Complex Ground-State and Excitation Energies in Coupled-Cluster Theory

TL;DR

This paper investigates the conditions under which coupled-cluster theory yields complex energies, analyzing mathematical properties and demonstrating how real or complex energies arise in different scenarios, with implications for physical accuracy.

Contribution

It provides a mathematical analysis of complex energies in coupled-cluster calculations and shows how symmetry and Hamiltonian properties influence their occurrence.

Findings

Ground-state energies are generally real for real Hamiltonians.

Complex excitation energies occur mainly at conical intersections.

Symmetry can prevent the emergence of complex energies.

Abstract

Since in coupled-cluster (CC) theory ground-state and excitation energies are eigenvalues of a non-Hermitian matrix, these energies can in principle take on complex values. In this paper we discuss the appearance of complex energy values in CC calculations from a mathematical perspective. We analyze the behaviour of the eigenvalues of Hermitian matrices that are perturbed (in a non-Hermitian manner) by a real parameter. Based on these results we show that for CC calculations with real-valued Hamiltonian matrices the ground-state energy generally takes a real value. Furthermore, we show that in the case of real-valued Hamiltonian matrices complex excitation energies only occur in the context of conical intersections. In such a case, unphysical consequences are encountered such as a wrong dimension of the intersection seam, large numerical deviations from full configuration-interaction (FCI) results, and the square-root-like behaviour of the potential surfaces near the conical intersection. In the case of CC calculations with complex-valued Hamiltonian matrix elements, it turns out that complex energy values are to be expected for ground and excited states when no symmetry is present. We confirm the occurrence of complex energies by sample calculations using a six-state model and by CC calculations for the H2O molecule in a strong magnetic field. We furthermore show that symmetry can prevent the occurrence of complex energy values. Lastly, we demonstrate that in most cases the real part of the complex energy values provides a very good approximation to the FCI energy.