Cherry-picking resolvents: A general strategy for convergent coupled-cluster damped response calculations of core-level spectra

Equation-of-motion coupled-cluster (EOM-CC) theory^1–6 provides a robust single-reference framework for computing multiple electronic states. EOM-CC affords a balanced description of states of different characters and a systematic improvement of results by incremental improvements in treating the electron correlation. The EOM-CC hierarchy of approximations is based on a standard hierarchy of CC models for the ground state, such as CC singles (CCS), approximate CC singles and doubles⁷ (CC2), CC with singles and doubles^2,8–10 (CCSD), CC with singles, doubles, and triples (CC3¹¹ and CCSDT^12–14), and so on. The EOM-CC formalism naturally extends to state and transition properties. Together, these features make EOM-CC an ideal framework for modeling spectroscopy. EOM-CC can be used to compute solvatochromic shifts,¹⁵ transition dipole moments,² spin–orbit^16–19 and non-adiabatic^20–22 couplings, photoionization cross sections,^23,24 and higher-order properties²⁵ such as two-photon absorption cross sections^26–29 and static and dynamic polarizabilities.^30–33 

The EOM-CC framework is being vigorously extended to the x-ray regime for modeling x-ray absorption (XAS), photoionization, and emission spectra,^34–39 as well as multi-photon phenomena such as resonant inelastic x-ray scattering^40–44 (RIXS) (Fig. 1). The successful extensions of EOM-CC to the x-ray domain exploit the core–valence separation (CVS) scheme,⁴⁵ which effectively addresses the key challenge in the theoretical treatment of core-level states: their resonance nature due to the coupling with the valence ionization continuum. Being embedded in the continuum, core-level states, strictly speaking, cannot be treated by methods developed for isolated bound states with L²-integrable wave functions.^46–48 Practically, the coupling with the continuum leads to erratic and often divergent behavior of the solvers, a lack of systematic convergence of results with the basis-set increase, and often unphysical solutions.^48,49

The CVS scheme allows one to separate the continuum of valence states from the core-level states through a deliberate pruning of the Fock space, i.e., by removing configurations that can couple core-excited (or core-ionized) configurations with the valence continuum (see the supplementary material for the analysis of different decay channels). CVS can be described as a diabatization procedure that separates the bound part of the resonance from the continuum, akin to the Feshbach–Fano treatment of resonances.^50–52 In contrast to other methods^46–48 for describing resonances, this approach, in which the continuum is simply projected out, does not involve state-specific parameterization (such as, for example, tuning the strength of complex absorbing potential for each state) and offers an additional benefit of eliminating the need to compute lower-lying electronic states.

Although separation of the full Hilbert space into bound and continuum parts is not well defined in general, it is possible in the case of core-level states because they are Feshbach resonances, which can only decay by a two-electron process in which one electron fills the core hole, liberating sufficient energy to eject another electron.^38,49 This special feature allows one to separate the continuum decay channels from the core-level states in the Fock space, by removing the configurations that couple the resonances with the continuum. Such pruning of the Fock space is equivalent to omitting the couplings between the core and valence excited (or ionized) determinants from the model EOM-CC Hamiltonian. The eigenstates of this reduced EOM-CC Hamiltonian are either purely valence or purely core excited (or ionized). The pure core-excited (or ionized) states become formally bound because of the decoupling from the valence determinants forming the continuum. Consequently, diagonalization of the core block of this CVS-EOM-CC Hamiltonian yields core-excited (or core-ionized) states without any convergence issues.

Initially used to describe transition energies and strengths of core-level states, the CVS scheme was recently extended into the response domain^41–43 to enable calculations of RIXS transition moments within damped linear response theory^53–56 and the EOM-CCSD method for excitation energies (EOM-EE-CCSD). In this approach, the response equations are solved in the truncated Fock space spanning the singly and doubly excited determinants in which at least one orbital belongs to the core,^41,42 in the same manner as done in CVS-EOM-CCSD calculations of core-excited or core-ionized states. Thus, all valence excited states are excluded from the response manifold, which can be justified by the resonant nature of the RIXS process. This truncation of the response manifold works well in many situations, but is expected to break down when off-resonance contributions to the RIXS cross section from the valence states become important. At least one class of systems where this happens is push–pull chromophores featuring low-lying charge-transfer states.⁴³ 

Here, we address this limitation of the previous formulation of RIXS theory within the CVS-EOM-CCSD framework and present a general strategy for including valence contributions to the RIXS signal while preserving smooth convergence of the RIXS response states. This strategy can also be applied to the modeling of other multi-photon x-ray processes, such as x-ray two-photon absorption. By using this approach, we provide a quantitative illustration of the significance of the valence contributions to RIXS spectra.

The derivation of the equations for RIXS transition moments between the initial (g) and final (f) states starts from the Kramers–Heisenberg–Dirac formula,^57,58 which translates to the following sum-over-states (SOS) expressions^40–44 within the EOM-EE-CCSD damped response theory:

and

Here, $T$ is an excitation operator containing the CCSD amplitudes, and $μ¯=e−TμeT$ is the similarity-transformed dipole moment. $Ln$ and $Rn$ are the EOM-CCSD left and right excitation operators, respectively,⁵⁹ for state n with energy $En$⁠. Their amplitudes and eigenenergies are found by diagonalizing the EOM-CCSD similarity-transformed Hamiltonian $H¯=e−THeT$ in the appropriate sector of the Fock space (i.e., singly and doubly excited determinants in EOM-EE-CCSD) as follows:

iε is the phenomenological damping (or inverse lifetime) term from damped response theory. ω_1x and ω_2y are the x-polarized absorbed and y-polarized emitted energies, respectively, satisfying the RIXS resonance condition,

We drop the Cartesian indices of the photon energies for brevity.

The Fock space of EOM-EE-CCSD comprises the Slater determinants Φ_ρ, where ρ ∈ {reference (0), CV, OV, COVV, CCVV, OOVV} and C, O, and V denote the core occupied, valence occupied, and unoccupied orbitals, respectively. Upon inserting the identity operator (1 = ∑_ρ|Φ_ρ⟩⟨Φ_ρ|) in Eqs. (1) and (2), we obtain

and

where the amplitudes of the intermediate $D̃yk$ and the first-order response wave function $Xx,ω1+iεk$ for state k are given by^26,41

and

The response wave function in Eq. (8) is complex-valued and given by a linear combination of all EOM-CCSD states. The amplitudes of this response wave function, expressed in the basis of Slater determinants, are solutions of the following response equation:

where the amplitudes of intermediate $Dxk$ are given by

Equation (9) is the direct result of using the identity operator $1=∑n|RnΦ0Φ0Ln|$ and the following resolvent:

In practice, response equations such as Eq. (9) are solved iteratively using standard procedures, e.g., the Direct Inversion in the Iterative Subspace⁶⁰ (DIIS), which rely on diagonal preconditioners. In the x-ray regime, such response equations often diverge for the following reasons.^41,42 The RIXS response wave function in Eq. (8) includes large contributions from high-lying core-excited states that are nearly resonant with the absorbed photon’s energy. These high-lying states are embedded in the valence ionization continuum and are Feshbach resonances that are metastable with respect to electron ejection. Mathematically, these core-excited states (and consequently, the response state) are strongly coupled to the 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 respective matrix elements are expected to be very small; this is discussed in the supplementary material). This coupling to the continuum leads to the oscillatory behavior of the leading response amplitudes with large magnitudes in the course of the iterative procedure. The problem is exacerbated by the coupling of these doubly excited determinants to the valence resonances, which leads to the erratic behavior of pure valence doubly excited response amplitudes. Moreover, for x-ray energies, the diagonal preconditioner for the valence doubly excited amplitudes is no longer a good approximation to the valence doubles–doubles block of $H¯−E0−ω+iε$⁠, which further cripples the convergence.

We note that lower-level theories such as CIS (configuration interaction singles)⁶¹ and TD-DFT (time-dependent density functional theory)^62–65 do not have valence double excitations in the excitation manifold, so that the issue with the convergence of response equations in the x-ray regime does not arise, simply because there is no coupling between Feshbach resonances and the valence continuum. Similarly, this issue does not arise for theories such as ADC(2)^66,67 and CC2 in which the doubly excited configurations are not coupled with each other^41,42 (the doubles–doubles block is diagonal), so the diagonal preconditioner for the doubles–doubles block is exact and the effect of double excitations is evaluated perturbatively 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 and implemented within the CVS-EOM-EE-CCSD framework. The target EOM states obtained by diagonalizing the similarity-transformed CVS-EOM-EE-CCSD Hamiltonian (⁠$H¯$⁠) are either purely valence excited or purely core excited. To distinguish between the full EOM-CCSD Hamiltonian, eigen- and response states from those of the CVS-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 is given by the following direct sum:

where ξ spans the reference and the valence excitation manifold (OV and OOVV configurations), λ spans the core-excitation manifold (CV, COVV, and CCVV configurations), and ρ denotes the entire manifold of the reference and singly and doubly excited determinants. The L and R in the sum over core-excited states are the left and right CVS-EOM-CCSD operators for the core-excited state n with energy E_n that diagonalize the core CVS-EOM-EE-CCSD similarity-transformed Hamiltonian (⁠$H¯core=e−THcoreeT$⁠). T is the CCSD operator expressed in the space of singly and doubly excited determinants (either full space or valence only if the frozen-core variant of the theory is used). Similarly, the sum over valence states (including the reference CCSD state with energy E_g) involves the EOM operators that diagonalize the valence Hamiltonian (⁠$H¯val$⁠) with the core states projected out. In terms of resolvents, the amplitudes of the response state in Eq. (12) are given by

where σ spans the reference and the valence excitation manifold and κ spans the core-excitation manifold. The core resolvent is given according to

where χ and τ span the reference and the core-excitation space.⁴¹ 

In what follows, we discuss several approximations to the resolvent or, equivalently, to the response wave function: (i) complete exclusion of all pure valence excited determinants from the response manifold (CVS-0 approximation); (ii) exclusion of the pure valence doubly excited determinants from the response manifold (CVS-uS approximation); (iii) approximating the valence resolvent by CCS (CVS-CCS approximation); and (iv) approximating the valence resolvent by CC2 (CVS-CC2 approximation). Configurational analysis of the response states and various contributions to the RIXS cross sections, given in the supplementary material, shows that singly excited valence determinants are expected to play a dominant role while the effect of doubles is indirect, i.e., they enter the response states and affect the amplitudes of single excitations, but their contribution to the RIXS moments is small. These new approaches—CVS-uS, CVS-CCS, and CVS-CC2—are implemented in a development version of the Q-Chem electronic structure package.^68,69 Below, we provide a detailed description of and describe the rationale behind each approximation and compare their ability to recover contributions from the valence states to the RIXS moments.

In the approach presented in Ref. 41 (we call it the CVS-0 approach here), the sum in Eq. (12) spans over core-excited states only. This truncation is justified because the contributions of nearly resonant 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

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

Whereas this approach is justified for cases in which the dominant contributions to RIXS moments arise from nearly resonant core-excited states (the most common scenario), Ref. 43 has identified cases in which valence states significantly contribute to the RIXS moments; these counterexamples are push–pull chromophores such as para-nitroaniline and 4-amino-4′-nitrostilbene. The CVS-EOM-CCSD calculation with the CVS-0 resolvent omits these contributions and is insufficient for modeling the RIXS spectra of such systems. The analysis of the various contributions to the response states, given in the supplementary material, identifies the terms that can become large for charge-transfer transitions and shows that these terms originate in singly excited valence configurations. Here, we provide a general strategy for including the contributions from valence states to the damped linear response in the x-ray regime, while preventing the direct coupling of response states with the continuum and thus preserving the robust convergence of response equations.

The response equation for X^k,val is given by

As explained above, the convergence of this response equation is compromised due to the coupling of valence resonances with the continuum and the use of a diagonal preconditioner for the OOVV–OOVV block of $H¯val$⁠. The continuum can be projected out by removing the singles–doubles and doubles–singles blocks from the CVS-EOM-EE-CCSD $H¯val$⁠. By further ignoring the doubles–doubles block of $H¯val$⁠, the corresponding preconditioner is no longer needed in the iterative procedure. Indices ξ and σ now span just the reference and the OV excited configurations, reducing the resolvent such that X^k,val is given as a linear combination of states obtained by the action of EOM operators R = r₀ + R₁ and L = L₁ on the CCSD reference. The response equation for this CVS plus valence uncoupled singles approach (we call it the CVS-uS approach), which effectively ignores the doubles amplitudes of X^k,val, is given by

Note that it is not sufficient to simply remove the OOVV block from $Dxk$ in Eq. (19)—this does not project out the valence resonances from the continuum nor does it exclude the use of the problematic doubles–doubles preconditioner. Therefore, a viable strategy for the inclusion of valence contribution to the RIXS moments must involve the modification of the response wave function and the model Hamiltonian (or the resolvent).

Another way to include the contributions from valence states approximated by single excitations is to express the response state in Eq. (13) in terms of the valence CVS-EOM-EE-CCS intermediate states with the CCS reference. This amounts to replacing the valence CVS-EOM-EE-CCSD resolvent in Eq. (13) by the valence CVS-EOM-EE-CCS resolvent (note the change from $H¯$⁠, E, and X to $H¯$⁠, E, and X),

where $H¯val$ is the CVS-EOM-EE-CCS similarity-transformed Hamiltonian, E₀ is the energy of the CCS reference, and X^k,val is the corresponding approximate X^k,val. Because the doubly excited configurations are not present in this low-level CVS-EOM-EE-CCS Hamiltonian, its eigenstates do not couple with the continuum, which, as discussed above, precludes the problematic convergence of RIXS response 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. The valence CVS-EOM-EE-CC2 Hamiltonian has a diagonal doubles–doubles block; therefore, the doubly excited configurations do not couple with each other and their contribution to the response is evaluated perturbatively. Indeed, as noted in Refs. 41 and 42, the CC2 linear response RIXS solutions do not diverge. We exploit this feature of CC2 to compute the valence response amplitudes in Eq. (22).

The strategy of replacing the valence CVS-EOM-EE-CCSD resolvent by a resolvent of a lower-level method that avoids the erratic convergence of RIXS response solutions can be extended to CIS, ADC(2), and DFT resolvents, for example. Importantly, the core and valence resolvents can be cherry-picked separately after enforcing CVS. This “cherry-picking-of-resolvents” strategy exploits the better convergence of x-ray response equations within the framework of the lower-level method, while using the better energies and wave functions of the initial and final states computed at the higher EOM-CCSD level of theory. We note that, because currently there is no method that can provide a higher-level treatment of RIXS within CC theory, there is no simple way to benchmark the different approximate treatments of the valence contributions to the RIXS moments. However, the variations between different models provide a rough estimate of the uncertainties in theoretical treatments and of the magnitude of valence contributions to the RIXS spectra. Moreover, the configurational analysis of response states given in the supplementary material provides theoretical justification for hierarchical expansion of the response Fock space in the CVS-0 → CVS-uS/CCS → CVS-CC2 series.

We begin by comparing the RIXS spectra computed using different resolvents for a well-behaved case: benzene. The RIXS spectrum of benzene with the CVS-0 approach is discussed in detail in Ref. 41. Figure 2 shows the spectra computed for resonant excitation of the two brightest XAS peaks, peak A corresponding to a core → π^* transition and peak B corresponding to a core → Ry transition. As one can see, the CVS-0 approach captures all the main features in the emission following peak-A excitation, and the inclusion of the valence excitations has negligible effect. For peak-B excitation, small differences appear in minor features relative to the CVS-0 results; the off-resonance valence contributions become more important due to the Rydberg character of the particle orbital and the smaller oscillator strength for the peak-B core excitation. The CVS-CCS and CVS-uS approaches yield similar peak intensities and differ slightly from the CVS-CC2 results. Another example (water) is given in the supplementary material. In this case, the response equations converge without approximations,^40–42 and one can compare the RIXS spectrum computed with the full untruncated EOM-CCSD resolvent against CVS-0, CVS-uS, CVS-CCS, and CVS-CC2. In agreement with the previous observation⁴¹ that CVS-0 yields a RIXS spectrum that is very similar to the one obtained in the full treatment, we observe close agreement between CVS-0, CVS-uS, CVS-CCS, and CVS-CC2; thus, in water, the effect of valence contributions is minor.

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

The RIXS spectrum computed with the CVS-0 approach shows a few small inelastic features. These features correspond to the XA₁ → 1B₂, XA₁ → 2A₂, XA₁ → 3B₁, and XA₁ → 4A₂ transitions at 4.68 eV, 5.91 eV, 6.42 eV, and 6.95 eV energy loss, respectively; XA₁ → 1B₂ being the dominant transition (see the supplementary material for raw data).

The spectra computed with the CVS-uS and CVS-CCS approaches are similar. However, these two approaches give different dominant features compared to the CVS-0 approach. Although the XA₁ → 1B₂ transition—dominant in the RIXS spectra with the CVS-0 approach—is still important, it is no longer the dominant feature with these approaches. Rather, the two dominant features arise from the XA₁ → 3B₁ and XA₁ → 2B₁ transitions at 6.46 eV and 5.96 eV, respectively. Other transitions, such as the XA₁ → 1B₁ at 4.64 eV, XA₁ → 4B₂ at 6.79 eV, and XA₁ → 5B₂ at 7.21 eV, also give non-negligible contributions to the RIXS spectra. The difference between CVS-0 and CVS-uS/CCS arise from the contributions from valence singly excited determinants to the response states, which result in large contributions to the RIXS moments in the case of charge-transfer final states (this is explained in the supplementary material).

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

The comparison of the RIXS spectra of para-nitroaniline in Fig. 3 highlights the significance of off-resonance contributions from the valence states. The RIXS spectra computed with the CVS-CC2 approach differs significantly from the CVS-CCS and CVS-uS approaches. This is not surprising, because, as it is well known from previous benchmarks,⁷¹ the valence two-photon absorption cross sections with CCS and CC2 response theory show significant differences. The choice of the valence resolvent within this framework, in addition to comparison with experiments, is subject to the ability of the model valence Hamiltonian to provide a balanced description of the full spectrum of states of the system.

In conclusion, we have presented a novel general strategy for including valence contributions into RIXS moments while preserving 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 regime due to the coupling of response states with the continuum, our strategy mitigates this issue by exploiting the CVS scheme that decouples the valence excitation block from the core-excitation block of the EOM-EE-CCSD Hamiltonian, which facilitates computing their contributions to the RIXS response separately. Refs. 41 and 42 have previously presented an EOM-EE-CCSD damped response 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 also include the contributions from valence excited states. This strategy involves the replacement of the valence CVS-EOM-EE-CCSD resolvent in the expression of the response state with a resolvent from a lower-level theory (such as CVS-EOM-CCS or CVS-EOM-CC2) for which the response equations do not diverge or the restriction of the valence CVS-EOM-EE-CCSD resolvent to the singly excited determinant space. We demonstrated the significance of including this off-resonance valence contribution to the RIXS cross section by comparing the RIXS emission spectra of para-nitroaniline computed with and without the different valence resolvents.

The supplementary material contains the configurational analysis of RIXS response states and their contributions to the RIXS moments; raw data for RIXS spectra of water, benzene, and para-nitroaniline; RIXS spectra for para-nitroaniline computed with different basis sets; XAS data for para-nitroaniline; and the geometries and basis sets used.

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

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