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Towards Balanced Description of Ground and Excited States with Transcorrelated F12 Methods

Conner Masteran, Bimal Gaudel, Edward F. Valeev

TL;DR

The paper tackles the imbalance in single-reference F12 methods that overemphasize ground-state correlation when using standard geminal ansatze. It introduces valence-constrained geminals (VVOs) and three active-space correlators (II, IP, PP) within a Hermitian unitary transcorrelated framework to produce a Hermitian two-body Hamiltonian that captures short-range dynamical correlation for ground and low-energy excited states. Through implementation in MPQC with SeQuant and benchmarking on BH, CH$_2$O, N$_2$, CH$_2$, and H$_2$O, the authors demonstrate that VVO-based IP and PP correlators offer the best balance between ground- and excited-state descriptions, reducing basis-set convergence errors by about one cardinal number for energies. Excited-state properties, including geometries and harmonic frequencies, are substantially improved by the F12 transcorrelation, though residual 1-particle basis-set incompleteness remains a target for future refinement and extension to response properties.

Abstract

By correlating only the 1-particle states occupied in the reference determinant the conventional design for the single-reference R12/F12 explicitly-correlated methods biases them towards the ground state description thereby making treatment of response properties of the ground state, and energies and other properties of excited states less robust. While the use of multireference methods and/or extensions of the standard SP projected geminals can achieve a more balanced description of ground and excited states, here we show that the same goals can be achieved by extending the action of F12 correlators to the occupied and valence unoccupied 1-particle states only. This design choice reflects the strong dependence of the optimal correlation lengthscale of the F12 ansatz on the orbital energies/structure, and helps to avoid the unphysical raising of the ground state energy if the F12 geminals are used to correlate pairs of all 1-particle states. The improved F12 geminal design is incorporated into the unitary transcorrelation framework to produce a unitary 2-body Hamiltonian that incorporates the short-range dynamical correlation physics for ground and low-energy excited states in a balanced manner. This explicitly-correlated effective Hamiltonian reduces the basis set requirement on the correlation-consistent basis cardinal number by 1 or more over the uncorrelated counterpart for the description of the ground state coupled-cluster singles and doubles (CCSD) energies, the vertical excitation energies and harmonic vibrational frequencies of equation-of-motion CCSD low-energy excited states.

Towards Balanced Description of Ground and Excited States with Transcorrelated F12 Methods

TL;DR

The paper tackles the imbalance in single-reference F12 methods that overemphasize ground-state correlation when using standard geminal ansatze. It introduces valence-constrained geminals (VVOs) and three active-space correlators (II, IP, PP) within a Hermitian unitary transcorrelated framework to produce a Hermitian two-body Hamiltonian that captures short-range dynamical correlation for ground and low-energy excited states. Through implementation in MPQC with SeQuant and benchmarking on BH, CHO, N, CH, and HO, the authors demonstrate that VVO-based IP and PP correlators offer the best balance between ground- and excited-state descriptions, reducing basis-set convergence errors by about one cardinal number for energies. Excited-state properties, including geometries and harmonic frequencies, are substantially improved by the F12 transcorrelation, though residual 1-particle basis-set incompleteness remains a target for future refinement and extension to response properties.

Abstract

By correlating only the 1-particle states occupied in the reference determinant the conventional design for the single-reference R12/F12 explicitly-correlated methods biases them towards the ground state description thereby making treatment of response properties of the ground state, and energies and other properties of excited states less robust. While the use of multireference methods and/or extensions of the standard SP projected geminals can achieve a more balanced description of ground and excited states, here we show that the same goals can be achieved by extending the action of F12 correlators to the occupied and valence unoccupied 1-particle states only. This design choice reflects the strong dependence of the optimal correlation lengthscale of the F12 ansatz on the orbital energies/structure, and helps to avoid the unphysical raising of the ground state energy if the F12 geminals are used to correlate pairs of all 1-particle states. The improved F12 geminal design is incorporated into the unitary transcorrelation framework to produce a unitary 2-body Hamiltonian that incorporates the short-range dynamical correlation physics for ground and low-energy excited states in a balanced manner. This explicitly-correlated effective Hamiltonian reduces the basis set requirement on the correlation-consistent basis cardinal number by 1 or more over the uncorrelated counterpart for the description of the ground state coupled-cluster singles and doubles (CCSD) energies, the vertical excitation energies and harmonic vibrational frequencies of equation-of-motion CCSD low-energy excited states.

Paper Structure

This paper contains 9 sections, 16 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Formalism
  3. Hermitian F12 Transcorrelation
  4. Valence-Constrained Geminals
  5. Technical Details
  6. Results
  7. Vertical Excitation Energies
  8. Excited State Properties
  9. Summary

Figures (2)

  • Figure 1: Schematic diagram of the single-particle state spaces used in this text.
  • Figure 2: Pseudo-eigenvalues of the unoccupied orbitals of BH in aug-cc-pVXZ basis sets. The valence orbitals (v) are shown on the left in green while the remaining non-valence (nv) unoccupied orbitals are seen on the right in purple. Notice that the size of the active space is independent of the chosen orbital basis set.