Direct Determination of Natural Orbitals and Natural Expansion Coefficients of Many-Electron Wavefunctions. I. Natural Orbitals in the Geminal Product Approximation

Abstract

The approximation of a many‐particle wavefunction as an antisymmetrized product of strongly orthogonal geminals (APSG) (geminal means two‐electron function) is analyzed in terms of its natural expansion. APSG wavefunctions have very simple and interesting properties. The ``generating spin geminals'' (GSG) are automatically natural spin geminals (NSG) of the system with Occupation Number 1 (in the Löwdin normalization); 1 is the upper bound for the eigenvalues of the two‐particle density matrix for wavefunctions of this type. The natural spin orbitals (NSO) of the GSG are automatically NSO of the total wavefunction and of all the NSG. A generalized ``paired orbital function'' is defined in terms of its natural expansion. It contains the APSG type as well as the Bardeen—Cooper—Schrieffer (BCS) functions as special cases. Integrodifferential equations determining the NSO and their expansion coefficients for an APSG function are derived for the special case of a singlet state by minimizing the total electronic energy. An iterative resolution procedure is proposed. When the state under consideration can be regarded as a closed‐shell state the first iteration step may already be a good approximation. In this case the equations become very simple and can be given a straightforward physical interpretation. Finally possible refinements of the theory are discussed.