Accurate local approximations to the triples correlation energy: formulation, implementation and tests of 5th-order scaling models

Local models for the triples part of the MP4 or CCSD(T) energy are formulated in terms of atom-labelled functions to describe the occupied and virtual orbital spaces. These models retain triple substitutions in which at most one of the three orbital replacements involves a change of atom. This reduces the number of triple substitutions from scaling with the 6th power of molecule size to scaling with the 4th power, and reduces the computational cost from 7th order to 5th order. Non-locality in the triple substitutions is dominated by terms in which an electron is scattered twice, while the other two singly scattered electrons exhibit non-locality that is similar to that seen in double substitutions. Two non-iterative computational models are designed around this observation. The first, ionic2, allows for non-locality only in the doubly-scattered electron, and recovers around 95% of the triples correlation energy (in the large-molecule limit). The second, ionic*, also approximately accounts for the effect of simultaneous non-locality of the doubly-scattered electron and the singly scattered electrons, and recovers over 99% of the triples energy. The latter yields a maximum error of 0.23 kcal mol(-1) and an RMS error of 0.05 kcal mol(-1) in the MP4/6-31G* triples energies of 179 closed shell molecules from the G3 database. A one-parameter empirical local model is introduced which recovers typically 99.7% of the MP4 triples correlation energy, and reduces the maximum error to 0.03 kcal mol(-1) and the RMS error to 0.01 kcal mol(-1). An implementation of these models is described which manifests the 5th-order scaling of cost with molecule size, without requiring storage of the local triples or the vvvo integrals.