Quantum theory of chemical valence concepts

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Abstract
Electron density operators and projection operators are fundamental quantities in the general physical interpretation of quantum theory. In this series of papers the aim is to explore their relevance to the definition and use of chemical valence concepts. Here, the problems involved in defining the charge on an atom in a molecule are discussed, and a new approach is formulated in terms of these operators. The approach depends on the result that, if a projection operator P is formed representing some subspace of a molecular Hilbert space, then the probability of occupancy of that subspace is Tr DP, where D is an appropriate electron density operator. In particular, the molecular one-electron Hilbert space is considered, and projection operators for atomic orbitals, atoms, pairs of atoms, atoms in threes, and so on, are found. The above result allows for the definition of corresponding occupation numbers. From these follow definitions of the charge on an atom in a molecule and of occupation numbers for electron density shared by pairs of atoms, atoms in threes, etc. The defined occupation numbers may take values subject to certain mathematically derived limits which are in accord with physical intuition. Furthermore, the occupation numbers are invariant to certain transformations commonly made in the course of molecular calculations. The situation within LCAO MO theory, including the rôle of the charge-and-bond-order matrix, is explored. A comparison with Mulliken's population analysis made. The new definitions include terms extra to the latter that are evidently necessary to maintain the limits referred to above. The different concepts involved in using the theory are illustrated by approximate applications to Hartree-Fock molecular orbital calculations.