CVS-IP-EOM-DSRG for Core-Level XPS

• CVS-IP-EOM-DSRG is a multireference computational method that accurately models core-ionization energies for XPS in strongly correlated molecular systems.
• It integrates a core-valence separation ansatz with a multireference ground-state MR-DSRG and an ionization-potential EOM formalism to target core-ionized states.
• The method offers three effective-Hamiltonian variants that balance computational cost with accuracy, featuring an efficient O(N^4) EOM step.

CVS-IP-EOM-DSRG denotes the core-valence separated multireference equation-of-motion driven similarity renormalization group method developed for simulating X-ray photoelectron spectra (XPS) of strongly correlated molecular systems. It combines a multireference MR-DSRG ground-state treatment with an ionization-potential EOM formalism and a CVS restriction that targets core-ionized states. In the reported implementation, the method is described as numerically robust and computationally efficient, delivering accurate core-ionization energies with $O(N^4)$ scaling relative to basis set size $N$ in the EOM step. The work also introduces a minimal modification of the ground-state MR-DSRG equations to guarantee rigorous core intensivity, and develops three effective-Hamiltonian variants: DSRG-MRPT2, DSRG-MRPT3, and MR-LDSRG(2) (Li et al., 25 Sep 2025).

1. Definition and scientific motivation

The method addresses the simulation of XPS for molecules that exhibit strong static correlation or broader multireference character. The stated motivation is that single-reference response methods such as EOM-CCSD struggle with core hole relaxation unless triples are included, while state-specific approaches such as $\Delta$SCF and $\Delta$RASSCF become costly when spectra must be computed over many geometries or many states. CVS-IP-EOM-DSRG is therefore positioned as a general, robust, and efficient multireference method that allows accurate and scalable computation of core-ionization energies while systematically incorporating dynamic and static correlation (Li et al., 25 Sep 2025).

The formal development builds on the earlier IP-EOM-DSRG framework for valence ionization, which formulates ionized states through a Hermitian generalized eigenvalue problem and combines naturally with three truncation schemes of the parent MR-DSRG theory: MR-LDSRG(2), DSRG-MRPT2, and DSRG-MRPT3 (Zhao et al., 16 Jun 2025). The CVS extension specializes that machinery to core-ionized sectors, where direct state targeting is otherwise hindered by the dense manifold of valence and continuum states.

A useful point of comparison is the broader core-level EOM literature. In single-reference coupled-cluster work, the CVS ansatz is motivated by the fact that core excitations and ionizations are buried in a dense continuum of valence states; restricting the EOM operator to configurations involving at least one core orbital both stabilizes the calculation and reduces computational cost (Matthews, 2020). CVS-IP-EOM-DSRG imports that same targeting principle into a multireference, similarity-transformed setting (Li et al., 25 Sep 2025).

2. Formal structure of the method

The ground state starts from a complete- or generalized-active-space reference wavefunction

$$|\Phi_0\rangle = \sum_{\mu=1}^d c_\mu |\phi_\mu\rangle .$$

The Hamiltonian is transformed according to

$$\bar{H}(s) = e^{-\hat{A}(s)} \hat{H} e^{\hat{A}(s)},$$

where $\hat{A}(s)$ is an anti-Hermitian operator built from excitation operators, and $s$ is a flow parameter that controls regularization by removing dangerous low-energy couplings and suppressing intruder states (Li et al., 25 Sep 2025).

For ionization potentials, the ionized $(N-1)$-electron states are written in EOM form as

$$|\Psi_\alpha\rangle = \bar{\mathcal{R}}_\alpha |\Psi_0\rangle,$$

with $N$0 truncated to one-hole and two-hole-one-particle terms,

$N$1

The resulting energies follow from the generalized eigenvalue problem

$N$2

with $N$3 and $N$4 (Li et al., 25 Sep 2025).

Within the broader MR-DSRG lineage, this generalized eigenproblem remains Hermitian, a structural feature emphasized already for valence IP-EOM-DSRG (Zhao et al., 16 Jun 2025). That property is consequential for numerical robustness because it avoids the non-Hermitian state manifolds common in some multireference EOM constructions. A plausible implication is that the CVS-IP formulation inherits much of the favorable eigensolver behavior of the underlying IP-EOM-DSRG formalism, while the CVS restriction further reduces the spectral congestion that typically complicates core-level calculations.

3. Core-valence separation and rigorous core intensivity

The need for core-valence separation arises because, in standard EOM-DSRG, core-ionized states are buried among many valence and continuum states, making convergence and state targeting difficult. The CVS approximation resolves this by restricting both the excitation operator and matrix elements so that only states with at least one core hole are described (Li et al., 25 Sep 2025).

Two practical CVS realizations are defined. In Full-CVS, all electrons are correlated and the EOM operator is restricted to remove continuum coupling. In fc-CVS, core-occupied orbitals are frozen in the ground-state reference, while the EOM operator is restricted in the same way; this is reported to offer improved accuracy through error cancellation between missing correlation and relaxation (Li et al., 25 Sep 2025). The general logic parallels that of CVS in EOM-CC, where only operators containing at least one core orbital index are retained in the target-space operator manifold (Matthews, 2020).

A central conceptual contribution of the work is the enforcement of core intensivity. Core intensivity is defined as the condition that a computed core-ionization energy for a fragment is independent of the number of non-interacting additional core or virtual orbitals. For unitary MR EOM approaches such as MR-DSRG, the formal requirement is that

$N$5

for all $N$6 in the hole space and $N$7 in the particle space (Li et al., 25 Sep 2025).

In truncated MR-DSRG at finite $N$8, these conditions are not all satisfied automatically. The proposed remedy is deliberately minimal: enforce the core-virtual block of the condition, $N$9, by directly solving

$\Delta$0

This modification is reported to make IP-EOM-DSRG rigorously core-intensive, while leaving ground-state energies essentially unchanged and avoiding other numerical instabilities (Li et al., 25 Sep 2025). This is methodologically significant because intensivity failures in internally contracted or truncated multireference response theories can otherwise lead to unphysical dependence on spectator orbitals.

4. Effective-Hamiltonian variants and computational characteristics

The CVS-IP-EOM-DSRG family is developed in three variants that differ in the approximation used for the effective Hamiltonian (Li et al., 25 Sep 2025).

Variant	Approximation	Characterization
DSRG-MRPT2	Second-order perturbative	Lowest computational cost
DSRG-MRPT3	Third-order perturbative	More dynamic correlation and relaxation than PT2
MR-LDSRG(2)	Non-perturbative, truncated to 1- and 2-body operators	Iterative, unitary, robust for strongly correlated cases

DSRG-MRPT2 is described as a perturbative expansion to second order. It is the most efficient of the three but is less accurate for strong correlation and relaxation effects. DSRG-MRPT3 extends the expansion to third order and is reported to capture more dynamic correlation and relaxation, with markedly improved potential energy curves and vibrational structure relative to PT2. MR-LDSRG(2) is an iterative, non-perturbative scheme truncated at 1- and 2-body operators; it offers the best accuracy, on par with PT3, but may face convergence problems at large $\Delta$1 (Li et al., 25 Sep 2025).

The dominant cost of the EOM step scales as $\Delta$2 because the excitation space is reduced by CVS, and the contribution from active orbitals is stated to be small unless the active space is large (Li et al., 25 Sep 2025). This quartic EOM scaling is lower than the $\Delta$3 EOM scaling reported for the earlier, non-CVS IP-EOM-DSRG formulation for valence ionizations (Zhao et al., 16 Jun 2025), reflecting the dimensionality reduction induced by the core-restricted manifold.

This computational profile places CVS-IP-EOM-DSRG in a specific position among core-level methods. For example, in approximate-triples single-reference EOM-CC, CVS can reduce excited-state scaling from $\Delta$4 to $\Delta$5 for several approaches, and in CVS-EOM-CCSD* the entire method scales rigorously as $\Delta$6 (Matthews, 2020). By contrast, the present multireference DSRG formulation emphasizes a lower-scaling EOM step together with explicit compatibility with CAS/GAS references (Li et al., 25 Sep 2025).

5. Benchmarks: vertical ionization energies, dissociation, and vibrational structure

The principal benchmark comprises vertical core-ionization energies for a representative molecular test set, compared against established single-reference and multireference methods. In this assessment, EOM-DSRG-PT3 and MR-LDSRG(2) yield mean absolute errors below 0.6 eV, whereas PT2 is somewhat less accurate at approximately 0.74 eV (Li et al., 25 Sep 2025).

The comparative conclusions are specific. The CVS-IP-EOM-DSRG methods are reported to outperform MR-ADC(2) and to match the accuracy of improved MR-ADC(2)-X. They remain slightly below CCSDT/CCSDTQ and state-specific methods which include full orbital relaxation (Li et al., 25 Sep 2025). This is consistent with the more general core-spectroscopy observation that linear-response or EOM approaches can miss relaxation unless higher excitations are incorporated; in the single-reference literature, approximate triples are often decisive for sub-eV accuracy in K-edge core energies (Matthews, 2020).

The method was also applied to potential energy curves and vibrationally resolved XPS. For $\Delta$7 and CO, all methods are described as qualitatively correct near equilibrium, but only PT3 and MR-LDSRG(2) show close agreement with accurate state-specific references across the dissociation region. The reported deficiencies of PT2 and MR-ADC(2)-X include larger errors in dissociation behavior, vibrational parameters, and even discontinuities in some cases (Li et al., 25 Sep 2025).

For vibrational spectra, the same hierarchy persists. PT3 and MR-LDSRG(2) reproduce experiment, including high-resolution vibrational structure and the correct trends in bond contraction or elongation, whereas PT2 and MR-ADC(2)-X can produce incorrect intensity profiles or vibrational constants, particularly for demanding edges such as O 1s in CO (Li et al., 25 Sep 2025). This strongly indicates that the improved treatment of relaxation and correlation in PT3 and MR-LDSRG(2) is not only reflected in vertical ionization energies but also in the curvature and displacement of core-ionized potential surfaces.

6. Ozone, relation to neighboring methods, and stated limitations

The XPS of ozone is identified as a particularly challenging application because the $\Delta$8 ground state is strongly multiconfigurational, and the splitting between terminal and central O K-edge ionizations is difficult for EOM methods that lack higher excitations. In this case, PT3 and MR-LDSRG(2) predict the energy splitting within 0.7 eV of experiment, which is reported to be comparable to MR-ADC(2)-X, but they exhibit larger shifts in absolute energies. The work states that state-specific multistate methods and RASPT2 are more precise for absolute energies in this system (Li et al., 25 Sep 2025).

The authors characterize the method’s strengths in four recurring terms: scalable, robust, accurate, and core-intensive. The robustness is attributed to DSRG regularization, which is described as intruder-state free; the flexibility derives from compatibility with GAS/CAS references; and the accuracy claims are concentrated on the DSRG-MRPT3 and MR-LDSRG(2) variants (Li et al., 25 Sep 2025). A plausible implication is that CVS-IP-EOM-DSRG is designed to fill the methodological space between lower-cost multireference response models and more expensive state-specific treatments.

The paper also states explicit limitations. Like other response methods, CVS-IP-EOM-DSRG is not fully state-specific, so it can miss relaxation effects unless effective higher excitations are present. Absolute energies can remain shifted for especially difficult targets such as ozone, even when relative gaps are described well. Finally, extension to XAS or UV/vis would require further development using particle-number-conserving excitation operators (Li et al., 25 Sep 2025).

Within the immediate research trajectory, CVS-IP-EOM-DSRG can be read as the core-level analogue of the previously introduced IP-EOM-DSRG for valence ionization (Zhao et al., 16 Jun 2025). Relative to single-reference CVS-EOM-CC developments, it addresses a different regime: systems where strong correlation and multireference character are central rather than peripheral. Relative to CVS-STEOM-CCSD-style approaches, which reduce convergence problems by diagonalizing an effective Hamiltonian in a smaller singles space (Ranga et al., 2020), CVS-IP-EOM-DSRG instead preserves a multireference EOM construction and regularized similarity transformation. This suggests a broader methodological pattern in core spectroscopy: CVS serves as the state-selection mechanism, while the decisive differences among frameworks lie in how orbital relaxation, dynamic correlation, and multiconfigurational reference effects are encoded.