Simulating X-ray Absorption Spectra WithLinear-Response Density Cumulant Theory

Ruojing Peng,† Andreas V. Copan,‡ and Alexander Yu. Sokolov∗,†

†Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio43210, United States

‡Chemical Sciences and Engineering Division, Argonne National Laboratory, Argonne,Illinois 60439, United States

E-mail: sokolov.8@osu.edu

Abstract

We present a new approach for simulating X-ray absorption spectra based on linear-responsedensity cumulant theory (LR-DCT) [Copan,A. V.; Sokolov, A. Yu. J. Chem. TheoryComput., 2018, 14, 4097–4108]. Our newmethod combines the LR-ODC-12 formulationof LR-DCT with core-valence separation ap-proximation (CVS) that allows to efficientlyaccess high-energy core-excited states. Wedescribe our computer implementation of theCVS-approximated LR-ODC-12 method (CVS-ODC-12) and benchmark its performance bycomparing simulated X-ray absorption spec-tra to those obtained from experiment for sev-eral small molecules. Our results demonstratethat the CVS-ODC-12 method shows a goodagreement with experiment for relative spacingsbetween transitions and their intensities, butthe excitation energies are systematically over-estimated. When comparing to results fromexcited-state coupled cluster methods with sin-gle and double excitations, the CVS-ODC-12method shows a similar performance for inten-sities and peak separations, while coupled clus-ter spectra are less shifted, relative to experi-ment. An important advantage of CVS-ODC-12 is that its excitation energies are computedby diagonalizing a Hermitian matrix, which en-ables efficient computation of transition inten-sities.

1 Introduction

Near-edge X-ray absorption spectroscopy(NEXAS) is a powerful and versatile experi-mental technique for determining the geomet-ric and electronic structure of a wide range ofchemical systems. The NEXAS spectra probeexcitations of core electrons into the low-lyingunoccupied molecular orbitals. Due to the lo-calized nature of core orbitals, these excitationsare very sensitive to the local chemical environ-ment, providing important information aboutmolecular structure. Recent advances in exper-imental techniques for generating and detectingX-ray radiation have spurred the developmentof NEXAS and its applications in chemistryand biology.1–7

Theoretical simulations of X-ray absorptionplay a critical role in interpretation of theNEXAS spectra.8 However, computations ofthe core-level excitations are very challeng-ing as they require simulating excited statesselectively in the high-energy spectral regionand a balanced treatment of electron corre-lation, orbital relaxation, and relativistic ef-fects, often combined with large uncontractedbasis sets. Many of the popular excited-statemethods have been adopted for simulationsof X-ray absorption spectra, including linear-response,9–14 real-time,15–17 and orthogonality-constrained18,19 density functional theory, con-figuration interaction,20–25 algebraic diagram-

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matic construction (ADC),5,26–30 as well aslinear-response (LR-) and equation-of-motion(EOM-) coupled cluster (CC) theories.31–44

Among these approaches, CC methods havebeen shown to yield particularly accurate re-sults for core excitation energies and intensitiesof small molecules.

Several techniques to compute the NEXASspectra within the framework of CC theoryhave been developed. These approaches usu-ally incorporate up to single and double excita-tions in the description of electron correlation(CCSD), but employ different strategies to ac-cess high energies required to excite core elec-trons. For example, in the complex polarizationpropagator-based CC theory (CPP-CC),34–38

core-level excitations are probed directly bycomputing the CC linear-response function overa grid of input frequencies in the X-ray re-gion. In the energy-specific EOM-CC approach(ES-EOM-CC),40 the high-energy excitationsare computed by using a frequency-dependentnon-Hermitian eigensolver. In practice, bothCPP-CC and ES-EOM-CC can only be appliedto narrow spectral regions that need to be se-lected a priori. This problem is circumventedin the time-dependent,43 multilevel,42 and core-valence-separated41,44 EOM-CC methods thatcan be used to compute NEXAS spectra forbroad spectral regions and a large number ofelectronic transitions.

In this work, we present an implementationof the recently developed linear-response den-sity cumulant theory (LR-DCT)45 for simulat-ing the NEXAS spectra of molecules. Althoughthe origin of LR-DCT is in reduced densitymatrix theory,46–51 it has a close connectionwith LR-CC methods, such as linear-responseformulations of linearized, unitary, and varia-tional CC theory.52–58 In our previous work,we have demonstrated that one of the LR-DCT methods (LR-ODC-12) provides very ac-curate description of electronic excitations inthe UV/Vis spectral region.45 In particular, fora set of small molecules, LR-ODC-12 showedmean absolute errors in excitation energies ofless than 0.1 eV, with a significant improve-ment over EOM-CC with single and double ex-citations (EOM-CCSD). We have also demon-

strated that LR-ODC-12 provides accurate de-scription of challenging doubly excited states inpolyenes.

Here, we test the accuracy of LR-ODC-12 forsimulations of core-level excitations. To effi-ciently access the X-ray spectral region, our newLR-ODC-12 implementation employs the core-valence separation (CVS) technique,59,60 origi-nally developed in the framework of ADC the-ory5,26–30 and later extended to other meth-ods.9,41,44 We test our new method (denotedas CVS-ODC-12) against LR-ODC-12 to as-sess the accuracy of the CVS approximationand benchmark its results for a set of smallmolecules.

2 Theory

2.1 Density Cumulant Theory(DCT)

We start with a short overview of density cu-mulant theory (DCT).61–67 The exact electronicenergy of a stationary state |Ψ〉 can be ex-pressed in the form

E = 〈Ψ|Ĥ|Ψ〉 =∑pq

hqpγpq +

1

4

∑pqrs

grspqγpqrs (1)

where the one- and antisymmetrized two-electron integrals (hqp and g

rspq) are traced with

the reduced one- and two-body density matri-ces (γpq and γ

pqrs ) over all spin-orbitals in a finite

one-electron basis set. Starting with Eq. (1),DCT expresses γpq and γ

pqrs in terms of the fully

connected contribution to γpqrs called the two-body density cumulant (λpqrs).

48,68–76 For γpqrs ,this is achieved using a cumulant expansion

γpqrs = γprγ

qs − γpsγqr + λpqrs (2)

where the first two terms represent a discon-nected antisymmetrized product of the one-body density matrices. To determine γpq fromλpqrs, the following non-linear relationship isused64 ∑

r

γprγrq − γpq =

∑r

λprqr (3)

2

where the r.h.s. of Eq. (3) contains a partialtrace of density cumulant. Eqs. (2) and (3)are exact, so that substituting in the exact λpqrsyields the exact electronic densities and energy.

In practice, DCT computes the electronic en-ergy by parametrizing and determining den-sity cumulant directly, circumventing computa-tion of the many-electron wavefunction. This isachieved by choosing a specific Ansatz for thewavefunction |Ψ〉 and expressing density cumu-lant as66

λpqrs = 〈Ψ|apqrs|Ψ〉c (4)

where apqrs ≡ a†pa†qasar is a two-body second-quantized operator and the subscript c indicatesthat only fully connected terms are retained.The most commonly used parametrization ofλpqrs, denoted as ODC-12,

64,65 consists of approx-imating Eq. (4) using a two-body unitary trans-formation of |Ψ〉 truncated at the second orderin perturbation theory

λpqrs ≈ 〈Φ|e−(T̂2−T̂†2 )apqrse

T̂2−T̂†2 |Φ〉c

≈ 〈Φ|apqrs|Φ〉c + 〈Φ|[apqrs, T̂2 − T̂†2 ]|Φ〉c

+1

2〈Φ|[[apqrs, T̂2 − T̂

†2 ], T̂2 − T̂

†2 ]|Φ〉c (5)

where T̂2 is the double excitation operatorwith respect to the reference determinant |Φ〉,whose parameters are determined to make theelectronic energy in Eq. (1) stationary. TheODC-12 energy is also made stationary withrespect to the variation of molecular orbitalsparametrized using the unitary singles opera-

tor eT̂1−T̂†1 .65 The parameters of the T̂1 and T̂2

operators determined from the stationarity con-ditions are used to compute the ODC-12 energy.

2.2 Linear-Response DCT (LR-DCT)

In conventional DCT, electronic energy andmolecular properties are determined for a sin-gle electronic state (usually, the ground state).To obtain access to excited states, we have re-cently combined DCT with linear-response the-ory that allows to compute excitation energiesand transition properties for a large numberof states simultaneously.45 In linear-response

DCT (LR-DCT), we consider the behavior of anelectronic system under a time-dependent per-turbation V̂ f(t), which can be described usingthe time-dependent quasi-energy function77,78

Q(t) = 〈Ψ(t)|Ĥ + V̂ f(t)− i ∂∂t|Ψ(t)〉 (6)

Here, |Ψ(t)〉 is the so-called “phase-isolated”wavefunction, which reduces to the usual time-independent wavefunction |Ψ〉 in the station-ary state limit. Importantly, for a periodictime-dependent perturbation, the quasi-energyaveraged over a period of oscillation ({Q(t)})is variational with respect to the exact time-dependent state.78 Such periodicity impliesthat the amplitude f(t) can be written in aFourier series

f(t) =∑ω

f(ω)e−iωt (7)

where the sum includes positive and negativevalues for all frequencies such that f(t) is real-valued.

To obtain information about excited states,{Q(t)} is made stationary with respect to all ofthe parameters that define the time-dependentwavefunction |Ψ(t)〉.79 We refer interested read-ers to our previous publication45 for derivationof the LR-DCT equations and summarize onlythe main results here. The LR-DCT excitationsenergies are computed by solving the general-ized eigenvalue problem

Ezk = Mzkωk (8)

In Eq. (8), ωk are the excitation energies,E is the LR-DCT Hessian matrix that con-tains second derivatives of the electronic energy{〈Ψ(t)|Ĥ|Ψ(t)〉} with respect to parameters ofthe T̂1 and T̂2 operators, and M is the metricmatrix that originates from second derivativesof the time-derivative overlap {〈Ψ(t)|iΨ̇(t)〉}.Importantly, the LR-DCT Hessian matrix E isHermitian, which ensures that the excitationenergies ωk have real values, provided that theHessian is positive semidefinite. The general-ized eigenvectors zk can be used to determine

3

the oscillator strength for each transition

fosc(ωk) =2

3ωk|〈Ψ|V̂ |Ψk〉|2 =

2

3ωk|z†kv′|2

z†kMzk(9)

where 〈Ψ|V̂ |Ψk〉 = 〈Ψ|µ̂|Ψk〉 is the transitiondipole moment matrix element and v′ is the so-called property gradient vector.80,81

In the linear-response formulation of theODC-12 method (LR-ODC-12),45 the E and Mmatrices in Eq. (8) have the following form:

E =

A11 A12 B11 B12A21 A22 B21 B22B∗11 B

∗12 A

∗11 A

∗12

B∗21 B∗22 A

∗21 A

∗22

(10)

M =

S11 0 0 00 1 0 00 0 −S∗11 00 0 0 −1

(11)where the A11 and B11 (A22 and B22) ma-trices contain second derivatives of the DCTenergy with respect to parameters of the T̂1(T̂2) operators, whereas A12 and B12 repre-sent mixed second derivatives. Each blockof E and M describe coupling between elec-tronic excitations or deexcitations of differentranks. Specifically, the A11, A12, and A22blocks correspond to interaction of single ex-citations with single excitations, single excita-tions with double excitations, and double ex-citations with each other, respectively. Simi-larly, the B11 block couples single excitationswith single deexcitations, whereas B12 and B22couple single-double and double-double excita-tion/deexcitation pairs. Finally, A∗11, A

∗12, and

A∗22 describe interaction of single and doubledeexcitations. The solution of the LR-ODC-12eigenvalue problem (8) has O(O2V 4) computa-tional scaling, where O and V are the numbersof occupied and virtual orbitals, respectively.

2.3 Core-Valence-Separated LR-ODC-12 (CVS-ODC-12)

The LR-ODC-12 generalized eigenvalue prob-lem (8) can be solved iteratively using one of

the multi-root variations of the Davidson algo-rithm.82,83 This iterative method proceeds byforming an expansion space for the generalizedeigenvectors zk starting with an initial (guess)set of unit trial vectors and progressively grow-ing this space until the lowest Nroot eigenvectorsare converged. While such algorithm is very ef-ficient for computing excitations of electrons inthe valence orbitals, it is not suitable for sim-ulations of the X-ray absorption spectra as itwould require converging thousands of roots si-multaneously to reach the energies necessary topromote core electrons.

A computationally efficient solution to thisproblem called core-valence separation (CVS)approximation has been proposed by Schirmerand co-workers within the framework of theADC methods.5,26–30 In this approach, the oc-cupied orbitals are divided into two sets: core,corresponding to the orbitals probed by core-level excitations, and valence, which contain theremaining occupied orbitals. The CVS approx-imation relies on the energetic and spatial sepa-ration of core and valence occupied orbitals59,60

and consists in retaining excitations that in-volve at least one core orbital while neglectingall excitations from valence orbitals. The CVSapproach can be combined with existing imple-mentations of excited-state methods based onthe Davidson or Lanczos eigensolvers, providingefficient access to core-level excitation energies.

We have implemented the CVS approxima-tion within our LR-ODC-12 program. The re-sulting CVS-ODC-12 algorithm solves the re-duced eigenvalue problem of the form

Ẽzk = M̃zkωk (12)

where Ẽ and M̃ are the reduced Hessianand metric matrices. These matrices areconstructed from the E and M matrices inEqs. (10) and (11) by selecting the matrix ele-ments corresponding to excitations and deexci-tations involving at least one core orbital andsetting all of the other elements to zero. Forexample, out of all matrix elements of the sin-gle excitation block A11 in Eq. (10), the CVS-ODC-12 Hessian Ẽ includes only the AIa,Jb ma-trix elements, where we use the I, J,K, . . . and

4

a, b, c, . . . labels to denote core and virtual or-bitals, respectively, and reserve i, j, k, . . . labelsfor the valence occupied orbitals. Similarly,for the single deexcitation block A∗11, only theAaI,bJ elements are included in the CVS-ODC-12 approximation.

For the matrix blocks involving double ex-citations or deexcitations (e.g., A22 or A

∗22),

we consider two different CVS schemes. Inthe first scheme, termed CVS-ODC-12-a, ma-trix elements corresponding to double excita-tions or deexcitations with only one core la-bel are included (e.g., AIjab,Klab or AIa,Klab),while they are set to zero if a double exci-tation/deexcitation contains two core indices(e.g., AIJab,KLab or AIJab,Klab). This CVSapproximation is similar to the one used inthe ADC methods.5,26–30 In the second CVSscheme, denoted as CVS-ODC-12-b, all ele-ments corresponding to double excitations ordeexcitations with either one or two core la-bels are included (e.g., AIjab,Klab, AIjab,KLab,etc.). This CVS approximation has been pre-viously used in combination with the EOM-CCSD method.41,44 Figure 1 illustrates the twoCVS approximations for the excitation blocksA11, A12, A21, and A22 of the LR-ODC-12Hessian matrix in Eq. (10). Other blocks of thereduced Ẽ and M̃ matrices can be constructedin a similar way. In Section 4.1, we will analyzethe accuracy of the CVS-ODC-12-a and CVS-ODC-12-b approximations by comparing theirresults with those obtained from the LR-ODC-12 method.

3 Computational Details

The CVS-ODC-12-a and CVS-ODC-12-b meth-ods were implemented in a standalone Pythonprogram. To obtain the one- and two-electronintegrals, our program was interfaced withPsi484 and Pyscf.85 Our implementation ofthe CVS eigenvalue problem (12) is based onthe multi-root Davidson algorithm,82,83 whereall vectors and matrix-vector products areconstructed as outlined in Section 2.3. Wevalidated our CVS-ODC-12 implementationsagainst a modified version of the LR-ODC-12

program,45 where the CVS approximation wasintroduced using the projection technique de-scribed by Coriani and Koch.41

In all of our computations, all electronswere correlated. For all systems, except ethy-lene and formic acid, we used the doubly-augmented core-valence d-aug-cc-pCVTZ basisset,86 where the second set of diffuse functions(d-) was included only for s- and p-orbitals. Werefer to this modified basis set as d(s,p)-aug-cc-pCVTZ. In a study of ethylene (C2H4), thed(s,p)-aug-cc-pCVTZ basis was used for car-bon atoms, while the aug-cc-pVTZ basis wasused for the hydrogen atoms. For formic acid(HCO2H), we used the d(s,p)-aug-cc-pCVTZbasis for the carbon and oxygen atoms and theaug-cc-pVDZ basis for the hydrogens.

The CVS-ODC-12 X-ray absorption spectrawere visualized by plotting the spectral function

T (ω) = − 1π

Im

[∑k

|〈Ψ|V̂ |Ψk〉|2

ω − ωk + iη

](13)

computed for a range of frequencies ω, where ωkare the CVS-ODC-12 excitation energies fromEq. (12), η is a small imaginary broadening,and the matrix elements |〈Ψ|V̂ |Ψk〉|2 are ob-tained according to Eq. (9). The simulatedspectra were compared to experimental spectrathat were digitized using the WebPlotDigitizerprogram.87 Electronic transitions were assignedbased on the spin and spatial symmetry of exci-tations, as well as the contributions to the gen-eralized eigenvectors zk in Eq. (12).

4 Results

4.1 Accuracy of the CVS Ap-proximations

We begin by comparing the accuracy of theCVS-ODC-12-a and CVS-ODC-12-b approxi-mations described in Section 2.3. Table 1 showscore excitation energies and oscillator strengthsof CO and H2O computed using the full LR-ODC-12 method and the two CVS methodswith small basis sets (STO-3G and 6-31G, re-spectively), for which it was possible to com-

5

LR-ODC-12

CVS-ODC-12-a

CVS

(a)

LR-ODC-12CVS-ODC-12-b

CVS

(b)

Figure 1: Illustration of two CVS approximations for the excitation blocks A11, A12, A21, andA22 of the LR-ODC-12 Hessian matrix E (Eq. (10)). Indices I, J,K, . . . and i, j, k, . . . denote coreand valence occupied orbitals, while a, b, c, . . . label virtual orbitals.

Table 1: Core excitation energies (in eV) and oscillator strengths (fosc) for three lowest singlet (Sn)and triplet (Tn) excited states computed using LR-ODC-12, as well as its CVS approximations(CVS-ODC-12-a and CVS-ODC-12-b). Also shown are mean absolute errors (∆MAE), standarddeviations (∆STD), and maximum absolute errors (∆MAX), relative to LR-ODC-12, computed usingdata for 12 lowest-energy states (see Supporting Information).

LR-ODC-12 CVS-ODC-12-a CVS-ODC-12-b

Excitation Energy fosc × 102 Energy fosc × 102 Energy fosc × 102CO (C-edge, STO-3G)

S1 (C1s → π∗) 289.13 8.716 289.16 6.565 289.11 8.206S2 295.55 0.004 295.54 0.012 295.54 0.014S3 295.84 295.84 295.84T1 (C1s → π∗) 288.14 288.16 288.14T2 294.17 294.15 294.15T3 294.77 294.76 294.75

CO (O-edge, STO-3G)S1 (O1s → π∗) 543.43 4.402 543.47 4.275 543.43 4.275S2 550.03 550.04 550.03S3 553.30 553.31 553.30T1 (O1s → π∗) 542.96 542.99 542.96T2 546.94 546.94 546.94T3 549.93 549.93 549.93

H2O (6-31G)S1 (O1s → 3s) 541.50 2.452 541.64 2.383 541.49 2.396S2 (O1s → 3p) 543.34 4.945 543.46 4.841 543.33 4.862S3 561.08 10.781 561.21 10.754 561.07 10.770T1 (O1s → 3s) 540.46 540.54 540.45T2 (O1s → 3p) 542.27 542.37 542.26T3 558.89 558.98 558.88∆MAE 0.04 0.078 0.01 0.031∆MAX 0.13 2.151 0.02 0.510∆STD 0.05 0.358 0.01 0.090

pute the LR-ODC-12 core-excited states usinga conventional Davidson algorithm. Out ofthe two CVS approximations, the best agree-ment with LR-ODC-12 is shown by CVS-ODC-

12-b that neglects excitations from valence or-bitals while retaining all excitations with atleast one core label. The superior performanceof CVS-ODC-12-b is reflected by its mean ab-

6

solute error (∆MAE) and standard deviation oferrors (∆STD) that do not exceed 0.01 eV, rel-ative to LR-ODC-12, for a combined set of 36electronic transitions (see Supporting Informa-tion for a complete set of data). The CVS-ODC-12-a approximation, which additionallyneglects double excitations from core to virtualorbitals (Figure 1), exhibits much larger ∆MAEand ∆STD values of 0.04 and 0.05 eV, respec-tively. Although for CO with the STO-3G basisset the CVS-ODC-12-a errors are in the rangeof 0.01-0.04 eV, they increase up to 0.13 eVfor H2O, where a larger 6-31G basis set wasused. These errors continue to grow with thesize of the one-electron basis set. For exam-ple, for transition from the carbon 1s orbital tothe π∗ molecular orbital of CO (C1s → π∗), theCVS-ODC-12-a and CVS-ODC-12-b excitationenergies computed using the aug-cc-pVTZ basisset are 289.1 and 288.1 eV, respectively, indicat-ing a large (∼ 1.0 eV) error of the CVS-ODC-12-a approximation, relative to CVS-ODC-12-b. Similar results are observed when analyzingperformance of the CVS approximations for os-cillator strengths, where CVS-ODC-12-b showsmuch smaller errors compared to CVS-ODC-12-a.

Overall, our results demonstrate that usingthe CVS-ODC-12-b approximation has a verysmall effect on the core-level excitation exci-tation energies and oscillator strengths, whilethe errors of the CVS-ODC-12-a approxima-tion are substantial, especially when large ba-sis sets are used. The small errors introducedby the CVS approximation in CVS-ODC-12-b are consistent with the CVS errors in theCVS-EOM-CCSD method developed by Cori-ani and Koch,41 which employs the same sin-gle and double excitation space when construct-ing the reduced eigenvalue problem. Since theCVS-ODC-12-a and CVS-ODC-12-b approxi-mations only differ in the treatment of doubleexcitations from core to valence orbitals and thenumber of those excitations is usually small, thecomputational cost of the CVS-ODC-12-b ap-proximation is similar to CVS-ODC-12-a. Forthis reason, we will use the CVS-ODC-12-b ap-proximation for our study of core-level excita-tion energies in Section 4.2 and will refer to it

as CVS-ODC-12 henceforth.

4.2 X-Ray Absorption of SmallMolecules

4.2.1 Excitations Energies

In this section, we use the CVS-ODC-12method to compute X-ray absorption spectraof small molecules. Table 2 shows the CVS-ODC-12 results for 36 core-level transitions of10 molecules. For comparison, we also showbest available theoretical results obtained fromvarious formulations of coupled cluster the-ory,34,39,41,43,44 as well as excitation energiesmeasured in the experiment.88–102 For all elec-tronic transitions, the CVS-ODC-12 methodcorrectly reproduces the order of peaks ob-served in the experimental spectra with tran-sitions shifted to higher energies. These sys-tematic shifts are exhibited by many electronicstructure methods9–14,34,43 and are usually at-tributed to basis set incompleteness error andincomplete description of dynamic correlation.As shown in Table 3, the magnitude of the com-puted shifts, given by the mean absolute errors(∆MAE) in the CVS-ODC-12 excitation ener-gies relative to experiment, increases with in-creasing energy of the K-edge transition. Inparticular, for C-, N-, and O-edge excitations,∆MAE increase in the following order: 2.5, 3.5,and 4.8 eV, respectively. Although ∆MAE de-pend on the type of K-edge excitation, the com-puted standard deviations of errors (∆STD) donot change significantly with increasing excita-tion energy and are relatively small (∼ 0.5 eV),indicating the systematic nature of the observedshifts. This is also supported by the percentageof ∆MAE relative to average excitation energyof each edge, which remains relatively constantfor C-, N-, and O-edge transitions (0.87 %, 0.87%, and 0.88 %, respectively).

The CVS-ODC-12 excitation energies are alsoshifted relative to reference theoretical resultsfrom various coupled cluster methods (Table 2).The ∆MAE values for the computed shifts are1.6, 2.5, and 3.9 eV for C-, N-, and O-edgetransitions, indicating that the shifts in the ref-erence coupled cluster excitation energies are

7

Table 2: Core excitation energies (in eV) and oscillator strengths (fosc) for selected K-edge transi-tions computed using CVS-ODC-12. Also shown are best available results from other theoreticalmethods and experiment. Experimental results are from Refs. 88–102.

CVS-ODC-12 Theory (reference) Experiment

Molecule Excitation Energy fosc × 102 Energy fosc × 102 EnergyCH4 C1s → 3s 290.6 2.19 288.2a 287.1

C1s → 3p 291.4 0.51 289.6a 288.0C2H4 C1s → π∗ 287.2 10.50 285.1b 9.86b 284.7

C1s → 3s 290.1 1.27 287.7b 0.91b 287.2C1s → 3p 290.7 4.08 288.4b 2.75b 287.9

C2H2 C1s → π∗ 288.2 9.89 287.1a 285.8C1s → 3s 290.6 1.44 289.5a 287.9C1s → 3p 291.8 0.70 289.8a 288.8

HCN C1s → π∗ 288.5 3.80 287.0c 286.4C1s → 3s 291.3 1.41 289.9c 289.1C1s → 3p 293.3 1.07 290.6

H2CO C1s → π∗ 287.7 6.63 286.8a 285.6C1s → 3s 292.6 1.04 291.9a 290.2C1s → 3p 293.6 3.04 292.3a 291.3

CO C1s → π∗ 288.6 7.14 288.0d 16.56d 287.3C1s → 3s 294.7 0.45 293.0b 0.38b 292.5C1s → 3p 295.7 0.43 294.0b 0.95b 293.4

H2O O1s → 3s 538.4 1.61 534.5d 1.28d 534.0O1s → 3p 540.1 3.55 536.4d 2.62d 535.9

H2CO O1s → π∗ 535.2 5.28 532.3a 530.8O1s → 3s 541.2 0.04 536.7a 535.4O1s → 3p 542.0 0.18 537.9a 536.3

CO O1s → π∗ 538.5 4.41 534.5d 8.13d 534.1O1s → 3s 543.6 0.14 539.7b 0.10b 538.8O1s → 3p 544.9 0.08 540.8b 0.10b 539.8

NH3 N1s → 3s 404.2 0.76 401.2b 0.63b 400.8N1s → 3p 405.9 2.96 402.9b 4.00b 402.5N1s → 3p 406.7 0.94 403.5b 0.63b 403.0

HCN N1s → π∗ 403.1 4.26 400.6c 399.7N1s → 3s 407.2 0.05 402.5

N2 N1s → π∗ 403.7 12.49 402.0e 22.84e 401.0N1s → 3s 409.4 0.56 407.6e 0.45e 406.1N1s → 3p 410.5 0.48 407.0

HF F1s → 4σ∗ 692.2 2.45 689.1e 2.26e 687.4F1s → 3pσ 696.0 0.74 692.8e 0.59e 690.8F1s → 3pπ 696.2 0.64 693.0e 1.09e 691.4

a IH-FSMRCCSD/aug-cc-pCVXZ (X = T, Q) from Ref. 39.b fc-CVS-EOM-CCSD/aug-cc-pCVTZ+Rydberg from Ref. 44.c TD-EOM-CCSD/aug-cc-pVTZ from Ref. 43.d CCSDR(3)/aug-cc-pCVTZ+Rydberg from Ref. 34.e CVS-EOM-CCSD/aug-cc-pCVTZ+Rydberg from Ref. 41.

less sensitive to the type of the K-edge tran-sition than those of CVS-ODC-12. These dif-ferences may originate from the different treat-ment of dynamic correlation and orbital relax-ation effects between CVS-ODC-12 and the ref-erence methods, as well as differences in theone-electron basis sets.

4.2.2 Peak Separations

We now discuss the accuracy of CVS-ODC-12for simulating energy separation between peaksin X-ray absorption spectra. Table 4 showspeak separations computed using CVS-ODC-12 and coupled cluster methods along with ex-perimental results. For most of the electronic

8

Table 3: Mean absolute errors (∆MAE), standard deviations (∆STD), and maximum absolute errors(∆MAX) of the CVS-ODC-12 method computed using excitation energies (in eV) from Table 2,relative to experiment. Experimental results are from Refs. 88–102.

Excitation energies Peak separations

Excitations ∆MAE ∆MAX ∆STD ∆MAE ∆MAX ∆STDC-edge 2.5 3.5 0.5 0.3 1.0 0.3N-edge 3.5 4.7 0.5 0.5 1.3 0.5O-edge 4.8 5.8 0.6 0.5 1.4 0.6All 3.5 5.8 1.1 0.4 1.4 0.4

Table 4: Peak separations (in eV) in the K-edge excitation spectra computed using CVS-ODC-12.Individual excitation energies are shown in Table 2. Also shown are best available results fromother theoretical methods and experiment. Experimental results are from Refs. 88–102.

Molecule Excitation CVS-ODC-12 Theory (reference) ExperimentCH4 C1s → (3p− 3s) 0.8 1.5a 1.0C2H4 C1s → (3s− π∗) 2.8 2.6b 2.6

C1s → (3p− 3s) 0.7 0.7b 0.6C2H2 C1s → (3s− π∗) 2.4 2.5a 2.1

C1s → (3p− 3s) 1.2 0.3a 0.9HCN C1s → (3s− π∗) 2.8 2.9c 2.7

C1s → (3p− 3s) 2.0 1.5H2CO C1s → (3s− π∗) 4.9 5.1a 4.6

C1s → (3p− 3s) 1.0 0.4a 1.1CO C1s → (3s− π∗) 6.1 5.7b 5.1

C1s → (3p− 3s) 1.0 1.0b 1.0H2O O1s → (3p− 3s) 1.7 1.8d 1.9H2CO O1s → (3s− π∗) 6.0 4.5b 4.6

O1s → (3p− 3s) 0.8 1.1b 0.9CO O1s → (3s− π∗) 5.2 4.7b 4.7

O1s → (3p− 3s) 1.3 1.1b 1.0NH3 N1s → (3p− 3s) 1.6 1.7b 1.7

N1s → (3p− 3p) 0.8 0.7b 0.5HCN N1s → (3s− π∗) 4.1 2.8N2 N1s → (3s− π∗) 5.7 5.6e 5.2

N1s → (3p− 3s) 1.1 0.9HF F1s → (3pσ − 4σ∗) 3.8 3.7e 3.4

F1s → (3pπ − 3pσ) 0.2 0.1e 0.6a IH-FSMRCCSD/aug-cc-pCVXZ (X = T, Q) from Ref. 39.b fc-CVS-EOM-CCSD/aug-cc-pCVTZ+Rydberg from Ref. 44.c TD-EOM-CCSD/aug-cc-pVTZ from Ref. 43.d CCSDR(3)/aug-cc-pCVTZ+Rydberg from Ref. 34.e CVS-EOM-CCSD/aug-cc-pCVTZ+Rydberg from Ref. 41.

transitions, the CVS-ODC-12 method showsa good agreement with experiment predictingpeak separations within 0.5 eV from experi-mental values. This is reflected by its ∆MAE of0.4 eV, relative to experiment (Table 3). Con-trary to excitation energies, the errors in theCVS-ODC-12 peak spacings exhibit little de-pendence on the type of the K-edge transition,showing a somewhat smaller ∆MAE for C-edge

excitations (0.3 eV) compared to that of the N-and O-edge transitions (0.5 eV). The accuracyof the CVS-ODC-12 peak separations is on parwith that of the reference coupled cluster meth-ods, which show ∆MAE of 0.3 eV with respect toexperiment. The coupled cluster methods showa similar ∆STD (0.3 eV) and a smaller ∆MAX(0.7 eV), in comparison to 0.4 eV and 1.4 eV ofCVS-ODC-12, respectively.

9

4.2.3 Simulated Spectra

In this section, we compare X-ray absorp-tion spectra computed using CVS-ODC-12 withthose obtained from experiment for three poly-atomic molecules: ethylene (C2H4), formalde-hyde (H2CO), and formic acid (HCO2H). Sincebasis sets employed in our study do not incor-porate Rydberg functions, we only consider re-gions of spectra dominated by core-level transi-tions into the low-lying π∗, 3s, and 3p orbitals.In addition to reporting the computed spec-tra, we provide the spectral data for all threemolecules in the Supporting Information.

Figure 2 shows the C-edge spectra of ethylenecomputed using CVS-ODC-12 for two broad-ening parameters along with an experimentalspectrum from Ref. 102. The simulated spec-tra were shifted by −2.57 eV to align the po-sition of the first peak with the one in the ex-perimental spectrum. The shifted CVS-ODC-12 spectra show a good agreement with exper-iment, reproducing the separation and relativeintensity of the C1s → π∗, 3s, and 3p transi-tions. The CVS-ODC-12 method overestimatesthe relative positions of the 3s and 3p peaks by∼ 0.2 eV, which is consistent with its ∆MAE of0.3 eV from Table 3.

Figure 3 reports the computed C-edge and O-edge spectra of formaldehyde. Aligning posi-tions of the C1s → π∗ and O1s → π∗ peaks withthose in the experimental spectrum93 requiresshifting the CVS-ODC-12 spectra by −2.00 and−4.524 eV, respectively, consistent with ∆MAEof 2.5 and 4.8 eV for excitation energies re-ported in Table 3. After the shift, the simu-lated C-edge spectrum reproduces position ofthe 3s and 3p peaks within 0.3-0.4 eV from ex-periment. The relative intensities of these tran-sitions also agree well with those observed inthe experimental spectrum. For oxygen edge,the agreement between CVS-ODC-12 and ex-periment is worse: the relative energies of the3s and 3p transitions are overestimated by ∼1.3 eV. In addition, in the simulated spectrumthe relative intensity of the 3s peak is signifi-cantly lower than the one obtained in the exper-iment. The energy spacing between the 3s and3p transitions (0.8 eV) is in a good agreement

with that from the experimental spectrum (0.9eV).

Finally, we consider formic acid as an exam-ple of a molecule with a more complicated X-ray absorption spectra. The computed CVS-ODC-12 spectra for carbon and oxygen edgeare shown in Figure 4. Aligning simulated andexperimental C-edge spectra requires a shift of−1.67 eV, while a large shift of −7.99 eV isneeded for the oxygen edge. For carbon edge,the CVS-ODC-12 method shows a reasonableagreement with experiment for the 3s transi-tion with an error of ∼ 0.5 eV, while a largererror of ∼ 0.8 eV is observed for the third peak(3p). When considering the oxygen edge, theexperimental spectrum103 shows two broad sig-nals at 532.1 and 535.3 eV attributed to theO1s → π∗ and O1s → 3s transitions, respec-tively. The CVS-ODC-12 O-edge spectrum re-veals that the second signal originates from sev-eral closely spaced peaks corresponding to exci-tations into 3s orbitals of all carbon and oxygenatoms, with significant contributions of excita-tions to 3p orbitals. When using a large broad-ening parameter, these transitions form a broadsignal with a maximum at 535.4 eV, in a verygood agreement with experimental spectrum.

5 Conclusions

In this work, we have presented a new ap-proach for simulations of X-ray absorption spec-tra based on linear-response density cumulanttheory (LR-DCT). Our new method combinesthe LR-ODC-12 formulation of LR-DCT withcore-valence separation approximation (CVS)that allows to efficiently access high-energycore-excited states. We considered two CVSapproximations of LR-ODC-12 (CVS-ODC-12)that incorporate different types of excitationsfrom core to virtual orbitals and comparedtheir results with core-level excitation energiesobtained from the full LR-ODC-12 method.Our results demonstrated that including dou-ble core-virtual excitations is crucial to main-tain high accuracy of the CVS approximationfor K-edge excitation energies, especially whenusing large one-electron basis sets.

10

CVS-ODC-12 (broad. = 0.05 a.u., shift = -2.57 eV)

Inte

nsity

(arb

. uni

ts)

CVS-ODC-12 (broad. = 0.4 a.u., shift = -2.57 eV)

284 285 286 287 288 289Excitation energy, eV

Experiment

π*

π*

3s3p

3s3p

3p3s

π*

Figure 2: C-edge X-ray absorption spectrum of ethylene computed using CVS-ODC-12. Resultsare shown for two broadening parameters (see Section 3 for details) and are compared toexperimental spectrum from Ref. 102. The CVS-ODC-12 spectrum was shifted by −2.57 eV toreproduce position of the C1s → π∗ peak in the experimental spectrum. See Table 2 and theSupporting Information for the CVS-ODC-12 excitation energies and oscillator strengths.

CVS-ODC-12 (broad. = 0.05 a.u., shift = -2.00 eV)

Inte

nsity

(arb

. uni

ts)

CVS-ODC-12 (broad. = 0.2 a.u., shift = -2.00 eV)

285 286 287 288 289 290 291 292 293Excitation energy, eV

Experiment

π*

3s

3p

π*

π*

3s

3p

3p

3p

3s3p

3p

(a)

CVS-ODC-12 (broad. = 0.05 a.u., shift = -4.524 eV)In

tens

ity (a

rb. u

nits

)

CVS-ODC-12 (broad. = 0.2 a.u., shift = -4.524 eV)

530 531 532 533 534 535 536 537 538 539Excitation energy, eV

Experiment

π*

3s 3p

π*

3s 3p

π*

3s 3p

(b)

Figure 3: C-edge (3a) and O-edge (3b) X-ray absorption spectra of formaldehyde computed usingCVS-ODC-12. Results are shown for two broadening parameters (see Section 3 for details) andare compared to experimental spectra from Ref. 93. The CVS-ODC-12 spectra were shifted toreproduce positions of the C1s → π∗ and O1s → π∗ peaks in the experimental spectra. See Table 2and the Supporting Information for the CVS-ODC-12 excitation energies and oscillator strengths.

We have used the CVS-ODC-12 method tocompute X-ray absorption spectra of severalsmall molecules and compared them to spectraobtained from experiment. The CVS-ODC-12method shows a good agreement with experi-ment for spacings between transitions and theirrelative intensities, but the computed spectraare systematically shifted to higher energies.The magnitude of these shifts increases with

increasing energy of the K-edge transition: forC-, N-, and O-edge transitions the CVS-ODC-12 excitation energies are shifted by ∼ 2.5, 3.5,and 4.8 eV, on average. However, the relativedistances between transitions depend much lesson the K-edge excitation energy, reproducingexperimental peak spacings within 0.3-0.5 eVfor most of the computed transitions. We havecompared the CVS-ODC-12 results with X-

11

CVS-ODC-12 (broad. = 0.05 a.u., shift = -1.67 eV)

Inte

nsity

(arb

. uni

ts)

CVS-ODC-12 (broad. = 0.3 a.u., shift = -1.67 eV)

287 288 289 290 291 292 293 294 295Excitation energy, eV

Experiment

π*

3s

π*

3s

π*

3s 3p

3p

3p

(a)

CVS-ODC-12 (broad. = 0.05 a.u., shift = -7.99 eV)

Inte

nsity

(arb

. uni

ts)

CVS-ODC-12 (broad. = 0.6 a.u., shift = -7.99 eV)

531 532 533 534 535 536 537 538Excitation energy, eV

Experimentπ*

3s

π*

3s

π*

3s

(b)

Figure 4: C-edge (4a) and O-edge (4b) X-ray absorption spectra of formic acid computed usingCVS-ODC-12. Results are shown for two broadening parameters (see Section 3 for details) andare compared to experimental spectra from Ref. 103. The CVS-ODC-12 spectra were shifted toreproduce positions of the C1s → π∗ and O1s → π∗ peaks in the experimental spectra. See theSupporting Information for the CVS-ODC-12 excitation energies and oscillator strengths.

ray absorption spectra computed using differ-ent formulations of excited-state coupled clus-ter theory with single and double excitations(CCSD). For peak spacings and intensities, theCVS-ODC-12 and CCSD methods show sim-ilar performance, but the CCSD spectra ex-hibit smaller shifts, particularly for N- and O-edge transitions. An important advantage ofCVS-ODC-12 over the CCSD methods is thatthe former is based on the diagonalization of aHermitian matrix, which enables efficient com-putation of transition intensities and guaran-tees that the resulting excitation energies havereal values, provided that the matrix is positivesemidefinite. Moreover, the CVS-ODC-12 andCCSD methods have the same O(N6) compu-tational scaling with the size of the one-electronbasis set N .

Overall, our results suggest that CVS-ODC-12 is a useful method for qualitative and semi-quantitative predictions of X-ray absorptionspectra of molecules. To improve the qualityof the CVS-ODC-12 excitation energies further,we plan to combine this method with frozen-core approximation, as suggested in the recentwork by Lopez et al.44 within the framework ofCVS-approximated EOM-CCSD. We also planto develop an efficient implementation of CVS-

ODC-12 that incorporates treatment of rela-tivistic effects and benchmark its performancefor open-shell molecules and transition metalcompounds.

Acknowledgement

This work was supported by start-up funds pro-vided by the Ohio State University.

Supporting Information Avail-

able

The following files are available free of charge.Table comparing the CVS-ODC-12 approxi-

mations with the full LR-ODC-12 method andtables with spectral data for ethylene, formalde-hyde, and formic acid.

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Graphical TOC Entry

X-ray absorption spectrum offormic acid (O-edge):

CVS

LR-ODC-12CVS-ODC-12

Linear-response density cumulant theory:

531 532 533 534 535 536 537 538

CVS-ODC-12

Experiment

Excitation energy, eV

20

Abstract1 Introduction2 Theory2.1 Density Cumulant Theory (DCT)2.2 Linear-Response DCT (LR-DCT)2.3 Core-Valence-Separated LR-ODC-12 (CVS-ODC-12)

3 Computational Details4 Results4.1 Accuracy of the CVS Approximations4.2 X-Ray Absorption of Small Molecules4.2.1 Excitations Energies4.2.2 Peak Separations4.2.3 Simulated Spectra

5 ConclusionsAcknowledgementSupporting Information AvailableReferences