Spin-coupled theory for 'N electrons in M orbitals' active spaces

Spin-coupled (SC) theory, an ab initio valence bond (VB) approach which uses a compact and an easy-to-interpret single-orbital product wave function comparable in quality to a ‘N in N’ complete-active-space self-consistent field CASSCF(N,N)] construction, is extended to ‘N in M’ (N ≠ M) active spaces. The SC(N,M) wave function retains the essential features of the original SC model: It involves just the products of nonorthogonal orbitals covering all distributions of N electrons between M orbitals in which as few orbitals as possible, |N – M|, are doubly occupied (for N > M) or missing (for N < M) and all other orbitals are singly occupied; each of these products is combined with a flexible spin function which allows any mode of coupling of the spins of the orbitals within the product. The SC(N,M) wave function remains much more compact than a CASSCF(N,M) construction; for example, the SC(6,7) wave function includes 35 configuration state functions (CSFs) as opposed to the 490 CSFs in the CASSCF case. The essential features of the SC(N,M) method are illustrated through a SC(6,5) calculation on the cyclopentadienyl anion, C5H5(–), and a SC(6,7) calculation on the tropylium cation, C7H7(+). The SC(6,5) and SC(6,7) wave functions for C5H5(–) and C7H7(+) are shown to provide remarkably clear modern VB models for the electronic structures of these aromatic cyclic ions which closely resemble the well-known SC model of benzene and yet recover almost all of the correlation energy included in the corresponding CASSCF(6,5) and CASSCF(6,7) wave functions: over 97% in the case of C5H5(–) and over 95% in the case of C7H7(+).