A pseudo-wavefunction description of time-dependent Hartree-Fock (TDHF) states is proposed and used to develop an analytic expression for derivative couplings between TDHF excited states based On the Hellmann-Feynman theorem. The resulting expression includes Pulay terms associated with using an atom-centered basis as well as a correction to ensure translational invariance. We demonstrate that our formalism recovers the well-known Chernyak-Mukamel expression near a crossing and in the limit of a complete basis, and thus our approach is consistent with time-dependent response theory. In a companion paper (DOI 10.1021/jp5057682), we investigate these derivative couplings near conical intersections and show that they behave correctly.