Low-order scaling local electron correlation methods. III. Linear scaling local perturbative triples correction (T)

A new method for the perturbative calculation of the correlation energy due to connected triple excitations (T) in the framework of local coupled cluster theory is presented, for which all computational resources scale linearly with molecular size. One notable complication in the formalism for connected triples introduced by the local approach is the nondiagonality of the Fock matrix in the localized MO (LMO) and projected AO (PAO) basis, which leads to couplings between individual triples amplitudes via the internal–internal and external–external blocks of the Fock matrix, respectively. Further complications and couplings arise due to the nonorthogonality of the PAOs. While the couplings via the external–external block can easily be dealt with, this is more difficult for the internal–internal couplings. In a previous paper we already published preliminary results of an approximation of the method, which neglects these internal–internal couplings entirely and recovers about 97% of the total local triples correlation energy. In the present work we implemented the “full” local (T) method, which involves the iterative solution of a system of linear equations for the triples amplitudes to take the internal–internal couplings fully into account. Moreover, a further variant of the method was implemented, which approximates the internal–internal couplings at the level of first-order perturbation theory with respect to the off-diagonal elements of the Fock matrix in LMO basis, thus avoiding the need for an iterative solution of the triples equations and storage of the triples amplitudes. The latter variant reliably recovers more than 99% of the full local triples energy. Test calculations with more than 1000 basis functions and over 300 correlated electrons are presented, showing a speedup of about ${10}^6$ relative to the estimated time of a corresponding conventional (T) calculation.