Fast computation of molecular RPA correlation energies using resolution-of-the-identity and imaginary frequency integration

Henk Eshuis, Julian Yarkony, and Filipp Furche Department of Chemistry, University of California, Irvine, CA 92697, USA (Dated: February 23, 2010) Abstract The random phase approximation (RPA) is an increasingly popular method to compute postKohn-Sham correlation energies, but its high computational cost has limited molecular applications to systems with few atoms so far. Here we present an efficient implementation of RPA correlation energies based on a combination of resolution-of-the-identity (RI) and imaginary frequency integration techniques. We show that the RI approximation to four-index electron repulsion integrals leads to a variational upper bound to the exact RPA correlation energy if the Coulomb metric is used. Auxiliary basis sets optimized for second-order Møller-Plesset (MP2) calculations are well suitable for RPA, as demonstrated for the HEAT [1] and MOLEKEL [2] benchmark sets. Using imaginary frequency integration rather than diagonalization to compute the matrix square root necessary for RPA, evaluation of the RPA correlation energy requires O(N4 log N) operations and O(N3) storage only; the price for this dramatic improvement over previous implementations is a numerical quadrature. We propose a numerical integration scheme that is exact in the two-orbital case and converges exponentially with the number of grid points. For most systems, 30-40 grid points yield μH accuracy in triple-zeta basis sets, but much larger grids are necessary for small-gap systems. The lowest-order approximation to the present method is a post-Kohn-Sham frequency domain version of opposite-spin Laplace-transform RI-MP2 [3]. Timings for polyacenes with up to 30 atoms show speed-ups of two orders of magnitude over previous implementations.