15⟩
|9⟩

1
3B1u, −1

16614.89 -10.73 −0.651 |11⟩ + 0.651 |9⟩ + 0.392i|12⟩
|10⟩

1
3B1u, 0

16614.89 -0.00 −0.995 |10⟩ + . . .
|11⟩

1
3B1u, 1

16614.89 0.00 −0.704 |9⟩ − 0.704 |11⟩ + . . .
|12⟩

1
1B2u, 0

16663.28 0.00 0.707e
−0.21πi |13⟩ + 0.707e
−0.21πi |15⟩
|13⟩

1
3B1g, −1

16663.28 0.02 1.000i|14⟩
|14⟩

1
3B1g, 0

16663.28 10.77 −0.920i|12⟩ + 0.277 |9⟩ − 0.277 |11⟩
|15⟩

1
3B1g, 1

16663.28 10.81 −0.650i|15⟩ + 0.650i|13⟩ + 0.393 |8⟩
|16⟩

1
1B1u, 0

16814.91 0.00 1.000e
+0.03πi |18⟩
|17⟩

1
3B2g, −1

16814.91 0.02 0.707 |17⟩ + 0.707 |19⟩
|18⟩

1
3B2g, 0

16814.91 1.92 −0.705 |19⟩ + 0.705 |17⟩ + . . .
|19⟩

1
3B2g, 1

16814.91 1.95 −0.997 |16⟩ + . . .
155

Table C.20: SOC-corrected states for (12); energies in cm−1
.
id EOM state Eex ∆ESOC SOC-corrected eigenvector
|0⟩

1
1A′
, 0

0.00 -0.18 0.997 |0⟩ + . . .
|1⟩

1
3A′
, −1

2.42 -0.36 0.997i|2⟩ + . . .
|2⟩

1
3A′
, 0

2.42 -0.36 0.707e
−0.90πi |3⟩ + 0.707e
−0.60πi |1⟩
|3⟩

1
3A′
, 1

2.42 0.00 0.707e
+0.33πi |1⟩ − 0.707 |3⟩
|4⟩

2
1A′
, 0

14209.76 -52.46 0.634e
+0.74πi |7⟩ + 0.634e
+0.05πi |5⟩ + . . .
|5⟩

2
3A′
, −1

14212.99 -7.70 0.989 |4⟩ + 0.124i|6⟩ + . . .
|6⟩

2
3A′
, 0

14212.99 -3.54 −0.992i|6⟩ + 0.123 |4⟩ + . . .
|7⟩

2
3A′
, 1

14212.99 -0.00 0.707e
−0.68πi |5⟩ − 0.707 |7⟩
|8⟩

1
3A′′
, −1

14387.20 -0.22 0.707e
+0.86πi |10⟩ + 0.707e
−0.86πi |8⟩
|9⟩

1
3A′′
, 0

14387.20 -0.00 −1.000 |9⟩
|10⟩

1
3A′′
, 1

14387.20 0.87 0.705e
+0.64πi |8⟩ + 0.705e
−0.64πi |10⟩ + . . .
|11⟩

1
1A′′
, 0

14388.01 51.76 0.895e
−0.61πi |11⟩ + . . .
|12⟩

3
1A′
, 0

16488.26 3.05 −0.997 |12⟩ + . . .
|13⟩

3
3A′
, −1

16496.32 0.00 0.707e
−0.86πi |13⟩ + 0.707e
−0.19πi |15⟩
|14⟩

3
3A′
, 0

16496.32 3.59 −0.999i|14⟩ + . . .
|15⟩

3
3A′
, 1

16496.32 4.11 0.706e
+0.06πi |15⟩ + 0.706e
+0.38πi |13⟩ + . . .
|16⟩

4
1A′
, 0

16656.83 0.43 −0.998 |16⟩ + . . .
|17⟩

4
3A′
, −1

16659.25 -0.00 0.707e
+0.42πi |19⟩ + 0.707e
−0.82πi |17⟩
|18⟩

4
3A′
, 0

16659.25 0.17 0.707e
−0.57πi |17⟩ + 0.707e
−0.33πi |19⟩
|19⟩

4
3A′
, 1

16659.25 0.83 0.996i|18⟩ + . . .
156

Table C.21: SOC-corrected states for (13); energies in cm−1
.
id EOM state Eex ∆ESOC SOC-corrected eigenvector
|0⟩

1
1A1, 0

0.00 -0.39 −0.707 |1⟩ − 0.707 |3⟩
|1⟩

1
3A1, −1

0.01 -0.38 0.707 |3⟩ − 0.707 |1⟩
|2⟩

1
3A1, 0

0.01 -0.12 1.000 |0⟩
|3⟩

1
3A1, 1

0.01 0.00 1.000 |2⟩
|4⟩

1
1B1, 0

13118.50 -33.59 −0.942 |6⟩ − 0.335i|8⟩
|5⟩

1
3B1, −1

13118.50 -2.33 −0.999 |4⟩ + . . .
|6⟩

1
3B1, 0

13118.50 -2.16 0.707 |5⟩ + 0.707 |7⟩ + . . .
|7⟩

1
3B1, 1

13118.50 -0.00 0.707 |7⟩ − 0.707 |5⟩
|8⟩

1
1B2, 0

13352.40 -0.15 1.000i|10⟩
|9⟩

1
3B2, −1

13352.40 0.00 0.707i|11⟩ + 0.707i|9⟩
|10⟩

1
3B2, 0

13352.40 0.02 0.707i|9⟩ − 0.707i|11⟩
|11⟩

1
3B2, 1

13352.40 31.00 −0.941i|8⟩ + 0.335 |6⟩ + . . .
|12⟩

2
1A1, 0

15457.49 2.23 1.000 |12⟩
|13⟩

2
3A1, −1

15465.56 0.00 1.000e
+0.63πi |14⟩
|14⟩

2
3A1, 0

15465.56 2.65 −0.707 |13⟩ − 0.707 |15⟩ + . . .
|15⟩

2
3A1, 1

15465.56 2.96 −0.707 |15⟩ + 0.707 |13⟩ + . . .
|16⟩

2
1B1, 0

16703.61 -0.00 0.707 |17⟩ − 0.707 |19⟩
|17⟩

2
3B1, −1

16703.61 0.01 1.000 |18⟩
|18⟩

2
3B1, 0

16703.61 0.03 −0.707 |19⟩ − 0.707 |17⟩
|19⟩

2
3B1, 1

16703.61 0.22 −1.000 |16⟩
157

Appendix D
Vibronic Coupling Effects in the Photoelectron Spectrum of Ozone:
A Coupled-Cluster Approach
D.1 Parameters for the KDC Hamiltonian
Below is an input for the xsim program, which contains all parameters used to define the KDC Hamiltonian.
Units
1
Dataset
1
States
2
Modes
3
Basis Functions
50 50 50
Lanczos
6000
Transition Moment
1. 1.35 ! Muller, Koppel, Cederbaum, et al. Chem. Phys. Lett. 197, 599 (1992)
Vertical Energies, cm-1
1 103456.7 ! EOMIP-CCSDT/pCVnZ/CBS n=5,6 + dQ/pwCVTZ + relativistic/CCSD/pwCVTZ
2 104449.7 ! EOMIP-CCSDT/pCVnZ/CBS n=5,6 + dQ/pwCVTZ + relativistic/CCSD/pwCVTZ
0 0 0 0 0 0 0 0 0 0 0 0
Harmonic Frequencies, cm-1 ! CCSDT/ANO1 for the neutral state
1 724.19
2 1168.58
3 1128.50
0 0 0 0 0 0 0 0 0 0 0 0
Linear, cm-1 ! Everything below: EOM-CCSDT/ANO1
1 1 2 1178.18
1 1 1 1261.02
2 2 2 -781.75
2 2 1 -1303.82
1 2 3 1393.82
0 0 0 0 0 0 0 0 0 0 0 0
158

Quadratic, cm-1
1 1 2 2 802.90
1 1 1 2 -185.50
1 1 1 1 779.81
2 2 2 2 1276.61
2 2 1 2 23.56
2 2 1 1 369.33
0 0 0 0 0 0 0 0 0 0 0 0
Cubic, cm-1
1 1 2 2 2 -202.85
1 1 1 2 2 74.39
1 1 1 1 2 -62.49
1 1 1 1 1 90.19
2 2 2 2 2 -282.04
2 2 1 2 2 75.56
2 2 1 1 2 -0.49
2 2 1 1 1 71.32
0 0 0 0 0 0 0 0 0 0 0 0
Quartic, cm-1
1 1 2 2 2 2 34.17
1 1 1 2 2 2 -26.65
1 1 1 1 2 2 3.16
1 1 1 1 1 2 -12.25
1 1 1 1 1 1 3.92
2 2 2 2 2 2 41.58
2 2 1 2 2 2 -35.55
2 2 1 1 2 2 -7.14
2 2 1 1 1 2 -6.88
2 2 1 1 1 1 11.34
0 0 0 0 0 0 0 0 0 0 0 0
D.2 Relevant Cartesian Coordinates
The CCSDT/ANO1 geometry of neutral ozone (Å) (The nuclear repulsion energy: 68.994611948499568 a.u.)
Enuc=68.994612
O 0.0000 0.0000 0.4431
O 0.0000 -1.0817 -0.2215
O 0.0000 1.0817 -0.2215
The KDC Hamiltonian parameterization corresponds to this point.
D.3 PES of the Ozone Cation
159

1 0 1
Q2
1
0
1
Q1
A1
1 0 1
Q2
B2
0 1000 2000 3000 cm 1
Figure D.1: Contour maps of the two diabatic PESs of the ozone cation (see Fig. 4 in the main text). Q1
and Q2 are two symmetric normal coordinates.
160

Appendix E
Photoswitching Molecules Functionalized with Optical Cycling Centers
Provide a Novel Platform for Studying Chemical Transformations in
Ultracold Molecules
E.1 DFT benchmarking
Initially, we planned to use Density Functional Theory (DFT) methods, benchmarked previously by some
of us using the PBE0 functional with the def2-TZVPPD basis set and D3 dispersion corrections [258, 259].
However, we soon realized that the types and the order of the excited states produced by the TD-DFT calculation did not match EOM-CC. Table E.1 presents the results of the screening of selected DFT functionals
for trans-pAB, showing the first four excited states, along with the respective transition orbitals, vertical
excitation energies, and oscillator strengths.
All DFT calculations were performed using the ORCA quantum chemistry package[312], version 6.
DFT and TD-DFT calculations used the RIJCOSX approximation; no Tamm–Dancoff approximation was
invoked in TD-DFT.
The only functionals found to produce results in qualitative agreement with EOM-CC are rangeseparated double-hybrid spin-opposite-spin scaled functionals, namely SOS-ωPBEPP86 and SOS-ωB88PP86.
Unfortunately, these functionals rely on MP2 calculations, which diminishes the appeal of using DFT methods.
E.2 Effect of the isomerization on the X→B transition
Fig. E.1 shows the natural transition orbitals for the X→B transition in bpAB.
161

Table E.1: Screening of selected DFT functionals with def2-TZVPP basis set. Excitation energy (in eV) and
oscillator strengths for the first four excited are reported. State characterization is omitted for non-physical
states. The reference EOM-CC calculation used aug-cc-pVDZ basis set.
Method ES1 ES2 ES3 ES4
PBE0 1.834 (0.0) 1.928 (0.18) 1.933 (0.20) 1.990 (0.00)
n → π
∗
s → p − like s → p − like π → π
∗
B3LYP 1.900 (0.0) 1.945 (0.04) 2.049 (0.00) 2.058 (0.19)
n → π
∗
* s → π
∗
* π → π
∗
* s → π
∗
TPSS0 1.862 (0.21) 1.866 (0.19) 1.871 (0.00) 1.908 (0.00)
s → π
∗
s → π
∗ n → π
∗
* π → π
∗
*
CAM-B3LYP 1.975 (0.0) 1.980 (0.0) 2.016 (0.19) 2.022 (0.19)
n → π
∗
* π → π
∗
s → π
∗
s → π
∗
wB2PLYP 3.373 (0.0) 2.058 (0.22) 2.079 (0.00) 2.076 (0.23)
s → π
∗ n → π
∗
* s → π
∗
wB2GP-PLYP 2.532 (0.0) 2.069 (0.23) 2.090 (0.25) 2.152 (0.0)
s → π
∗
s → π
∗ n → π
∗
*
wPBEPP86 2.679 (0.01) 2.067 (0.24) 2.090 (0.25) 2.122 (0.0)
s → π
∗
s → π
∗ n → π
∗
*
DSD-BLYP/2013 D3BJ 4.228 (0.00) 2.310 (0.27) 2.335 (0.28) 2.985 (0.33)
s → π
∗
s → π
∗
s → dz
2
SOS-WPBEPP86 2.059 (0.25) 2.078 (0.26) 2.278 (0.00) 2.278 (0.24)
s → p s → p s → d n → π
∗
*
SOS-wB88PP86 2.051 (0.25) 2.051 (0.25) 2.276 (0.00) 2.257 (0.24)
s → p s → p s → d n → π
∗
*
EOM-EA-CCSD 1.964 (0.26) 1.983 (0.27) 2.271 (0.25) 2.943 (0.00)
s → p s → p s → d s → d
Figure E.1: Natural transition orbitals (with weights ≥0.25) for the X→B transition in cis-bpAB.
162

Table E.2: FCFs for tpAB A→X fluorescence.
FCF transition
0.858 A(0) → X(0)
0.037 A(0) → X(1v10)
0.027 A(0) → X(1v2)
0.024 A(0) → X(1v6)
0.013 A(0) → X(1v8)
0.009 A(0) → X(1v17)
0.005 A(0) → X(1v4)
0.004 A(0) → X(1v50)
0.005 A(0) → X(1v29)
0.003 A(0) → X(1v22)
0.001 A(0) → X(1v2,1v10)
0.001 A(0) → X(1v6,1v10)
E.3 Franck–Condon factors
With potential future applications in mind, we computed Franck–Condon factors (FCFs). FCFs are overlaps
between initial and final vibrational states:
fνµ =
D
χ
f
ν

χ
i
µ
E
. (E.1)
We computed the FCFs using the vertical gradient approximation as implemented in the ezFCF package [310]. At the optimized EOM-EA-CCSD/cc-pVDZ geometry for the ground state, we computed the
harmonic frequencies and normal modes for the ground state. At the same geometry with the same method
we then calculated the gradient for the excited state. From these, we estimated the equilibrium geometry of the excited state. We then use that excited state geometry for the parallel-mode double-harmonic
calculation of the FCFs.
The FCFs for the tpAB molecule are collected in Table E.2. Mode 10 (366 cm−1
) is a Ca-O stretch. Mode
2 (38 cm−1
) is the Ca-O-AB bend in plane. Mode 6 (186 cm−1
) is a mixture of a C-N-N-C squeezing and a
Ca-O stretching. Mode 8 (266 cm−1
) is similar to mode 6.
The FCF for cpAB emission from the ground vibrational state of the A state are collected in Table E.3.
Mode 9 (298 cm−1
) is in a leading way the Ca-O stretch but it also involves motion in the rest of the
163

Table E.3: FCFs for cpAB A→X fluorescence.
FCF transition
0.806 A(0) → X(0)
0.055 A(0) → X(1v1)
0.047 A(0) → X(1v9)
0.016 A(0) → X(1v2)
0.008 A(0) → X(1v7)
0.006 A(0) → X(1v3)
0.005 A(0) → X(1v16)
0.004 A(0) → X(1v17)
0.004 A(0) → X(1v14)
0.004 A(0) → X(1v50)
0.003 A(0) → X(1v29)
0.003 A(0) → X(1v4)
0.003 A(0) → X(1v1,1v9)
0.002 A(0) → X(1v0)
0.002 A(0) → X(1v5)
0.002 A(0) → X(2v1)
0.002 A(0) → X(1v10)
0.002 A(0) → X(1v13)
0.002 A(0) → X(1v15)
0.002 A(0) → X(1v30)
0.001 A(0) → X(1v1,1v2)
0.001 A(0) → X(2v9)
0.001 A(0) → X(1v23)
0.001 A(0) → X(1v54)
molecule. Mode 1 (43 cm−1
) is a Ca-O-C bend, but it also involves rotation of the remote phenyl ring.
Mode 2 is also a bend of the Ca-O-C part that is coupled to the motion of the reminder of the molecule.
E.4 Relevant Cartesian coordinates
25
tpAB, X, EOM-EA-CCSD/cc-pVDZ, Nuclear Repulsion Energy = 1011.183962 hartree
C 5.3832949554 1.1268023932 -0.0000000000
C 3.9857848754 1.0242652883 -0.0000000000
C 6.1864366870 -0.0311896649 0.0000000000
C 5.5825967915 -1.2986615016 0.0000000000
C 4.1810701040 -1.4061755307 0.0000000000
C 3.3823631861 -0.2507216815 -0.0000000000
N 1.9614847270 -0.4835236846 -0.0000000000
N 1.2812134824 0.5744215944 -0.0000000000
C -0.1339985068 0.3649235430 -0.0000000000
C -0.9257923961 1.5257693382 -0.0000000000
C -2.3247269471 1.4417831343 0.0000000001
164

C -2.9688065782 0.1818510535 0.0000000000
C -2.1571876382 -0.9857954463 -0.0000000000
C -0.7640421169 -0.8992048201 -0.0000000001
H 5.8537616426 2.1167460216 -0.0000000000
H 3.3494475397 1.9129885234 -0.0000000001
H 7.2785746364 0.0584934720 0.0000000000
H 6.2000128737 -2.2038310242 0.0000000001
H 3.6827083791 -2.3812968442 0.0000000001
H -0.4209780821 2.4983115557 0.0000000000
H -2.9389457145 2.3492456808 0.0000000001
H -2.6524563713 -1.9638216298 -0.0000000000
H -0.1447973927 -1.8007567674 -0.0000000001
O -4.3059650637 0.0857744134 0.0000000000
Ca -6.4134084937 -0.0683092669 0.0000000000
25
cpAB, X, EOM-EA-CCSD/cc-pVDZ, Nuclear Repulsion Energy = 1070.687319 hartree
C 2.6781911187 -1.7756773868 -1.2763959690
C 2.4057235373 -0.4046638393 -1.1512733827
C 3.4588997986 -2.4380817126 -0.3118390466
C 3.9917334619 -1.7144238678 0.7690647817
C 3.7479526134 -0.3358675487 0.8827525478
C 2.9263566315 0.3099010754 -0.0563771273
N 2.7996008146 1.7528834600 0.0526783944
N 1.6825562332 2.3225413788 0.0523575229
C 0.4341513897 1.5835562463 0.0752758955
C -0.5897558499 2.0732921462 -0.7541272382
C -1.8560458532 1.4742368630 -0.7567668371
C -2.1528722867 0.4053131171 0.1236385311
C -1.1297503516 -0.0370198315 0.9989495106
C 0.1475839200 0.5360767829 0.9720632202
H 2.2747285899 -2.3305092641 -2.1309847067
H 1.7914396841 0.1112218491 -1.8961246432
H 3.6619771828 -3.5101034326 -0.4104190551
H 4.6149450286 -2.2194968328 1.5156033542
H 4.1824488983 0.2511047765 1.6992698212
H -0.3735716924 2.9261719430 -1.4075532441
H -2.6441103785 1.8434881736 -1.4229081613
H -1.3553981905 -0.8449428646 1.7044090349
H 0.9173910708 0.1730687125 1.6600598445
O -3.3697643668 -0.1613755506 0.1412332454
Ca -5.2816560827 -1.0567832255 0.1711274484
26
trans-bpAB, X, EOM-SF-CCSD/cc-pVDZ, Nuclear Repulsion Energy = 1294.263016 hartree
N -0.3420305297 -0.5282212074 0.0000000000
N 0.3420892032 0.5284770910 0.0000000000
C -1.7570868236 -0.3055700695 0.0000000000
165

C 1.7570905269 0.3057352686 0.0000000000
C -2.3765288730 0.9629748805 0.0000000000
C -2.5590997234 -1.4590706309 -0.0000000000
C 2.3764213087 -0.9628732575 0.0000000000
C 2.5592092426 1.4591698524 -0.0000000000
C -3.7702655116 1.0620727324 0.0000000000
C -3.9582052229 -1.3631196093 -0.0000000000
C 3.7701319876 -1.0620990062 0.0000000000
C 3.9582914276 1.3630962797 -0.0000000000
C -4.5865776836 -0.0982770135 -0.0000000000
C 4.5866055545 0.0981844819 -0.0000000000
H -1.7496016471 1.8590216858 0.0000000000
H -2.0624483455 -2.4356512007 -0.0000000000
H 1.7494092693 -1.8588648518 0.0000000000
H 2.0626421764 2.4357961439 -0.0000000000
H -4.2568698354 2.0442001609 0.0000000000
H -4.5803368187 -2.2649536161 -0.0000000000
H 4.2566585798 -2.0442633513 0.0000000000
H 4.5805116926 2.2648670630 -0.0000000000
O -5.9517762260 0.0107822454 -0.0000000000
Ca -8.1286608483 0.1772514427 0.0000000000
O 5.9516859442 -0.0109945681 -0.0000000000
Ca 8.1287411755 -0.1776709460 -0.0000000000
26
cis-bpAB, X, EOM-SF-CCSD/cc-pVDZ, Nuclear Repulsion Energy = 1382.875084 hartree
Ca 5.3866142952 0.1436922329 2.8385157117
O 4.0227292881 0.0781350219 1.1396407606
C 3.1740867799 0.0490540576 0.0631975122
C 3.3104847626 -0.9343595776 -0.9426039650
H 4.1142648631 -1.6740139194 -0.8555623751
C 2.4529428524 -0.9431659667 -2.0521036765
H 2.5826471999 -1.6811202467 -2.8518819539
C 1.4026255211 -0.0157359562 -2.1524339573
C 1.2689787786 0.9801548663 -1.1662841441
H 0.4793434901 1.7336324845 -1.2512979194
C 2.1486929605 1.0139586367 -0.0765745203
H 2.0505161817 1.7940138967 0.6868457780
N 0.6261485492 -0.0390497739 -3.3787156263
N -0.6261485492 0.0390497739 -3.3787156263
C -1.4026255211 0.0157359562 -2.1524339573
C -2.4529428524 0.9431659667 -2.0521036765
H -2.5826471999 1.6811202467 -2.8518819539
C -3.3104847626 0.9343595776 -0.9426039650
H -4.1142648631 1.6740139194 -0.8555623751
C -1.2689787786 -0.9801548663 -1.1662841441
H -0.4793434901 -1.7336324845 -1.2512979194
C -2.1486929605 -1.0139586367 -0.0765745203
166

H -2.0505161817 -1.7940138967 0.6868457780
C -3.1740867799 -0.0490540576 0.0631975122
O -4.0227292881 -0.0781350219 1.1396407606
Ca -5.3866142952 -0.1436922329 2.8385157117
167

Abstract (if available)

Abstract This dissertation presents a study of molecular candidates for direct laser cooling. The main results of this work are: 1) extending the set of promising candidates for producing cold and ultracold samples of molecular gases, 2) introducing new resources, intrinsic to polyatomic molecules, to the field of ultracold matter, 3) developing rigorous, quantum-mechanical protocols for description of laser-coolable molecules. These contributions resulted from simulations of vibrationally resolved electronic spectra, which included also more subtle (but crucial to the experiments) effects such as the spin-orbit and vibronic interactions. The systems studied in this work range in size from diatomics to polyatomic, organic and aromatic molecules decorated with a functional group of a bivalent metal atom or, more generally, an optical cycling center (OCC).

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Wójcik, Paweł (author)

Core Title Quantum mechanical description of electronic and vibrational degrees of freedom in laser-coolable molecules

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Degree Conferral Date 2025-05

Publication Date 02/25/2025

Defense Date 02/25/2025

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

Tag computational chemistry,electronic spectroscopy,laser cooling,molecular spectroscopy,molecules,OAI-PMH Harvest,quantum mechanics

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Krylov, Anna (committee chair), Benderskii, Alexander (committee member), Di Felice, Rosa (committee member)

Creator Email pawel.wojcik@usc.edu,pawel.wojcik5@gmail.com

Unique identifier UC11399HPNF

Identifier etd-WjcikPawe-13829.pdf (filename)

Legacy Identifier etd-WjcikPawe-13829

Document Type Dissertation

Format theses (aat)

Rights Wójcik, Paweł

Internet Media Type application/pdf

Type texts

Source 20250227-usctheses-batch-1242 (batch), 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 author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.

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

Repository Email cisadmin@lib.usc.edu