---
title: CVS-IP-EOM-DSRG for Core-Level XPS
url: https://www.emergentmind.com/topics/cvs-ip-eom-dsrg
type: topic
---

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

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)** [2509.21646].

## 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 [2509.21646].

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** [2506.13693]. 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 [2001.09218]. CVS-IP-EOM-DSRG imports that same targeting principle into a multireference, similarity-transformed setting [2509.21646].

## 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 [2509.21646].

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 \(\hat{\mathcal{R}}_\alpha\) truncated to **one-hole** and **two-hole-one-particle** terms,
\[
\hat{\mathcal{R}}_\alpha
= \sum_i r_\alpha^i \{\hat{a}_i\}
+ \frac{1}{2}\sum_{ij,a} r^{ij}_{a,\alpha}\{\hat{a}^{a}_{ij}\}.
\]
The resulting energies follow from the generalized eigenvalue problem
\[
\sum_q \langle \Phi_0 | \hat{\rho}_p^\dagger \bar{H} \hat{\rho}_q | \Phi_0 \rangle r^q_\alpha
=
E_\alpha
\sum_q \langle \Phi_0 | \hat{\rho}_p^\dagger \hat{\rho}_q | \Phi_0 \rangle r^q_\alpha,
\]
with \(E_0 = \langle \Phi_0|\bar{H}|\Phi_0\rangle\) and \(\omega_\alpha = E_\alpha - E_0\) [2509.21646].

Within the broader MR-DSRG lineage, this generalized eigenproblem remains **Hermitian**, a structural feature emphasized already for valence IP-EOM-DSRG [2506.13693]. 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 [2509.21646].

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 [2509.21646]. 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 [2001.09218].

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
\[
\mathcal{S}_a^i
=
\langle \Phi_0 | \{ \hat{a}_a^i \} \bar{H} | \Phi_0 \rangle
= 0
\]
for all \(i\) in the hole space and \(a\) in the particle space [2509.21646].

In truncated MR-DSRG at finite \(s\), these conditions are not all satisfied automatically. The proposed remedy is deliberately minimal: enforce the **core-virtual block** of the condition, \(\mathcal{S}_e^m = 0\), by directly solving
\[
\bar{H}_e^m = 0 .
\]
This modification is reported to make **IP-EOM-DSRG rigorously core-intensive**, while leaving ground-state energies essentially unchanged and avoiding other numerical instabilities [2509.21646]. 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 [2509.21646].

| 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 \(s\) [2509.21646].

The dominant cost of the **EOM step** scales as \(\mathcal{O}(N^4)\) 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 [2509.21646]. This quartic EOM scaling is lower than the \(O(N^5)\) EOM scaling reported for the earlier, non-CVS IP-EOM-DSRG formulation for valence ionizations [2506.13693], 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 \(\mathscr{O}(n^7)\) to \(\mathscr{O}(n^6)\) for several approaches, and in CVS-EOM-CCSD* the entire method scales rigorously as \(\mathscr{O}(n^6)\) [2001.09218]. By contrast, the present multireference DSRG formulation emphasizes a lower-scaling EOM step together with explicit compatibility with CAS/GAS references [2509.21646].

## 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** [2509.21646].

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** [2509.21646]. 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 [2001.09218].

The method was also applied to **potential energy curves** and **vibrationally resolved XPS**. For **\(\mathrm{N_2}\)** 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 [2509.21646].

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** [2509.21646]. 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 **\(\mathrm{O_3}\)** 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 [2509.21646].

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 [2509.21646]. 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** [2509.21646].

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 [2506.13693]. 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 [2010.08932], 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.

Source: https://www.emergentmind.com/topics/cvs-ip-eom-dsrg