Many-body methods for excited state potential energy surfaces. I. General theory of energy gradients for the equation-of-motion coupled-cluster method

The formal theory is presented for calculating the analytic first derivative of the energy with respect to arbitrary perturbations within the equation-of-motion coupled-cluster (EOM-CC) approximation. Through use of the Dalgarno–Stewart interchange theorem (Z-vector method), terms involving derivatives of the ground state cluster amplitudes are eliminated, leading to the definition of a new quasiparticle de-excitation operator which simplifies the theory and significantly reduces the expected cost associated with studying potential energy surfaces for excited electronic states. For both illustrative and pragmatic reasons, the final equations are cast in a form similar to that developed for ground state CC energy derivatives, involving contraction of effective one- and two-particle density matrices with matrix elements of the differentiated Hamiltonian. Some aspects regarding calculation of the gradient are discussed with particular attention devoted to similarities between the structure of the present formulas and those which have been previously implemented for the ground state problem.