Unexpected constraint implicit in the time-dependent Hartree-Fock equations

It is shown that the time-dependent Hartree-Fock equations have an unexpect ed constraint implicitly inherent in them. For the case of two electrons in teracting via the Hamiltonian H(1,2) = H-0(1) + H-0(2) + V(1,2), and descri bed by the time-dependent Schrodinger equation i Psi(1,2) = H(1,2)Psi(1, 2) , (where the over-dot stands for time derivative) it will be shown that the time-dependent orbitals defined by the ansatz Psi(1,2) = phi(1)(1)phi(2)(2 ) +/- phi(1)(2)phi(2)(1), not only must satisfy the two coupled orbital equ ations with the two time-dependent mean-field potentials, but also implicit ly must satisfy their own one-particle equations i phi(k)(j) = H-0(j)phi(k) (j). Unlike stationary-state methods, this represents a constraint which is not consistent with a time-dependent ab initio theory, but may be appropri ate for a perturbative theory. (C) 2000 Elsevier Science B.V. All rights re served.