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Modeling x-ray spectroscopy in condensed phase
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Modeling x-ray spectroscopy in condensed phase
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Content
MODELING X-RAY SPECTROSCOPY IN CONDENSED PHASE
by
Sourav Dey
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)
December 2023
Copyright 2023 Sourav Dey
Acknowledgements
I would like to take this opportunity to express my sincere gratitude to all those who have contributed to the completion of this doctoral thesis. First and foremost, I would like to thank my
advisor, Prof. Anna Krylov for her guidance, wisdom and support throughout my PhD. I am
grateful for her patience and understanding especially during the tougher times. Her feedback and
suggestions have helped me grow as a better researcher over the years. Her enthusiasm for science
combined with her work ethics with a touch of realism is definitely something I hope to cultivate
within me.
I would like to thank Professors Curt Wittig, Andrey Vilesov, Oleg Prezhdo, Alexander Benderskii, Aiichiro Nakano, Daniel Lidar and Rosa Di Felice for the courses they taught. They have
aided my research in various capacities and I will draw upon them in my future endeavors. I
am thankful to Prof Chi Mak, Alexander Benderskii, Shaama Sharada, Aiichiro Nakano and late
Prof Sri Narayan who have taken out time to be on my Defense and Qualification exam committees and provided me with their constructive feedback. In addition, I’m grateful to Prof Madhav
Ranganathan of IITK, my Master’s thesis advisor, who advised me to apply to USC.
I would like to thank Dr. Musahid Ahmed from the Lawrence Berkeley National Laboratory
for the initial motivation for our work on water in Chapter 2. I would like to thank our collaborator
Prof Henrik Koch and Prof Sarai Dery Folkestad who helped us with multi-level coupled-cluster
calculations. It was a pleasure working with them.
ii
I would like to thank the past and present members of iOpenShell for creating not only an
intellectualy stimulating and fun environment in the lab: 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. Maxim Ivanov, Dr. Pavel Pokhilko, Dr. Sarai
Dery Folkestad, Dr. Yongbin Kim, Madhubani Mukherjee, Goran Giudetti, Nayanthara Jayadev,
Pawel Wojcik, George Baffour Pipim and Kyle Alexander Tanovitz. I am especially grateful to
Dr. Tirthendu Sen for his help when I first arrived to the United States. I also made some great
friends outside the lab: Dr. Swetha Erukala, Michael Guile, Dr. Anwesha Maitra, Dr. Pratyusha
Das, Shivalee Dey and Sraddha Agrawal. All of you made this journey, far away from home, a
little less difficult and more rewarding.
Last but no way the least, I’m extremely indebted to my parents Samaresh Dey and Rupna Dey,
my brother Saikat Dey, my grandparents and family and friends in India whose blessings and love
are a constant source of inspiration.
iii
Table of Contents
Acknowledgements ii
List of Tables vi
List of Figures vii
Abbreviations xi
Abstract xiii
Chapter 1:Introduction and overview 1
1.1 X-ray photoelectron spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Core-level ionization energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Modelling spectroscopy in condensed phase . . . . . . . . . . . . . . . . . . . . . 7
Chapter 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Chapter 2:Core-ionization spectrum of liquid water 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 Equilibrium simulations of bulk water . . . . . . . . . . . . . . . . . . . . 24
2.2.2 Calculations of core IEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 Protocols for selecting the QM subsystem for IE calculations . . . . . . . . 30
2.2.4 Extrapolation to bulk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3.1 Analysis of structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter 2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Chapter 3:Future work 54
3.1 Using better sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Understanding water-glycerol solution . . . . . . . . . . . . . . . . . . . . . . . . 56
Chapter 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
iv
Chapter A:Supplementary information for Chapter 2 62
A.1 Data deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
A.2 Equilibrium sampling: Molecular dynamics versus ab initio molecular dynamics . . 62
A.3 Convergence with respect to the QM size and number of snapshots: Additional
results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
A.4 Analysis of the computed spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.5 Recalculating the spectra using different hydrogen-bond distributions . . . . . . . . 67
A.6 Q-Chem input for CVS-EOM-CCSD calculations for 20 QM waters . . . . . . . . 69
Chapter A References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
v
List of Tables
2.1 Experimental values of the 1sO level shift (∆IE) in bulk water. The best values are
shown in bold (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 1sO IE of isolated water molecule. . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Effect of triple excitations on IE (eV) from multi-level calculations. . . . . . . . . 35
2.4 Shift (∆IE) of the 1sO IE of liquid water relative to the gas-phase water and the
width (FWHM) of the band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A.1 Structural properties of liquid water from the O–O radial distribution function,
gOO(r), obtained from the AIMD and MD simulations around the central water
molecule of their respective simulation boxes and experimental data.a
. . . . . . . 63
A.2 The fraction of water molecules with double-donor (DD), single-donor (SD), and
non-donor (ND) configurations from different computational methods. . . . . . . 64
vi
List of Figures
1.1 Excited-state processes in fluorescent dyes. The main relaxation channel is fluorescence. Radiationless relaxation, a process in which the chromophore relaxes to
the ground state by dissipating electronic energy into heat, reduces the quantum
yield of fluorescence. Other competing processes, such as transition to a triplet
state via intersystem crossing (not shown), excited-state chemistry, and electron
transfer, alter the chemical identity of the chromophore, thus leading to temporary
or permanent loss of fluorescence (blinking and bleaching) or changing its color
(photoconversion). Adapted from Ref. 1. . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Photoelectron spectrum of liquid water. Panel A: Theoretical PES computed with
EOM-IP-CCSD using pentamers extracted from equilibrium MD simulations of
liquid water. The 1b1 band (gray) is defined as any state in the 9.5-12.5 eV ionization energy range, the 3a1 band (yellow) is defined as any state in the 12.5-15
eV range, and the 1b2 band (red) is defined as any state in the 16.5-18.5 eV range.
Panel B: Deconvolution fit of the experimental PES (symbols) obtained at hν =
265 eV and θ = 0. The fit gives peaks corresponding to ionization from the 1b1,
3a1, and 1b2 orbitals of water for gas phase (green peaks) and liquid (blue peaks).
Adapted from Ref. 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 (a) Sketch of the liquid-jet vacuum chamber for photoelectron spectroscopy measurements using soft X-ray photons from a synchrotron light facility. Photo inset:
Conical aperture of the electron analyzer (EA) and the glass capillary forming the
jet. Also shown are the (1) differential pumping section to introduce the high-flux
X-ray beam, (2) interaction chamber and turbomolecular pumps, (3) liquid nitrogen (LN2) trap for collecting the frozen solution, and (4) electron energy analyzer.
(b) Schematic energy-level diagram relevant for the photoionization experiment.
Upward arrows: Electron ejection of a valence and a core-level electron, when
applying lower and higher photon energies, respectively. Measured electron kinetic energy (eKE) and inferred electron binding energy (BE) are also indicated.
Adapted from Ref. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 A cluster from the center of the box of one water molecule and its first solvation
shell representing a minimal QM subsystem. The Dyson orbital associated with
the 1sO ionization of the central water is shown in blue. . . . . . . . . . . . . . . . 7
vii
1.5 Glycerol in a waterbox within a QM/MM description. The glycerol (sticks and
balls) is treated with quantum mechanics and the waters (points) treated as point
charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Energy-level diagram for photoelectron experiments in a gas phase (right panel)
and a liquid jet (left panel). Abbreviations: Evac, vacuum level; EF, Fermi energy;
φ, work function; ana, analyzer; KE, kinetic energy; IE, ionization energy. The
vacuum level (i.e., the level corresponding to the ejected electrons with zero kinetic
energy) in the liquid jet is higher than the vacuum level of an isolated molecule in
gas phase due to the presence of the field created by the water surface. This field
also affects the gas-phase molecules in the vicinity of the jet. Whereas in the gasphase experiment the vacuum level of the gas matches that of the analyzer, in the
aqueous solution experiment the vacuum level of the gas is pinned to the vacuum
levels of the liquid and of the analyzer, which results in an electric field between the
two (depicted as sloped energy levels of the gas). The Fermi levels of the jet and
the analyzers match by design (they are both grounded), but their vacuum levels do
not because of the difference in the respective workfunctions. Zero kinetic energy
is defined for both experiments as the vacuum level of the analyzer. Reproduced
from Ref. 13 with permission from the Royal Society of Chemistry. . . . . . . . . 15
2.2 Molecular density (ρ(N)), charge density (ρ(e)), and potential (denoted as φ in this
figure and as ϕin in the text) along the interface normal of the vacuum-water system
computed using several water models. Reprinted from Ref. 23 with permission of
AIP Publishing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Experimental spectra from Pellegrin et al. (at 0.003 mbar and room temperature),12
Liu et al.,24 and Olivieri et al. (at 0 V bias).13 The spectra were aligned by the
position of the gas-phase peak (the narrow feature) by applying global shifts of
+0.59 eV, +4.9 eV, and +0.87 eV, respectively. . . . . . . . . . . . . . . . . . . . . 22
2.4 Model system for liquid water. Top: simulation box with 392 water molecules.
Bottom: A cluster from the center of the box of one water molecule and its first
solvation shell representing a minimal QM subsystem. The Dyson orbital associated with the 1sO ionization of the central water is shown in blue. . . . . . . . . . . 28
2.5 Panels (a) and (b) show in bright colors 6-water QM systems comprising the central
water, its first solvation shell, and a 6th water selected from the second solvation
shell. In panel (a), the 6th water molecule accepts a hydrogen bond from a water
molecule from the first solvation shell. In panel (b), the 6th water molecule donates
a hydrogen bond to a water molecule from the first solvation shell. The points in
panels (c) and (d) represent the shift in IE of the central water molecule (δ) due
to adding the 6th water. Each point corresponds to a different selection of the 6th
water sampled over two snapshots from the TIP3P waterbox. The points are colorcoded to show whether the 6th water acts as a donor (pink), acceptor (yellow), or
both (blue). Panel (c) shows δ versus the shortest Ocen(Hcen)···H(O) distance and
panel (d) shows δ versus the shortest O1
stshell(H1
stshell)···H(O) distance. . . . . . . 31
viii
2.6 IE of the central water molecule for a single snapshot computed using different
protocols for selecting the QM system in the CVS-EOM-IP-CCSD calculations.
The rest of the waters are described by point charges. . . . . . . . . . . . . . . . . 32
2.7 Convergence of the shift in IE in between protocols 2 and 3 (δbasis) with the 20 QM
waters. The estimated shift δbasis=0.114 eV, with a 0.035 eV standard deviation. . . 33
2.8 The number of hydrogen bonds in the first solvation shell as a function of the QM
system size for a single snapshot. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.9 Convergence of the shift in IE due to approximate triples (δtriples) through the CVSEOM-IP-MLCC3 approach. The estimated effect of triples on the shift in ∆IE.
δtriples=-0.34 eV, with a 0.07 eV standard deviation. . . . . . . . . . . . . . . . . . 36
2.10 Theoretical 1sO spectra of liquid water computed using the minimal QM system
(5 waters) and assembled using the IE of the central water molecule. Top: Spectra
constructed from the MD (TIP3P) and AIMD trajectories (400 snapshots). Bottom:
Spectra constructed from the MD trajectory using different number of snapshots.
All spectra were obtained using gaussians with FWHM = 0.2 eV, except for the
black trace in the right panel which was produced using 0.05 eV FWHM to show
the intrinsic roughness of the spectrum produced from the 3,000 snapshots. . . . . 37
2.11 Theoretical 1sO spectra of liquid water computed using small (5) and large (20)
QM systems constructed from the AIMD trajectory (400 snapshots). The spectra
were constructed using the 1sO IEs of either one central water or five QM waters. . 39
2.12 Experimental12, 13, 24 and theoretical 1sO XPS spectra of water. The spectra of
Pellegrin et al.,12 Liu et al.,24 and Olivieri et al.13 spectra has been shifted by
+0.59 eV, +4.9 eV, and +0.87 eV, respectively, to match the theoretical gas phase
peak. The computationally constructed spectrum is shown by dashed line. The
computed spectra of liquid water includes triples, basis-set, and Born corrections. . 40
2.13 The 1sO spectra of liquid water constructed from the central water and the first
shell (total 5 waters) broken into the contributions from structures with different
hydrogen-bonding patterns (raw spectrum without corrections). Top panel shows
the contributions from structures with different hydrogen-bonding patterns around
the central water and bottom panel shows the contributions of different hydrogenbonding patterns around the first solvation shell. The spectra were computed from
the AIMD snapshots trajectory using QM with 20 waters and treating the rest of
the waters as point charges. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.14 The 1sO spectra of liquid water (raw spectra without corrections) constructed from
the a) central water molecule and b) 5 water molecules computed using AIMD
snapshots (black) and synthetic spectra obtained by re-weighting the contributions
from the dominant hydrogen-bonding patterns to match the distributions from other
simulations (see text). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
ix
3.1 a) gOO(r) obtain from NVT simulations at the experimental carried out at T =
298.15 K with the q-TIP4P/f (blue), TTM3-F (cyan), BLYP (green), BLYP-D3
(orange), and MB-pol (red) potentials. b) gOH(r) calculated from both classical
(MD, green) and quantum (PIMD, red) NVT simulations with the MB-pol potential
at T = 298.15 K. Adapted from Ref. 3. . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Schematic diagram of glycerol and water mixtures at different relative concentrations showing bulk water, solvated water, trapped water and bulk glycerol. Adapted
from Ref. 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3 X-ray photoelectron spectra at different glycerolwater concentrations. (A) C 1s
spectra measured at a photon energy of 315 eV. Gray inset at 100.0 mol % is the
gas phase spectrum of glycerol, used to disentangle the gas phase (292 eV) and
condensed phase (290.5 eV) contributions. (B) O 1s spectra measured at a photon
energy of 560 eV. A spectrum is superimposed on the 1.0 mol % spectrum showing
gas phase (540 eV) and condensed phase (538 eV) peaks of pure water. Solid and
dashed lines indicate the condensed phase and gas phase BE positions, respectively.
Adapted from Ref. 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 Condensed phase (filled circles) and gas phase (open circles) XPS intensities as
a function of glycerol mol % (100 mol % glycerol intensity is not reported here
since its aerosol generation mechanism is different from glycerolwater mixtures).
(A) C 1s XPS peak intensities and (B) O 1s XPS peak intensities. Three regions
corresponding to glycerolwater networks are indicated in colored regions: red (X1)
is solvated water, green (X2) is confined water, and blue (X3) is the bulk glycerol
network. Adapted from Ref. 21. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
A.1 The O-O radial distribution function computed from the AIMD and MD trajectories compared with the experimental gOO(r)
2
. . . . . . . . . . . . . . . . . . . . 63
A.2 The O edge ionization spectra computed 5 and 6 water QM systems. . . . . . . . . 65
A.3 Convergence of CVS-EOM-IP-CCSD shift (from eT calculations) with respect to
the number of snapshots. ∆IE=1.15 eV, with standard deviation of 0.55 eV. . . . . 66
A.4 The O edge ionization spectra of the AIMD/400, 5w/20QM system convoluted
with gaussians of width (FWHM) 0.5, 0.2 and 0.05 eV respectively. . . . . . . . . 67
A.5 The 1sO spectra of liquid water constructed from the MD trajectory and re-weighting
the contributions from the dominant hydrogen-bonding patterns to match the distributions from the MD trajectory. . . . . . . . . . . . . . . . . . . . . . . . . . . 68
x
Abbreviations
AIMD ab initio molecular dynamics
ARPES angle-resolved photoelectron spectroscopy
CC coupled-cluster
CCSD coupled-cluster method with singles and doubles
CNTO correlated natural transition orbital
CVS core-valence separation
DA single donor-single acceptor
DAA single donor-double acceptor
DDA double donor-single acceptor
DDAA double donor-double acceptor
DFT density functional theory
EA electron analyzer
EOM-CCSD equation-of-motion coupled-cluster method with singles and doubles
FTIR fourier transformed infrared
FWHM full width half maximum
xi
eKE electronic kinetic energy
HF Hartree-Fock
KE kinetic energy
fc frozen-core
IE ionization energy
IP ionization potential
MD molecular dynamics
MC Monte Carlo
MLCC3 multilevel coupled-cluster method with triple excitations
MLCC multilevel coupled-cluster method
PES photoelectron spectroscopy
PAO projected atomic orbital
PIMD Path integral molecular dynamics
QM quantum mechanical
MM molecular mechanical
QM/MM quantum mechanics/molecular mechanics
THz-TDS Terahertz time domain spectroscopy
TS transition state
UPS ultraviolet photoelectron spectroscopy
XPS x-ray photoelectron spectroscopy
ZPE zero point energy
xii
Abstract
This thesis addresses challenges associated with studying excited-state processes in the condensed
phase, particularly in liquid solutions. Theory plays essential role in interpreting the experimental
measurements. While ground-state properties can be accurately modeled using standard quantum
chemistry methods, excited-states calculations remain challenging. Excited states play a crucial
role in various applications, such as solar energy harvesting, vision, artificial sensors, and photovoltaics. Excited states are also exploited in many spectroscopies, which provide important
information about molecular structure and properties. The main obstacle in a quantitative description of light-induced processes is incorporating the influence of the environment on excited
states. Since many important processes occur in solution or in solids, it is essential to develop
methods that can account for the effects of the surrounding medium on molecular chromophores.
The core-excited/ionized states in condensed phase are particularly challenging to model owing to
their high sensitivity to the local environment. This thesis explores the application of equation-ofmotion coupled-cluster (EOM-CCSD) methods with molecular dynamics simulation to model the
core-ionization energy in condensed phase. In Chapter 1, I discuss the essential features of x-ray
spectroscopy.
In Chapter 2, I present state-of-the-art calculations of the core-ionization spectrum of water.
Despite a significant progress in procedures developed to mitigate various experimental complications and uncertainties, the experimental determination of ionization energies (IEs) of solvated
xiii
species involves several non-trivial steps such as assessing the effect of the surface potential, electrolytes, and finite escape depths of photoelectrons. This provides a motivation to obtain robust
theoretical values of the intrinsic bulk ionization energy and the corresponding solvent-induced
shift. Towards this goal, we developed theoretical protocols based on coupled-cluster theory and
electrostatic embedding. Our value of the intrinsic solvent-induced shift of the core 1sO ionization
energy of water is -1.79 eV. The computed absolute position and the width of the 1sO in photoelectron spectrum of water are 538.47 eV and 1.44 eV, respectively, agree well with the best
experimental values.
In Chapter 3, I discuss future directions in modeling condensed-phase spectroscopy.
xiv
Chapter 1: Introduction and overview
Light-matter interactions is a basis of spectroscopy, a technique that delivers fundamental information about matter. When a molecule absorps a photon, it becomes excited. Depending on
the wavelength of light, the excitation can be in the rotational, vibrational, or electronic. When
molecules absorb light in the visible, ultraviolet, or x-ray range of the electromagnetic spectrum,
they undergo transition into electronically excited states. Electronically excited states play important roles in biological systems and photoactive materials (eg. OLEDs, solar cells etc). FIG. 1.1
illustrates a variety of excited-state processes that can happen in electronically excited molecules,
such as common dyes.
1.1 X-ray photoelectron spectroscopy
Photoelectron spectroscopy (PES) is a powerful analytical technique that is used to study the electronic structure of molecules and materials. It works by measuring the kinetic energy of electrons
that are emitted from a sample when it is exposed to light. By analyzing the energy distribution of
the emitted electrons, researchers can gain insights into the electronic properties of the substance,
including its valence band structure, density of electronic states, and chemical composition.2 PES
has a wide range of applications in fields such as materials science, chemistry, and physics. It
can be used to study a variety of materials, including metals, semiconductors, polymers, and organic compounds.3, 4 Additionally, PES can be used to study materials in a variety of different
1
Figure 1.1: Excited-state processes in fluorescent dyes. The main relaxation channel is fluorescence. Radiationless relaxation, a process in which the chromophore relaxes to the ground
state by dissipating electronic energy into heat, reduces the quantum yield of fluorescence.
Other competing processes, such as transition to a triplet state via intersystem crossing (not
shown), excited-state chemistry, and electron transfer, alter the chemical identity of the chromophore, thus leading to temporary or permanent loss of fluorescence (blinking and bleaching) or changing its color (photoconversion). Adapted from Ref. 1.
states, including solid, liquid, and gas phases. FIG. 1.2 shows the experimental and theoretical
photoelectron spectra of water in gas and condensed phase.
There are several types of PES, including ultraviolet photoelectron spectroscopy (UPS), Xray photoelectron spectroscopy (XPS), and angle-resolved photoelectron spectroscopy (ARPES),
among others. Each technique has its own strengths and is suited to different types of samples and
applications. In this thesis, we focus on X-ray photoelectron spectroscopy (XPS), which is one of
the most commonly used PES techniques. The importance of this technique was recognized by the
Nobel Prize in Physics to Kai M. Seigbahn in 1981.
FIG. 1.3 shows schematic diagrams of an experimental technique used to measure XPS spectra
in solutions. In the liquid jet technique (FIG. 1.3 (a)), a liquid sample is focused into a thin stream
and then ejected through a small orifice into a vacuum chamber. The ejected liquid forms a stable
2
Figure 1.2: Photoelectron spectrum of liquid water. Panel A: Theoretical PES computed
with EOM-IP-CCSD using pentamers extracted from equilibrium MD simulations of liquid
water. The 1b1 band (gray) is defined as any state in the 9.5-12.5 eV ionization energy range,
the 3a1 band (yellow) is defined as any state in the 12.5-15 eV range, and the 1b2 band (red) is
defined as any state in the 16.5-18.5 eV range. Panel B: Deconvolution fit of the experimental
PES (symbols) obtained at hν = 265 eV and θ = 0. The fit gives peaks corresponding to
ionization from the 1b1, 3a1, and 1b2 orbitals of water for gas phase (green peaks) and liquid
(blue peaks). Adapted from Ref. 5.
microjet, which is irradiated with X-rays to produce XPS spectra.7 FIG. 1.3 (b) shows a schematic
energy-level diagram showing ionization of a valence and core-level electron when irradiated by
a photon of lower and higher energy respectively. There also exists other experimental techniques
like ’dip and pull’8
and velocity map imaging from aqueous aerosol nanoparticles.9 XPS has
been used to study the effect of DNA damage by radiation in aqueous media,10 redox potential of
nucleotides6
and hydrogen bonding network in glycerol aerosols.
3
Figure 1.3: (a) Sketch of the liquid-jet vacuum chamber for photoelectron spectroscopy measurements using soft X-ray photons from a synchrotron light facility. Photo inset: Conical
aperture of the electron analyzer (EA) and the glass capillary forming the jet. Also shown
are the (1) differential pumping section to introduce the high-flux X-ray beam, (2) interaction chamber and turbomolecular pumps, (3) liquid nitrogen (LN2) trap for collecting the
frozen solution, and (4) electron energy analyzer. (b) Schematic energy-level diagram relevant for the photoionization experiment. Upward arrows: Electron ejection of a valence and
a core-level electron, when applying lower and higher photon energies, respectively. Measured electron kinetic energy (eKE) and inferred electron binding energy (BE) are also indicated. Adapted from Ref. 6.
1.2 Core-level ionization energy
To model XPS, one needs to be able to compute IEs of core electrons. There are a number of
electronic structure methods that provide accurate description of ionized states. The Koopmans
theorem provides the simplest way to describe ionization energies and their corresponding canonical orbital gives the spatial description from where the electron is ejected. However, these singledeterminant models lack the electron correlation effects. Many methods have been developed over
the past half century to include electron correlation in the excited or ionized states. One of the
most successful methods to study these states is the equation-of-motion coupled-cluster (EOMCC) suite of methods.11–13 They provide a balanced description of the excited and ionized states
and can also produce relevant interstate properties.
4
X-ray spectroscopy is superficially similar to UV-Vis spectroscopy. The latter exploits transition of low-energy valence electrons while the former exploits transition of deep lying core electrons. However, the core-excited/ionized states are very high in energy. For example, they lie
around 300 eV, 400 eV and 540 eV for 1s electrons of C, N and O respectively. This means that
core-level states are embedded in ionization continuum, i.e., they are metastable with respect to
electron ejection. This poses convergence and numerical stability problems. A solution to this
challenge is provided by the core-valence separation (CVS) strategy suggested by Cederbum and
co-workers in 1980.14
The CVS scheme is based on the assumption that the core-level and valence excitations can be
decoupled because of their energy difference. In this scheme, the continuum is projected out and
the core states are artificially stabilized. EOM methods is a suitable method which accounts for
the orbital relaxation in target states. The frozen-core (fc) CVS-EOM-CC method allows one to
efficiently describe core-level states at the CC level by exploiting the CVS approximation.15
The IP version of EOM-CCSD is used to model XPS spectroscopy. The target-state wave
functions are constructed as follows
|ΨR⟩ = e
Tˆ
Rˆ|Φ0⟩ (1.1)
⟨ΨL| = ⟨Φ0|Leˆ −Tˆ
(1.2)
Tˆ = ∑
µ
tµτµ = Tˆ
1 +Tˆ
2 +...+TˆN (1.3)
where |Φ0⟩ is a reference Slater determinant and Tˆ is the cluster operator. The excitation operators
Rˆ and Lˆ †
act on the coupled-cluster wave function e
Tˆ
|Φ0⟩ to yield the target states.
5
In fc-CVS-EOM-CC ansatz the core-valence correlation is neglected at the CCSD step. The CVS
scheme is invoked by restricting the excitation operator to contain at least one core orbital. Thus,
the excitation operators are given by
RˆCVS
IP = ∑
I
rIaˆI +
1
2 ∑
IJa
r
a
IJaˆ
†
aaˆJaˆI + ∑
I jva
r
a
I jv
aˆ
†
aaˆjv
aˆI (1.4)
LˆCVS
IP = ∑
I
l
I
aˆ
†
I +
1
2 ∑
IJa
l
IJ
a aˆ
†
I
aˆ
†
J
aˆa + ∑
I jva
l
I jv
a aˆ
†
I
aˆ
†
jv
aˆa (1.5)
The EOM equation is
H¯ RˆCVS
IP = ERˆCVS
IP (1.6)
where H¯ is the effective Hamiltonian in the space of single and double excited determinants within
the frozen-core approximation. The eigenvalue of Eq. (1.6) give the energies of core-level sates.
Dyson orbitals represent the orbitals from which an electron leaves during photoionization.
These can be described as overlaps between the initial N-electron and final N −1-electron states.
φ
Dyson(x1) = √
N
Z
Ψ
N
i
(x1, x2,..., xN)Ψ
N−1
f
(x1, x2,..., xN)dx2 ... dxN (1.7)
If ΨN
i
is a Hartree-Fock wave function, then φ
Dyson is the canonical orbital from which an electron
is ejected. Dyson orbitals computed using EOM-IP-CCSD wave functions represent the correlated
orbital from where electron leaves at the CCSD level of theory. The Dyson orbitals for 1s coreionized states are localized and look similar to canonical Hartree-Fock orbitals.16 They show where
the electron is removed from and there can be used to identify the site of ionization in condensed
6
phase. For example, Figure 2.4 shows the Dyson orbital corresponding to the 1sO ionization from
a water molecule in a water box.
Figure 1.4: A cluster from the center of the box of one water molecule and its first solvation
shell representing a minimal QM subsystem in a water box. The Dyson orbital associated
with the 1sO ionization of the central water is shown in blue.
1.3 Modelling spectroscopy in condensed phase
Computational modeling of spectroscopy in condensed phase is a challenging task for electronic
structure theory. In order to accurately model the properties of molecules in condensed phase, one
needs theoretical models that can capture the complex interplay between electronic and molecular
structure. The modeling can be separated into two parts: obtaining proper conformational sampling
and electronic-structure calculation of relevant transitions.
7
In our studies, we use EOM-CCSD, which is an excellent method to model excited and ionized
states. However, EOM-CCSD scales as O(N
6
) and, therefore, is prohibitive for larger systems.
Highly efficient implementations17, 18 of EOM-CCSD have been developed in the last decade to
make the method more affordable.
To circumvent this problem, quantum chemical methods are often used within a multiscale
modeling framework designed to describe extended systems. In multiscale modeling, different
methods of required accuracy are used to describe different parts of the system. For example,
the region closely associated with the excitation/ionization is usually described with a high-level
electronic structure method such as EOM-CCSD and the other regions are described with less
expensive method. One such popular method is electrostatic embedding. Here, the system is
partitioned into a quantum mechanical (QM) subsystem and a molecular mechanical (MM) subsystem described by a classical force field. The point charges of the MM subsystem are included
in the one-electron Hamiltonian of the QM subsystem. The interaction Hamiltonian is given by
the equation
Hint =
n
∑
i
M
∑
A
qA
|RA −ri
|
+
N
∑
i
M
∑
A
qAZI
|RA −ri
|
(1.8)
where n is the number of electrons, N the number of nuclei in the QM subsystem, M the number
of nuclei in the MM subsystem. qA and ZI are the electronic and nuclear charge respectively and
RA and ri are the nuclear and electronic coordinates respectively. FIG. 1.5 shows a representative
example of QM/MM framework where the QM subsystem contains the glycerol molecule and the
waters are in the MM subsystem. The water point charges polarize the glycerol molecule and
provides a solvation effect.
To successfully model spectra in condensed phase, one needs to have good equilibrium sampling. Thermal fluctuations are important in condensed phase, especially aqueous solutions, and
8
Figure 1.5: Glycerol in a waterbox within a QM/MM description. The glycerol (sticks and
balls) is treated with quantum mechanics and the waters (lines) treated as point charges.
play important role in several physical, chemical and biological processes. These thermal fluctuations affect the transient structure of the liquid through the motions and collisions of molecules
leading to breaking and forming of non-covalent bonds i.e., hydrogen bonds, π- π stacking etc.
These fluctuations give rise to spectral broadening called inhomogeneous broadening. The width
of the spectra can be used to gain insight about the solution at a molecular level such as hydrogenbonding network, solute aggregation, molecular diffusion, etc.
Molecular dynamics (MD) or Monte-Carlo (MC) techniques are commonly employed to sample different conformations of gaseous molecules, clusters, or condensed phase. According to the
ergodic hypothesis, over long periods of time, the time spent by a system in some region of the
phase space of microstates with the same energy is proportional to the phase volume of this region,
i.e., that all accessible microstates are equiprobable over a long period of time. Therefore, the
conformational space of a system can be adequately sampled with a long enough MD simulation.
9
Various sampling techniques exist which in combination with molecular dynamics to efficiently
the phase space of complicated systems. In classical MD, the interaction between molecules are
described by empirical force fields.
However, classical MD simulations cannot describe inherently quantum phenomenon such as
bond breaking. The ab initio MD (AIMD) methodology can be used to better represent the quantum nature of molecules during the simulation. In AIMD, potential energy surface is described
by quantum chemistry methods, i.e., by solving the Schrodinger equation at each nuclear arrange- ¨
ment. This makes it more reliable than MD, but also more expensive.
In the past few decades, experimentalists have developed techniques to measure XPS spectra
of aqueous solutions. The developments have made it possible to study biomolecules in condensed
phase. This presents a opportunity for the theoreticians to validate there theoretical methods and
consequently gain a molecular-level understanding of the complex systems. In this thesis, we
present protocols to model XPS spectra in condensed phase. Chapter 2 gives a concise description
about the on-going efforts and challenges to experimentally measure the core-ionization energy of
liquid water and develop protocols to reliably model its O1s spectrum.
10
Chapter 1 References
[1] Acharya, A.; Bogdanov, A. M.; Bravaya, K. B.; Grigorenko, B. L.; Nemukhin, A. V.;
Lukyanov, K. A.; Krylov, A. I. Photoinduced chemistry in fluorescent proteins: Curse or
blessing? Chem. Rev. 2017, 117, 758–795.
[2] Kostko, O.; Bandyopadhyay, B.; Ahmed, M. Vacuum ultraviolet photoionization of complex
chemical systems Annu. Rev. Phys. Chem. 2016, 67, 19–40.
[3] Shutthanandan, V.; Nandasiri, M.; Zheng, J.; Engelhard, M. H.; Xu, Wu; Thevuthasan, S.;
Murugesan, V. Applications of xps in the characterization of battery materials JESRP 2019,
231, 2–10.
[4] McArthur, S. L. Applications of xps in bioengineering Surf Interface Anal 2006, 38, 1380–
1385.
[5] Gozem, S.; Seidel, R.; Hergenhahn, U.; Lugovoy, E.; Abel, B.; Winter, B.; Krylov, A. I.;
Bradforth, S. E. Probing the electronic structure of bulk water at the molecular length scale
with angle-resolved photoelectron spectroscopy J. Phys. Chem. Lett. 2020, 11, 5162–5170.
[6] Seidel, R.; Winter, B.; Bradforth, S. E. Valence electronic structure of aqueous solutions:
Insights from photoelectron spectroscopy Annu. Rev. Phys. Chem. 2006, 67, 283–305.
[7] Perez Ramirez, L.; Boucly, A.; Saudrais, F.; Bournel, F.; Gallet, J.-J.; Maisonhaute, E.;
Milosavljevic, A. R.; Nicolas, C.; Rochet, F. The fermi level as an energy reference in liquid
jet x-ray photoelectron spectroscopy studies of aqueous solutions Phys. Chem. Chem. Phys.
2021, 23, 16224–16233.
[8] Liu, J.; Han, Y.; Liu, C.; Yang, B.; Liu, Z. Origin of the liquid/gaseous water binding energy
splitting measured via x-ray photoelectron spectroscopy J. Phys. Chem. Lett. 2023, 14, 863–
869.
[9] Weeraratna, C.; Amarasinghe, C.; Lu, W.; Ahmed, M. A direct probe of the hydrogen bond
network in aqueous glycerol aerosols J. Phys. Chem. Lett. 2021.
[10] Hahn, Marc Benjamin; Dietrich, Paul M.; Radnik, Jorg In situ monitoring of the influence of
water on dna radiation damage by near-ambient pressure x-ray photoelectron spectroscopy
Commun. Chem 2021.
[11] Stanton, J. F.; Bartlett, R. J. The equation of motion coupled-cluster method. A systematic
biorthogonal approach to molecular excitation energies, transition probabilities, and excited
state properties J. Chem. Phys. 1993, 98, 7029–7039.
11
[12] Levchenko, S. V.; Krylov, A. I. Equation-of-motion spin-flip coupled-cluster model with
single and double substitutions: Theory and application to cyclobutadiene J. Chem. Phys.
2004, 120, 175–185.
[13] Krylov, A. I. Equation-of-motion coupled-cluster methods for open-shell and electronically
excited species: The hitchhiker’s guide to Fock space Annu. Rev. Phys. Chem. 2008, 59,
433–462.
[14] Cederbaum, L. S.; Domcke, W.; Schirmer, J. Many-body theory of core holes Phys. Rev. A
1980, 22, 206.
[15] Vidal, M. L.; Feng, X.; Epifanovsky, E.; Krylov, A. I.; Coriani, S. A new and efficient
equation-of-motion coupled-cluster framework for core-excited and core-ionized states J.
Chem. Theory Comput. 2019, 15, 3117–3133.
[16] Vidal, M. L.; Krylov, A. I.; Coriani, S. Dyson orbitals within the fc-CVS-EOM-CCSD framework: theory and application to X-ray photoelectron spectroscopy of ground and excited
states Phys. Chem. Chem. Phys. 2020, 22, 2693–2703.
[17] Kaliman, I.; Krylov, A. I. New algorithm for tensor contractions on multi-core CPUs, GPUs,
and accelerators enables CCSD and EOM-CCSD calculations with over 1000 basis functions
on a single compute node J. Comput. Chem. 2017, 38, 842–853.
[18] Epifanovsky, E.; Wormit, M.; Kus, T.; Landau, A.; Zuev, D.; Khistyaev, K.; Manohar, P. U.; ´
Kaliman, I.; Dreuw, A.; Krylov, A. I. New implementation of high-level correlated methods
using a general block-tensor library for high-performance electronic structure calculations J.
Comput. Chem. 2013, 34, 2293–2309.
=
12
Chapter 2: Core-ionization spectrum of
liquid water
2.1 Introduction
Water is the most important substance on Earth. It is essential for life to exist—the working
definition of a planet capable of sustaining life includes the presence of liquid water.1 Water
is a natural environment for biochemical, geophysical, environmental, and many technological
processes. Hence, understanding properties of water on a molecular level is a prerequisite for
understanding how water influences and drives chemistry. Yet, despite a plethora of experimental
and theoretical studies, water continues to puzzle scientists.2–4
In this contribution, we focus on the most basic property of water, that is, its electronic structure. The key element of electronic structure is the shapes and energies of molecular orbitals,
which describe the states of electrons and ultimately determine chemical properties of a substance.
Molecular orbitals can be probed by photoelectron spectroscopies, connecting theory with the experiment.5, 6 An important question is how molecular orbitals are affected by the environment, that
is, in which way a water molecule in bulk water differs from an isolated water molecule.7 Because
molecular orbitals are sensitive to the intermolecular interactions, their energies can also provide
information about local solvent structure and its fluctuations.8
13
Photoelectron spectroscopy using microjets is a tool to interrogate electronic structure of solvated species, including that of the bulk solvent itself.9
It has been applied to study various aqueous solutions, both in the UV-Vis and X-ray regimes.7, 10–16 Because the light beam has a finite
width, it ionizes both molecules in the microjets and gaseous molecules around it, giving rise to
the spectra containing two well-separated features—a narrow gas-phase peak and a broader band
corresponding to the liquid. Superficially, these experiments appear to be a straightforward extension of the gas-phase photoelectron experiments17–21 in which the ionization energy (IE) is given
by the difference between the energy of the photon (hν) and a measured kinetic energy (KE) of the
ejected electrons:
IEgas = hν−KE. (2.1)
However, the quantitative interpretation of these experiments is more difficult, as explained in
detail by Olivieri et al.13 and more recently by Thurmer et al. ¨
16 The essential difference is that in
bulk measurements, the kinetic energy of ejected photoelectrons is affected by the interactions with
the surface. In addition, the interpretation of experimental spectra is affected by various parasitic
fields, which are not present in the gas-phase experiments, and uncertainties in calibration.16
The energy diagram in Fig. 2.1 explains this issue. Whereas the Fermi level (EF) of the liquid
sample (e.g., microject) and the analyzer are matched by design, their vacuum levels (defined as
zero kinetic energy of photoelectrons) do not match. The net result is that in order to extract the
true IE from microjet experiments,13 one needs to know the difference between the workfunctions
of the liquid and the analyzer, φwat and φana,
IEwat = hν−KE+ (φwat −φana). (2.2)
Proper accounting for this difference is not trivial22 and different approaches have been developed
to address referencing and calibration issues.
14
This journal is © the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 29506 --29515 | 29507
(strongly) on the chemical composition of the solution. These
results unambiguously reveal that the two assumptions invoked
a priori to interpret PES measurements from aqueous solutions
using soft X-ray radiation are not universally valid and may
introduce errors in the reported ionization energies. Quantitative
ionization energies from an aqueous solution are only realized
after accounting for the vacuum level offset between the
solution and the photoelectron analyzer, something heretofore
never reported. From the vacuum level offset we derive the work
function (electrochemical potential) of the aqueous solution
and show that it too varies substantially with the chemical
composition of the solution.
Ionization energies from photoelectron spectroscopy
Gas phase water has a well-known IE18,19 that can be measured
by PES,
IEmeas
gas = hn ! KE, (1)
where hn is the photon energy and KE the measured kinetic
energy of the photoelectron. Zero KE is defined at the vacuum
level of the analyzer. The measurement is straightforward to
interpret in absence of any external influence because the
vacuum level (Evac) of the gas is equilibrated with that of the
analyzer (Fig. 1, right hand side). In this case the measured
ionization energy of the gas (IEmeas
gas ) is also the real ionization
energy (IEreal
gas ), that is, the exact difference in energy between
the occupied orbital under study and vacuum level of the gas.
Photoelectron spectroscopy from a liquid jet of aqueous
solution combines a gas phase environment (the Knudsen layer
of evaporating/condensing water molecules that surround the jet)
with that of a (liquid) surface (Fig. 1, left hand side). The vacuum
level of the gas is no longer equilibrated with that of the analyzer
everywhere but is instead now pinned to both the vacuum levels
of the analyzer and aqueous solution20 with a linear gradient of
the electric field between. The (liquid) surface and the analyzer
are equilibrated through their Fermi energies, ensured during
the PES experiment by measuring only conductive solutions that
are in electrical contact with the analyzer (the liquid is grounded
with the analyzer). The real ionization energies of the aqueous
solution are given by,
IEreal
water = hn ! KE + (fwater ! fana), (2)
in absence of a streaming potential where fwater and fana are
the work functions of the aqueous solution and analyzer,
respectively, and zero KE is again defined at the vacuum level
of the analyzer. The measured ionization energies—those
reported in the literature under the assumption the vacuum
level, and not the Fermi energy of the aqueous solution is
equilibrated with the analyzer—are given by,
IEmeas
water = hn ! KE, (3)
which deviates from the real value by the vacuum level offset
(fwater ! fana). While eqn (2) is straightforward to interpret, it
requires quantifying the vacuum level offset between an aqueous
solution and the analyzer, something heretofore never reported.
Without accounting for this offset, IEs for the same aqueous
solution will vary between laboratories (as is evident from the
literature)11–16 because KE depends on the vacuum level of the
analyzer used to record the spectra. This effect is evident from
the example schematic energy diagram of Fig. 1, which depicts
the case where the work function of the aqueous solution is
greater than that of the analyzer [(fwater ! fana) 4 0]. Under these
conditions IEmeas
water is lower than IEreal
water; however, because no
adequate reference IEs exist in the literature for aqueous solutions this effect would not be immediately obvious to the experimenter. A quantifiable observable is IEmeas
gas , which under these
conditions is also lower than IEreal
gas (for which adequate reference
is available18,19), because the gas resides in a gradient of the
electric field that affects all its energy levels (vacuum level and O
1s core level equally, see Fig. 1).20 One can, therefore, immediately
determine if the work function of an aqueous solution is greater
(IEmeas
gas o IEreal
gas ) or smaller (IEmeas
gas 4 IEreal
gas ) than that of the
analyzer by comparing the IE of gas phase water in absence and
in the presence of the aqueous solution surface. If the vacuum
levels of the aqueous solution and analyzer are equilibrated, as is
traditionally assumed for the interpretation of liquid jet PES
measurements, then IEmeas
gas = IEreal
gas and IEmeas
water = IEreal
water.
Experimental observation of a vacuum level offset between an
aqueous solution and the photoelectron analyzer
The ionization energy of the O 1s orbital of gas phase water is
measured under near ambient pressure photoemission (NAPP)
conditions at 1.5 mbar (Fig. 2, red trace). After calibrating the
photon energy using hn and 2hn (see Experimental methods) we
obtain the real ionization energy, IEreal
gas = 539.82("0.02) eV,
a result that agrees well with the literature.18,19 Exchanging the
NAPP conditions of the gas phase measurement for a liquid jet
Fig. 1 Energy level diagram for a gas phase (right hand side) and liquid jet
of aqueous solution (left hand side) photoelectron spectroscopy experiment. Abbreviations: Evac, vacuum level; EF, Fermi energy; f, work function;
ana, analyzer; KE, kinetic energy; IE, ionization energy. In the gas phase
experiment the vacuum level of the gas is equilibrated with that of the
analyzer, whereas in the aqueous solution experiment the vacuum level of
the gas is pinned to the vacuum levels of the liquid and of the analyzer,
which results in an electric field between the two (depicted as sloped energy
levels of the gas). Zero kinetic energy is defined for both experiments as the
vacuum level of the analyzer.
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Figure 2.1: Energy-level diagram for photoelectron experiments in a gas phase (right panel)
and a liquid jet (left panel). Abbreviations: Evac, vacuum level; EF, Fermi energy; φ, work
function; ana, analyzer; KE, kinetic energy; IE, ionization energy. The vacuum level (i.e.,
the level corresponding to the ejected electrons with zero kinetic energy) in the liquid jet
is higher than the vacuum level of an isolated molecule in gas phase due to the presence
of the field created by the water surface. This field also affects the gas-phase molecules in
the vicinity of the jet. Whereas in the gas-phase experiment the vacuum level of the gas
matches that of the analyzer, in the aqueous solution experiment the vacuum level of the gas
is pinned to the vacuum levels of the liquid and of the analyzer, which results in an electric
field between the two (depicted as sloped energy levels of the gas). The Fermi levels of the
jet and the analyzers match by design (they are both grounded), but their vacuum levels do
not because of the difference in the respective workfunctions. Zero kinetic energy is defined
for both experiments as the vacuum level of the analyzer. Reproduced from Ref. 13 with
permission from the Royal Society of Chemistry.
Importantly, the jet also affects the energy levels of the gas-phase molecules near it, so the
measured IEs of the gas-phase molecules in microjet experiments differ from the true gas-phase
IEs:13
IEgas = hν−KE−c(φwat −φana). (2.3)
15
Here c is a geometric factor, which depends on the details of the experimental setup—i.e., which
area around the microjet is probed by the beam—usually, c varies between 1 and 0.5.13
This contribution due to the workfunction differences, which gives rise to the vacuum levels
mismatch between the jet and the analyzer and which was neglected in early experiments, is responsible for noticeable discrepancies in the reported IEs (as illustrated by the water core-level IEs
below). Olivieri et al.13 introduced a procedure designed to extract the true IEwat from the microjet
experiments by applying a variable bias to the microjet. The bias shifts the energy levels of the
liquid and by measuring the dependence of the measured IEs on the applied bias the true IEs can
be extracted. By applying this technique to core ionization of water (specifically, 0.05 M solution
of NaCl), the authors13 determined that in their experiment (φwat −φana) equals +0.57 (±0.07) eV
and the geometric factor is c = 0.70 (±0.05) eV·V
−1
. This means that the bulk and gas-phase peak
positions from the zero-bias measurement need to be shifted by 0.57 eV and by 0.40 eV, respectively. The value of 0.57 eV is consistent with the optimal bias of +0.5 eV determined in Ref.
13.
Thus, the workfunctions difference affects not only the absolute value of the bulk IE, but also
the shift of the bulk IE relative to the gas-phase IE (∆IE). Therefore, the true ∆IE cannot be taken as
a difference between the gas-phase and liquid peaks’ maxima from an unbiased microjet spectrum.
This effect on gas-phase molecules around the jet is clearly seen from the shift of the gas-phase
peak relative to the true gas-phase IE—in Ref. 13 this shift equals 0.41 eV; the microjet value is
red-shifted, as expected from Fig. 2.1 and from the theoretical value of the interface potential23
(illustrated in Fig. 2.2).
Thurmer et al. developed an alternative approach to eliminate referencing the bulk IEs to ¨
the perturbed gas-phase solvent peak16—they determine the absolute bulk IEs as the difference
between the peaks in the photoelectron spectrum (i.e., measured photoelectron kinetic energy)
relative to the cutoff value defined as the slowest photoelectrons emerging from the liquid jet.
16
These low-energy photoelectrons are photoelectrons that lost nearly all their kinetic energy due to
inelastic scattering and have minimal energy to escape the liquid. To reveal this intrinsic onset of
the photoelectron spectrum, a negative accelerating bias is applied.16 This approach does not rely
on the gas-phase peak and does not require quantification of various parasitic fields. Thurmer et al. ¨
also introduced an additional procedure aiming at determining the solution workfunction.16They
reported16 the following values of the 1sO IE and φwat: 538.10±0.05 eV and 4.73±0.09 eV, to
be compared with 538.21±0.07 eV and 4.65±0.09 eV by Olivieri et al.13 Despite using different
protocols, the two sets of values agree with each other within the reported error bars.
A recent paper24 focused on another technique for measuring bulk IEs based on ambientpressure experiments using water films (dip-and-pull method) instead of microjets and discussed
the application of bias for determining the true bulk IEs from the photoelectron measurements.
The authors estimated bias required to align the vacuum levels of the liquid and the analyzer to be
equal +0.435 V (versus the standard hydrogen electrode), which is close to the estimated optimal
bias in the microjet experiment.13 However, the extracted values of the core IEs of water were
different.
In microscopic terms, the real IE of solvated species includes contribution of the interface potential, ϕin (Galvani potential). The difference between ϕin and ϕout (potential outside the sample
called Volta potential) is called surface potential χ. For idealized neutral solutions, the outer potential is zero, such that χ = ϕin; however, the exact value of ϕout depends on the position of the
surface defining the interface. The net result is that in order to obtain the intrinsic, bulk IE, one
should account for the interface potential:
IEwat = hν−KE−ϕin. (2.4)
17
Experimentally, the real IE is measured, i.e., including contributions of ϕin. Thus, a special care
is needed to make meaningful comparisons with the theoretical values (which may or may not
include the interface potential).
Fig. 2.2 shows the results of simulations of water illustrating the electrostatic potential arising solely due to the interface. Hence, in order to reach the vacuum level (lowest free-electron
state), the electrons need to overcome the electric field created by the liquid/gas interface. Surface
potential also contributes to the solvation free energy, e.g., for a particle with charge q
∆Greal = ∆Gintr −qϕin, (2.5)
where ∆Greal and ∆Gintr are real and intrinsic (i.e., bulk value without interface) Gibbs free energies
of solvation (we use the label “real” for consistency with previous work23).
position of the oxygen atom for its periodic image will lie
within the unit cell and this image is thus chosen as part of
the bulk phase. The resultant potential from this neat liquid
simulation is labeled TIP3P-NEATa. Similarly, the assignment of boundary waters or their image to the bulk phase can
be made based on the geometric center !TIP3P-NEATb" or
the bisector point closest to the hydrogen atoms !TIP3PNEATc" as in the center assignment in the IP-TIP3b and
IP-TIP3c models, respectively. The corresponding interface
potentials are plotted in Fig. 3. The potentials from these
simulations correspond perfectly to their analogs in the IP
models.
To compare with a more realistic simulation of a
vacuum-water interface, the IP model whose center coincides
with the geometric center of the TIP3P force field !i.e., IPTIP3Pb" is chosen. This choice is based on the simple rational that molecules at the interface prefer to be solvated by
surrounding molecules and will therefore tend to minimize
their exposed surface area to vacuum. The realistic vacuumwater interface system is generated from a MD simulation of
500 TIP3P water molecules forming a liquid slab surrounded
by vacuum. The molecule and charge density from the simulation and IP-TIP3b model are presented in the top panels of
Fig. 4. Panel 3 shows the corresponding potential profile.
The potential drop for IP-TIP3Pb is 400 mV, in reasonable
agreement with the MD simulation value of 520 mV.
It is clear from Fig. 3 that varying the center of the IP
model between the geometric center and the oxygen atom
position can lead to precise agreement with the interface potential calculated from the MD simulation. However, this
does not imply that the orientational structure of water molecules at a vacuum-water interface system is purely isotropic, as it is assumed by construction in the IP model. Experiments have shown that water molecules adopt a subtle
orientational order in the interface region between the liquid
and vacuum.36 This orientational structure does not go away
with an appropriate choice of molecular center but the cumulative contribution to the IP does go to zero.
Missing from these atomistic force fields of water and
methane are atomic charge densities, Zi
. This core atom electron density has a benign affect on the intermolecule electrostatic interactions between molecules and are justifiably excluded from the electrostatic description of these force field
models. However, the inclusion of this contribution can have
a dramatic affect on the interface potential !see Sec. IV A".
Taking water as an example, we consider the IP-TIP3Pb
model with the addition of an atomic potential that is reasonably consistent with the atomic properties of an oxygen atom
!IP-TIP3Pb!
". The atom potential is centered on the oxygen
atom !R1" with a charge of Z1= 8e and a negative charge
density width that is !1= 0.5 Å. The resultant potential,
shown in panel 4 of Fig. 4, is approximately 7 V. This is
nearly an order of magnitude larger in magnitude and of
opposite sign to the potentials calculated from the IP-TIP3P
model or the associated TIP3P force field. Such a large posi-
-20 0 20
Z (Å)
-800
-600
-400
-200
0
φ (mV)
IP-TIP3Pa
IP-TIP3Pb
IP-TIP3Pc
TIP3P-NEATa
TIP3P-NEATb
TIP3P-NEATc
FIG. 3. !Color" The potential drop along the interface
normal for three IP models of the TIP3P water force
field. The center of the IP-TIP3Pa corresponds to the
position of the oxygen atom. The center of the IPTIP3Pb model is the geometric center of the molecule
and the center of the IP-TIP3Pc model is along the
H–O–H bisector closest to the hydrogen atoms. The potential in the bulk region of the interface model is negative with respect to the vacuum for all three models.
Also included are the interfacial potential computed
from simulation of neat liquid water in PBCs with the
TIP3P force field. Image atoms are replaced by vacuum
leaving only the molecules of the reference unit cell.
Boundary conditions are applied to positions on the
molecule that are analogous to the choice for molecular
center in the IP models.
-20 -10 0 10 20
Z (Å)
-600
-500
-400
-300
-200
-100
0
φ (mV)
-20 -10 0 10 20
0
0.01
0.02
0.03
0.04
ρ (N/Å3
)
-20 0 20
-0.04
0
0.04
ρ (e/Å3
)
-20 0 20
0
2000
4000
6000
8000
TIP3P
IP-TIP3Pb
IP-TIP3Pb*
1 2
3 4
FIG. 4. !Color online" Shown are the molecule density, charge density, and
potential along the interface normal of the vacuum-water system. The molecule density is plotted in the top left panel, the charge density is plotted in
the top right panel and the interface potential is plotted in the bottom panel.
The data for the TIP3P model are from a simulation of a liquid water slab
comprised of 500 molecules surrounded by vacuum. The IP-TIP3Pb model
is the IP model translated from the TIP3P water force field with molecular
center chosen to coincide with the geometric center of the molecule. The
IP-TIP3Pb! model includes an atomic potential contribution.
234706-6 E. Harder and B. Roux J. Chem. Phys. 129, 234706 !2008"
Figure 2.2: Molecular density (ρ(N)), charge density (ρ(e)), and potential (denoted as φ in
this figure and as ϕin in the text) along the interface normal of the vacuum-water system
computed using several water models. Reprinted from Ref. 23 with permission of AIP Publishing.
The microscopic origin of the interface potential and different ways to compute it have been
extensively discussed.23, 25–28 Importantly, contrary to earlier proposals, surface potential does
18
not arise due to orderd dipoles, but is dominated by quadrupole terms. For example, by using
atomistic simulations Harder and Roux23 have illustrated that the interface potential arises not due
to orientational ordering of molecules on the surface, but due to the intramolecular asymmetry
of the charge distribution. By using molecular dynamics (MD) simulations, they estimated the
interface potential of water to be equal to -510 meV. An ab initio simulation by Mundy and coworkers25 yielded a smaller value, -18 meV. The negative sign means that the solvation energy of
anions becomes less negative and the solvation energy of cations becomes more negative due to the
interface (as per Eq. (2.5)), and that the apparent IE of the solvated neutral species is red-shifted
relative to the intrinsic bulk IE (as per Eqs. (2.4)).
We note that the definition and the value of the interfacial potential depend on the type of experiment, as has been explained in Ref. 26: i.e., the magnitude and even the sign of water’s surface
potential differs between electrochemical experiments (which are relevant to the photoionization
experiments) and high-energy electron holography measurements. Despite these difficulties faced
by theory and experiment, the literature seems to converge on the existence of a negative effective
potential for a single ion moving from the gas phase into liquid water of roughly -0.4 V27 or less.
In addition to the effect of the interface, the measured IEs of bulk water can also be affected
by the presence of solutes. Both microjet and dip-and-pull experiments are carried out using electrolyte solutions (i.e., NaCl, KOH) to mitigate the effect of creating charges in the jet (due to
streaming through the capillary and ionization) that can affect the subsequent ionization. The possible effect of ions present in these solutions on the IE of the bulk solvent was shown to be small29
but non-negligible. Importantly, that the ions affect both the intrinsic bulk IE and the surface potential. Finally, the measured IEs always contain contributions from both the surface and the bulk
species because the photoelectrons have finite escape depth (determined by the inelastic mean free
19
path). The contribution of the bulk increases at higher photon energies. The simulations24 suggest that the apparent peak position of liquid water can vary by as much as 0.3 eV as the inelastic
mean-free path changes from 3 to 50 A. ˚
In this contribution, we focus on the intrinsic 1sO IE of bulk water. The gas-phase value is
well-known:13, 30, 31 539.82 (±0.02) eV (value taken from Ref. 13). The question is then what
is the magnitude of the shift of the 1sO level in the bulk relative to the gas-phase IE (∆IE). As in
the valence domain, the IEs of the solvated neutral species are expected to red-shift relative to the
gas phase due to the strong solvent stabilization of the resulting cationic state. The magnitude of
the shift quantifies the overall effect of the solvent stabilization of the core-hole state, whereas the
width of the peak gives a measure of the solvent’s structural fluctuations. In contrast to the valence
domain, the photoelectron spectra of the core-level ionization do not contain multiple overlapping
bands and, therefore, are somewhat easier to interpret. Core-level states are also known to be very
sensitive to the local environment, which can provide a handle for connecting the spectroscopic
measurements with the structure.
Table 2.1 summarizes experimental values of ∆IE for 1sO ionization of liquid water from different experimental setups (microjets, clusters, dip-and-pull). These papers, which span the time
range from 1986 till 2023, report values from -1.3 to -2.8 eV. To further illustrate the discrepancies
between different experiments, Fig. 2.3 shows three spectra from Refs. 12, 13, and 24. These
discrepancies are quite substantial, providing the illustration of the challenges in experimental determination of IEs due to the factors discussed above. They also illustrate the progress made by
the experimental community in addressing these challenges. In particular, the best experimental
estimates derived from microjet experiments are -1.61 eV (Ref. 13) and -1.72 eV (Ref. 16) agree
with each other within the specified error bars. In contrast to the microjet experiments, the best
estimate from “dip-and-pull” experiments is -2.2 eV (Ref. 24).
20
Table 2.1: Experimental values of the 1sO level shift (∆IE) in bulk water. The best values are
shown in bold (see text).
∆IE, eV Photon Energy, eV Details Source
-1.3 560 Large water cluster (∼1000 molecules) 32
-1.3 720 water layer on SiO2 film 33
-1.5 800 liquid jet, 3 M NaNO2/NaNO2 34
-1.5 560 liquid nanoparticles, 0.01 M glycerol 35
-1.58 420 liquid jet, 0.05 M NaCl, zero bias 13
-1.61 420 liquid jet, 0.05 M NaCl, corrected 13
-1.6 750 liquid jet, 10−4 M NaCl, 0.003 mbar, 25◦C 12
-1.72 650 liquid jet, 0.05 M NaCl, from absolute IEsa 16
-1.77 600 liquid jet 11
-1.8 735 ice 36
-1.9 1486 7 mol% LiCl, thin film 37
-1.9 1253 liquid jet, 1 M KCl 14
-1.9 750 liquid jet, 10−4 M NaCl, 1.4 mbar, 4◦C 12
-1.91 1012 liquid jet, 0.14 M NaCl 15
-2.0 4000 dip-and-pull, Pt/1 M KOH 38
-2.0 4000 dip-and-pull, Pt/1 M KOH and 0.1 M KF 39
-2.0 4000 dip-and-pull, Pt/0.1 M KOH 40
-2.0 4000 dip-and-pull, Co/0.1 M KOH 41
-2.2 4000 dip-and-pull, CoxO/1 M KOH 42
-2.2 5400 dip-and-pull, Pt/1 M KF 24
-2.3 4000 dip-and-pull, NiFe/0.1 M KOH 43
-2.4 4000 dip-and-pull, NiFecoCeOx/1 M KOH and 0.1 M KF 44
-2.8 4000 dip-and-pull, Pt/6 M KF 45
a
Computed using the reported absolute water 1sO IE (538.10 eV) and the gas-phase value (539.82
eV).
Given the challenges of the experimental determination of the absolute values of bulk IEs
and the surface potential, and persistent disagreements13, 16, 46, 47 about the details of experimental
protocols, accurate theoretical modeling of the core-ionization spectrum of water is important for
providing a robust theoretical reference.
21
Figure 2.3: Experimental spectra from Pellegrin et al. (at 0.003 mbar and room temperature),12 Liu et al.,24 and Olivieri et al. (at 0 V bias).13 The spectra were aligned by the
position of the gas-phase peak (the narrow feature) by applying global shifts of +0.59 eV,
+4.9 eV, and +0.87 eV, respectively.
Previous theoretical calculations of core-level ionization of liquid water have been limited to
density functional theory (DFT) and Hartree–Fock methods,11, 24, 34, 48 and varied greatly in terms
of model structures and sampling of equilibrium dynamics (many were carried out on model clusters rather than bulk).
Here we employ high-level quantum chemistry methods to compute intrinsic core-level IE of
bulk water. In addition to providing the best theoretical estimate of the bulk IE, we also aim to
carefully investigate the convergence of the spectrum with respect to the details of computational
protocol, to aid future theoretical studies. We use MD simulations with classical forcefields and
ab initio potentials to simulate bulk water, and then use the snapshots from the MD simulations to
22
carry out QM/MM (quantum mechanics/molecular mechanics) calculations of IEs using equationof-motion coupled-cluster (EOM-CC) methods49 adapted for calculations of core-level states50, 51
by core-valence separation (CVS).52 EOM-CC is a state-of-the art technique capable of treating
electronically excited and ionized species.49 Here we go beyond EOM-CC with single and double
excitations (EOM-CCSD) and also evaluate the effect of triple excitations by using the MLCC3
method (multilevel coupled-cluster method with triple excitations).53–56 The main challenge in
applying these methods to modeling condensed-phase phenomena is how to properly account for
the effect of the solvent via embedding. Here we show that in calculations of core-level IEs simple electrostatic embedding (QM/MM)57, 58 converges slowly with respect to the size of the QM
system, which illustrates the high sensitivity of the core-level states to the environment. We also
investigate the contribution of different types of structures present in bulk water in the overall spectrum. Our calculations represent the most ambitious simulations that provide a reliable ab initio
estimate of the core-ionization spectrum of water.
The structure of the paper is as follows: the next section describes the details of computational
protocols and Section 2.3 presents the results of the simulations of core-level IE of bulk water.
In Conclusions, we outline the limitations of the current simulations and provide suggestions for
future studies.
2.2 Computational details
The simulations of core-ionization spectra include two steps: equilibrium simulations of bulk water
and the calculations of the IEs using the snapshots from the equilibrium simulations. Electronic
structure and AIMD calculations were carried out using the Q-CHEM and eT electronic structure
packages,59–61 and MD simulations were carried out using GROMACS.
62
23
2.2.1 Equilibrium simulations of bulk water
The equilibrium simulations were carried out using classical MD with TIP3P63 waters and with ab
initio molecular dynamics (AIMD). In AIMD simulations, we used QM/MM scheme with the QM
waters described by ωB97X-D/6-31G* and MM waters described by TIP3P.
The MD simulations were set up as follows. First, we optimized the structure of a single water
molecule with ωB97M-V/aug-cc-pVTZ. We then used this structure to create a cubic water box 22
A˚ ×22 A˚ ×22 A to serve as the starting structure for MD simulations. The simulation box consisted ˚
of 392 water molecules, giving rise to density of 997 kg/m3
. The system was then equilibrated
using the NVT ensemble at 300 K for 2 ns (the time step for the thermostat was 100 fs). Following
the equilibration, we ran a 3 ns production trajectory (with time step of 2 fs) from which snapshots
for the spectra calculations were collected. This trajectory was also used to compute structural
parameters of bulk water.
The QM/MM AIMD simulations were initiated from 40 structures taken from the equilibrium
MD trajectory. The QM region included all waters within 6.5 A radius from the central water ˚
(this selection criterion resulted in the 35-40 QM water molecules, depending on a snapshot).
The QM part was treated by ωB97X-D/6-31G* and the MM part by TIP3P. From each starting
structure, a 2 ps long trajectory was propagated with a time step of 42 a.u. (1.016 fs) using the
NVT ensemble at T = 300 K (the thermostat was applied every 100 fs). The first picosecond of each
trajectory was treated as equilibration and the second ps was treated as a production run. Hence,
the total simulation time in AIMD was 40 ps; these trajectories were used to collect snapshots for
the QM/MM simulations of the ionization spectrum and to compute structural parameters of bulk
water. We note that this time is shorter than in the MD simulations, however, the convergence of
the radial distribution function, gOO(r), shows that this simulation time is sufficient (e.g., gOO(r)
computed from the full MD trajectory of 3 ns and from a 40 ps segment look the same).
The link to downloadable tar files with 3,000 MD and 400 AIMD snapshots is given in the SI.
24
The simulation of bulk water properties is known to be difficult. As discussed in several recent
papers,64–66 the results are sensitive to the interaction potentials used, the size of the simulation
box, the exact details of dynamics (e.g., thermostat), as well as on whether nuclear quantum effects
are included.
Although we used one of the best functionals (ωB97X-D,67 which includes long-range
Coulomb exchange and dispersion correction) in our AIMD simulations, the analysis of structural parameters (gOO(r), the number of hydrogen bonds formed) shows that our AIMD water
is somewhat over-structured relative to TIP3P and state-of-the-art simulations,65, 66 similar to the
simulations using less accurate density functionals. However, the structure of the second solvation
shell appears to be reproduced better with AIMD than with TIP3P, as compared to the experimental gOO(r). The detailed analysis of structural parameters extracted from the MD and AIMD
simulations as well as comparison with other simulations are given in the SI. Overall, we find that
the differences between the spectra computed with MD and AIMD snapshots are small and are
washed out by the statistical averaging. However, it is desirable to improve the sampling in future
work by using, for example, path-integral simulations with accurate many-body potentials, as was
done in Ref. 66.
Here we define hydrogen bonds by the criterion of Luzar and Chandler,68 i.e., when the O-O
distance RO−O < 3.5 A and ˚ ∠ O...O−H <30◦
. We use this definition in the analysis of structures
from the equilibrium simulations and in comparisons between different protocols of building up
QM.
2.2.2 Calculations of core IEs
IEs were computed using the CVS-EOM-IP-CCSD method50, 69, 70 and the 6-311+G(3df) basis set
fully uncontracted on oxygen, denoted below as u6-311+G(3df). Uncontracted Pople’s basis sets
25
have been shown to be effective in describing strong orbital relaxation effects common in corehole states.71 The effect of triple excitations was accounted by additional CVS-EOM-IP-MLCC3
calculations for a smaller number of snapshots.
Table 2.2: 1sO IE of isolated water molecule.
Method IE, eV ∆ vs exp
fc-CVS-EOM-IP-CCSDa 540.26 +0.44
CVS-EOM-IP-CCSDa 541.32 +1.50
fc-CVS-EOM-IP-CC3a 537.68 -2.14
CVS-EOM-IP-CC3a 538.72 -1.10
CVS-EOM-IP-CCSD/AC5Zb 541.78 +1.96
CVS-EOM-IP-CCSDT/AC5Zb 539.81 -0.01
Exp.c 539.82±0.02
a u6-311+G(3df). b From Ref. 72, with scalar relativistic corrections; AC5Z denotes
aug-cc-pCV5Z. c From Ref. 13.
There are two variants of CVS-EOM-CC approach, one in which the core is frozen at the CCSD
step (fc-CVS-EOM-CCSD50) and the one in which the core is active (CVS-EOM-CCSD69, 70).
Table 2.2 summarizes the results for the isolated water molecule (using ωB97X-D/aug-cc-pVTZ
optimized structure). As one can see, the two versions of CVS-EOM-IP-CCSD differ by about
1 eV. At the CCSD level, fc-CVS-EOM-IP-CCSD is closer to the experiment due to fortuitous
cancellation of errors, however, at the CC3 level, CVS-EOM-IP-CC3 yields a smaller error than
fc-CVS-EOM-IP-CC3. The effect of triple excitations is -2.6 eV for both methods. Remarkably,
even when triples are included (at the CC3 level), the computed IEs are red-shifted with respect to
the experiment by 1-2 eV. This large discrepancy can be attributed to the slow convergence of the
core IEs with respect to the correlation treatment. According to a detailed benchmark study,72 for
molecules comprising first-row elements, quantitative agreement with experimental IEs is achieved
at the CVS-EOM-IP-CCSDTQ level whereas CVS-EOM-IP-CCSDT is within 0.3 eV from the
experimental values. The results for water from this study are shown in Table 2.2—as one can see,
26
full inclusion of triple excitations brings the computed IE within 0.01 eV from the experiment.
Whereas this calculation used a very large basis set and included scalar relativistic correction,
the comparison between the respective CVS-EOM-IP-CCSD and CVS-EOM-IP-CCSDT values
shows that the main source of the errors in our CC3 calculations is due to an insufficient correlation
treatment. Despite these discrepancies in the absolute value of gas-phase IE computed with CVSEOM-IP-CC3, we anticipate much higher accuracy in ∆IE, as the errors should cancel out. In
the simulations of the spectrum, we use fc-CVS-EOM-IP-CCSD and evaluate the effect of triple
excitations to the shift by computing the difference between the ∆IE from CVS-EOM-IP-CCSD
and CVS-EOM-IP-MLCC3.
In order to estimate the effect of the structure on the IE, we also computed IEs (with fcCVS-EOM-IP-CCSD) for the structures optimized with ωB97X-D/6-31G* (the level at which the
AIMD simulation was performed) and the TIP3P water structure. The respective IEs are 540.26
eV and 540.25 eV, respectively, which is close to 540.26 eV (for the ωB97X-D/aug-cc-pVTZ
structure). Hence, small differences of water structures due to different levels of theory used to
simulate the bulk are not expected to affect the computed IEs.
To compute bulk spectra, we considered several protocols designed to mitigate potential issues
due to the description of waters on the boundary between the QM and MM parts. Our results
indicate that electrostatic embedding in which the MM waters are described by point charges is
not sufficient for describing core-level IEs and that the IEs of the water molecules on the boundary
are not accurate. Therefore, we used the following multi-layer scheme for the calculation of the
spectra. We first computed all core IEs in the QM part—for example, for a calculation with 5
waters in the QM part, we computed 5 IEs. We then analyzed the respective Dyson orbitals51
to assign the computed IEs to particular water molecules. Fig. 2.4 shows a structure of 5 water
molecules embedded in the MM part with the Dyson orbital on the central water molecule. The
27
Figure 2.4: Model system for liquid water. Top: simulation box with 392 water molecules.
Bottom: A cluster from the center of the box of one water molecule and its first solvation shell
representing a minimal QM subsystem. The Dyson orbital associated with the 1sO ionization
of the central water is shown in blue.
assignment of IEs can also be accomplished by considering the Mulliken charges from the natural
orbital analysis73 (we used this approach in the production simulations). We then constructed
the bulk spectra by taking the IEs corresponding to a specified number of water molecules. For
example, in the calculations with 5 QM waters, one can construct the spectrum by taking all 5 IEs
28
or by taking only the IE of the central water molecule. The overall spectra were constructed as
histograms (with 0.05 eV bins) by collecting the IEs from the snapshots and then convoluted with
gaussians. The spectra shown in the manuscript were produced using gaussians with 0.2 eV width
(FWHM). Discarding some of the computed IEs in the construction of the spectra slows down the
convergence with respect to the number of snapshots, but removes the artifacts due to waters on
the QM/MM interface. As we show below, the best protocol (in terms of balancing accuracy of
IEs and convergence with respect to sampling) entails using 20 QM waters and assembling the
spectra by taking the IEs of the 5 central waters.
Triple excitations are important for obtaining accurate core excitation and ionization energies
within the CC/EOM-CC framework.74–76 The CC3 model includes the effects of triple excitations
in a perturbative manner.53 This method generally scales as O(N
7
). However, for core excitations
using the CVS scheme, the scaling is O(N
6
) in an optimized implementation.55 Nevertheless,
the ground-state calculation is rather expensive, even for systems where the CCSD calculation
is routine. The costs can be significantly reduced using the multilevel coupled-cluster approach
(MLCC) in which the higher-order excitations in the cluster operator are restricted to an active
orbital space. In the MLCC3 model, the triple excitation operator is restricted while the single and
double excitation operators are unrestricted and act in the entire orbital space. The MLCC models
target intensive properties, such as excitation or ionization energies. Additional reduction in cost
can be achieved by restricting the entire cluster operator to an active space. In this way, one obtains
the CC-in-HF models, where the inactive orbitals are not correlated but contribute to the energy via
the Fock matrix.61, 77, 78 This approach, which can be described as a type of electronic embedding,
has been demonstrated to work well for modeling solvent effects in spectroscopy.79
The multilevel models rely on a physically appropriate selection of the active orbital space.
Localized orbitals are an obvious option when the property of interest is localized. Core IEs is
29
an example of such local properties. Another option is to use correlated natural transition orbitals
(CNTOs),80 which use information from the excitation amplitudes of a lower-level model to generate an active space for the MLCC calculation, similarly to other approaches using virtual orbital
spaces truncated on the basis of lower-level natural orbitals.81, 82
Here, we use a hybrid active orbital selection strategy where CNTOs are used for the occupied
orbital space and projected atomic orbitals (PAOs)83 on the five central water molecules determine
the virtual orbital space. In the present MLCC calculations, the active orbital space contains 25
occupied and 245 virtual orbitals. A detailed description of the orbital selection procedure with
CNTOs and PAOs for the MLCC models can be found in Ref. 84.
2.2.3 Protocols for selecting the QM subsystem for IE calculations
The first solvation shell of a water molecule comprises 4-5 water molecules85–87 (Fig. 2.4 shows a
typical structure from the equilibrium simulations). Thus, our minimal QM system for computing
bulk IEs comprises 5 water molecules. As we show below, the convergence of the core IEs with
respect to the QM size is slow and much larger QM systems are needed for converged results.
To determine an optimal protocol for building up a larger QM system, we compared two different approaches. In the first approach, we used the distance from the oxygen or hydrogen of
the central water Ocen(Hcen)···H(O) to select the next water molecule to be added to the QM system. In the second approach, we used the distance from the waters in the first solvation shell
O1
stshell(H1
stshell)···H(O) to select the next water. Fig. 2.5 (c)-(d) shows the shifts in the IE of the
central water (δ) upon increasing the QM size from 5 to 6 waters (with the rest of the waters described by point charges) for two MD snapshots (the 5 waters are chosen to represent minimal QM
in the center of the box and the 6th water is taken at random from the simulation box). For each
snapshot, we determine whether the 6th water molecule acts as a donor or acceptor (or both). As
30
Figure 2.5: Panels (a) and (b) show in bright colors 6-water QM systems comprising the
central water, its first solvation shell, and a 6th water selected from the second solvation shell.
In panel (a), the 6th water molecule accepts a hydrogen bond from a water molecule from the
first solvation shell. In panel (b), the 6th water molecule donates a hydrogen bond to a water
molecule from the first solvation shell. The points in panels (c) and (d) represent the shift in
IE of the central water molecule (δ) due to adding the 6th water. Each point corresponds to
a different selection of the 6th water sampled over two snapshots from the TIP3P waterbox.
The points are color-coded to show whether the 6th water acts as a donor (pink), acceptor
(yellow), or both (blue). Panel (c) shows δ versus the shortest Ocen(Hcen)···H(O) distance and
panel (d) shows δ versus the shortest O1
stshell(H1
stshell)···H(O) distance.
expected, the shifts in IE (δ) are mostly positive when the added water molecule acts as hydrogenbond donor and negative when the added water molecule acts as hydrogen-bond acceptor. For the
structures used to construct Fig. 2.5, the δIE for hydrogen-bond acceptor structures, hydrogenbond donor structures, and non-hydrogen bonded waters range between -0.016 and -0.150 eV,
-0.024 and 0.109 eV and -0.032 and 0.039 eV, respectively. The waters that act as both hydrogen
bond acceptor and donor result in δ between 0 and -0.05 eV. The magnitude of the shifts shown
versus minimum Ocen(Hcen)···H(O) distance (Fig. 2.5c) exhibit no systematic trend. In contrast,
when δ are shown against the minimum O1
stshell(H1
stshell)···H(O) distance (Fig. 2.5d), we observe
a smoother behavior—as the distance increases, the shifts become less negative for the structures
31
with hydrogen-bond acceptors and less positive for the structures with hydrogen-bond acceptors.
Thus, water molecules that form hydrogen bonds with the waters in the first solvation shell have a
greater impact on the IE of the central water molecule.
Figure 2.6: IE of the central water molecule for a single snapshot computed using different
protocols for selecting the QM system in the CVS-EOM-IP-CCSD calculations. The rest of
the waters are described by point charges.
The above analysis shows that waters that are hydrogen-bonded to the first solvation shell have
the strongest effect on the IE of the central water. Hence, we can use this criterion of building up
the QM system instead of the distance from the central water. Fig. 2.6 shows the convergence
of the core IE of the central water with respect to the QM system size using these two criteria for
growing the QM system. The smallest QM comprises 5 water molecules (we pick a water molecule
at the center of the simulation box and chose 4 closest waters). We then increase the QM size by
adding more waters. In protocol 1, we add the next water based on the Ocen(Hcen)···H(O) distance.
In protocols 2 and 3, we add waters based on their O1
stshell(H1
stshell)···H(O) distance. The central
water and the first solvation shell are described with the fully uncontracted 6-311+G(3df) basis on
oxygen and the 6-311G basis on hydrogens. The rest of the waters in the QM system are treated
with a smaller basis—6-311G* in protocols 1 and 2, and 6-31G in protocol 3. The rest of the
waters are described by point charges. We freeze all the oxygen cores in the QM system in the
32
CVS-EOM-IP calculations and request the number of IEs equal to the number of oxygen atoms in
the QM system. We then select the IE of the central water.
As Fig. 2.6 shows, increasing the QM size results in a red shift of the 1sO IE of the central
water. The minimal QM is clearly not sufficient—increasing the size from 5 to 20 waters leads
to the red shift of 0.3-0.4 eV. The convergence is not monotonous and quite slow—we observe
small fluctuations (∼0.1 eV) even beyond 25 waters. Fig. 2.6 also shows that selecting waters
by the distance from the first solvation shell is more effective and results in faster and smoother
convergence. For the QM size of 20 water molecules, the difference in the IE between protocol 1
and 2 is 0.06 eV and between protocol 2 and 3 (basis-set effects)—0.107 eV. We note that the difference due to using a smaller basis increases with the system size (5-12) and then becomes nearly
constant (about 0.10 eV for the QM sizes of 12-20). Hence, in our production-level simulations
we use protocol 3 and estimate the basis-set correction by taking a difference between protocols
2 and 3 for a small number of snapshots. Figure 2.7 shows the convergence of the shift in 1sO IE
of the central water for the simulation with the 20 QM waters between protocol 2 and protocol 3.
The average value of the shift (δbasis) is 0.105 eV.
Figure 2.7: Convergence of the shift in IE in between protocols 2 and 3 (δbasis) with the 20
QM waters. The estimated shift δbasis=0.114 eV, with a 0.035 eV standard deviation.
33
To further analyze the convergence with respect to the QM size, Fig. 2.8 shows the number
of hydrogen bonds formed by the water molecules of the first solvation shell as a function of the
QM size for the same snapshot as used in Fig. 2.6. As one can see, beyond the QM size of 10, the
protocol based on the distance from the first solvation shell captures more hydrogen bonds than
the protocol based on the distance from the central water. We also see that all bonds are captured
for a smaller QM size when QM is selected using the former protocol: saturation is reached at 15
waters versus 18. On the basis of this analysis, we conclude that the QM size of 20 waters should
be sufficient to capture all hydrogen bonds formed by the first solvation shell and we use QM size
of 20 waters in our production-level calculations.
Figure 2.8: The number of hydrogen bonds in the first solvation shell as a function of the QM
system size for a single snapshot.
Table 2.3 shows the effect of triple excitation and different types of embedding evaluated for
the same snapshot as used in Fig. 2.6. The u6-311+G(3df) basis was used for 5 waters and
6-31G for the rest of quantum waters. Although the absolute values of IEs vary between the
methods, the value of the shift, ∆IE, is rather insensitive to the type of embedding, CVS scheme,
or correlation treatment. The effect of MM charges (beyond 20 quantum waters) is about 0.1 eV
34
Table 2.3: Effect of triple excitations on IE (eV) from multi-level calculations.
Method Gas-phase IE Bulk IE ∆IE
CVS-EOM-IP-CCSD (20)a 541.32 540.36 -0.96
CVS-EOM-IP-CCSD-in-HF (5/15)b 541.32 540.45 -0.87
CVS-EOM-IP-CCSD (20) in MMc 541.32 540.24 -1.08
CVS-EOM-IP-CCSD-in-HF (5/15) in MM d 541.32 540.33 -0.99
CVS-EOM-IP-MLCC3 (20)e 538.72 537.38 -1.34
CVS-EOM-IP-MLCC3 (20) in MMf 538.72 537.25 -1.47
fc-CVS-EOM-IP-CCSD in MMg 540.26 539.17 -1.09
a 20 water molecules are treated by CCSD, MM water molecules are ignored.
b 5 water molecules are treated by CCSD, 15 water molecules are treated by HF, MM water
molecules are ignored.
c Same as a, but including MM charges.
d Same as b, but including MM charges.
e 20 water molecules are treated by MLCC3. CNTO/PAOs used for active orbital selection.
f Same as e, but including MM charges.
g 20 water molecules are treated by CCSD, the rest by MM charges
and the effect of freezing the second solvation shell (15 water molecules) at the Hartree–Fock
level of theory is approximately 0.1 eV (decreasing the magnitude of the red shift). The effect
of including triple excitations with MLCC3 is almost 0.4 eV (increasing the magnitude of the red
shift) for this snapshot.
The multilevel framework makes it possible to use approximate triples for a study such as this,
however, the cost remains a limitation and performing such calculation for hundreds of snapshots
is impractical. We therefore use a subset of 40 snapshots to estimate the effect of approximate
triples; the results are shown in Figure 2.9. The average shift is -0.34 eV, with a 0.07 eV standard
deviation. This correction will be applied to the final averaged fc-CVS-EOM-CCSD result to
provide our best estimate for the ∆IE.
35
Figure 2.9: Convergence of the shift in IE due to approximate triples (δtriples) through the
CVS-EOM-IP-MLCC3 approach. The estimated effect of triples on the shift in ∆IE. δtriples=-
0.34 eV, with a 0.07 eV standard deviation.
2.2.4 Extrapolation to bulk
To obtain bulk IE, one needs to extrapolate the results from finite-size simulations, such as our
simulation box used in IE calculations, to the infinite system size. The importance of this step
has been recently illustrated by Tazhigulov and Bravaya88—they have shown that the computed
energies converge very slowly with the simulation size. Based on the linear dependence of the
energies on the inverse system size, they derived a simple correction using Born solvation model.88
For vertical IEs of the solvated neutral species, the Born correction is88
VIE∞ = VIER −
1
2
(1−
1
εopt
)
1
R
, (2.6)
where VIE∞ is intrinsic vertical IE in the bulk, VIER
is VIE computed in a finite cluster of radius
R, and εopt is solvent’s optical dielectric constant (1.78 for water); all quantities are in atomic units.
For our simulation box (R ≈11 A), the correction results in an additional red shift in IE of 0.29 eV. ˚
36
2.3 Results and Discussion
Figure 2.10: Theoretical 1sO spectra of liquid water computed using the minimal QM system
(5 waters) and assembled using the IE of the central water molecule. Top: Spectra constructed from the MD (TIP3P) and AIMD trajectories (400 snapshots). Bottom: Spectra
constructed from the MD trajectory using different number of snapshots. All spectra were
obtained using gaussians with FWHM = 0.2 eV, except for the black trace in the right panel
which was produced using 0.05 eV FWHM to show the intrinsic roughness of the spectrum
produced from the 3,000 snapshots.
37
One of the two goals of this study is to understand the effect of the simulation protocol on the
computed spectra. We begin with comparing the results of the MD and AIMD simulations. Fig.
2.10 (top panel) compares the spectra constructed using the MD and AIMD snapshots. In this
calculation, we used the smallest QM size (5 waters only, the rest being treated by point charges)
and constructed the spectra using the IE of the central water molecule. The two simulations yield
essentially identical spectra, both in terms of the band maximum and in terms of the band width
(inhomogeneous broadening). There are small differences in fine structure, but, as we show below,
these features become smoothed out when the sampling is increased. Thus, we conclude that
for this system the core IE spectra are not sensitive to the differences between TIP3P and AIMD
sampling of the equilibrium dynamics. The bottom panel of Fig. 2.10 shows the convergence of the
spectrum with respect to the number of snapshots. As one can see, while the band maximum and
its width are captured by the simulation with 400 snapshots, the finer details continue to evolve and
the spectrum converges only around 3,000 snapshots (although the remaining roughness remains
visible). We note that the convergence can be accelerated when using more than one IEs, as we do
in the simulations with larger QM sub-systems.
We also point out that the double-peak structure, which is not visible in the experiments, persists and remains visible even for the simulation with 3,000 snapshots. It becomes more pronounced when thinner gaussians are used. This feature is reminiscent of the double-peak structure
observed on the x-ray emission spectra in liquid water.66
Fig. 2.11 shows the spectra constructed from either one IE (of the central water) or 5 IEs
(of the cluster of 5 waters). As one can see, for the small QM (5 waters), the spectrum changes
qualitatively when the IEs of all 5 waters are used, leading to a much smaller red shift relative to
the gas-phase peak than the calculation using the IEs of 1 central water (the difference between
the two calculations is 0.25 eV). This illustrates the limitations of the electrostatic embedding,
which apparently is not able to correctly describe the waters on the QM/MM boundary. Using the
38
Figure 2.11: Theoretical 1sO spectra of liquid water computed using small (5) and large
(20) QM systems constructed from the AIMD trajectory (400 snapshots). The spectra were
constructed using the 1sO IEs of either one central water or five QM waters.
QM of 20 waters, results in a larger red shift (by about 0.5 eV), consistent with the benchmark
calculations for a single snapshot (Fig. 2.6). Moreover, the calculations with the large QM yield
the same spectra (in terms of the peak maximum and its width) whether using the IEs of one or
five waters, suggesting that the boundary is sufficiently far in this case. The spectrum computed
with the IEs of 5 waters is smoother, as in such calculation the sampling is effectively improved by
a factor of 5. Hence, we construct our best spectrum by using the IEs of the five water molecules
embedded in the QM cluster of 20 waters (with the rest of waters described by point charges).
The effect of triples is evaluated using the MLCC3 method with 20 CCSD waters and the triples
operator restricted to orbitals within the first solvation shell of the central water molecule.
39
Figure 2.12: Experimental12, 13, 24 and theoretical 1sO XPS spectra of water. The spectra of
Pellegrin et al.,12 Liu et al.,24 and Olivieri et al.13 spectra has been shifted by +0.59 eV, +4.9
eV, and +0.87 eV, respectively, to match the theoretical gas phase peak. The computationally
constructed spectrum is shown by dashed line. The computed spectra of liquid water includes
triples, basis-set, and Born corrections.
Fig. 2.12 shows our best spectrum and compares it with the available experimental spectra.12, 13, 24 We note that the reported spectrum of Olivieri et al.13 is for zero bias and, therefore,
does not include the correction due to the workfunctions difference. The corrected values from
this experiment are given in Table 2.4, which also lists the computed peak positions and widths
for different protocols and compares them with the experimental values. To extract the value of
the peak maxima and the width from the computed spectra, we convoluted the computed spectra
with broad gaussians (FWHM=0.5 eV) to remove the noise due to finite sampling. The effect of
this broadening is analyzed in the SI: Figure S4 in the SI shows the sample spectra obtained by
convoluting the raw data with gaussians of different widths. As one can see, using broad gaussians
does not introduce noticeable change in the band width.
40
Our best value of the shift is -1.79 eV—it is computed from the spectrum based on 400 AIMD
snapshots with 20 QM waters and using the IEs of 5 waters (AIMD/400, 5w/20QM scheme) treated
with fc-CVS-EOM-CCSD, to which we add the triples, basis-set, and Born corrections. This
protocol yields the absolute value of the bulk IE of 538.47 eV. Our results agree well with the best
experimental values, i.e., by Olivieri et al.,13 who reported the bulk water IE of 538.21±0.07 eV
and the shift of 1.61±0.09 eV, and with Thurmer et al., ¨
16 who reported IE of 538.10±0.05 eV.
Table 2.4: Shift (∆IE) of the 1sO IE of liquid water relative to the gas-phase water and the
width (FWHM) of the band.
Setupa ∆IE, eV FWHM, eV
AIMD/400, 1w/5QM -0.84 1.40
AIMD/400, 5w/5QM -0.45 1.55
MD/400, 1w/5QM -0.74 1.43
MD/400, 5w/5QM -0.41 1.50
MD/3000, 1w/5QM -0.75 1.42
MD/3000, 5w/5QM -0.40 1.51
AIMD/400, 1w/20QM -1.37 1.41
AIMD/400, 5w/20QM -1.27 1.44
AIMD/40, 5w/20QMb
-1.15
Best estimatec
-1.79 1.44
Exp. (Ref. 13), zero bias -1.77 1.53
Exp. (Ref. 13), corrected -1.61 1.53
Exp. (Ref. 16) -1.72
Exp. (Ref. 24) -2.2 1.93
a Sampling/snapshots, number of IEs/size of QM; fc-CVS-EOM-IP-CCSD/u6311+G(3df).
b CVS-EOM-IP-CCSD/u6311+G(3df).
c AIMD/400, 5w/20QM/fc-CVS-EOM-IP-CCSD value with the CC3 (δtriples=-0.340 eV) and
basis-set (δbasis=+0.114 eV) corrections, as well as Born correction (-0.29); see text.
41
Figure 2.13: The 1sO spectra of liquid water constructed from the central water and the first
shell (total 5 waters) broken into the contributions from structures with different hydrogenbonding patterns (raw spectrum without corrections). Top panel shows the contributions
from structures with different hydrogen-bonding patterns around the central water and bottom panel shows the contributions of different hydrogen-bonding patterns around the first
solvation shell. The spectra were computed from the AIMD snapshots trajectory using QM
with 20 waters and treating the rest of the waters as point charges.
2.3.1 Analysis of structures
The spectra of the bulk species reports on the structure of the solvent around the solute and its
dynamic fluctuations. In particular, the spectrum of water reflects the contributions from different
hydrogen-bonding patterns. Computationally, it is possible to break down the total computed spectra into different contributions, similarly to the analysis in other theoretical studies.11, 66 Fig. 2.13
42
shows the breakdown of the total spectrum into the contributions from structures with different
hydrogen-bonding patterns. Panel (a) shows contributions from snapshots where the central water
molecule forms a single donor-single acceptor (DA), single donor-double acceptor (DAA), double
donor-single acceptor (DDA), and double donor–double acceptor (DDAA) hydrogen bonds. Collectively, these types of structures add up to 97.2% of the structures sampled in the simulation. The
shape of the band is dominated by the DDAA pattern, which has the largest population. The structures in which the central water water acts as hydrogen-bond donor yield red-shifted IEs and the
structures in which the central water acts as hydrogen-bond acceptor are blue shifted. The DDAA
motif, in which there is an equal number of both kinds of hydrogen bonds, features a high-energy
and a low-energy peaks. The DDA and DAA motifs have peaks at lower and higher energies,
respectively.
The bottom panel of Fig. 2.13 shows the analysis of the spectrum in terms of the contributions
from different hydrogen-bonding patterns around the first solvation shell. The motifs shown in
the figure add up to 68.9% of the structures sampled in by the simulation. The 6D6A motif,
which corresponds to the DDAA motif of the central water, has the highest contribution. Due to
many different motifs of hydrogen bonding around the first shell, the overall effect is that these
structures cannot be mapped into particular spectral features but rather they collectively contribute
to the inhomogeneous broadening and smoothening of the spectrum.
Finally, we use the contributions of different hydrogen-bonding patterns to estimate the effect
of different equilibrium sampling on the computed spectra using the approach from Ref. 66. Fig.
2.14 shows our original spectrum computed using 400 AIMD snapshots and several synthetic
spectra obtained by rescaling the relative contributions of the structures with double donor (sum of
contributions of the DDA and DDAA structures) and single-donor (sum of the contributions of the
DA and DAA structures) motifs to match the results from other simulations (see Table S2 in the
SI). We see that the effect on the overall shape of the band, its maximum and width, are negligible,
43
Figure 2.14: The 1sO spectra of liquid water (raw spectra without corrections) constructed
from the a) central water molecule and b) 5 water molecules computed using AIMD snapshots (black) and synthetic spectra obtained by re-weighting the contributions from the dominant hydrogen-bonding patterns to match the distributions from other simulations (see text).
especially, when sampling is adequate (such as in the simulations using the IEs of the 5 waters).
Hence, imperfections in the equilibrium water structures seem to be washed out by the averaging
and are not expected to affect the computed value of the ∆IE.
2.4 Conclusion
We presented a state-of-the-art simulation of the 1sO ionization of liquid water. We employed
highly accurate EOM-CC methods adapted to core-vacancy states by using the CVS scheme. Equilibrium sampling was carried out using MD and AIMD simulations. We carefully analyzed the
effect of the embedding on the computed IEs and show that the convergence of the result with
respect to the size of the QM system is slow. Our production-level calculations were carried out
44
using the QM system of 20 water molecules embedded in the MM charges. We also evaluated the
effect of triple excitations using the MLCC3 framework. Our calculations yield the value of the
intrinsic bulk IE of 538.47 eV and the FWHM of the bulk peak of 1.44 eV; the computed shift
relative to the gas-phase IE is -1.79 eV (including MLCC3 and basis-set corrections).
These results agree well with the best experimental values from the most recent liquid-jet experiments of the bulk water IE of 538.21±0.07 eV and the shift of 1.61±0.09 eV (Ref. 13) and
538.10±0.05 eV (Ref. 16). Given the remaining uncertainties in the experimental determination
of the true (intrinsic) bulk IEs (surface potential, presence of solvated species, etc), our results
provide an important reference value.
We conclude with listing the aspects of the present theoretical treatment that need improvement. First, improved equilibrium sampling is highly desirable, i.e., using higher-quality ab initio
treatment, larger QM sizes, and including nuclear quantum effects. Second, one can further increase the size of the QM system in the IE calculations, which can be achieved using multi-level
methodologies, to achieve full convergence. Third, improving correlation treatment beyond CC3
is desirable. Fourth, the effect of electrolytes both on the bulk IE and on the surface potential
needs to be studied. Fifth, the effect of the finite-depth probed in microjet experiments needs to be
investigated by simulations. We hope to address these issues in future studies. The availability of
accurate intrinsic bulk IE and reliable experimental value (real IE) can provide an estimate of the
surface potential (ϕin) of water.
45
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53
Chapter 3: Future work
This thesis focuses on developing protocols to model core-level spectroscopy in condensed phase.
Chapter 2 presented a protocol that successfully reproduces the XPS spectra of water. In addition,
it reveals interesting hydrogen-bonding patterns in liquid water and their effect on the spectra. In
this Chapter, we discuss remaining issues and outline future directions.
3.1 Using better sampling
The quality of spectra depends on the quality of sampling. The radial distribution function is
an important indicator of the sampling quality and provides information about the solvation shell
structure in a liquid. The Appendix of Chapter 2 shows the oxygen-oxygen radial distribution
function (gOO(r)) of water. The AIMD sampling with QM/MM strategy produces too structured
first solvation shell. On the other hand, TIP3P simulation produces a better first solvation shell
but the second solvation shell is flatter than the experimental one. Further work is needed to
obtain a better sampling of liquid water. New density functionals1
and many-body expansion
models2, 3 have been used to reproduce more accurate structural properties of liquid water. FIG.
3.1 a) compares the gOO(r) of water simulated using different density functionals and the MB-pol
potential energy function. The MB-pol potential energy function accurately reproduces the gOO(r)
from experiments.4–6
In Chapter 2, we treated the dynamics of water nuclei using classical equations of motion.
However, nuclear quantum effects are important because of nuclear zero-point energy, tunneling
54
Figure 3.1: a) gOO(r) obtain from NVT simulations at the experimental carried out at T =
298.15 K with the q-TIP4P/f (blue), TTM3-F (cyan), BLYP (green), BLYP-D3 (orange), and
MB-pol (red) potentials. b) gOH(r) calculated from both classical (MD, green) and quantum
(PIMD, red) NVT simulations with the MB-pol potential at T = 298.15 K. Adapted from Ref.
3.
and exchange effects.7 The quantum effects are important for lighter atoms like hydrogen and
may affect the overall structure of water. Path integral molecular dynamics (PIMD) can be used to
include quantum nuclear effects in molecular dynamics simulation. FIG. 3.1 b) shows the comparison between MD and PIMD simulation of water. The gOH(r) calculated using PIMD shows less
structure than MD and is closer to the experiment.3 Therefore, it is important to explore incorporating nuclear quantum effects using PIMD to obtain a better sampling of water and consequently
lead to calculating more accurate core-level excitation/ionization energy.
Depending on the photon energy, the XPS setup may probe the structure of the solution at
different depths. A recent study performed a depth profiling of liquid water.8 However, they used
a water cluster of 60 water molecules to represent liquid water. It would be important to simulate
a water-vacuum interface with better sampling. MB-pol has been shown to reproduce the watervacuum interface accurately.9 Depth profiling of liquid water with better sampling and high-level
electron structure should reveal interesting features about the liquid-vapor interface of water.
55
3.2 Understanding water-glycerol solution
Glycerol is extensively used as a cryoprotector, i.e. it is used to prevent or reduce cell damage
during cryopreservation or dehydration.10, 11 Glycerol disrupts the hydrogen bond network in
water therefore prevents ice formation. Glycerol-water mixtures have been a subject of numerous
MD simulations12–17 as well as experiments.18, 19 A recent study combining MD simulations and
fourier transformed infrared spectroscopy (FTIR) data showed that as the mole fraction of water
decreases in the water-glycerol mixture, water gets trapped into small clusters.20 The water clusters
aren’t large enough to support crystalline structure and therefore inhibit ice formation.
Core-level spectroscopy is a very sensitive probe of local structure. Hence, it can be used
to study hydrogen bond networks in condensed phase. Ahmed and co-workers used XPS spectroscopy to study glycerol-water aerosols.21 Using a combination of XPS and FTIR spectroscopic
data, they claim that hydrogen-bonding network in glycerol-water mixtures can be defined as bulk
water, solvated water, confined water and bulk glycerol depending on the relative concentration of
the mixture. The energy range of the photons used in the study selectively probed the surface of the
glycerol-water aerosols. To investigate the bulk properties, the authors used terahertz time domain
spectroscopy (THz-TDS) and infrared-attenuated total reflection (IR-ATR) spectroscopy. Notably
the above mentioned techniques have been previously employed to study the hydrogen bonding
network of glycerol-water solutions.22, 23 The change in the infrared wavenumbers of the OH,
CHOH, and CH2 stretching modes and the relative absorption coefficient follow the same trend as
the XPS intensity. They indicated a solvated water hydrogen-bond network upto 8 mol% and the
presence of confined water clusters upto 20-30 mol%. A schematic diagram presenting this result
is shown in FIG. 3.2. This also indicates the surface and bulk of the glycerol water aerosol has a
similar composition at all concentrations.
56
Figure 3.2: Schematic diagram of glycerol and water mixtures at different relative concentrations showing bulk water, solvated water, trapped water and bulk glycerol. Adapted from
Ref. 21.
In Ref. 21 the authors fitted the 1sC peaks with 2 and 3 gaussians and the 1sO peaks with 2
gaussians respectively. Where the XPS was fitted with two gaussians, the authors attributed the
high energy peaks to gaseous glycerol and the low energy peak to solvated glycerol. The solvated
glycerol peak amplitude (both C 1s and O 1s) vs mol % of glycerol plot shows a small peak around
9.5 mol% and a more prominent peak around 23 mol% as shown in FIG 3.4
We plan to construct both the O-edge and C-edge spectra of glycerol in water and bulk glycerol. Adequate sampling will reveal the nature of hydrogen-bonding network of water at different
concentrations of glycerol. FIG. 3.3 shows the experimental XPS spectra fitted with solvated and
unsolvated glycerol. The intensity of the peaks change but not the energy with different glycerol
concentrations. We can study the effect of hydrogen bonds on the core-level IEs of glycerol with
the protocol developed in Chapter 2. This will give a more detailed understanding of hydrogenbonding interactions between glycerol and water. There are no experimental results resolving the
O-edge XPS signals between the glycerol and water. Our protocol can distinguish between the
two. We will also perform AIMD calculations of glycerol. Since most of the molecular dynamics
57
Figure 3.3: X-ray photoelectron spectra at different glycerolwater concentrations. (A) C 1s
spectra measured at a photon energy of 315 eV. Gray inset at 100.0 mol % is the gas phase
spectrum of glycerol, used to disentangle the gas phase (292 eV) and condensed phase (290.5
eV) contributions. (B) O 1s spectra measured at a photon energy of 560 eV. A spectrum is
superimposed on the 1.0 mol % spectrum showing gas phase (540 eV) and condensed phase
(538 eV) peaks of pure water. Solid and dashed lines indicate the condensed phase and gas
phase BE positions, respectively. Adapted from Ref. 21.
simulation studies of glycerol in the literature has been performed with classical force-fields, our
AIMD simulations can be used to verify their accuracy.
58
Figure 3.4: Condensed phase (filled circles) and gas phase (open circles) XPS intensities as
a function of glycerol mol % (100 mol % glycerol intensity is not reported here since its
aerosol generation mechanism is different from glycerolwater mixtures). (A) C 1s XPS peak
intensities and (B) O 1s XPS peak intensities. Three regions corresponding to glycerolwater
networks are indicated in colored regions: red (X1) is solvated water, green (X2) is confined
water, and blue (X3) is the bulk glycerol network. Adapted from Ref. 21.
59
Chapter 3 References
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and dynamical properties of liquid water by ab initio molecular dynamics based on SCAN
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chemical accuracy for water simulations through a density-corrected many-body formalism
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flexible monomers. iii. liquid phase properties J. Chem. Theory Comput. 2014, 10, 2906–
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trehalose and glycerol Biophys. J. 2004, 86, 3836–3845.
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cryoprotective properties of glycerol/water mixtures J. Phys. Chem. B 2006, 110, 13670–
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solutions: A molecular dynamics simulation study J. Mol. Liq. 2009, 146, 23–28.
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glycerol–water liquid mixtures J. Phys. Chem. B 2011, 115, 14572–14581.
[17] Seyedi, S.; Martin, D. R.; Matyushov, D. V. Dynamical and orientational structural crossovers
in low-temperature glycerol Phys. Rev. E 2016, 94, 012616.
[18] Behrends, R.; Fuchs, K.; Kaatze, U.; Hayashi, Y.; Feldman, Y. Dielectric properties of glycerol/water mixtures at temperatures between 10 and 500
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[19] Novo, L. P.; Gurgel, L. V. A.; Marabezi, K.; d.S. Curvelo, A. A. Delignification of sugarcane bagasse using glycerol–water mixtures to produce pulps for saccharification Bioresour.
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[21] Weeraratna, C.; Amarasinghe, C.; Lu, W.; Ahmed, M. A direct probe of the hydrogen bond
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61
Appendix A: Supplementary information
for Chapter 2
A.1 Data deposition
The equilibrium dynamics trajectory files for the MD and AIMD simulations as well as raw data
for the 1sO IE spectrum from the AIMD/400, 1w/20QM simulations were deposited in Zenodo
archive.1
A.2 Equilibrium sampling: Molecular dynamics versus ab initio molecular dynamics
Fig. A.1 and Table A.1 compare the structural properties of water computed from the MD (TIP3P)
and AIMD trajectories. For adequate comparison, in both simulations we computed gOO(r) (O-O
radial distribution function, RDF) between the selected central water molecule and the rest of the
waters, so that in AIMD simulations the short-range structure of gOO(r) comes from the QM part.
We note that water molecules within 6.5 A radius of the central water were included in the QM ˚
region in the AIMD simulation. It covers part of the third shell and has more structure than TIP3P
and experiment. Below we compare the RDF from our simulations with that generated from the
neutron diffraction data refined with X-ray data (experimental).2
62
Figure A.1: The O-O radial distribution function computed from the AIMD and MD trajectories compared with the experimental gOO(r)
2
Table A.1: Structural properties of liquid water from the O–O radial distribution function,
gOO(r), obtained from the AIMD and MD simulations around the central water molecule of
their respective simulation boxes and experimental data.a
gOO(r) r
min
1
, A˚ r
min
2
, A˚ n1 n2
MD 3.54 5.76 4.91 23.41
AIMD 3.28 5.51 4.53 23.46
Expt2 3.39 5.58 4.73 23.31
r
min
1
and r
min
2
are the first and the second minima of gOO(r), which define the first and second
coordination shells; n1 and n2 are the number of water molecules inside the first and the second
coordination shells.
The first peak of the radial distribution function for TIP3P water is closer to the experimental
one than that generated from the AIMD simulation. However, the average coordination number
generated from the two calculation and from the experiment are not very different and lie between
4.5 and 4.9. These results indicate that the first shell is just more structured in case of water
described by ωB97X-D/6-31G* than TIP3P water and experiment. As discussed in Section 2.3.1
of the main manuscript, an important aspect which determines the XPS spectra of water is its
hydrogen bonding environment. It is only indirectly captured by the radially averaged structure of
gOO(r). Therefore, we calculated the average number of hydrogen bonds around the central water
63
molecule and compared it to the experiment which uses a three dimensional distribution of water
molecules.3 The later uses the same definition of hydrogen bonding as used to construct Fig. 2.13
of the main draft. The experiments show that the total number of hydrogen bonds formed by a
water molecule is 3.58, which is same as that obtained from the AIMD trajectory while we get an
average of 3.28 hydrogen bonds from the MD trajectory.
The gOO(r) calculated from the three methods have first peak at the same radial distance. While
both the first minima and second minima (r
min
1
and r
min
1
) is systematically shifted to larger r values
for the MD (TIP3P) simulation, MD yields much flatter 2nd peak. The ωB97X-D functional4
reproduces the second shell accurately, but produces over-structured 1st solvation shell. Table A.2
shows the breakdown of different hydrogen-bonding motifs in our simulations and in previous
studies.
Table A.2: The fraction of water molecules with double-donor (DD), single-donor (SD), and
non-donor (ND) configurations from different computational methods.
Method DD SD ND
AIMD/ωB97X-Da 81 16 3
TIP3P 65 27 8
AIMD/MP2b,c 53 40 7
PIMD/MBPOLd 58 36 6
CPMDc 79 20 1
TIP3PFc 75 22 3
Expc 15±25 80±20 5±5
a This work.
b Fragment-based method parameterized by MP2.
c Quoted from Ref. 5.
d From Ref. 6.
64
A.3 Convergence with respect to the QM size and number of
snapshots: Additional results
Figure A.2: The O edge ionization spectra computed 5 and 6 water QM systems.
Including 5 or 6 water molecules in the QM region produces similar spectra because an water
molecule can have 4-5 molecules in its first solvation shell. The energy peaks are similar when
taking just the central water to construct the spectra but the shape may differ. However, it is almost
exactly similar when including all the first shell water in the QM region. The snapshots are from
the AIMD trajectory.
65
10 20 30 40
N snapshots
0.95
1.00
1.05
1.10
1.15
1.20
IE [e
V]
Figure A.3: Convergence of CVS-EOM-IP-CCSD shift (from eT calculations) with respect to
the number of snapshots. ∆IE=1.15 eV, with standard deviation of 0.55 eV.
A.4 Analysis of the computed spectra
Table 2.4 in the main paper gives the peak position and width of the 1sO spectra of water calculated using different setups. The peak and the FWHM of the spectra were calculated by convoluting
the core IEs with gaussians of FWHM=0.5 eV. Figure A.4 shows the 1sO spectra of AIMD/400,
5w/20QM system convoluted with gaussians with FWHM equals 0.5, 0.2, and 0.3 eV, respectively.
The peak for FWHM = 0.5 eV is 538.996 eV and the peak for FWHM = 0.2 eV is 538.929 eV,
whereas the spectrum with FWHM = 0.05 eV is noisy and its peak maximum is difficult to determine. The width of the spectra for FWHM = 0.5, 0.2 and 0.05 eV is 1.44, 1.345, and 1.331 eV,
respectively.
66
Figure A.4: The O edge ionization spectra of the AIMD/400, 5w/20QM system convoluted
with gaussians of width (FWHM) 0.5, 0.2 and 0.05 eV respectively.
A.5 Recalculating the spectra using different hydrogen-bond
distributions
Figure A.5 shows the 1sO spectra MD/400, 1w/5QM system and the AIMD/400, 1w/5QM spectra
constructed by re-weighting with the dominant hydrogen-bond distributions from the trajectory of
TIP3P water.
67
Figure A.5: The 1sO spectra of liquid water constructed from the MD trajectory and reweighting the contributions from the dominant hydrogen-bonding patterns to match the distributions from the MD trajectory.
68
A.6 Q-Chem input for CVS-EOM-CCSD calculations for 20
QM waters
$REM
BASIS = MIXED
SCF_GUESS = CORE
SCF_CONVERGENCE = 8
MAX_SCF_CYCLES = 200
THRESH = 14
METHOD = EOM-CCSD
CVS_IP_STATES = [20]
MEM_TOTAL = 230000
N_FROZEN_CORE = 20
SYM_IGNORE = TRUE
NO_REORIENT = TRUE
CC_TRANS_PROP = 1
STATE_ANALYSIS = TRUE
MOLDEN_FORMAT = TRUE
CC_BACKEND = VM
$END
69
Chapter A References
[1] Core-ionization spectrum of liquid water. Dey, S.; Folkestad, S. D.; Paul, A.; Koch, H.; Krylov,
A. I. https://doi.org/10.5281/zenodo.7978306, 2023.
[2] Soper, A. K. The radial distribution functions of water as derived from radiation total scattering
experiments: Is there anything we can say for sure? ISRN Phys. Chem. 2013 2013, 7, 074506.
[3] Soper, A. K.; Bruni, F.; Ricci, M. A. Site-site pair correlation functions of water from 25 to
400◦—the methanol hydration case J. Mol. Liq. 2020, 300, 112258.
[4] Chai, J.-D.; Head-Gordon, M. Systematic optimization of long-range corrected hybrid density
functionals J. Chem. Phys. 2008, 128, 084106.
[5] Liu, J.; He, X.; Zhang, J. Z. H. Structure of liquid water—a dynamical mixture of tetrahedral
and ‘ring-and-chain’ like structures Phys. Chem. Chem. Phys. 2017, 19, 11931–11936.
[6] Cruzeiro, V. W. D.; Wildman, A.; Li, X.; Paesani, F. Relationship between hydrogen-bonding
motifs and the 1b1 splitting in the x-ray emission spectrum of liquid water J. Phys. Chem. Lett.
2021, 12, 3996–4002.
70
Abstract (if available)
Abstract
This thesis addresses challenges associated with studying excited-state processes in the condensed phase, particularly in liquid solutions. Theory plays essential role in interpreting the experimental measurements. While ground-state properties can be accurately modeled using standard quantum chemistry methods, excited-states calculations remain challenging. Excited states play a crucial role in various applications, such as solar energy harvesting, vision, artificial sensors, and pho- tovoltaics. Excited states are also exploited in many spectroscopies, which provide important information about molecular structure and properties. The main obstacle in a quantitative de- scription of light-induced processes is incorporating the influence of the environment on excited states. Since many important processes occur in solution or in solids, it is essential to develop methods that can account for the effects of the surrounding medium on molecular chromophores. The core-excited/ionized states in condensed phase are particularly challenging to model owing to their high sensitivity to the local environment. This thesis explores the application of equation-of- motion coupled-cluster (EOM-CCSD) methods with molecular dynamics simulation to model the core-ionization energy in condensed phase.
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Dey, Sourav
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Modeling x-ray spectroscopy in condensed phase
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2023-12
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