Linear scaling local coupled cluster theory with density fitting.: Part I:: 4-external integrals

The density fitting approximation is applied to the most expensive class of 2-electron integrals in local CCSD, i.e., to those integrals that involve four virtual orbitals ( or projected AOs). The fitting error in the correlation energy is systematic and considerably smaller than the deviation between the local and the canonical CCSD energy. In order to restore O(N) scaling locality must be exploited for the fitting functions as well as for orbitals. Local fitting domains specified for individual centre pairs provide an adequate basis for such a local description, however, Dunlap's robust formula for the approximate integrals then no longer simplifies to the usual expression known as the V approximation. A symmetric formula is proposed as an alternative, which, although formally non-robust, yields virtually the same results as the robust formalism. The additional fitting error due to the introduction of local fitting domains is considerably smaller than the original fitting error itself ( by at least an order of magnitude). Test calculations demonstrate O( N) scaling for the new LDF-LCCSD method. The approximate calculation of the 4-external integrals via density fitting in LDF-LCCSD is 10-100 times faster than the exact calculation via the O(N) 4-index transformation in LCCSD.