Statistical Theory of Many-Electron Systems. General Considerations Pertaining to the Thomas-Fermi Theory

The density matrix for a many-electron system has been examined with a view toward extending the statistical theory of Thomas and Fermi. The single-particle density matrix is expanded in terms of a power series in $\hslash$, according to a procedure developed here for operators consisting of a series of powers of the classical Hamiltonian, the zeroth-order terms corresponding to the Thomas-Fermi theory. Second-order terms in the expansion are identifiable in the expression for the energy: they correspond to exchange energy, the Weizsäcker inhomogeneity correction for the kinetic energy and, possibly, correlation energy. Their presence follows from the expansion procedure, rather than from an a priori insertion into the theory.