Metastable electronic states can be characterized by a complex Siegert energy,
where the width, , is proportional to the inverse lifetime of the state: . Complex coordinate methods aim to compute this complex energy as an eigenvalue of an effective non-Hermitian Hamiltonian. One such method is the method of complex basis functions (CBFs) where a basis of Gaussians with complex exponents is used in conjunction with a symmetric (not complex-conjugated) inner product to effictively produce a finite-basis representation of a non-Hermitian operators.770, 1204, 1205, 1203 In cases, such as temporary anions, where the decay channel is of 1-electron character, a mean-field theory can provide approximate Siegert energies for a many-electron system.
The simplest such approximation is the static-exchange approximation. In this approximation the Siegert energies of an ()-electron state are computed by diagonalizing a Fock operator computed from the density of an -electron state.1204 This approximation neglects orbital relaxation effects which can be included by a non-Hermitian self-consistent-field (NH-SCF) procedure.770, 1205 In practice the NH-SCF energy functional is the same as the Holomorphic Hartree-Fock energy functional (Eq. 4.64), though it is used for a different purpose. Both static-exchange and NH-SCF theories using complex basis functions (CBFs) are available in Q-Chem. Specification of the complex basis set is described in Section 8.7.