General technique for analytical derivatives of post-projected Hartree-Fock

TL;DR

This paper develops a general method for calculating analytical derivatives in symmetry-projected Hartree-Fock methods, specifically extending to post-PHF approaches like ECISD, enabling efficient geometry optimization and property calculations.

Contribution

It introduces a novel strategy for post-PHF analytical gradients, bridging the gap between single-reference and multi-reference methods, demonstrated through numerical examples.

Findings

Analytical gradients for post-PHF methods are successfully derived.

The method accurately predicts geometries comparable to multi-reference CI.

Numerical examples confirm the approach's effectiveness for ozone and cyclobutadiene.

Abstract

In electronic structure theory, the availability of analytical derivative is one of the desired features for a method to be useful in practical applications, as it allows for geometry optimization as well as computation of molecular properties. With the recent advances in the development of symmetry-projected Hartree-Fock (PHF) methods, we here aim at further extensions by devising the analytic gradients of post-PHF approaches with a special focus on spin-extended (spin-projected) configuration interaction with single and double substitutions (ECISD). Just like standard single-reference methods, the mean-field PHF part does not require the corresponding coupled-perturbed equation to be solved, while the correlation energy term needs the orbital relaxation effect to be accounted for, unless the underlying molecular orbitals are variationally optimized in the presence of the correlation energy. We present a general strategy for post-PHF analytical gradients, which closely parallels that for single-reference methods, yet addressing the major difference between them. The similarity between ECISD and multi-reference CI not only in the energy but also in the optimized geometry is clearly demonstrated by the numerical examples of ozone and cyclobutadiene.