Cherry-picking resolvents: A general Cite as: J. Chem....

J. Chem. Phys. 153, 141104 (2020); https://doi.org/10.1063/5.0020843 153, 141104

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Cherry-picking resolvents: A generalstrategy for convergent coupled-clusterdamped response calculations of core-levelspectraCite as: J. Chem. Phys. 153, 141104 (2020); https://doi.org/10.1063/5.0020843Submitted: 04 July 2020 . Accepted: 10 September 2020 . Published Online: 09 October 2020

Kaushik D. Nanda , and Anna I. Krylov

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Cherry-picking resolvents: A general strategyfor convergent coupled-cluster damped responsecalculations of core-level spectra

Cite as: J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843Submitted: 4 July 2020 • Accepted: 10 September 2020 •Published Online: 9 October 2020

Kaushik D. Nandaa) and Anna I. Krylova)

AFFILIATIONSDepartment of Chemistry, University of Southern California, Los Angeles, California 90089-0482, USA

a)Authors to whom correspondence should be addressed: [email protected] and [email protected]

ABSTRACTDamped linear response calculations within the equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) framework usuallydiverge in the x-ray regime. This divergent behavior stems from the valence ionization continuum in which the x-ray response states areembedded. Here, we introduce a general strategy for removing the continuum from the response manifold while preserving important spectralproperties of the model Hamiltonian. The strategy is based on decoupling the core and valence Fock spaces using the core–valence separation(CVS) scheme combined with separate (approximate) treatment of the core and valence resolvents. We illustrate this approach with thecalculations of resonant inelastic x-ray scattering (RIXS) spectra of benzene and para-nitroaniline using EOM-CCSD wave functions andseveral choices of resolvents, which differ in their treatment of the valence manifold. The method shows robust convergence and extends thepreviously introduced CVS-EOM-CCSD RIXS scheme to systems for which valence contributions to the total cross section are important,such as the push–pull chromophores with charge-transfer states.

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Equation-of-motion coupled-cluster (EOM-CC) theory1–6 pro-vides a robust single-reference framework for computing multipleelectronic states. EOM-CC affords a balanced description of statesof different characters and a systematic improvement of results byincremental improvements in treating the electron correlation. TheEOM-CC hierarchy of approximations is based on a standard hier-archy of CC models for the ground state, such as CC singles (CCS),approximate CC singles and doubles7 (CC2), CC with singles anddoubles2,8–10 (CCSD), CC with singles, doubles, and triples (CC311

and CCSDT12–14), and so on. The EOM-CC formalism naturallyextends to state and transition properties. Together, these featuresmake EOM-CC an ideal framework for modeling spectroscopy.EOM-CC can be used to compute solvatochromic shifts,15 transi-tion dipole moments,2 spin–orbit16–19 and non-adiabatic20–22 cou-plings, photoionization cross sections,23,24 and higher-order proper-ties25 such as two-photon absorption cross sections26–29 and staticand dynamic polarizabilities.30–33

The EOM-CC framework is being vigorously extended to thex-ray regime for modeling x-ray absorption (XAS), photoionization,and emission spectra,34–39 as well as multi-photon phenomena suchas resonant inelastic x-ray scattering40–44 (RIXS) (Fig. 1). The suc-cessful extensions of EOM-CC to the x-ray domain exploit the core–valence separation (CVS) scheme,45 which effectively addresses thekey challenge in the theoretical treatment of core-level states: theirresonance nature due to the coupling with the valence ionizationcontinuum. Being embedded in the continuum, core-level states,strictly speaking, cannot be treated by methods developed for iso-lated bound states with L2-integrable wave functions.46–48 Practi-cally, the coupling with the continuum leads to erratic and oftendivergent behavior of the solvers, a lack of systematic conver-gence of results with the basis-set increase, and often unphysicalsolutions.48,49

The CVS scheme allows one to separate the continuum ofvalence states from the core-level states through a deliberate pruning

J. Chem. Phys. 153, 141104 (2020); doi: 10.1063/5.0020843 153, 141104-1

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FIG. 1. In the coherent RIXS process, an x-ray photon of energy ω1 (resonantwith a core-excited state) is absorbed and another x-ray photon of energy ω2is emitted. The difference between the incoming and outgoing photon energiesequals the excitation energy of the final valence state f relative to the initial stateg. The process is often described in terms of a transition via a virtual state (blackdashed line), which represents collective contributions from all electronic statesof the system, including valence bound states (solid black), valence resonances(green), core-excited states (blue), and valence continuum states (ultrafine graydashes).

of the Fock space, i.e., by removing configurations that can couplecore-excited (or core-ionized) configurations with the valence con-tinuum (see the supplementary material for the analysis of differentdecay channels). CVS can be described as a diabatization proce-dure that separates the bound part of the resonance from the con-tinuum, akin to the Feshbach–Fano treatment of resonances.50–52

In contrast to other methods46–48 for describing resonances, thisapproach, in which the continuum is simply projected out, does notinvolve state-specific parameterization (such as, for example, tuningthe strength of complex absorbing potential for each state) and offersan additional benefit of eliminating the need to compute lower-lyingelectronic states.

Although separation of the full Hilbert space into bound andcontinuum parts is not well defined in general, it is possible inthe case of core-level states because they are Feshbach resonances,which can only decay by a two-electron process in which one elec-tron fills the core hole, liberating sufficient energy to eject anotherelectron.38,49 This special feature allows one to separate the contin-uum decay channels from the core-level states in the Fock space,by removing the configurations that couple the resonances withthe continuum. Such pruning of the Fock space is equivalent toomitting the couplings between the core and valence excited (orionized) determinants from the model EOM-CC Hamiltonian. Theeigenstates of this reduced EOM-CC Hamiltonian are either purelyvalence or purely core excited (or ionized). The pure core-excited(or ionized) states become formally bound because of the decou-pling from the valence determinants forming the continuum. Con-sequently, diagonalization of the core block of this CVS-EOM-CC

Hamiltonian yields core-excited (or core-ionized) states without anyconvergence issues.

Initially used to describe transition energies and strengthsof core-level states, the CVS scheme was recently extended intothe response domain41–43 to enable calculations of RIXS transi-tion moments within damped linear response theory53–56 and theEOM-CCSD method for excitation energies (EOM-EE-CCSD). Inthis approach, the response equations are solved in the truncatedFock space spanning the singly and doubly excited determinantsin which at least one orbital belongs to the core,41,42 in the samemanner as done in CVS-EOM-CCSD calculations of core-excitedor core-ionized states. Thus, all valence excited states are excludedfrom the response manifold, which can be justified by the resonantnature of the RIXS process. This truncation of the response mani-fold works well in many situations, but is expected to break downwhen off-resonance contributions to the RIXS cross section fromthe valence states become important. At least one class of systemswhere this happens is push–pull chromophores featuring low-lyingcharge-transfer states.43

Here, we address this limitation of the previous formula-tion of RIXS theory within the CVS-EOM-CCSD framework andpresent a general strategy for including valence contributions tothe RIXS signal while preserving smooth convergence of the RIXSresponse states. This strategy can also be applied to the modelingof other multi-photon x-ray processes, such as x-ray two-photonabsorption. By using this approach, we provide a quantitative illus-tration of the significance of the valence contributions to RIXSspectra.

The derivation of the equations for RIXS transition momentsbetween the initial (g) and final (f ) states starts from the Kramers–Heisenberg–Dirac formula,57,58 which translates to the followingsum-over-states (SOS) expressions40–44 within the EOM-EE-CCSDdamped response theory:

M f←gxy (ω1x + iε,−ω2y − iε)

= −∑n(⟨Φ0L f ∣μ̄y∣RnΦ0⟩⟨Φ0Ln∣μ̄x∣RgΦ0⟩

En − Eg − ω1x − iε

+⟨Φ0L f ∣μ̄x∣RnΦ0⟩⟨Φ0Ln∣μ̄y∣RgΦ0⟩

En − Eg + ω2y + iε) (1)

and

Mg←fxy (−ω1x + iε,ω2y − iε)

= −∑n(⟨Φ0L g ∣μ̄x∣RnΦ0⟩⟨Φ0Ln∣μ̄y∣RfΦ0⟩

En − Eg − ω1x + iε

+⟨Φ0L g ∣μ̄y∣RnΦ0⟩⟨Φ0Ln∣μ̄x∣RfΦ0⟩

En − Eg + ω2y − iε). (2)

Here, T is an excitation operator containing the CCSD ampli-tudes, and μ̄ = e−TμeT is the similarity-transformed dipole moment.Ln and Rn are the EOM-CCSD left and right excitation opera-tors, respectively,59 for state n with energy En. Their amplitudesand eigenenergies are found by diagonalizing the EOM-CCSDsimilarity-transformed Hamiltonian H = e−THeT in the appropriatesector of the Fock space (i.e., singly and doubly excited determinants

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in EOM-EE-CCSD) as follows:

HRn = EnRn and LnH = LnEn. (3)

iε is the phenomenological damping (or inverse lifetime) term fromdamped response theory. ω1x and ω2y are the x-polarized absorbedand y-polarized emitted energies, respectively, satisfying the RIXSresonance condition,

ω1 − ω2 = Ef − Eg . (4)

We drop the Cartesian indices of the photon energies for brevity.The Fock space of EOM-EE-CCSD comprises the Slater deter-

minants Φρ, where ρ ∈ {reference (0), CV, OV, COVV, CCVV,OOVV} andC,O, andV denote the core occupied, valence occupied,and unoccupied orbitals, respectively. Upon inserting the identityoperator (1 =∑ρ|Φρ⟩⟨Φρ|) in Eqs. (1) and (2), we obtain

Mf←gxy (ω1 + iε,−ω2 − iε)

= −∑ρ(⟨D̃ fy ∣Φρ⟩⟨Φρ∣X

gx,ω1+iε⟩ + ⟨D̃

fx ∣Φρ⟩⟨Φρ∣X

gy,−ω2−iε⟩) (5)

and

Mg←fxy (−ω1 + iε,ω2 − iε)

= −∑ρ(⟨D̃ gx ∣Φρ⟩⟨Φρ∣X

fy,ω1−iε⟩ + ⟨D̃

gy ∣Φρ⟩⟨Φρ∣X

fx,−ω2+iε⟩), (6)

where the amplitudes of the intermediate D̃ ky and the first-orderresponse wave function X kx,ω1+iε for state k are given by

26,41

⟨D̃ ky ∣Φρ⟩ = ⟨Φ0L k∣μ̄y∣Φρ⟩ (7)

and

⟨Φρ∣X kx,ω+iε⟩ =∑n⟨Φρ∣RnΦ0⟩

⟨Φ0Ln∣μ̄x∣RkΦ0⟩En − Eg − (ω + iε)

. (8)

The response wave function in Eq. (8) is complex-valuedand given by a linear combination of all EOM-CCSD states. Theamplitudes of this response wave function, expressed in the basisof Slater determinants, are solutions of the following responseequation:

∑ρ⟨Φν∣H − Eg − (ω + iε)∣Φρ⟩⟨Φρ∣X kx,ω+iε⟩ = ⟨Φν∣D kx ⟩, (9)

where the amplitudes of intermediate D kx are given by

⟨Φν∣D kx ⟩ = ⟨Φν∣μ̄x∣RkΦ0⟩. (10)

Equation (9) is the direct result of using the identity operator1 = ∑n ∣RnΦ0⟩⟨Φ0Ln∣ and the following resolvent:

∑n

⟨Φρ∣RnΦ0⟩⟨Φ0Ln∣Φν⟩En − Eg − ω − iε

= ⟨Φρ∣(H − Eg − ω − iε)−1∣Φν⟩. (11)

In practice, response equations such as Eq. (9) are solvediteratively using standard procedures, e.g., the Direct Inversionin the Iterative Subspace60 (DIIS), which rely on diagonal pre-conditioners. In the x-ray regime, such response equations often

diverge for the following reasons.41,42 The RIXS response wave func-tion in Eq. (8) includes large contributions from high-lying core-excited states that are nearly resonant with the absorbed photon’senergy. These high-lying states are embedded in the valence ion-ization continuum and are Feshbach resonances that are metastablewith respect to electron ejection. Mathematically, these core-excitedstates (and consequently, the response state) are strongly coupled tothe continuum via a subset of doubly excited determinants, {OOVV}(formally, singly excited {OV} determinants can also couple core-excited states to the continuum, but one can show that the respectivematrix elements are expected to be very small; this is discussed in thesupplementary material). This coupling to the continuum leads tothe oscillatory behavior of the leading response amplitudes withlarge magnitudes in the course of the iterative procedure. The prob-lem is exacerbated by the coupling of these doubly excited determi-nants to the valence resonances, which leads to the erratic behaviorof pure valence doubly excited response amplitudes. Moreover, forx-ray energies, the diagonal preconditioner for the valence doublyexcited amplitudes is no longer a good approximation to the valencedoubles–doubles block of H − E0 − (ω + iε), which further cripplesthe convergence.

We note that lower-level theories such as CIS (configurationinteraction singles)61 and TD-DFT (time-dependent density func-tional theory)62–65 do not have valence double excitations in the exci-tation manifold, so that the issue with the convergence of responseequations in the x-ray regime does not arise, simply because thereis no coupling between Feshbach resonances and the valence con-tinuum. Similarly, this issue does not arise for theories such asADC(2)66,67 and CC2 in which the doubly excited configurationsare not coupled with each other41,42 (the doubles–doubles block isdiagonal), so the diagonal preconditioner for the doubles–doublesblock is exact and the effect of double excitations is evaluatedperturbatively rather than iteratively.

The first practical solution to these convergence problems,which stem from the physics of core-level states and plague higher-level many-body approaches, was introduced in Refs. 41 and 42 andimplemented within the CVS-EOM-EE-CCSD framework. The tar-get EOM states obtained by diagonalizing the similarity-transformedCVS-EOM-EE-CCSD Hamiltonian (H) are either purely valenceexcited or purely core excited. To distinguish between the full EOM-CCSD Hamiltonian, eigen- and response states from those of theCVS-EOM-CCSD method, we use different notations: H, R, L, E,and X for full EOM-CCSD and H, R, L, E, and X for CVS-EOM-CCSD. Equation (8) in terms of these CVS-EOM-EE-CCSD states isgiven by the following direct sum:

⟨Φρ∣X kx,ω+iε⟩ ≈ ⟨Φξ ∣Xk,valx,ω+iε⟩⊕ ⟨Φλ∣Xk,corex,ω+iε⟩; (12)

⟨Φξ ∣Xk,valx,ω+iε⟩ =

val

∑n⟨Φξ ∣R

nΦ0⟩⟨Φ0Ln∣μ̄x∣RkΦ0⟩En − Eg − (ω + iε)

; (13)

⟨Φλ∣Xk,corex,ω+iε⟩ =

core

∑n⟨Φλ∣R

nΦ0⟩⟨Φ0Ln∣μ̄x∣RkΦ0⟩En − Eg − (ω + iε)

, (14)

where ξ spans the reference and the valence excitation manifold(OV and OOVV configurations), λ spans the core-excitation man-ifold (CV, COVV, and CCVV configurations), and ρ denotes the

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entire manifold of the reference and singly and doubly excited deter-minants. The L and R in the sum over core-excited states are theleft and right CVS-EOM-CCSD operators for the core-excited staten with energy En that diagonalize the core CVS-EOM-EE-CCSDsimilarity-transformed Hamiltonian (Hcore = e−THcoreeT). T is theCCSD operator expressed in the space of singly and doubly exciteddeterminants (either full space or valence only if the frozen-corevariant of the theory is used). Similarly, the sum over valence states(including the reference CCSD state with energy Eg) involves theEOM operators that diagonalize the valence Hamiltonian (Hval) withthe core states projected out. In terms of resolvents, the amplitudesof the response state in Eq. (12) are given by

⟨Φξ ∣Xk,valx,ω+iε⟩ =∑

σ⟨Φξ ∣(H

val− Eg − ω − iε)

−1∣Φσ⟩⟨Φσ ∣Dkx⟩, (15)

⟨Φλ∣Xk,corex,ω+iε⟩ =∑

κ⟨Φλ∣(H

core− Eg − ω − iε)

−1∣Φκ⟩⟨Φκ∣Dkx⟩, (16)

where σ spans the reference and the valence excitation manifold andκ spans the core-excitation manifold. The core resolvent is givenaccording to

⟨Φλ∣(Hcore− Eg − ω − iε)

−1∣Φκ⟩ =

core

∑n>0

⟨Φλ∣RnΦ0⟩⟨Φ0Ln∣Φκ⟩En − Eg − ω − iε

= ⟨Φχ ∣(H − Eg − ω − iε)−1∣Φτ⟩⊖

⟨Φ0∣Φ0⟩⟨Φ0(1 + Λκ)∣Φτ⟩E0 − Eg − ω − iε

,

(17)

where χ and τ span the reference and the core-excitation space.41In what follows, we discuss several approximations to the

resolvent or, equivalently, to the response wave function: (i) com-plete exclusion of all pure valence excited determinants from theresponse manifold (CVS-0 approximation); (ii) exclusion of the purevalence doubly excited determinants from the response manifold(CVS-uS approximation); (iii) approximating the valence resolventby CCS (CVS-CCS approximation); and (iv) approximating thevalence resolvent by CC2 (CVS-CC2 approximation). Configura-tional analysis of the response states and various contributions tothe RIXS cross sections, given in the supplementary material, showsthat singly excited valence determinants are expected to play a dom-inant role while the effect of doubles is indirect, i.e., they enterthe response states and affect the amplitudes of single excitations,but their contribution to the RIXS moments is small. These newapproaches—CVS-uS, CVS-CCS, and CVS-CC2—are implementedin a development version of the Q-Chem electronic structure pack-age.68,69 Below, we provide a detailed description of and describethe rationale behind each approximation and compare their abil-ity to recover contributions from the valence states to the RIXSmoments.

In the approach presented in Ref. 41 (we call it the CVS-0approach here), the sum in Eq. (12) spans over core-excited statesonly. This truncation is justified because the contributions of nearlyresonant core-excited states are dominant for most RIXS transitions,owing to the resonant nature of RIXS. The numerical convergence of

the response state, now approximated as

⟨Φρ∣X kx,ω+iε⟩ ≈ ⟨Φλ∣Xk,corex,ω+iε⟩, (18)

is smooth because the continuum has been projected out by CVS.Numeric benchmarks have shown that for systems for which theresponse equations could be converged with the standard EOM-EE-CCSD resolvent, this scheme results in the RIXS spectra that agreewell with the full, untruncated calculation.41

Whereas this approach is justified for cases in which the dom-inant contributions to RIXS moments arise from nearly resonantcore-excited states (the most common scenario), Ref. 43 has identi-fied cases in which valence states significantly contribute to the RIXSmoments; these counterexamples are push–pull chromophores suchas para-nitroaniline and 4-amino-4′-nitrostilbene. The CVS-EOM-CCSD calculation with the CVS-0 resolvent omits these contribu-tions and is insufficient for modeling the RIXS spectra of suchsystems. The analysis of the various contributions to the responsestates, given in the supplementary material, identifies the terms thatcan become large for charge-transfer transitions and shows thatthese terms originate in singly excited valence configurations. Here,we provide a general strategy for including the contributions fromvalence states to the damped linear response in the x-ray regime,while preventing the direct coupling of response states with the con-tinuum and thus preserving the robust convergence of responseequations.

The response equation for Xk ,val is given by

∑ξ⟨Φσ ∣H

val− Eg − (ω + iε)∣Φξ⟩⟨Φξ ∣X

k,valx,ω+iε⟩ = ⟨Φσ ∣D

kx ⟩. (19)

As explained above, the convergence of this response equation iscompromised due to the coupling of valence resonances with thecontinuum and the use of a diagonal preconditioner for the OOVV–OOVV block of Hval. The continuum can be projected out byremoving the singles–doubles and doubles–singles blocks from theCVS-EOM-EE-CCSD Hval. By further ignoring the doubles–doublesblock of Hval, the corresponding preconditioner is no longer neededin the iterative procedure. Indices ξ and σ now span just the refer-ence and the OV excited configurations, reducing the resolvent suchthat Xk ,val is given as a linear combination of states obtained by theaction of EOM operators R = r0 + R1 and L = L1 on the CCSD ref-erence. The response equation for this CVS plus valence uncoupledsingles approach (we call it the CVS-uS approach), which effectivelyignores the doubles amplitudes of Xk ,val, is given by

0,OV

∑ξ⟨Φσ ∣H

val− Eg − (ω + iε)∣Φξ⟩⟨Φξ ∣X


kx ⟩. (20)

Note that it is not sufficient to simply remove the OOVV blockfrom D kx in Eq. (19)—this does not project out the valence reso-nances from the continuum nor does it exclude the use of the prob-lematic doubles–doubles preconditioner. Therefore, a viable strat-egy for the inclusion of valence contribution to the RIXS momentsmust involve the modification of the response wave function and themodel Hamiltonian (or the resolvent).

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Another way to include the contributions from valence statesapproximated by single excitations is to express the response statein Eq. (13) in terms of the valence CVS-EOM-EE-CCS intermedi-ate states with the CCS reference. This amounts to replacing thevalence CVS-EOM-EE-CCSD resolvent in Eq. (13) by the valenceCVS-EOM-EE-CCS resolvent (note the change from H, E, and X toH, E, and X),

⟨Φξ ∣Xk,valx,ω+iε⟩ ≈ ⟨Φξ ∣X

k,valx,ω+iε⟩, (21)

0,OV

∑ξ⟨Φσ ∣H

val− E0 − (Eg − E0) − (ω + iε)∣Φξ⟩⟨Φξ ∣X


kx ⟩,

(22)

where Hval is the CVS-EOM-EE-CCS similarity-transformed Hamil-tonian, E0 is the energy of the CCS reference, and Xk ,val is the corre-sponding approximate Xk ,val. Because the doubly excited configura-tions are not present in this low-level CVS-EOM-EE-CCS Hamil-tonian, its eigenstates do not couple with the continuum, which,as discussed above, precludes the problematic convergence of RIXSresponse states. We call this the CVS-CCS approach.

Yet another alternative entails replacing the valence CVS-EOM-EE-CCSD resolvent by the valence resolvent from the CVS-EOM-EE-CC2 model; we call this the CVS-CC2 approach. Thevalence CVS-EOM-EE-CC2 Hamiltonian has a diagonal doubles–doubles block; therefore, the doubly excited configurations do notcouple with each other and their contribution to the response isevaluated perturbatively. Indeed, as noted in Refs. 41 and 42, theCC2 linear response RIXS solutions do not diverge. We exploitthis feature of CC2 to compute the valence response amplitudesin Eq. (22).

The strategy of replacing the valence CVS-EOM-EE-CCSDresolvent by a resolvent of a lower-level method that avoids theerratic convergence of RIXS response solutions can be extended toCIS, ADC(2), and DFT resolvents, for example. Importantly, thecore and valence resolvents can be cherry-picked separately afterenforcing CVS. This “cherry-picking-of-resolvents” strategy exploitsthe better convergence of x-ray response equations within the frame-work of the lower-level method, while using the better energies andwave functions of the initial and final states computed at the higherEOM-CCSD level of theory. We note that, because currently thereis no method that can provide a higher-level treatment of RIXSwithin CC theory, there is no simple way to benchmark the differ-ent approximate treatments of the valence contributions to the RIXSmoments. However, the variations between different models pro-vide a rough estimate of the uncertainties in theoretical treatmentsand of the magnitude of valence contributions to the RIXS spec-tra. Moreover, the configurational analysis of response states givenin the supplementary material provides theoretical justification forhierarchical expansion of the response Fock space in the CVS-0 →CVS-uS/CCS→ CVS-CC2 series.

We begin by comparing the RIXS spectra computed usingdifferent resolvents for a well-behaved case: benzene. The RIXSspectrum of benzene with the CVS-0 approach is discussed in detailin Ref. 41. Figure 2 shows the spectra computed for resonant exci-tation of the two brightest XAS peaks, peak A corresponding to acore → π∗ transition and peak B corresponding to a core → Rytransition. As one can see, the CVS-0 approach captures all the

FIG. 2. RIXS emission spectra for benzene computed using different resol-vents within the CVS-EOM-EE-CCSD framework with (top) pumping peak A(285.97 eV) and (bottom) pumping peak B (287.80 eV). Scattering angle θ = 0○,6-311(2+,+)G∗∗(uC) basis.41,70 The spectra are convoluted using normalizedGaussians (FWHM = 0.25 eV).

main features in the emission following peak-A excitation, and theinclusion of the valence excitations has negligible effect. For peak-B excitation, small differences appear in minor features relative tothe CVS-0 results; the off-resonance valence contributions becomemore important due to the Rydberg character of the particle orbitaland the smaller oscillator strength for the peak-B core excitation.The CVS-CCS and CVS-uS approaches yield similar peak intensi-ties and differ slightly from the CVS-CC2 results. Another exam-ple (water) is given in the supplementary material. In this case, theresponse equations converge without approximations,40–42 and onecan compare the RIXS spectrum computed with the full untrun-cated EOM-CCSD resolvent against CVS-0, CVS-uS, CVS-CCS, andCVS-CC2. In agreement with the previous observation41 that CVS-0yields a RIXS spectrum that is very similar to the one obtained in thefull treatment, we observe close agreement between CVS-0, CVS-uS, CVS-CCS, and CVS-CC2; thus, in water, the effect of valencecontributions is minor.

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FIG. 3. RIXS emission spectra for para-nitroaniline for the incident photon res-onant with the lowest core excitation (XA1 → B2) computed within the CVS-EOM-EE-CCSD framework using different resolvents. Scattering angle θ = 0○,6-311++G∗∗(uC) basis. The spectra are convoluted using normalized Gaussianfunctions (FWHM = 0.25 eV).

Figure 3 compares the RIXS emission spectra for para-nitroaniline41 computed using the CVS-EOM-EE-CCSD frameworkwith different resolvents. Here, the incident photon’s energy is res-onant with the lowest XA1 → B2 core excitation at 285.88 eV,which corresponds to the dominant x-ray absorption peak (see thesupplementary material).

The RIXS spectrum computed with the CVS-0 approach showsa few small inelastic features. These features correspond to the XA1→ 1B2, XA1 → 2A2, XA1 → 3B1, and XA1 → 4A2 transitions at4.68 eV, 5.91 eV, 6.42 eV, and 6.95 eV energy loss, respectively;XA1 → 1B2 being the dominant transition (see the supplementarymaterial for raw data).

The spectra computed with the CVS-uS and CVS-CCSapproaches are similar. However, these two approaches give differ-ent dominant features compared to the CVS-0 approach. Althoughthe XA1 → 1B2 transition—dominant in the RIXS spectra with theCVS-0 approach—is still important, it is no longer the dominantfeature with these approaches. Rather, the two dominant featuresarise from the XA1 → 3B1 and XA1 → 2B1 transitions at 6.46 eVand 5.96 eV, respectively. Other transitions, such as the XA1 → 1B1at 4.64 eV, XA1 → 4B2 at 6.79 eV, and XA1 → 5B2 at 7.21 eV, alsogive non-negligible contributions to the RIXS spectra. The differencebetween CVS-0 and CVS-uS/CCS arise from the contributions fromvalence singly excited determinants to the response states, whichresult in large contributions to the RIXS moments in the case ofcharge-transfer final states (this is explained in the supplementarymaterial).

Next, we compare the spectra obtained with the CVS-0 andCVS-CC2 approaches. Whereas the former spectrum is dominatedby the XA1 → 1B2 transition, the latter shows additional transitionssuch as XA1→ 1B1, XA1→ 2B1, XA1→ 4B2, XA1→ 3A2 (at 6.85 eV),and XA1 → 5B2. We note that the relative cross sections of theseRIXS transitions also differ, with the cross section of the XA1 → 2B1transition being the largest.

The comparison of the RIXS spectra of para-nitroaniline inFig. 3 highlights the significance of off-resonance contributionsfrom the valence states. The RIXS spectra computed with the CVS-CC2 approach differs significantly from the CVS-CCS and CVS-uSapproaches. This is not surprising, because, as it is well known fromprevious benchmarks,71 the valence two-photon absorption crosssections with CCS and CC2 response theory show significant dif-ferences. The choice of the valence resolvent within this framework,in addition to comparison with experiments, is subject to the abilityof the model valence Hamiltonian to provide a balanced descriptionof the full spectrum of states of the system.

In conclusion, we have presented a novel general strategy forincluding valence contributions into RIXS moments while pre-serving the smooth convergence of the response states in the x-ray regime. Whereas the iterative procedure for computing EOM-EE-CCSD response states typically diverges in the x-ray regimedue to the coupling of response states with the continuum, ourstrategy mitigates this issue by exploiting the CVS scheme thatdecouples the valence excitation block from the core-excitationblock of the EOM-EE-CCSD Hamiltonian, which facilitates com-puting their contributions to the RIXS response separately. Refs. 41and 42 have previously presented an EOM-EE-CCSD dampedresponse theory approach that employs a damped CVS-EOM-EE-CCSD resolvent for computing the contribution from the core-excited states. Here, we introduce a more general strategy to alsoinclude the contributions from valence excited states. This strat-egy involves the replacement of the valence CVS-EOM-EE-CCSDresolvent in the expression of the response state with a resol-vent from a lower-level theory (such as CVS-EOM-CCS or CVS-EOM-CC2) for which the response equations do not diverge orthe restriction of the valence CVS-EOM-EE-CCSD resolvent tothe singly excited determinant space. We demonstrated the sig-nificance of including this off-resonance valence contribution tothe RIXS cross section by comparing the RIXS emission spectra ofpara-nitroaniline computed with and without the different valenceresolvents.

The supplementary material contains the configurational anal-ysis of RIXS response states and their contributions to the RIXSmoments; raw data for RIXS spectra of water, benzene, and para-nitroaniline; RIXS spectra for para-nitroaniline computed with dif-ferent basis sets; XAS data for para-nitroaniline; and the geometriesand basis sets used.

This work was supported by the U.S. National Science Founda-tion (Grant No. CHE-1856342).

A.I.K. is the president and a part-owner of Q-Chem, Inc.

DATA AVAILABILITY

The data that support the findings of this study are availablewithin the article and its supplementary material.

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