Configuration interaction in a basis of biorthogonal states

This article develops techniques for configuration interaction in a nonorthogonal basis. Hamiltonian operator matrix elements such as 〈φ̃u‖Ĥ‖φv〉 are easily evaluated when biorthogonal orbitals are used for the bra and ket configuration states. The ket configuration state ‖φv〉 is constructed from a basis of orbitals {Xi} and the bra configuration state 〈φ̃u‖ employs the dual orbitals {X̃i}. The dual basis is defined by contraction of the original basis with the inverse overlap matrix: X̃i(r)= ∑jXj(r)S−1ji. A configuration interaction wave function ‖ψ〉 then corresponds to a stationary point of the modified energy functional E′=〈ψ̃‖Ĥ‖ψ〉/〈ψ̃‖ψ〉. This energy functional becomes exact as either ‖ψ〉 or ‖ψ̃〉 approaches an exact eigenstate of the Schrödinger equation. The wave function ‖ψ̃〉 is expanded with respect to the dual basis, whereas the original configuration state basis is employed for expansion of ‖ψ〉. The energy E′ satisfies the variation theorem if and only if the dual basis is related to the original basis by a linear transformation. Such a transformation exists when a full CI is done with respect to the manifold defined by nonorthogonal active orbitals. However, one can retain nonorthogonality among those orbitals which are doubly occupied in all configurations without invalidating the variational stability. Configuration interaction in a nonorthogonal basis can be efficiently done in direct CI mode. Modification of the second-quantization formalism shows that the structure constants of the direct CI are identical with those that would pertain to a nonorthogonal basis. The dual basis that is used for the bra states modifies the integral transformation, but otherwise has no effect.