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 analyse 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 close to the conical intersection, 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 ${\mathrm{H}}_2\mathrm{O}$ molecule in a strong magnetic field. We furthermore show that symmetry can prevent the occurrence of complex energy values. Lastly, we demonstrate that in case of complex Hamiltonians the real part of the complex energy value provides a very good approximation to the FCI energy.

GRAPHICAL ABSTRACT

Acknowledgments

This paper is dedicated to Professor John Stanton on the occasion of his 60th birthday. One of the authors (J.G.) thanks him for more than 30 years of friendship and intense scientific collaborations which led to the development of the CFOUR program package and about 90 joint publications. The authors thank Professor Martin Hanke-Bourgeois (Johannes Gutenberg-Universität Mainz) for fruitful discussions concerning eigenvalue theory and acknowledge helpful discussions with Marios-Petros Kitsaras (Johannes Gutenberg-Universiät Mainz), Dr. Simen Kvaal (University of Oslo), and Professor Lan Cheng (Johns Hopkins University).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work has been supported by the Deutsche Forschungs-gemeinschaft via grant STO-1239/1-1.