Simulating X-ray absorption spectra with linear-response density cumulant theory
SSimulating 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: [email protected]
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. , , , 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. 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.
Theoretical simulations of X-ray absorptionplay a critical role in interpretation of theNEXAS spectra. 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, real-time, and orthogonality-constrained density functional theory, con-figuration interaction, algebraic diagram-1 a r X i v : . [ phy s i c s . c h e m - ph ] F e b atic construction (ADC), as well aslinear-response (LR-) and equation-of-motion(EOM-) coupled cluster (CC) theories. 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), 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), 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, multilevel, and core-valence-separated 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) for simulat-ing the NEXAS spectra of molecules. Althoughthe origin of LR-DCT is in reduced densitymatrix theory, it has a close connectionwith LR-CC methods, such as linear-responseformulations of linearized, unitary, and varia-tional CC theory. 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. 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, origi-nally developed in the framework of ADC the-ory and later extended to other meth-ods.
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.
We start with a short overview of density cu-mulant theory (DCT).
The exact electronicenergy of a stationary state | Ψ (cid:105) can be ex-pressed in the form E = (cid:104) Ψ | ˆ H | Ψ (cid:105) = (cid:88) pq h qp γ pq + 14 (cid:88) pqrs g rspq γ pqrs (1)where the one- and antisymmetrized two-electron integrals ( h qp and g rspq ) are traced withthe reduced one- and two-body density matri-ces ( γ pq and γ pqrs ) over all spin-orbitals in a finiteone-electron basis set. Starting with Eq. (1),DCT expresses γ pq and γ pqrs in terms of the fullyconnected contribution to γ pqrs called the two-body density cumulant ( λ pqrs ). For γ pqrs ,this is achieved using a cumulant expansion γ pqrs = γ pr γ qs − γ ps γ qr + λ pqrs (2)where the first two terms represent a discon-nected antisymmetrized product of the one-body density matrices. To determine γ pq from λ pqrs , the following non-linear relationship isused (cid:88) r γ pr γ rq − γ pq = (cid:88) r λ prqr (3)2here 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 λ pqrs yields 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 (cid:105) and expressing density cumu-lant as λ pqrs = (cid:104) Ψ | a pqrs | Ψ (cid:105) c (4)where a pqrs ≡ a † p a † q a s a r is a two-body second-quantized operator and the subscript c indicatesthat only fully connected terms are retained.The most commonly used parametrization of λ pqrs , denoted as ODC-12, consists of approx-imating Eq. (4) using a two-body unitary trans-formation of | Ψ (cid:105) truncated at the second orderin perturbation theory λ pqrs ≈ (cid:104) Φ | e − ( ˆ T − ˆ T † ) a pqrs e ˆ T − ˆ T † | Φ (cid:105) c ≈ (cid:104) Φ | a pqrs | Φ (cid:105) c + (cid:104) Φ | [ a pqrs , ˆ T − ˆ T † ] | Φ (cid:105) c + 12 (cid:104) Φ | [[ a pqrs , ˆ T − ˆ T † ] , ˆ T − ˆ T † ] | Φ (cid:105) c (5)where ˆ T is the double excitation operatorwith respect to the reference determinant | Φ (cid:105) ,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 e ˆ T − ˆ T † . The parameters of the ˆ T and ˆ T operators determined from the stationarity con-ditions are used to compute the ODC-12 energy. 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. In linear-response DCT (LR-DCT), we consider the behavior of anelectronic system under a time-dependent per-turbation ˆ
V f ( t ), which can be described usingthe time-dependent quasi-energy function Q ( t ) = (cid:104) Ψ( t ) | ˆ H + ˆ V f ( t ) − i ∂∂t | Ψ( t ) (cid:105) (6)Here, | Ψ( t ) (cid:105) is the so-called “phase-isolated”wavefunction, which reduces to the usual time-independent wavefunction | Ψ (cid:105) 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. Such periodicity impliesthat the amplitude f ( t ) can be written in aFourier series f ( t ) = (cid:88) 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 ) (cid:105) . We refer interested read-ers to our previous publication 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 Ez k = Mz k ω k (8)In Eq. (8), ω k are the excitation energies, E is the LR-DCT Hessian matrix that con-tains second derivatives of the electronic energy {(cid:104) Ψ( t ) | ˆ H | Ψ( t ) (cid:105)} with respect to parameters ofthe ˆ T and ˆ T operators, and M is the metricmatrix that originates from second derivativesof the time-derivative overlap {(cid:104) Ψ( t ) | i ˙Ψ( t ) (cid:105)} .Importantly, the LR-DCT Hessian matrix E isHermitian, which ensures that the excitationenergies ω k have real values, provided that theHessian is positive semidefinite. The general-ized eigenvectors z k can be used to determine3he oscillator strength for each transition f osc ( ω k ) = 23 ω k |(cid:104) Ψ | ˆ V | Ψ k (cid:105)| = 23 ω k | z † k v (cid:48) | z † k Mz k (9)where (cid:104) Ψ | ˆ V | Ψ k (cid:105) = (cid:104) Ψ | ˆ µ | Ψ k (cid:105) is the transitiondipole moment matrix element and v (cid:48) is the so-called property gradient vector. In the linear-response formulation of theODC-12 method (LR-ODC-12), the E and M matrices in Eq. (8) have the following form: E = A A B B A A B B B ∗ B ∗ A ∗ A ∗ B ∗ B ∗ A ∗ A ∗ (10) M = S − S ∗
00 0 0 − (11)where the A and B ( A and B ) ma-trices contain second derivatives of the DCTenergy with respect to parameters of the ˆ T ( ˆ T ) operators, whereas A and B repre-sent mixed second derivatives. Each blockof E and M describe coupling between elec-tronic excitations or deexcitations of differentranks. Specifically, the A , A , and A blocks 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 B block couples single excitationswith single deexcitations, whereas B and B couple single-double and double-double excita-tion/deexcitation pairs. Finally, A ∗ , A ∗ , and A ∗ describe interaction of single and doubledeexcitations. The solution of the LR-ODC-12eigenvalue problem (8) has O ( O V ) computa-tional scaling, where O and V are the numbersof occupied and virtual orbitals, respectively. 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.
This iterative method proceeds byforming an expansion space for the generalizedeigenvectors z k starting with an initial (guess)set of unit trial vectors and progressively grow-ing this space until the lowest N root 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. 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 orbitals 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 ˜Ez k = ˜Mz k ω k (12)where ˜E and ˜M 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 A in Eq. (10), the CVS-ODC-12 Hessian ˜E includes only the A Ia,Jb ma-trix elements, where we use the
I, J, K, . . . and4 , 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 ∗ , only the A aI,bJ elements are included in the CVS-ODC-12 approximation.For the matrix blocks involving double ex-citations or deexcitations (e.g., A or A ∗ ),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., A Ijab,Klab or A Ia,Klab ),while they are set to zero if a double exci-tation/deexcitation contains two core indices(e.g., A IJab,KLab or A IJab,Klab ). This CVSapproximation is similar to the one used inthe ADC methods.
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., A Ijab,Klab , A Ijab,KLab ,etc.). This CVS approximation has been pre-viously used in combination with the EOM-CCSD method.
Figure 1 illustrates the twoCVS approximations for the excitation blocks A , A , A , and A of the LR-ODC-12Hessian matrix in Eq. (10). Other blocks of thereduced ˜E and ˜M 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. 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 with
Psi4 and Pyscf . Our implementation ofthe CVS eigenvalue problem (12) is based onthe multi-root Davidson algorithm, 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, where the CVS approximation wasintroduced using the projection technique de-scribed by Coriani and Koch. 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, 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 (C H ), 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(HCO H), 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 ) = − π Im (cid:34)(cid:88) k |(cid:104) Ψ | ˆ V | Ψ k (cid:105)| ω − ω k + iη (cid:35) (13)computed for a range of frequencies ω , where ω k are the CVS-ODC-12 excitation energies fromEq. (12), η is a small imaginary broadening,and the matrix elements |(cid:104) Ψ | ˆ V | Ψ k (cid:105)| are ob-tained according to Eq. (9). The simulatedspectra were compared to experimental spectrathat were digitized using the WebPlotDigitizerprogram. Electronic transitions were assignedbased on the spin and spatial symmetry of exci-tations, as well as the contributions to the gen-eralized eigenvectors z k in Eq. (12). 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 H O 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
R-ODC-12 CVS-ODC-12-aCVS (a)
LR-ODC-12 CVS-ODC-12-bCVS (b)
Figure 1: Illustration of two CVS approximations for the excitation blocks A , A , A , and A 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 ( f osc ) for three lowest singlet (S n )and triplet (T n ) 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-bExcitation Energy f osc × Energy f osc × Energy f osc × CO (C-edge, STO-3G)S (C s → π ∗ ) 289.13 8.716 289.16 6.565 289.11 8.206S (C s → π ∗ ) 288.14 288.16 288.14T (O s → π ∗ ) 543.43 4.402 543.47 4.275 543.43 4.275S (O s → π ∗ ) 542.96 542.99 542.96T O (6-31G)S (O s → s ) 541.50 2.452 541.64 2.383 541.49 2.396S (O s → p ) 543.34 4.945 543.46 4.841 543.33 4.862S (O s → s ) 540.46 540.54 540.45T (O s → p ) 542.27 542.37 542.26T MAE
MAX
STD 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-6olute 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 ∆
MAE and ∆
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 H O, 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 1 s orbital tothe π ∗ molecular orbital of CO (C s → 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 ( ∼ 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. 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, as well as excitation energiesmeasured in the experiment.
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 methods 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 ( ∼ 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 are7able 2: Core excitation energies (in eV) and oscillator strengths ( f osc ) 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) ExperimentMolecule Excitation Energy f osc × Energy f osc × EnergyCH C s → s a s → p a H C s → π ∗ b b s → s b b s → p b b H C s → π ∗ a s → s a s → p a s → π ∗ c s → s c s → p CO C s → π ∗ a s → s a s → p a s → π ∗ d d s → s b b s → p b b O O s → s d d s → p d d CO O s → π ∗ a s → s a s → p a s → π ∗ d d s → s b b s → p b b N s → s b b s → p b b s → p b b s → π ∗ c s → s N s → π ∗ e e s → s e e s → p s → σ ∗ e e s → p σ e e s → p π e e 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.
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 electronic8able 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 separationsExcitations ∆
MAE ∆ MAX ∆ STD ∆ MAE ∆ MAX ∆ STD
C-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) ExperimentCH C s → (3 p − s ) 0.8 1.5 a H C s → (3 s − π ∗ ) 2.8 2.6 b s → (3 p − s ) 0.7 0.7 b H C s → (3 s − π ∗ ) 2.4 2.5 a s → (3 p − s ) 1.2 0.3 a s → (3 s − π ∗ ) 2.8 2.9 c s → (3 p − s ) 2.0 1.5H CO C s → (3 s − π ∗ ) 4.9 5.1 a s → (3 p − s ) 1.0 0.4 a s → (3 s − π ∗ ) 6.1 5.7 b s → (3 p − s ) 1.0 1.0 b O O s → (3 p − s ) 1.7 1.8 d CO O s → (3 s − π ∗ ) 6.0 4.5 b s → (3 p − s ) 0.8 1.1 b s → (3 s − π ∗ ) 5.2 4.7 b s → (3 p − s ) 1.3 1.1 b N s → (3 p − s ) 1.6 1.7 b s → (3 p − p ) 0.8 0.7 b s → (3 s − π ∗ ) 4.1 2.8N N s → (3 s − π ∗ ) 5.7 5.6 e s → (3 p − s ) 1.1 0.9HF F s → (3 p σ − σ ∗ ) 3.8 3.7 e s → (3 p π − p σ ) 0.2 0.1 e 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. 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 .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 (C H ), formalde-hyde (H CO), and formic acid (HCO H). 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 ∗ , 3 s , and 3 p 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 − s → π ∗ , 3 s , and 3 p transi-tions. The CVS-ODC-12 method overestimatesthe relative positions of the 3 s and 3 p peaks by ∼ 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 C s → π ∗ and O s → π ∗ peaks withthose in the experimental spectrum requiresshifting the CVS-ODC-12 spectra by − − MAE of 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 3 s and 3 p 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 the3 s and 3 p transitions are overestimated by ∼ s peak is signifi-cantly lower than the one obtained in the exper-iment. The energy spacing between the 3 s and3 p 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 − − s transi-tion with an error of ∼ ∼ p ). When considering the oxygen edge, theexperimental spectrum shows two broad sig-nals at 532.1 and 535.3 eV attributed to theO s → π ∗ and O s → s 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 3 s orbitals of all carbon and oxygenatoms, with significant contributions of excita-tions to 3 p 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. 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
VS-ODC-12 (broad. = 0.05 a.u., shift = -2.57 eV) I n t en s i t y ( a r b . un i t s ) CVS-ODC-12 (broad. = 0.4 a.u., shift = -2.57 eV)
284 285 286 287 288 289Excitation energy, eV
Experiment π*π* 3s 3p3s 3p3p3sπ*
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 − s → 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) I n t en s i t y ( a r b . un i t s ) 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 3pπ*π* 3s 3p 3p3p3s 3p 3p (a)
CVS-ODC-12 (broad. = 0.05 a.u., shift = -4.524 eV) I n t en s i t y ( a r b . un i t s ) 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 3pπ* 3s 3pπ* 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 C s → π ∗ and O s → 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 ∼ VS-ODC-12 (broad. = 0.05 a.u., shift = -1.67 eV) I n t en s i t y ( a r b . un i t s ) 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* 3sπ* 3sπ* 3s 3p 3p3p (a)
CVS-ODC-12 (broad. = 0.05 a.u., shift = -7.99 eV) I n t en s i t y ( a r b . un i t s ) CVS-ODC-12 (broad. = 0.6 a.u., shift = -7.99 eV)
531 532 533 534 535 536 537 538Excitation energy, eV
Experiment π* 3sπ* 3sπ* 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 C s → π ∗ and O s → 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 ( N ) 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. 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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CVSLR-ODC-12 CVS-ODC-12
Linear-response density cumulant theory:
531 532 533 534 535 536 537 538