Hybrid coupled-cluster and perturbation method for extended systems of one-dimensional periodicity

A hybrid of the coupled-cluster singles and doubles (CCSD) and second-order Møller-Plesset perturbation (MP2) methods [M. Nooijen, J. Chem. Phys. 111, 10815 (1999); A. D. Bochevarov and C. D. Sherrill, ibid. 122, 234110 (2005); A. D. Bochevarov et al., ibid. 125, 054109 (2006)] is formulated and implemented for one-dimensional periodic extended systems, in which the excitation (T) amplitudes of active bands are determined iteratively by CCSD, while the T amplitudes of mixed active/inactive bands are held fixed at the first-order Møller-Plesset perturbation values. The occupied and virtual bands near the Fermi level, which can cause instability in MP2 when they are (quasi-)degenerate, are selected as active bands to be treated by CCSD, which can, in principle, resist such instability. Two contraction schemes of the T amplitudes (Contractions A and B) are considered. Contraction A is the one proposed for molecules and used also for extended systems because it is efficient for CCSD, but not necessarily so for the hybrid CCSD/MP2. Contraction B is introduced to be more optimally efficient for the hybrid CCSD/MP2 by maximizing the number of intermediate quantities made of the inactive T amplitudes and molecular integrals, which do not vary during CCSD iterations and are computed only once, stored, and reused. In an application to trans-polyacetylene, a smooth transition of the results of the hybrid CCSD/MP2 is observed toward those of CCSD and MP2 by increasing and decreasing, respectively, the number of active bands. With the smallest active space, the hybrid CCSD/MP2 with Contractions A and B achieves a speedup by a factor of 360 and 520, respectively, relative to CCSD. When all of the occupied bands and about half of the virtual bands are active, the hybrid CCSD/MP2 can recover 98% of the CCSD correlation energy or half of the difference between CCSD and MP2 at less than a tenth of the usual CCSD cost.