In molecular theory the wave function is usually constructed from antisymmetrized products, or ‘Slater determinants’, of one-electron ‘orbitals’. A single determinant of suitably chosen,
doubly
occupied orbitals is often a fair approximation to a singlet ground state; but when more general products are admitted, as in ‘configuration interaction’ calculations, it is first necessary to resolve a high ‘spin degeneracy’ by constructing spin eigenfunctions (SE’s). In §1, the fundamental basis of recent methods (McWeeny 1954
b
) is clarified by a group theoretical approach. Next, in §2, the energy expression, using as wave function an arbitrary mixture of similar SE’s, is written very simply in terms of the reduced density matrices for one and two particles, and formulae for the calculation of these matrices are given. The remaining problem is to get a ‘best’ wave function, usually with limited configuration interaction, by (i) variation of SE coefficients and (ii) variation of the orbitals appearing in the SE’s; this problem is formally solved in §3. (i) is the usual configuration interaction process; but (ii) is new and leads, when the orbitals are expressed in terms of a standard basic set (e.g. of atomic orbitals), to a complete generalization of the Roothaan 1951) equations. These (matrix) equations are simple in appearance, but their numerical solution calls for new techniques; and it is possible that the Roothaan (i.e. Hartree–Fock) approach, followed by configuration interaction, provides about the best working compromise between (i) and (ii). In §4, some points of contact between one- and many-configuration theories are noted. In particular, certain density matrix elements provide appropriate generalizations of the ‘charges’ and ‘bond orders’ of Coulson and Longuet-Higgins and continue to describe the response of a system to changes in its ‘Coulomb’ and ‘resonance’ integrals.
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