Third- and fourth-order perturbation corrections to excitation energies from configuration interaction singles

Complete third-order and partial fourth-order Rayleigh-Schrodinger perturbation corrections to excitation energies from configuration interaction singles (CIS) have been derived and termed CIS(3) and CIS(4)(P). They have been implemented by the automated system TENSOR CONTRACTION ENGINE into parallel-execution programs taking advantage of spin, spatial, and index permutation symmetries and applicable to closed- and open-shell molecules. The consistent use of factorization, first introduced by Head-Gordon et al. in the second-order correction to CIS denoted CIS(D), has reduced the computational cost of CIS(3) and CIS(4)(P) from O(n(8)) and O(n(6)) to O(n(6)) and O(n(5)), respectively, with n being the number of orbitals. It has also guaranteed the size extensivity of excited-state energies of these methods, which are in turn the sum of size-intensive excitation energies and the ground-state energies from the standard Møller-Plesset perturbation theory at the respective orders. The series CIS(D), CIS(3), and CIS(4)(P) are usually monotonically convergent at values close to the accurate results predicted by coupled-cluster singles and doubles (CCSD) with a small fraction of computational costs of CCSD for predominantly singly excited states characterized by a 90%-100% overlap between the CIS and CCSD wave functions. When the overlap is smaller, the perturbation theory is incapable of adequately accounting for the mixing of the CIS states through higher-than-singles sectors of the Hamiltonian matrix, resulting in wildly oscillating series with often very large errors in CIS(4)(P). Hence, CIS(3) and CIS(4)(P) have a rather small radius of convergence and a limited range of applicability, but within that range they can be an inexpensive alternative to CCSD.