Some Theorems Concerning Symmetry, Angular Momentum, and Completeness of Atomic Geminals with Explicit r12 Dependence

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Abstract
Several results are presented which have important implications for the calculation of many-electron wavefunctions. Electron-pair functions are constructed to be eigenfunctions of the two-electron inversion, permutation, and angular-momentum operators with quantum numbers L and M. For a given L and M there exist 2L+1 particular linear combinations of binary products of spherical harmonics called generators. These 2L+1 generators multiplied by members of a complete set of S-type geminals constitute a complete set of geminal basis functions with that angular momentum. This approach establishes a connection between a formulation in terms of Eulerian angles developed by Wigner and others, and analyses based on configuration interaction expansions. One of several possible forms for the complete set of S-type geminals employs only 1s-type orbitals and a correlation factor, R(r12). Mixing of symmetry may occur when a strongly orthogonal geminal is projected out from a symmetry-adapted one. This problem is easily circumvented. It is shown that any geminal is annihilated by an operator which projects out the component strongly orthogonal to a particular geminal with exponential r12 dependence.