  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.
