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

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.

Unitary coupled-cluster based self-consistent polarization propagator theory: a quadratic unitary coupled-cluster singles and doubles scheme

“…CC calculations in combination with complex Hamiltonians, e.g., the Hamiltonian in magnetic fields and/or including spin-orbit coupling have been shown to produce complex ground-state energies. 30 This is a non-trivial formal problem of the standard CC theory, although the real part of the complex CC energy is expected to serve as an accurate approximation to the full configuration interaction energy. Further, EOM-CC calculations have been demonstrated to have incorrect crossing conditions for intersections between electronic states of the same symmetry (known as "same-symmetry conical intersection").…”

“…Further, EOM-CC calculations have been demonstrated to have incorrect crossing conditions for intersections between electronic states of the same symmetry (known as "same-symmetry conical intersection"). [30][31][32][33] To enable CC calculations of same-symmetry conical intersections, Köhn and Tajti 32 have proposed a simple correction to obtain physically meaningful potential energy surfaces around conical intersections.…”