Analytic energy gradients for general coupled-cluster methods and fourth-order many-body perturbation theory

Energy gradient equations are presented for the coupled-cluster model with all possible excitations. By taking advantage of the equations for the coupled-cluster amplitudes, the gradient formulas may be expressed without explicit reference to the first-order changes in the amplitudes, in contrast to all earlier work. The coupled-cluster doubles (CCD) and coupled-cluster singles, doubles, and triples (CCSDT) models are treated as special cases of the general theory. Finally, by limiting the model to finite orders in perturbation theory, the gradient equations for the full fourth-order many-body perturbation energy are derived. Like the fourth-order energy itself, the gradient procedure is shown to be an n7 process in the number of basis functions. The computational implementation of this fourth-order energy gradient is discussed in detail.