Critical point search and linear response theory for computing electronic excitation energies of molecular systems. Part II. CASSCF

TL;DR

This paper extends the geometric formalism of CASSCF to derive linear response equations for excited states, introducing a state-specific method and demonstrating its effectiveness on molecular systems.

Contribution

It develops a geometrical framework for CASSCF excited state calculations, deriving new linear response equations and a state-specific method based on first-order derivatives.

Findings

The state-specific method is effective for molecular excited states.

Numerical tests on water, formaldehyde, and ethylene validate the approach.

Identifying excited states reliably remains challenging due to nonlinearity.

Abstract

The computation of excited states within the Complete Active Space Self-Consistent Field (CASSCF) framework remains a significant challenge in quantum chemistry, both theoretically and algorithmically. In this work, we extend the K\"ahler manifold formalism introduced in Part I of this series to the CASSCF theory, and draw a geometrical connection from the time-dependent CASSCF equations to state-specific and linear response methodologies for excited states. This is achieved by first investigating the underlying CASSCF manifold and identifying its K\"ahler structure, which is complicated by the nontrivial coupling of CI and orbital degrees of freedom. Building on these theoretical findings, we derive the CASSCF linear response equations in a straightforward manner, and develop a robust state-specific method that relies solely on first-order derivatives of the CASSCF energy functional. Numerical results on representative molecular systems-water, formaldehyde, and ethylene-demonstrate the effectiveness of the proposed state-specific method, while revealing the difficulty of reliable identification of excited states due to nonlinearity induced by the CASSCF theory.