The density matrix in many-electron quantum mechanics I. Generalized product functions. Factorization and physical interpretation of the density matrices

Many-electron wave functions are usually constructed from antisymmetrized products of one-electron orbitals (determinants) and energy calculations are based on the matrix element expressions due to Slater (1931). In this paper, the orbitals in such a product are replaced by ‘group functions’, each describing any number of electrons, and the necessary generalization of Slater’s results is carried out. It is first necessary to develop the density matrix theory of N -particle systems and to show that for systems described by ‘generalized product functions’ the density matrices of the whole system may be expressed in terms of those of the component electron groups. The matrix elements of the Hamiltonian between generalized product functions are then given by expressions which resemble those of Slater, the ‘coulomb’ and ‘exchange’ integrals being replaced by integrals containing the one-electron density matrices of the various groups. By setting up an ‘effective’ Hamiltonian for each electron group in the presence of the others, the discussion of a many-particle system in which groups or ‘shells’ can be distinguished (e. g. atomic K, L, M , ..., shells) can rigorously be reduced to a discussion of smaller subsystems. A single generalized product (cf. the single determinant of Hartree—Fock theory) provides a convenient first approximation; and the effect of admitting ‘excited’ products (cf. configuration interaction) can be estimated by a perturbation method. The energy expression may then be discussed in terms of the electon density and ‘pair’ functions. The energy is a sum of group energies supplemented by interaction terms which represent (i) electrostatic repulsions between charge clouds, (ii) the polarization of each group in the field of the others, and (iii) ‘dispersion’ effects of the type defined by London. All these terms can be calculated, for group functions of any kind, in terms of the density matrices of the separate groups. Applications to the theory of intermolecular forces and to π -electron systems are also discussed.