Perturbative corrections to coupled-cluster and equation-of-motion coupled-cluster energies: A determinantal analysis

We develop a combined coupled-cluster (CC) or equation-of-motion coupled-cluster (EOM-CC) theory and Rayleigh–Schrödinger perturbation theory on the basis of a perturbation expansion of the similarity-transformed Hamiltonian $\mathrm{H̄=}\exp \mathrm{(-T)H }\exp \mathrm{(T).}$ This theory generates a series of perturbative corrections to any of the complete CC or EOM-CC models and hence a hierarchy of the methods designated by $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ or $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n).}$ These methods systematically approach full configuration interaction (FCI) as the perturbation order $\mathrm{(n)}$ increases and/or as the cluster and linear excitation operators become closer to complete $\mathrm{(m}$ increases), while maintaining the orbital-invariance property and size extensivity of CC at any perturbation order, but not the size intensivity of EOM-CC. We implement the entire hierarchy of $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ into a determinantal program capable of computing their energies and wave functions for any given pair of m and n. With this program, we perform $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ calculations of the ground-state energies and vertical excitation energies of selected small molecules for all possible values of m and $\text{0⩽n⩽5.}$ When the Hartree–Fock determinant is dominant in the FCI wave function, the second-order correction to CCSD [CC(2)PT(2)] reduces the differences in the ground-state energy between CCSD and FCI by more than a factor of 10, and thereby significantly outperforms CCSD(T) or even CCSDT. The third-order correction to CCSD [CC(2)PT(3)] further diminishes the energy difference between CC(2)PT(2) and FCI and its performance parallels that of some CCSD(TQ) models. $\mathrm{CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ for the ground state with some multideterminantal character and $\mathrm{EOM-CC}\mathrm{(m)}\mathrm{PT}\mathrm{(n)}$ for the excitation energies, however, appear to be rather slowly convergent with respect to n.