In Hartree-Fock theory and its various generalizations, it is customary to solve an eigenvalue problem involving an effective one-body Hamiltonian. The eigenvectors determine the Fock-Dirac density matrix, which also appears in the effective Hamiltonian, and solution proceeds iteratively until self-consistency is achieved.

An alternative (necessary and sufficient) condition for a solution is that the density matrix ($\rho$) is idempotent and commutes with the Hamiltonian (h). The change in $\rho$, accompanying a change $\Delta$ in h, can then be expressed as a perturbation series. Formulas for the perturbation, to all orders, are obtained in terms of the unperturbed Hamiltonian and density matrix. It is also shown that the whole perturbation may be obtained directly, without separating the orders, and that the approach is related to earlier steepest-descent methods.

• Received 16 October 1961

DOI:https://doi.org/10.1103/PhysRev.126.1028