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

Complete third-order and partial fourth-order Rayleigh-Schrödinger 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.