Transformation Properties of Antisymmetric Spin Eigenfunctions under Linear Mixing of the Orbitals

Abstract

After recalling the duality between the general linear group GL(m), represented by its N-fold inner product, and the permutation group SN, we have given a survey of its quantum chemical consequences. It causes the one-to-one correspondence between the total spin quantum number and the permutation symmetry of N-electron spin functions, and, via the Pauli principle which imposes permutation symmetry on the spatial part also, it leads to specific properties of antisymmetric spin eigenfunctions under orbital transformations. Such functions can be classified according to the irreducible representations of GL(m). For special orbital transformations, often occurring in quantum chemistry, which mix only orbitals in different subsets among each other, we have derived how the transformation of the N-electron wavefunctions simplifies, by a reduction of the representations of GL(m). The theory is illustrated by an example and some applications are discussed.