Cluster perturbation theory. VI. Ground-state energy series using the Lagrangian

We have extended cluster perturbation (CP) theory to comprehend the Lagrangian framework of coupled cluster (CC) theory and derived the CP Lagrangian energy series (L_CP) where the 2n + 1/2n + 2 rules for the cluster amplitudes and multipliers are used to get the energy corrections. We have also developed the variational CP (L_CP) series, where the total cluster amplitudes and multipliers are determined through the same orders as in the L_CP series, but the energy is obtained by inserting the total cluster amplitudes and multipliers in the Lagrangian. The energies of the L_CP series have errors that are bilinear in the errors of the total cluster amplitudes and multipliers. Test calculations have been performed for S(D) and SD(T) orbital excitation spaces. With the exception of molecular systems that have a low lying doubly excited state compared to the electronic ground state configuration, we find that the fourth order models L_{CPS (D-4)}, L_{CPSD (T-4)}, and L_CPSD(T-4) give energies of CC target state quality. For the L_{CPS (D-4)} model, CC target state quality is obtained as the L_{CPS (D-4)} calculation determines more than 99.7% of the coupled cluster singles and doubles (CCSD) correlation energy as the numerical deviations of the L_{CPS (D-4)} energy from the CCSD energy were more than an order of magnitude smaller than the triples correlation contribution. For the L_{CPSD (T-4)} and L_CPSD(T-4) models, CC target state quality was obtained, given that the L_{CPSD (T-4)} and L_CPSD(T-4) calculations recover more than 99% of the coupled cluster singles doubles and triples (CCSDT) correlation contribution and as the numerical deviations of the L_{CPSD (T-4)} and L_CPSD(T-4) energies from the CCSDT energy were nearly and order of magnitude smaller than the quadruples correlation contribution. We, thus, suggest that the fourth order models may replace the full target CC models with no or very limited loss of accuracy.