Theory of projected probabilities on non-orthogonal states: Application to electronic populations in molecules

Often it is important to consider the expansion of a quantum state ) in terms of physically meaningful basis states. For example, molecular orbitals can be expressed as linear combinations of atomic orbitals, or vibrational states can be expressed as super positions of local or normal mode eigenstates. In such expansions, it then becomes desirable to determine how much character a quantum state has in one of these basis states. One way of accimplishing this task is to calculate the projected probability of |) on basis state |j). In this paper, we consider this general quantum mechanical problem. If the basis states are orthonormal, then the projected probability of|) on |j) is of course |<|j>|². However, if the basis states are not orthogonal, then this result is no longer valid and one must develop a more general theory to calculate these projected probabilities. An earlier paper used one-dimensional projection operators to initiate this theory and gave closed form results for the case of two non-orthogonal basis states [1]. One- and many-dimensional projection operators, together with linear algebraic techniques, are used to extend this theory to the n non-orthogonal basis state case. Explicit closed form results are given for the two- and three-state cases, and a general algorithm is developed for the case of four or more basis states. Application of the theory is made to atomic populations in three- to six-atom molecules, and comparisons are made to the related work of Mulliken.