The density matrix in self-consistent field theory I. Iterative construction of the density matrix

Abstract
The approximate self-consistent field equations of Roothaan (1951) are studied from the standpoint of density matrix theory. Their direct solution, which entails repeated determination of the individual occupied orbitals, is shown to be quite unnecessary. All information about the system under consideration can be obtained in terms of a one-particle density matrix; and this can be constructed by an iterative method which involves only matrix algebra. As an illustration of the method, the ground state of the beryllium atom is calculated. The solution, which is approximated in a basis of four Slater functions, surpasses in accuracy all other analytical one-determinant wave functions so far available, with one exception (Holoien, 1955). And although it approaches the accurate solution of the Hartree-Fock equations the computation required is relatively small. Extensions of the method, in order to admit configuration-interaction calculations, are also discussed. An appendix introduces a 'steepest descent' method of solving eigenvalence problems, which may be regarded as the prototype of the iterative construction described in the paper.