First principles derivation of the effective valence shell Hamiltonian for large molecules

We construct an effective Hamiltonian for the valence shell of a large electronic system. The procedure begins by classifying the complete one electron space according to core, valence, and excited orbitals. An N−electron subspace 𝒯N spanned by Slater determinants with a fixed set of Nc core orbitals and different sets of Nv = (N − Nc) valence orbitals is defined. A canonical transformation on the Coulomb Hamiltonian is used to eliminate the interaction between 𝒯N and its orthogonal complement, ?N, thereby defining an effective N−electron Hamiltonian. This effective Hamiltonian is expanded in a cluster development of linked one−, two−, three−,⋅⋅⋅body operators in terms of which the conditions of having vanishing matrix elements between 𝒯N and ?N can be explicitly formulated. Then starting with this effective N−electron Hamiltonian we construct an equivalent Hamiltonian to operate in the space of antisymmetrized products of valence orbitals. To within a constant (times the identity on the valence space) this equivalent Hamiltonian—dubbed the valence shell Hamiltonian—has the same matrix elements between antisymmetrized products of valence orbitals as does the effective N−electron Hamiltonian on 𝒯N. This formalism is then applied to generate an effective pi−electron Hamiltonian for planar conjugated molecules. We conclude by deriving explicit formulas for the transformed dipole transition operators.