Orbital Self-Consistent-Field Theory. I. General Theory for Multiconfigurational Wave Functions

A general formulation of multiconfigurational self-consistent-field theory is given. The fundamental condition to be satisfied by the $M$ occupied orbitals ${\psi }_{\mu }$ of an $N\left(\le M\right)$-body system, in order that the total energy of one state be extremalized, is expressed in terms of the fundamental invariant $\mathbf{\varrho }=\Sigma {\mu =1\mathrm{}}^M\left|{\psi }_{\mu }〉〈{\psi }_{\mu }\right|$. The condition placed on $\mathbf{\varrho }$ is of the same form as the condition on the fundamental invariant of the Hartree-Fock theory. Setting $M=N$, we derive the unrestricted Hartree-Fock equations from the condition on $\mathbf{\varrho }$. For $M>N$ there are three freedoms one may take with the fundamental condition on $\mathbf{\varrho }$. Exploiting any or all of these freedoms yields alternative forms of the fundamental condition. This enables us to derive an effective one-body Hamiltonian which is the sum of a Hartree-type Hamiltonian and a correlation and exchange operator. For finite $M>N$, the one-body Hamiltonian contains a nonlocal exchange and correlation operator. This operator is defined in terms of the one- and two-body density matrices. The connection between the orbitals of this theory and those of the Hartree-Fock theory is explored. The theory as outlined here is applicable to any system of $N$ identical particles, but our discussion is oriented towards electronic systems. The theory contains most self-consistent-field theories as special cases, and gives a basis for the self-consistent-field formulation of others.