Variational formulation of perturbative explicitly-correlated coupled-cluster methods

We present a variational formulation of the recently-proposed CCSD(2)R(12) method [Valeev, Phys. Chem. Chem. Phys., 2008, 10, 106]. The centerpiece of this approach is the CCSD(2)(R12) Lagrangian obtained via Lowdin partitioning of the coupled-cluster singles and doubles (CCSD) Hamiltonian. Extremization of the Lagrangian yields the second-order basis set incompleteness correction for the CCSD energy. We also developed a simpler Hylleraas-type functional that only depends on one set of geminal amplitudes by applying screening approximations. This functional is used to develop a diagonal orbital-invariant version of the CCSD(2)((R12) over bar) method in which the geminal amplitudes are fixed at the values determined by the first-order cusp conditions. Extension of the variational d-CCSD(2)((R12) over bar) method to include perturbatively the effect of connected triples produces the d-CCSD(T)((R12) over bar) method that approximates the complete basis-set limit of the standard CCSD plus perturbative triples [CCSD(T)] method. For a set of 20 small closed-shell molecules, the d-CCSD(T)((R12) over bar) method recovered at least 94.5/97.3% of the CBS CCSD(T) correlation energy with the aug-cc-pVDZ/aug-cc-pVTZ orbital basis set. For 12 isogyric reactions involving these molecules, combining the aug-cc-pVTZd-CCSD(T)((R12) over bar) correlation energies with the aug-cc-pVQZ Hartree-Fock energies produces the electronic reaction energies with a mean absolute deviation of 1.4 kJ mol(-1) from the experimental values. The d-CCSD(T)((R12) over bar) method has the same number of optimized parameters as the corresponding CCSD(T) model, does not require any modi. cation of the coupled-cluster computer program, and only needs a small triple-zeta basis to match the precision of the considerably more expensive standard quintuple-zeta CCSD(T) computation.