S O F TWARE F OCU S

The ezSpectra suite: An easy-to-use toolkit forspectroscopy modeling

Samer Gozem1 | Anna I. Krylov2

1Department of Chemistry, Georgia StateUniversity, Atlanta, Georgia, USA2Department of Chemistry, University ofSouthern California, Los Angeles,California, USA

CorrespondenceSamer Gozem, Department of Chemistry,Georgia State University, Atlanta, GA30303, USA.Email: sgozem@gsu.edu

Anna I. Krylov, Department of Chemistry,University of Southern California, LosAngeles, CA 90089, USA.Email: krylov@usc.edu

Funding informationAmerican Chemical Society PetroleumResearch Fund; National ScienceFoundation; U.S. Department of Energy

Edited by: Peter R. Schreiner,Editor-in-Chief

Abstract

A molecule's spectrum encodes information about its structure and electronic

properties. It is a unique fingerprint that can serve as a molecular

ID. Quantum chemistry calculations provide key ingredients for interpreting

spectra, but modeling the spectra rarely ends there; it requires additional steps

that entail combined treatments of electronic and nuclear degrees of freedom

and account for specifics of the experimental setup (light energy, polarization,

averaging over molecular orientations, temperature, etc.). This Software Focus

article describes the ezSpectra suite, which currently comprises two stand-

alone open-source codes: ezFCF and ezDyson. ezFCF calculates Franck–Condon factors, which yield vibrational progressions for polyatomic molecules,

within the double-harmonic approximation. ezDyson calculates absolute cross-

sections for photodetachment/photoionization processes and photoelectron

angular distributions using Dyson orbitals computed by a quantum chemistry

program.

This article is categorized under:

Electronic Structure Theory > Ab Initio Electronic Structure Methods

Theoretical and Physical Chemistry > Spectroscopy

Software > Simulation Methods

KEYWORD S

Electronic structure, Franck–Condon factors, Photoelectron spectroscopy, Spectroscopy,Transition dipole moment

1 | INTRODUCTION

The spectrum is a message from the molecule.Unknown

Spectroscopy is the most powerful tool for probing matter. Despite the mind-boggling sophistication of modern lightsources and spectroscopic techniques, conceptually it remains just as simple as in the time of Newton's experiments: weshine light at a sample and back comes the spectrum with information about the internal structure of the sample.

Spectroscopy allows us to communicate with molecules. Each spectrum is a message from the molecule, reportingon its structure. However, the message is coded. To translate an intricate pattern of spectral lines and intensities intowhat the nuclei and electrons are doing, we need a cipher. Theory provides the key to decipher the message (seeFigure 1). Through quantum chemical calculations, we can obtain essential parameters of the molecular structure.

Received: 1 March 2021 Revised: 24 April 2021 Accepted: 28 April 2021

DOI: 10.1002/wcms.1546

WIREs Comput Mol Sci. 2021;e1546. wires.wiley.com/compmolsci © 2021 Wiley Periodicals LLC. 1 of 22

https://doi.org/10.1002/wcms.1546

Using these parameters as an input, we can simulate the spectra. By comparing the computed and experimentally mea-sured spectra, we can then confirm molecular identity, assign spectral features, and relate the spectra to such conceptsas equilibrium bond lengths, vibrational frequencies, molecular orbitals, and so on.

Different spectroscopies probe different types of transitions and report on different molecular properties. To appreci-ate the scope and variety of modern spectroscopy, consider the vast range of energies and intensities of light used,1 inaddition to nonlinear,2 angle-resolved,3,4 and time-resolved techniques.5 Experimental innovations constantly driveelectronic-structure developments, pushing quantum chemistry into new domains. But modeling spectra rarely ends atthe electronic structure calculations and often requires additional steps that entail combined treatments of electronicand nuclear degrees of freedom and account for the specifics of the experimental setup (light polarization, averagingover molecular orientations, temperature, etc.).

This Software Focus article describes the ezSpectra suite, which bridges quantum-chemistry calculations and experi-mental observables (spectra). Our focus is on spectroscopies that probe the transitions between different electronicstates, such as traditional one-photon absorption (UV/Vis, XAS), photoionization/photodetachment (UPS, XPS, ARPES,etc.), as well as nonlinear spectroscopies (2PA, RIXS, resonance Raman, etc.). Despite the vast differences betweenthese techniques, the common thread is the need to combine electronic factors (relevant energy differences and transi-tion moments) with vibrational structure. In what follows, we describe the theory of linear electronic absorption spec-troscopy and photoionization/photodetachment spectroscopies, followed by a brief outline of the extensions to othertypes of spectra.

By treating light as a weak time-dependent perturbation of molecular eigenstates,6 we encounter singularities in thefirst-order perturbed amplitudes at frequencies resonant with energy differences between the (unperturbed) molecularstates. The singularities mean that the molecule can undergo transitions between different states when the frequencymatches the energy gap. Within the dipole approximation (i.e., assuming that the molecular size is smaller than thewavelength of the light), the probability of transition between an initial state (i) and a final state (f) is proportional tothe square of the transition dipole matrix element:

Pif /ðΨi r,Rð Þ μ Ψf r,Rð Þ dr dR

� �2

, ð1Þ

FIGURE 1 Spectroscopic measurements (red panel) yield encrypted messages, which can be decoded using electronic structure

calculations (blue panels) to reveal detailed information about molecular structure (black panel)

2 of 22 GOZEM AND KRYLOV

where μ is the electronic dipole moment operator, R and r are the collective nuclear and electronic coordinates, respec-tively, and Ψi(r,R) and Ψf (r,R) are the wave functions of the two states. By using the separable ansatz7 (either withinthe Born–Oppenheimer approximation8,9 or within the exact treatment10), the full wave functions can be representedas products of the electronic (ψ(r;R)) and nuclear (χ(R)) wave functions. Within the Born–Oppenheimerapproximation,8 the electronic energies and wave functions are determined by the electronic Hamiltonian (full molecu-lar Hamiltonian minus the nuclear kinetic energy operator), such that they depend only parametrically on the nuclearcoordinates (as signified by “;” instead of “,”):

Pi0f 00 /ðψ i r;Rð Þχi0 Rð Þ μ ψ f r;Rð Þχf 00 Rð Þ dr dR

� �2

: ð2Þ

Here, indices i0 and f 00 denote vibrational levels of the two electronic states (which, within the Born–Oppenheimerapproximation, are determined by the nuclear problem with the potential defined by the electronic Schrödinger equa-tion). Integration over the electronic coordinates yields the electronic transition dipole moment for the i! f transition

μif Rð Þ¼ðψ i r;Rð Þ μ ψ f r;Rð Þ dr, ð3Þ

so that the probability of transition can be written as

Pi0f 00 /ðχi0 Rð Þ μif Rð Þ χf 00 Rð Þ dR

� �2

: ð4Þ

Equation (4) is the starting point for understanding the spectrum. It contains electronic factor μif(R), which encodesthe information about the electronic and vibrational wave functions of the two states. Within the Condonapproximation,11 it is assumed that μif(R) depends weakly on the nuclear coordinates and, therefore, can be evaluatedat a fixed nuclear geometry (e.g., at the equilibrium geometry Re of the initial state). In this case, the probability of thetransition simply becomes:

Pi0f 00 / μif Reð Þ2ðχi0 Rð Þ χf 00 Rð Þ dR

� �2

: ð5Þ

The overlap integrals between the nuclear wave functions of the two states are called the Franck–Condon factors(FCFs)11–13—their squares are the intensities due to vibrational progressions (sometimes the intensities are calledFCFs). The stick spectrum can then be assembled from the individual FCFs as follows:

I ωð Þ¼Xi0, f 00

Pi0f 00 �δ ω�Ei0f 00� �

, ð6Þ

where δ is the Kronecker δ-function and Ei0f 00 is the energy difference between the two vibronic states. The macroscopicobservables, such as cross sections, Einstein coefficients, and so forth, can be computed from I(ω) (see, for example,Ref. 14).

Figure 2 illustrates how Equations (5) and (6) give rise to the spectral profile due to transitions from the initial state(red) to two final states (blue and green). The μif for each state determines the magnitude of the electronic contributionto the cross-section, and the FCFs determine the probability of transition to different vibrational levels within each elec-tronic state. As per Equation (5), FCFs are simply the overlap integrals between the initial-state vibrational wave func-tion (red translucent curve in Figure 2) and the final-state vibrational wave functions (blue or green translucentcurves). A large overlap leads to a large FCF, giving rise to a high probability of transition to that vibrational level. Fortwo states represented by two identical one-dimensional harmonic potentials, only the diagonal transitions (Δν=0) areallowed by virtue of the orthogonality of the Hermite polynomials, leading to a spectrum with no vibrational progres-sion (e.g., for a vibrationally cold molecule, only 00 transitions would be seen). Transitions to multiple vibrational statesbecome allowed when the two potentials are offset with respect to each other (i.e., when ΔQ≠0) or when their

GOZEM AND KRYLOV 3 of 22

curvatures are different (i.e., when the frequency changes upon the transition). The modes that are affected by thechanges in the geometry or frequencies upon excitation are called Franck–Condon active modes. At finite temperatures,when higher vibrational states of the initial state are populated, “hot bands” appear in the spectra, such as transitionsfrom ν = 1 shown in Figure 2, in addition to the main progression from the lowest ν = 0 vibrational state.

Figure 2 also shows two important (and often misused) quantities: the vertical (Ev) and adiabatic (Eaee ) electronic

excitation energies. The vertical excitation energy is simply the electronic energy difference between the two potentialenergy surfaces at the equilibrium geometry of the initial (e.g., ground) state. The adiabatic electronic excitation energyis the energy difference between the two potential energy surfaces at their respective equilibrium geometries(in spectroscopic notations, it is denoted by Tee); obviously, Ea

ee is always smaller than Ev. Although Ev and Eaee are easy

to define theoretically and are straightforward to compute by quantum-chemistry programs, they are not directlyrelated to experimental observables, because they do not correspond to energy differences between vibronic states.

By taking into account energies of the vibrational levels, one can define the energy difference between the lowestvibrational states—this is the adiabatic excitation energy Ea

00 (T00 in spectroscopic notations). The adiabatic excitationenergy includes the electronic energy difference (Ea

ee ) and the zero-point energy correction (i.e., the difference in thezero-point vibrational energies of the two states). For a vibrationally cold molecule, Ea

00 corresponds to the onset ofthe spectral band, provided that the corresponding FCF ( χ00 χ000

�� ��) is sufficiently large. If the geometry relaxation in the

target state is too large, the FCF for the 00 transition can become too small to be observed.The vertical excitation energy is not an experimental observable. Using a semiclassical treatment of a one-

dimensional harmonic model, one can argue that the largest FCF should correspond to the vertical transition, as shownfor the ν = 0 to ν 0 = 2 transition in Figure 2. Consequently, it is tempting to interpret the band maximum (λmax) as thevertical excitation energy. While this is a reasonable assumption for many diatomics, calculations for polyatomic mole-cules show that the maximum intensity often corresponds to the 00 transition, such that λmax represents the adiabatic

FIGURE 2 Factors determining the spectral line shape for electronic transitions. The ground state (red), the first excited state (blue),

and the second excited state (green) are represented by one-dimensional harmonic potentials along a normal mode Q. The change in the

excited-state equilibrium geometry relative to the ground state is denoted by ΔQ0 and ΔQ0 0. The main progression is determined by the

overlap of the initial state ν = 0 wave function (red translucent area curve) and each of the excited-state vibrational wave functions (blue or

green translucent area curves). If T≠ 0, hot bands may appear, such as peaks resulting from the ν = 1 transitions indicated by dotted lines

on the right. The total spectrum is an overlay of the main progression and hot bands. Instrumental broadening and interactions with the

environment (e.g., inhomogeneous broadening in condensed phase) results in a broadened spectrum like the one shown by black dashed

lines on the right. The definition of electronic vertical (Ev0 ) and adiabatic (Ea0ee) excitation energies are shown for the first excited state

4 of 22 GOZEM AND KRYLOV

excitation energy15 (more on this below). Hence, the most reliable way to compare theory and experiment is to comple-ment quantum-chemistry calculations of the electronic energy differences by calculation of the FCFs.

The above treatment describes both the transitions between the electronically bound states (i.e., the ground andexcited states of a molecule) and the transitions to the continuum, when the energy of light (Ehν) is sufficient to eject anelectron. In the latter case, the spectra are measured as the yield of photoelectrons at different kinetic energies (Ek), asper the following energy balance:

Ek ¼Ehν�ΔEi0f 00 : ð7Þ

In the experimental community, it is common to distinguish between photoionization (electrons ejected from a neutralmolecule) and photodetachment (electrons ejected from an anion), but, for the sake of brevity, we will use the termphotoionization to refer to both.

For the transitions between bound electronic states, Equation (5) yields the total absorption cross section, and μif isan electronic transition dipole moment element. For the transitions into the continuum, the probability of ejecting anelectron is also determined by Equation (5), but μif is replaced by the photoelectron dipole matrix element μi∞

16:

μi∞ Rð Þ¼ðψni r1,r2,…,rn;Rð Þ μ A ψn�1

f r1,r2,…,rn�1;Rð Þψ elk rnð Þ

h idr1dr2…drn, ð8Þ

where n is the number of electrons in the initial state, A is an anti-symmetrizing operator, ψn�1f is the wave function of

the ionized (detached) target, and ψ elk is the wave function of the ejected electron with wave vector k.

The ezSpectra suite currently comprises two stand-alone open-source codes (distributed under the GPL license):ezFCF and ezDyson. The former computes FCFs (the overlap integrals in Equation (5)), and the latter computes proper-ties related to the photoelectron dipole matrix element in Equation (8). Both are written in C++ and use a built-inparser to read xml-formatted input files. The source code, precompiled executables, manuals, and examples of inputand output files are available for download at http://iopenshell.usc.edu/downloads/.17 We also provide Python scriptsfor automatic preparation of the xml input files from raw outputs of quantum-chemistry programs. The scripts currentlysupport Q-Chem, Cfour, ACESII, Molpro, ORCA, GAMESS, and Gaussian outputs.

2 | ezFCF

2.1 | Software capabilities and features

2.1.1 | Overview

ezFCF, formerly known as ezSpectrum, computes vibrational progressions (stick spectra) for electronic excitation orphotoelectron spectra for polyatomic molecules within the double-harmonic approximation.

2.1.2 | Required input

ezFCF requires equilibrium geometries, harmonic frequencies, and normal-mode vectors for each electronic state.These quantities can be computed using an ab initio electronic structure package (Figure 3). Additionally, the usershould specify the temperature (which depends on experimental conditions) and the adiabatic excitation or ionizationenergy (which can be either computed or taken from the experiment).

2.1.3 | What the code computes

ezFCF computes FCFs, the overlaps between the multidimensional vibrational wave functions of the initial (usually,ground) state and the final (excited or ionized) states, as per Equation (5). In the current version, both the initial andthe final states are described by harmonic potentials (hence, double-harmonic approximation).

GOZEM AND KRYLOV 5 of 22

For a one-dimensional case (diatomics or a molecule with only one Franck–Condon active mode), the FCFs are sim-ply the overlaps between the eigenstates of the two shifted harmonic oscillators.18 For a multidimensional case, thevibrational wave functions are products of the eigenstates of one-dimensional harmonic oscillators

χ0ν01,ν02,…,ν0N Q0ð Þ ¼YNa¼1

χ0a,ν0a Q0a

, ð9Þ

χ 00ν001 ,ν002 ,…,ν00N Q00ð Þ ¼YNb¼1

χ00b,ν00b Q00b

: ð10Þ

Here χ0 and χ00 denote the wave functions for the initial and final states, respectively, each expressed in its own normalcoordinates Q0 and Q00. Indices a and b run over all normal modes and strings ν1, ν2, …, νN give the number of quanta ineach mode.

How can we evaluate the multidimensional overlap integral between χ0(Q0) and χ 00(Q00), which are expressed in dif-ferent coordinate systems? One possibility is to transform one of the states into the normal coordinates of the otherstate; this can be achieved by using Duschinsky rotations19 (named after Duschinsky who first described the effect ofnormal-mode rotation in polyatomic molecules20). Alternatively, the task can be significantly simplified by assumingthat the normal coordinates of the initial and final states are the same, such that along each normal coordinate onestate is offset (by ΔQ) with respect to the other and the curvature of the potentials is different. This gives rise to theparallel-mode approximation in which the multidimensional integral becomes a product of one-dimensional integralsfor each normal mode:

χ 0ν01,ν02,…,ν0N Qð Þ��χ00ν0 01,ν0 02,…,ν0 0N QþΔQð ÞD E

¼YNa¼1

χ0a,ν0a Qað Þ��χ00a,ν0 0a QaþΔQað ÞD E

: ð11Þ

In this expression, normal modes of the initial state are used, so that the final state is offset with respect to the initialstate, but one can also use normal modes of the target state and treat the initial state as being offset. The one-dimensional integrals are evaluated analytically using the expressions from Ref. 18. The effect of normal modes rota-tions and parallel-mode approximation are illustrated in Figure 4, which also explains the impact of choosing normalmodes of the initial versus the final state.

The validity of the parallel-mode approximation (and the alignment of the normal modes) can be assessed by check-ing the normal-modes overlap matrix, which should be close to diagonal. The order of the normal modes can be flipped,which is manifested by near-unity off-diagonal values in the overlap matrix; this does not mean that the parallel-modeapproximation is invalid, as the diagonal form of the overlap matrix can be simply restored by the manual reordering ofthe normal modes to match. If the normal modes are rotated (i.e., when the values of the diagonal elements are signifi-cantly less than one, even after the modes are properly ordered), it is recommended to use the normal modes of the tar-get state instead of the normal modes of the initial state. As explained in Figure 4, the initial vibrational state(ν01 ¼ ν02…¼ ν0N ¼ 0), which gives rise to the main vibrational progressions, has a fully symmetric wave function. Conse-quently, projecting such a state into the rotated coordinate system (normal modes of the target state Q

0 0) does not affect

its nodal structure and, therefore, introduces relatively small errors in the overlaps. In contrast, projecting the

FIGURE 3 ezFCF input parameters, options, and output. The gray box indicates selected features implemented in the code. The output

contains information listed in yellow box

6 of 22 GOZEM AND KRYLOV

vibrationally excited state into another coordinate system can significantly affect the overlaps because of the incorrectrepresentation of the nodal structure of the target state.

To further reduce the cost of the calculations, the maximum number of vibrational excitations in the target state aretruncated at a user-defined threshold (this is justified because the overlaps for highly excited states eventually becomesmall) and by excluding some modes (such as those that have small ΔQ and small change in frequencies) from the cal-culation. The number of quanta in the initial state depends on the temperature: for T = 0 K, ν0max = 0, but for finite Tmore states need to be included to describe hot bands. The populations of higher vibrational states are given by theBoltzmann factors.

Figure 2 illustrates what is computed in ezFCF. If there are multiple final states (as in the figure), ezFCF reads theinput parameters for each state and computes the overall spectrum by adding up the FCFs for all states, properly shiftedby the respective excitation or ionization energies. Using the equilibrium geometries, harmonic frequencies, and normalmode vectors of the initial and final state(s) provided by ab initio calculations, ezFCF first aligns the equilibrium geome-try of the initial and final states to maximize the overlap of the normal modes. The displacements between the initialstate and each final state (ΔQs in Figure 2) along each normal mode are then determined, using either the normalmodes of the initial or the final state, as specified by the user (see Figure 4). If T = 0 K, only excitations from the ν = 0vibrational level of the initial state are considered. At finite temperatures, the thermal (Boltzmann) distribution is usedto populate higher vibrational levels of the initial state (up to νmax) and hot bands are computed as well. ezFCF printsthe energies and the FCFs (and their squares) for each vibronic transition, so that the result can be plotted as a stickspectrum, as per Equation (6). The overall spectrum (printed in the ezFCF output) contains the superposition of thetransitions from ν = 0 and all hot bands (scaled by Boltzmann factors), as shown in Figure 2, and can be directly com-pared with the vibrational progressions in experimental spectra.

The details of the algorithm, which uses a recurrence relation to evaluate the overlap integrals accounting forDuschinsky rotations,19 is described in the ezFCF manual and in more details in Refs. 22, 23. Such calculations aremore expensive than parallel-mode approximation and scale unfavorably with the system size. The cost can be con-trolled by limiting the number of Franck–Condon active modes and vibrational excitation levels in each mode. We notethat double-harmonic approximation fails in systems with large-amplitude soft modes, such as molecular clusters; insuch cases more sophisticated treatments are needed.24

ezFCF is not the only code capable of computing FCFs. Other software tools with similar functionality areFCFAST,25 CDECK,26 hotFCHT,27 molFC,28 VIBRON,29 and FCClasses.30 In addition, some quantum chemistry codesoffer built-in capabilities for computing FCFs for selected methods. For example, Gaussian and Q-Chem can computeFCFs within double-harmonic approximation and accounting for Herzberg-Teller effects.30–32 The distinguishing fea-tures of ezFCF are its generality and flexibility. In contrast to built-in codes, ezFCF can be used with any electronic

FIGURE 4 The effect of rotations of the normal coordinates on FCFs within the parallel-mode approximation. (a) The correct overlap

between wave functions of the lower (Q0) and upper (Q

0 0) surfaces. (b) The overlap when the lower normal coordinates are rotated to

coincide with the upper coordinates. (c) The overlap when the upper normal coordinates are rotated to coincide with the lower coordinates.

Source: Reprinted with permission from Ref. 21. Copyright (2009) American Chemical Society

GOZEM AND KRYLOV 7 of 22

structure method while allowing the user to control most of the simulation parameters. ezFCF is an open-source pro-gram with interfaces to Q-Chem, Cfour, ACESII, Molpro, ORCA, GAMESS, and Gaussian. The code design allows theuser to combine not only the results of the calculations by different electronic structure methods, but even the results ofdifferent quantum chemistry packages.

2.2 | Examples

To illustrate the theory, we consider several representative examples from the literature. Figure 5 compares the experi-mental photoelectron spectrum of the phenolate anion with the stick spectrum computed with ezFCF.34 The computedspectrum reproduces the overall shape of the spectrum well, including the intensity pattern. At higher energies, thecomputed spacings between the peaks are slightly overestimated relative to the experimental ones, which is expecteddue to the harmonic approximation (the levels of the harmonic oscillator are equidistant whereas for the anharmonicMorse potential the separation between the levels decreases with energy). The most intense peak in the spectrum is the00 transition (the smaller peak just under 2.2 eV is a hot band from the first vibrationally excited level of the mainprogression-forming normal mode). For this system, the difference between the vertical and adiabatic detachmentenergy is relatively small (0.05 eV), yet it is significant for quantitative comparisons of the theoretical results with theexperimental values.

The same study34 also reported the photoelectron spectrum of the HBDI anion, a model system of the chromophoreof the green fluorescent protein. For this larger molecule, the spectrum is broader and more congested due to multipleFranck–Condon active modes. The structural relaxation (the difference between Ev and Ea) is also larger in HBDI rela-tive to phenolate—0.11 eV versus 0.05 eV; yet, the band maximum still corresponds to the 00 transition and not to thevertical detachment energy. A similar trend was observed35 for the excitation spectrum of the HBDI anion—here againthe largest intensity corresponds to the 00 transition. This theoretical prediction was later confirmed by Weber andcoworkers, who reported a vibrationally resolved action spectrum of the HBDI anion.36

In the condensed phase, the spectra are broadened due to the interactions with the solvent (in addition to instru-mental broadening); yet, they often retain discernible features of the underlying vibrational structure. An example,shown in Figure 6, is flavin. The excitation and emission spectra of flavin were measured at high resolution by Dickand coworkers for lumiflavin in helium nanodroplets (Figure 6a).37 The accompanying calculations reproduced reason-ably well the Franck–Condon progression and ascribed the strongest progression-forming mode to an in-plane bendingvibration.37 The UV/Vis spectrum in aqueous solution is broad and featureless, but in many proteins the spectra arebetter resolved and show three distinct bands in the S0!S1 transition (black spectrum in Figure 6b). The computedspectrum (FCFs broadened by empirically fit Gaussians, red spectra in Figure 6b) reproduces these features and assigns

FIGURE 5 Photoelectron stick spectrum of the phenolate anion computed in double-harmonic parallel normal mode approximation at

T = 300 K (red sticks). The experimental33 photoelectron spectrum is shown in black. The structure and displacement vectors indicate the

primary Franck–Condon active mode responsible for the well-resolved vibrational peaks.

Source: Reprinted with permission from Ref. 34. Copyright (2013) American Chemical Society

8 of 22 GOZEM AND KRYLOV

them to the main progression-forming mode.38 The spectrum features two intense vibronic bands at about 2.6 eV and2.8 eV, as well as a third, less-intense peak at about 2.95 eV. Neither of the two more intense bands is close to the verti-cal one; they are shifted by about 0.15–0.40 eV relative to Ev.38 The magnitude of the zero-point energy correction,�0.065 eV, is also noticeable. The large difference between Ev and a properly computed λmax has been pointed out inseveral studies on flavins using various levels of theory.38–41

These examples illustrate that a common practice of interpreting the band maxima in photoelectron or UV/Visabsorption spectra as vertical ionization or excitation energies is not often justified for polyatomic molecules, in whichλmax often corresponds to the 00 transition, and is, therefore, much closer to the adiabatic ionization/excitation ener-gies. In molecules discussed here, the adiabatic and vertical energy differ by as much as 0.1–0.4 eV, which is clearlyimportant for quantitative comparisons between theory and experiment. The reason that the conclusions based on theone-dimensional model (often referred to as the Franck–Condon principle) do not hold up in polyatomic molecules isthat small structural relaxation along many degrees of freedom adds up to a large energy value.15,34 Hence, despite alarge energetic difference between Ev and Ea, the actual displacements along Franck–Condon active modes are suffi-ciently small, so that the 00 transition often remains the most intense one. We note that the smallness of structuralchanges can be attributed to the delocalized nature of molecular orbitals in polyatomic molecules.15

Obviously, the best strategy for benchmark studies is to complement electronic structure calculations with calcu-lations of FCFs, to unambiguously determine the position of the highest intensity relative to Ev (or Ea

ee). If this is notpossible, the next best approach is to compare experimental band maxima with the computed Ea

00 (or Eaee ). These

observations further highlight the importance of assessing the performance of electronic structure methods to describepotential energy surfaces. Even when comparing theory with theory, it is not sufficient to rely on vertical excitationenergies alone. As was recently illustrated by Szalay and coworkers42 for the coupled-cluster family of methods (rang-ing from CC2 to CCSDT), already at the Franck–Condon geometries, the nuclear gradients differ considerably amongmethods, indicating that the shapes of the resulting potential energy surfaces and, consequently, FCFs would be verydifferent. One consequence is that the ability of low-level methods (such as CC2 or ADC(2)) to produce accurate verti-cal excitation energies (as compared with the high-level theoretical reference) is an artifact due to a fortuitous cancel-lation of errors and that the errors for the adiabatic excitation energies are much larger.42 Figure 7 further illustratesthis point by comparing three excited-state potential energy surfaces that have the same Ev but different shapes, giving

FIGURE 6 (a) Experimental and computed (stick spectra) Franck–Condon pattern of the excitation and emission spectra of flavin.37

(c) Experimental (black, for flavin mononucleotide in iLOV41) and computed (red) spectra for flavin in condensed phase. The FCFs were

computed for a gas-phase lumiflavin model using ezFCF with B3LYP/cc-pVTZ (red stick spectra and solid red line broadened spectrum) and

by Davari et. al using a slightly different approach (dashed red line).41 The adiabatic and vertical excitation energies are indicated with

vertical dashed lines.

Source: Panel (a) reprinted with permission from Ref. 37. Copyright (2012) Elsevier B.V. Panel (b) adapted with permission from Ref. 41.

Copyright (2016) American Chemical Society

GOZEM AND KRYLOV 9 of 22

rise to very different FCFs. Thus, calculation of FCFs should be an integral part of benchmarking excited-statemethods.25,43

ezFCF can be used to generate requisite data to carry out more sophisticated calculations, such as inclusion of non-Condon effects35 and modeling spectral line-shapes in nonlinear spectroscopies, such as two-photon absorption,35 reso-nance Raman and resonant inelastic X-ray scattering.44

3 | ezDyson

3.1 | Software capabilities and features

3.1.1 | Overview

ezDyson computes absolute photoionization cross sections, photoelectron angular distributions (PADs), and anisotropyparameters (β) using Dyson orbitals computed by a quantum-chemistry program.

3.1.2 | Required input

ezDyson requires molecular geometry and orbital data from a quantum-chemistry package, as well as several experi-mental details (Figure 8). The user should specify the ionization energy and the range of electron kinetic energies forwhich the cross sections will be computed. The user also should specify whether the calculation is for randomly ori-ented molecules or for a molecule aligned and fixed in space. In the latter case, the user should indicate the direction ofthe laser polarization in the molecular frame.

The quality of the calculations is sensitive to the treatment of the photoelectron wave function. For absolute crosssections, correct spin and orbital degeneracy factors also need to be provided by the user.

3.1.3 | What the code computes

ezDyson computes absolute photoionization cross sections, PADs, and the β anisotropy parameter, which are all func-tions of the photoelectron dipole matrix element from Equation (8). In what follows we briefly introduce these quanti-ties and explain how they are computed in ezDyson.

Equation (8) defines the photoelectron dipole matrix element μi∞ as an integral over coordinates of all n electrons(represented by r). Because electrons are indistinguishable and assuming strong orthogonality (i.e., that the photoelec-tron wave function is orthogonal to all orbitals of the molecule), the integration can be done in two steps, giving rise to

FIGURE 7 A schematic illustration of FCFs revealing the differences between excited-state methods that are not captured by vertical

energy calculations. Different excited-state geometries (compare panels a and b) give rise to different FCF intensity pattern and band

position. Different excited-state curvatures (compare panels a and c) give rise to different spacings between the vibrational peaks

10 of 22 GOZEM AND KRYLOV

μi∞ ¼ðϕd r1ð Þμψ el

k r1ð Þdr1 � ϕd μ ψ elk

�� �,

��� ð12Þ

where ϕd is obtained by integrating the initial n-electron and final n�1-electron wave functions:

ϕd r1ð Þ¼ffiffiffiffiN

p ðψni r1,…,rnð Þψn�1

f r2,…,rnð Þdr2…drn: ð13Þ

For the sake of brevity, here we have dropped the implicit dependence on nuclear coordinates R.ϕd(r), which is a function of only one electron, contains all the information about the two many-body wave func-

tions ψni and ψn�1

f needed to evaluate μi∞; it is called a Dyson orbital or generalized overlap.45,46 Within the Koopmans'framework, ϕd is simply the Hartree–Fock canonical orbital vacated by the electron upon ionization. When computedusing correlated wave functions (such as EOM-CC,47–50 configuration interaction,51 Green's functions and propagatormethods,52,53 CASSCF,54 and CASPT255), Dyson orbitals account for electron correlation and orbital relaxation effectsthat are missing in Koopmans' picture of ionization while retaining the conceptual simplicity of a one-electron picture.Detailed pedagogical overviews of Dyson orbitals and their applications can be found in recent perspectives by Ortiz46

and by Krylov.45

Among numerous correlated approaches, EOM-CC56–60 is uniquely suitable for computing Dyson orbitals (and, con-sequently, photoionization and photodetachment cross sections) in a broad range of systems. The Fock-space formula-tion enables access to different electronic structure patterns including electronically excited and open-shell species aswell as states with multi-configurational character48,61,62 whereas the single-reference formulation makes the deriva-tions of working equations rather straightforward. Among recent extensions of EOM-CC, we note the implementationof Dyson orbitals for electronically metastable states (resonances)50 and core-level states.49,63 The versatility of EOM-CChas recently been exploited in extending the concept of Dyson orbitals to two-body Dyson orbitals suitable to describeAuger spectra64,65 in the framework of EOM-CC theory for core-level states.66–68

Equation (12) shows that the shape of the Dyson orbital determines the state of the ejected electron, by virtue of thedipole selection rules. For photoionization from an s orbital, for instance, the integral in Equation (12) is only non-zerowhen the photoelectron wave function ψ el

k has a p-like shape aligned with the direction of the light polarization vector.The photoelectron is said to be a p-wave. Likewise, when photoionization occurs from a p (or p-like) orbital, the photo-electron could be ejected as an s-wave or a d-wave. Because the direction in which the electron is ejected is related tothe shape of the orbital from which it was ionized, it is informative to measure the PAD, which describes the angulardistribution (or the anisotropy) of the ejected photoelectron with respect to the polarization of light. PAD is related tothe differential cross section (dσ/dΩk) that describes the dependence of the cross section (σ) on the solid angle of photo-electron emission (Ωk).

Cooper and Zare showed that for an ensemble of randomly oriented molecules, this anisotropy persists and has thefollowing simple form69:

dσdΩk

¼ σ

4π1þβP2 cos θkð Þð Þ½ �, ð14Þ

FIGURE 8 ezDyson input parameters, options, and output. The gray box indicates options and features implemented in the code, which

the user controls. The main information printed in the output are in yellow

GOZEM AND KRYLOV 11 of 22

where P2 is a second order Legendre polynomial and θk is the direction of the ejected photoelectron with respect to thepolarization of the incident light. β is the aforementioned anisotropy parameter that characterizes the angular distribu-tion of the photoelectrons: it can vary between �1 and 2 (�1 is perfectly perpendicular, 0 is perfectly isotropic, +2 isperfectly parallel).

The plot of β as a function of the energy of ionizing radiation (or photoelectron kinetic energy) encodes the informa-tion about the shape (i.e., angular momentum and the diffuseness) of the orbital from which the electron is ejected.Figure 9 shows the β values for H� and F� as a function of the photoelectron kinetic energy computed with ezDyson.While β is constant for H� at all energies (consistent with a p-wave that is parallel to the polarization of light), F� has amore complex energy-dependent β profile. At low energies, ionization of fluoride's p-orbital results in a photoelectronwave function dominated by an s-wave (and, therefore, isotropic). In contrast, at high energies, the photoelectron wavefunction is dominated by a d-wave. At intermediate energies, cross-terms arising from the combination of s-waves andd-wave result in a perpendicular distribution of electrons, with β =� 1 at some point.70

The absolute photoionization cross section is a measure of the total probability of an electron being ejected in anydirection. It is obtained by integrating the differential cross section over all solid angles (Ωk). The calculation of energy-dependent photoionization cross sections is useful for assigning the electronic origins of photoelectron spectra (e.g., toidentify contributions from direct ionization and auto-ionizing metastable states). The ability to accurately computeabsolute cross sections is also important in applications of photoelectron spectroscopy as an analytic tool for quantita-tive chemical analysis.71

The differential cross section is computed from μi∞ using16,72

dσdΩk

¼ 4π2kEc

ϕdjμjψ elk

� �2 ¼ 4π2kEc

μi∞ θ,ϕð Þj j2, ð15Þ

where k is the magnitude of the photoelectron wave vector k, E is the energy of the ionizing radiation, and c is thespeed of light. θ and ϕ are the polar and azimuthal angles of the photoelectron relative to the polarization of light.

Equation (15) is the starting point for computing total cross sections, PADs, and β anisotropy. ezDyson evaluates theintegral in Equation (15) by numerical integration on a grid using Dyson orbitals computed by a quantum chemistryprogram (Dyson orbitals are specified as an expansion over the atom-centered Gaussian basis functions).

The treatment of the photoelectron wave function ψ elk is one of the key challenges for quantitative modeling of pho-

toionization and photodetachment. Currently, ezDyson uses two simple models—plane wave and Coulomb wave. Thelimitations of these models are well documented and can be addressed by more sophisticated approaches.73

In the plane-wave representation, the photoelectron wave function is

ψ elk rð Þ¼ 1

2πð Þ3=2eik�r , ð16Þ

where (2π)�3/2 is the continuum normalization for plane waves.74 ezDyson provides two implementations for comput-ing the respective matrix elements: one uses Equation (16) as-is and another uses a partial-wave expansion in whichthe plane wave is expressed as a sum of spherical waves75:

eik�r ¼ 4πX∞l¼0

Xl

m¼�l

ilRl krð ÞYlm rð ÞYlm kð Þ: ð17Þ

By using the partial-wave expansion, one can analyze the wave function of the ejected photoelectron in terms ofangular momentum quantum numbers. Each spherical wave is characterized by its energy (E¼ k2

2m ) and angularmomentum (l, m), and is a product of a radial function Rl(kr) and spherical harmonic functions Ylm. Ylm(r) depends onthe position vector r and, therefore, determines the shape of the plane wave in position space, while Ylm kð Þ determinesthe direction in which the plane wave travels.75 By providing the coefficients of each partial wave, ezDyson enablesquantitative application of the mixed s& p model of Sanov to polyatomic molecules.70,76

In practice, the expansion in Equation (17) is truncated at some lmax (specified by the user). In atoms, the truncationis justified because strict selection rules preclude contributions from high angular momentum waves to the photoelec-tron wave function. In molecules the truncation may be justified when Dyson orbitals resemble atomic orbitals, whichis a main rationale behind the mixed s& p model.70,76 Moreover, the contributions from large angular momentum aresmall at low photoelectron kinetic energies.

12 of 22 GOZEM AND KRYLOV

Using a plane wave is most appropriate for photodetachment, where there is no electrostatic interaction betweenthe ejected photoelectron and the remaining neutral core. In contrast, in photoionization the core is charged and theelectrostatic interaction between the photoelectron and the core cannot be easily ignored. In this case, the Coulombwave may be used instead of a plane wave. The Coulomb wave can also be expanded into a sum of Coulomb partialwaves using the same expression as Equation (17).6 The details of the implementation and the analytic expressions forcomputing the total cross section, PADs, and β using plane or Coulomb waves can be found in the ezDyson manual.77

When using sufficiently high energy, multiple ionized target states f become accessible and one needs to account fortheir contributions to the total cross section. In addition, the calculation should account for electronic degeneracies.The contributions can be added up as follows:

dσdΩk

¼ 4π2Ec

�Xf

kf μfi∞ θ,ϕð Þ

��� ���2: ð18Þ

Because different target states may give rise to different photoelectron kinetic energies, kf is a function of thestate f. Different states also give rise to different Dyson orbitals, according to Equation (13). For non-degeneratetarget states, kf and μfi∞ differ, and one can compute cross sections for the individual states separately and add themup. For systems with symmetry-imposed degeneracies, kf and μfi∞ are identical for the degenerate states and, therefore,the cross sections may be computed for only one of the states and then multiplied by the number of the states in themanifold. Users also need to account for spin-degeneracy of the target states. For ionization of closed-shell singletstates, the cross section should be multiplied by 2.77 For example, the cross section for photodetachment from F� to theF(2P) channel is

σ 2P ¼ 3�2�σ ϕd � pz

, ð19Þ

where the factor of 3 is for orbital degeneracy and the factor of 2 is for spin degeneracy. Likewise, for ionization from πorbitals in CO or acetylene, the cross sections should be multiplied by 4 (2 for orbital degeneracy and 2 for spin degener-acy). These degeneracy factors need to be specified by the user in the ezDyson input.

3.1.4 | Averaging over molecular orientations

Experiments are most often performed on isotropic samples comprising randomly oriented molecules. ezDyson codeprovides two averaging procedures: analytic and numeric. The analytic averaging algorithm largely follows the

FIGURE 9 β values for H� and F� as a function of energy of the photoelectron kinetic energy

GOZEM AND KRYLOV 13 of 22

approach described in Ref. 78 and is documented in the ezDyson manual. The analytic averaging is fast and employsthe partial-wave expansion from Equation (17). The numeric averaging approach does not rely on the partial-waveexpansion and, therefore, does not entail the truncation at some lmax; this is useful at high energies where the trunca-tion of the partial-wave expansion becomes problematic because the contributions from high angular momentum termsare no longer negligible.

3.1.5 | Accounting for FCFs

The shapes of photoionization spectra are also affected by FCFs, as per Equation (5). The electronic μi∞ term is com-puted in ezDyson with the appropriate averaging approach, while the second term is the FCF for a given vibronic transi-tion computed by ezFCF. The absolute total cross section can be computed for any vibronic transition using thefollowing equation:

σνν0k ¼ Pνσkhχ i,νjχf ,ν0 i2, ð20Þ

where Pν, the population of the initial state vibrational level ν, accounts for the thermal population of vibrational statesat non-zero temperatures. The sum of all Pν must be normalized to 1 to preserve the meaning of the absolute cross sec-tions. Note that σk depends on the energy of the ionizing radiation E (see Equation (15)), which is equal to the energyof the vibronic transition plus the kinetic energy of the photoelectron (Ek). The latter can be computed from the elec-tron 00 transition and the vibrational energy levels:

Ek ¼ k2νν02

¼E� IEIF00� EF

vib νð Þ�EIvib ν0ð Þ ¼E� IEIF

00�ΔEvib ν,ν0ð Þ, ð21Þ

where IEIF00 is the adiabatic ionization/detachment energy (including the zero-point energy correction) of the initial

state, and Eivib νð Þ and Ef

vib ν0ð Þ are the excess vibrational energies of the initial and final states above ν = 0 and ν0 = 0,respectively. To compute the total cross section at a given energy, one needs to sum over all vibronic transitions accessi-ble at that energy, in a fashion similar to Equation (18). Because σk is only computed for discrete values of Ek, theezDyson distribution includes a script (xsecFCF) to compute the result of Equation (20) for discrete energies using theoutput of ezFCF (see ezDyson manual77 for details).

3.1.6 | Single-center versus multi-center treatment of the photoelectron wave function

Because of the dipole moment operator in Equation (8), the origin of the free-electron state ψ elk matters. Consider, for example,

an s-type Dyson orbital: if the partial-wave expansion has the same origin as the orbital, then only p-waves give rise toa non-zero matrix element, however, if the expansion center is offset, then many partial waves survive. Obviously, inatoms the expansion should be atom-centered, however, in polyatomic systems (and especially in clusters) the choice isno longer trivial.

One possibility is to use the centroid of the Dyson orbital as the origin79: for atoms, this results in atom-centered expansions,whereas for relatively localized molecular orbitals this entails placing the origin where the density of the Dyson orbital is thelargest. This choice also implicitly ensures the orthogonality of ψ el

k to the Dyson orbital.72,79 While working reasonably wellfor molecules, this choice becomes problematic in certain situations due to the limitations of our theoretical model.

To understand the issue, imagine two atoms at infinite separation; the total photoionization cross sections should simply bethe sum of the atomic values computed using the atom-centered expansions. However, the Dyson orbital is delocalized by sym-metry, such that its centroid falls in between the two atoms. As illustrated by numeric examples in Ref. 80, for two infinitely sep-arated fragments the results of the two calculations (one using atom-centered expansion and one using the expansion at themidpoint) are vastly different, with atom-centered expansion yielding the correct description. However, at short inter-atomic dis-tances, the correct result is obtained when the expansion is placed in between the atoms. This can be understood in the contextof Sanov's molecular interferometer,81 where the short-time evolution of time-resolved photoelectron images from dissociatingI�2 was discussed in terms of the interference between the two waves coming from the individual atoms. This

14 of 22 GOZEM AND KRYLOV

interference can be described as ionization of a single entangled state spanning both iodine atoms at short I�2 bondlengths. However, at large inter-atomic separations the variation in β ceases and it approaches the asymptotic valuecorresponding to the PAD of I�. The distance at which the atoms become independent and the coherences die off isrelated to the energy of the photoelectrons. In principle, the exact treatment should correctly describe both regimes, butthe single-centered model currently used in ezDyson would fail at large separations because it does not account for theinterfering contributions of photoelectrons coming off the two degenerate orbitals, such as in-phase and out-of-phasecombinations of p orbitals in Sanov's experiment.81

This limitation provides a motivation for developing a multi-center model for photoionization. A simple variant ofthis model was used in Ref. 80 to describe photoionization of bulk water. In a multi-center model, the Dyson orbital issplit into fragments (molecular or atomic) and the cross-section is computed as the sum of the fragments' contributions.If the coherences between the centers are accounted for (as done, e.g., in Refs. 82, 83), the multi-center treatmentshould be able to smoothly cover both short-distance and long-distance cases (or, equivalently, low-energy and highenergy regimes). In the current implementation in ezDyson, the coherences are neglected, so that the multi-centermodel is only appropriate for high-energy regime, when the de Broglie wavelength of photoelectrons is shorter than thedistance between the centers. The inclusion of coherences is under development.

3.2 | Examples

3.2.1 | Absolute photoionization cross sections

Figure 10 shows computed and experimental absolute cross sections for H� (photodetachment) and He (photoioniza-tion). For hydride, photodetachment results in a neutral atom (hydrogen) that does not exert electrostatic potential onthe ejected electron. Therefore, the plane-wave treatment of the photoelectron is a reasonable approximation, as con-firmed by the computed photoionization cross sections: the black curve in Figure 10a falls within the experimentalerror (red dots and associated error bars). Figure 10a also illustrates the impact of the quality of the Dyson orbital onthe cross-section. The dashed line shows the cross section computed using the canonical Hartree–Fock orbital(Koopmans' theorem)—it is rather different from the one computed using the correlated Dyson orbital and falls outsidethe experimental spectrum.

FIGURE 10 (a) Absolute photodetachment cross section from H� using a plane wave description of the photoelectron and the Dyson

orbital computed with EOM-IP-CCSD/aug-cc-pVTZ. The cross section computed using the Hartree–Fock canonical molecular orbital is

shown with a dashed line. (b) Absolute photoionization cross section from He using a plane wave (black line) and a Coulomb wave (blue

line) descriptions of the photoelectron and the Dyson orbital computed with EOM-IP-CCSD/aug-cc-pVTZ. The plane-wave treatment

neglects interactions between the photoelectron and ionized core, which is a reasonable approximation for photodetachment. The Coulomb

wave accounts for an interaction with a +1 charge placed at the orbital centroid, which is a reasonable description for atomic

photoionization.

Source: Adapted with permission from Ref. 85. Copyright (2015) American Chemical Society

GOZEM AND KRYLOV 15 of 22

Helium becomes positively charged upon photoionization. Hence, ignoring the interaction between the ejected elec-tron and the positive He+ core yields incorrect cross sections—compare the black curve with the red dots in Figure 10b.Using a Coulomb wave with +1 charge yields much better agreement with the experimental spectrum (blue curve inFigure 10b). Because ezDyson simulates direct photoionization, the computed cross section does not capture resonances,such as those observed in He around 60 eV. The contributions from resonances can be estimated following the proce-dure described in Ref. 84.

Figure 11 shows two examples of absolute photoionization cross section calculations in molecules. The simplestmolecules, such as dihydrogen or small molecules with only one heavy atom,85 show the same trend as helium: theplane-wave calculations are inaccurate, whereas the Coulomb-wave calculations agree quantitatively with the experi-mental photoionization spectra.

In larger molecules (e.g., formaldehyde in Figure 11b), we often find that neither the plane wave nor the Coulombwave with +1 charge yield quantitatively accurate photoionization cross sections. However, using a Coulomb wavewith some partial effective charge (Zeff) leads to a photoionization spectrum that matches both the shape and the abso-lute magnitude of the experimental near-threshold photoionization efficiency spectrum. How to determine Zeff a prioriis not clear. We have shown that the effective charge effectively tunes the shape of the photoelectron wave function toprovide the correct overlap with the Dyson orbital in the interaction region and proposed a variational approach fordetermining optimal charges.85 Effective charges can be estimated using simple atomic models (e.g., Rydberg formula),giving rise to Belki�c's86 and Slater's rules.87 The utility of these effective charges within the Coulomb-wave approxima-tion have been recently illustrated in the context of modeling Auger spectra.64,65

3.2.2 | Photoelectron angular distributions

PADs and the associated βs are more sensitive to the treatment of the photoelectron wave function than absolute crosssections. Therefore, quantitative calculations of PADs are more challenging, especially in molecules. In the condensedphase, additional complications arise due to elastic scattering of photoelectrons.88–90 These challenges have been dis-cussed in a recent study on photoionization of bulk water.80

Since photoelectron anisotropies depend on orbital shapes, PADs report on electronic structure changes associatedwith chemical (reaction or conformational dynamics) or physical processes, such as solvation. Photoelectron spectros-copy and imaging have been extensively used to monitor dynamics5,91,92 and to understand molecular conformations.93

Figure 12 shows experimental and theoretical (computed with ezDyson) anisotropies for several para-substituted

FIGURE 11 Absolute photoionization cross section from molecular hydrogen (a) and formaldehyde (b) using a plane-wave (black line)

and a Coulomb-wave (blue line) descriptions of the photoelectron and Dyson orbitals computed with EOM-IP-CCSD/aug-cc-pVTZ. In

formaldehyde, neither the plane wave nor Coulomb wave reproduce the experimental total cross section, but a Coulomb wave with a partial

effective charge of Zeff = 0.25 (orange) does. The solid and dashed orange lines in panel (b) show the calculations with and without FCFs,

respectively. By including the FCFs, the calculation reproduces the small rise in cross section around 11 eV.

Source: Adapted with permission from Ref. 85. Copyright (2015) American Chemical Society

16 of 22 GOZEM AND KRYLOV

phenolate anions. For rigid anions, the anisotropies computed for a single structure (global minimum) agree well withthe experiment, whereas for the flexible anion (para-ethyl phenolate, panel b), averaging over the torsional motion isneeded. This example illustrates the sensitivity of PADs to molecular conformations and the utility of modeling in inter-preting the experimental measurements.

Angular-resolved imaging can also be used to understand intermolecular interactions. By comparing PADs in thegas and liquid phases, it is possible to ask interesting questions, for example, how intermolecular interactions affect theshape of the molecular orbitals of a solute. This was the focus of a recent study,80 which attempted to quantify howwater's p-like orbitals are altered when moving from the gas to the liquid phase by re-hybridization, loss of symmetry,scattering, and delocalization.

The PADs of water and the associated β have been experimentally measured at different energies of the ionizingradiation both in the gas80,94,95 and liquid80,90,96 phases using the liquid microjet technique.97,98 Figure 13 shows the

FIGURE 12 Plots of the experimental (circles) and computed (solid lines) β as a function of the electron kinetic energy (eKE) for the

lowest detachment channel for (a) para-methyl phenolate, pMP�; (b) para-ethyl phenolate, pEP�; and (c) para-vinyl phenolate, pVP�. ForpEP�, two computed PADs are shown, corresponding to global (red) and local (blue) minimum-energy structures. The respective Dyson

orbitals are shown on the right.

Source: Reprinted with permission from Ref. 93. Copyright (2017) American Chemical Society

FIGURE 13 Experimental (symbols) and computed (solid lines) β values for the ionization of 3a1 orbitals of gas-phase (red) and liquid-

phase (blue) water. Gas-phase data are from Truesdale et al.94 (red triangles), Banna et al.95 (red rhombuses), and from Gozem, Seidel et al.80

(red circles), liquid-phase experimental data are from Gozem, Seidel et al.80 (blue circles). Computed anisotropies are shown as lines using a

Coulomb wave with Belki�c's charges for the description of the photoelectron. Liquid β calculations were performed on pentamer structures

extracted from the molecular dynamics simulation of water using: (1) a multi-center model with expansion centers on individual water

molecules (blue solid line); (2) the multi-center model including empirical scattering corrections (blue dashed line); and (3) a single-center

model with the expansion center placed at the centroid of the pentamer cluster Dyson orbital (blue dotted line)

GOZEM AND KRYLOV 17 of 22

experimental and computed gas-80,94,95 and liquid-phase80 β values for the 3a1 orbital of water. Gas-phase calculationswere performed using a Coulomb wave with Zeff = 2.06 (charge computed by Belki�c's rule86). Whereas in the low-energy regime, where slow outgoing electrons interact with the perturbing molecular potential for a longer time, thePADs are very sensitive to the magnitude of the effective charge at large energies the differences between plane- andCoulomb-wave calculations with different charges are rather small, because fast photoelectrons are less sensitive to theshape of the perturbing cationic potential.

The computed energy-dependent β values agree only qualitatively with the experiment, showing discrepancies inthe low energy regime. Although the agreement for other valence orbitals of water (1b1 and 1b2) is slightly worse thanfor 3a1, the gas-phase model is able to capture the variations of the angular distribution from different ionized states,meaning that it can be used to model changes in anisotropy as a result of perturbation due to solvent.

To quantify the effect of solvation of individual water molecules, β values were computed using pentamer structuresextracted from a molecular dynamics simulation of neat water. The corresponding Dyson orbitals of these snapshots(Figure 13, bottom right) are typically delocalized. β values computed using these delocalized orbitals and a single-center Coulomb wave placed at the centroid of the pentamer cluster orbital were almost isotropic (β≈ 0), suggesting asubstantial loss of anisotropy (dotted line in Figure 13). However, the multi-center approach yielded β values almostidentical to the ones computed in the gas phase. As explained above, in this regime the multi-center treatment is moreappropriate, because X-ray ionization produces high-energy photoelectrons with de Broglie wavelengths smaller thanthe intermolecular distance between the water molecules. The experiment showed some loss of anisotropy of water inthe liquid phase relative to the gas phase (blue versus red dots in Figure 13), which, through comparison with water1sO measurements,99 was attributed almost exclusively to electron scattering, rather than an intrinsic change in water'sorbital shape. Empirically correcting the computed localized liquid β values for scattering using a scaling factor derivedfrom the 1sO experiment99 yields the dashed line in Figure 13, which agrees well with the experiment. Surprisingly, thisfinding means that water's molecular gas-phase orbitals are not sufficiently perturbed in the liquid to cause noticeablechange in the β anisotropy.

In sum, although quantitative computations of PADs require a careful treatment of the photoelectron wave func-tion, phase shifts79 and elastic scattering, and therefore remain elusive, simple models, such as those implemented inezDyson, deliver valuable qualitative (and often semi-quantitative) results. Such calculations can also provide insightinto the evolution of the electronic structure in the course of dynamics. ezDyson has been incorporated in the excited-state molecular dynamics program system Newton-X to enable simulation of time-resolved photoelectronspectroscopies.100,101

4 | CONCLUSION AND OUTLOOK

Absorption and photoionization spectra provide valuable information about the electronic and vibrational energy levelsof the system. In addition, the intensity pattern, absolute values of cross sections, and anisotropies encode informationabout the electronic wave functions. The information contained in the spectra can be deciphered by using quantum-chemistry simulations. The ezSpectra suite provides a set of tools for computing spectra from electronic-structure calcu-lations. Future developments include improving the theoretical models (e.g., multi-center treatments of photoioniza-tion), time-dependent modeling of FCFs for treating larger systems, inclusion of vibronic effects, and extensions toother spectroscopies and magnetic systems. We also plan to integrate these tools into Python workflows to create a con-venient and flexible platform for spectroscopy modeling. We envision that ezSpectra tools can be integrated into othermodeling softwares.

ACKNOWLEDGMENTSThe ezSpectrum code was originally developed by Dr. Vadim Mozhayskiy and Anna I. Krylov. The original version ofezDyson was developed by Dr. Melania Oana and Anna I. Krylov. Dr. Evgeny Epifanovsky and Dr. Ilya Kaliman havehelped to parallelize ezDyson. Dr. Kadir Diri, Dr. Anastasia Gunina, and Sahil Gulania contributed to the ezDyson man-ual. Dr. Takatoshi Ichino, Prof. John Stanton, Prof. Steven Bradforth, and Dr. David Osborn helped to validate theimplementation for the calculation of photoionization total cross sections and angular distributions in ezDyson. PawelWojcik contributed to ezFCF. We thank all users who provided feedback and additional conversion scripts. Develop-ment and support of ezFCF and ezDyson has been funded by the National Science Foundation and the Department of

18 of 22 GOZEM AND KRYLOV

Energy (Anna I. Krylov). Some of the applications described were made possible thanks to funding from the AmericanChemical Society Petroleum Research Fund and the National Science Foundation (Samer Gozem).

CONFLICT OF INTERESTAnna I. Krylov is the president and a part-owner of Q-Chem, Inc.

AUTHOR CONTRIBUTIONSSamer Gozem: Conceptualization; data curation; formal analysis; funding acquisition; investigation; methodology;project administration; software; validation; visualization; writing-original draft; writing-review & editing. AnnaKrylov: Conceptualization; data curation; formal analysis; funding acquisition; investigation; methodology; projectadministration; software; validation; visualization; writing-review & editing.

DATA AVAILABILITY STATEMENTData sharing is not applicable to this article as no new data were created or analyzed in this study.

ORCIDSamer Gozem https://orcid.org/0000-0002-6429-2853Anna I. Krylov https://orcid.org/0000-0001-6788-5016

RELATED WIREs ARTICLES‘Q-Chem: An Engine for innovation’ by Gill and KrylovElectron propagator theory: an approach to prediction and interpretation in quantum chemistryNewton-X: a surface-hopping program for nonadiabatic molecular dynamicsThe virtual multifrequency spectrometer: a new paradigm for spectroscopyGoing beyond the vertical approximation with time-dependent density functional theory

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How to cite this article: Gozem S, Krylov AI. The ezSpectra suite: An easy-to-use toolkit for spectroscopymodeling. WIREs Comput Mol Sci. 2021;e1546. https://doi.org/10.1002/wcms.1546

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