Application of complete space multireference many-body perturbation theory to N2: Dependence on reference space and H0
Authors: Finley James P. , Freed Karl F. 
Published: July 2002

We study the wide latitude available in choosing the reference space and the zeroth order Hamiltonian H0 for complete reference space multireference perturbation theory. This effective Hamiltonian Heff method employs a general one-body form of H0 which is varied by using different molecular orbitals and orbital energies. An energy gap is imposed between the zeroth order reference and secondary space states by forcing the valence orbitals to be degenerate. The forced valence orbital degeneracy removes the occurrence of detrimentally small perturbation energy denominators. Extensive computations are provided for the nitrogen molecule, where calculated ground state spectroscopic constants are compared with full configuration interaction computations and calculated vertical excitation energies are compared with multireference coupled cluster computations. It is demonstrated that the forced reference space degeneracy can lead to certain perturbation denominators becoming too small for practical convergence. This characteristic is illustrated by a simple two-orbital model which stresses the need for important zeroth order excitation energies (equivalent to the factors appearing in the perturbation energy denominators) to exceed those in an Epstein–Nesbet perturbation partitioning. This simple model illustrates the general behavior found in all the more extensive Heff computations.
In many cases where zeroth order excitation energies are too small for satisfactory third order results, improvements are obtained by using an H0 which redefines the orbital energies in order to increase problematic zeroth order excitation energies. The necessary orbital energy shifts are identified by examining the first order wave functions from larger reference spaces and the zeroth order energies. Frequently, fractional occupancy Fock-type operators are employed to provide the requisite orbital energy shifts. Some of the reference spaces investigated deviate extremely from quasidegeneracy and, thus, appear to be beyond the range of applicability of the forced degeneracy Heff method. Novel techniques are employed for properly treating some of these cases, including the use of orbitals which optimize the quasidegeneracy of the reference space and minimize energy denominator problems. By considering reference spaces of varying sizes, we describe the tradeoff between employing large reference spaces, which provide excellent first order descriptions, and the difficulties imposed by the fact that larger reference spaces severely violate the quasidegeneracy constraints of the Heff method. The same tradeoff exists when the optimal first order CASSCF orbitals are compared with orbitals generated by a VN−1 potential.
The VN−1 potential orbitals, which produce relatively quasidegenerate reference spaces, are equivalent to the sequential SCF orbitals used in previous Heff computations, but are more simply obtained by a unitary transformation. The forced degenerate valence orbital energy εv̄ is computed from an averaging scheme for the valence orbital energies. The ground state N2 computations contrast two averaging schemes—populational and democratic. Democratic averaging weighs all valence orbitals equally, while populational averaging weighs valence orbitals in proportion to their ground state populations. Populational averaging is determined to be useful only in situations where core–core and core–valence correlation are unimportant. A Fock-type operator used by Roos and co-workers is employed to uniquely define CASSCF orbitals within their invariant subspaces. This operator is found to be more compatible with populational than democratic averaging, especially when the reference space contains high lying orbitals.