The density matrix that leads to a minimum kinetic energy for a given density is considered as a convex superposition of pure states. It is shown that the conditions of stationarity of the kinetic energy and collapse to the given density require that each of the pure state wave functions involved be a single determinant in the same eigenspace of a particular, n-electron Hamiltonian and that all of the orbitals are eigenfunctions of the same effective one-electron Hamiltonian. The potential function arises originally as a Lagrange multiplier associated with the density constraint. In some cases it can (at least in principle) be determined. The role of electron-electron interactions and possible treatment of excited states are considered. © 1986 American Institute of Physics.