Applications of Löwdin’s Metric Matrix: Atomic Spectral Methods for Electronic Structure Calculations

Abstract

Löwdin’s metric matrix constructed in the non-orthogonal antisymmetrized atomic spectral-product basis of Moffitt is employed in developing a new theoretical approach to ab initio calculations of the adiabatic (Born-Oppenheimer) electronic wave functions and potential energy surfaces of molecules and other atomic aggregates. The metric matrix is used to demonstrate that Momtt’s basis in the absence of prior antisymmetrization contains the totally antisymmetric irreducible representation of the symmetric group of aggregate electron coordinate permutations once and only once, and so is suitable for calculations of physical eigenstates. The unphysical representations of the symmetric group also spanned by the simple spectral-product basis are eliminated from the Hamiltonian matrix, and its physical block isolated, after its construction in this representation employing the unitary transformation obtained from diagonalization of the metric matrix. Löwdin’s observations on restrictions to ordered electron configurations in configuration-interaction expansions help to clarify the origins of linear dependence in the Moffitt basis, and to demonstrate the equivalence of its canonical orthogonalization with isolation of the totally antisymmetric representation in the corresponding orthonormal spectral-product basis. Convergence of the new approach accordingly follows in the limit of closure to wave functions and energies identical with those obtained employing prior antisymmetrization in the absence of linear dependence. The particularly simple form of the spectral-product Hamiltonian, composed of atomic and atomic pair-interaction energy matices which can be determined once and for all for repeated use, suggests the method has potential merit as a viable computational approach. Aspects of its implementation are explored with numerical examples to illustrate the convergence of the method.