  In this paper a method to obtain a critical point of the discretized
Hartree-Fock functional from an approximate critical point is given. The method
is based on Newton's method on the Grassmann manifold. We apply Newton's method
regarding the discretized Hartree-Fock functional as a function of a density
matrix. The density matrix is an orthogonal projection in the linear space
corresponding to the discretization onto a subspace whose dimension is equal to
the number of electrons. The set of all such matrices are regarded as a
Grassmann manifold. We develop a differential calculus on the Grassmann
manifold introducing a new retraction (a mapping from the tangent bundle to the
manifold itself) that enables us to calculate all derivatives. In order to
obtain reasonable estimates, we assume that the basis functions of the
discretization are localized functions in a certain sense. As an application we
construct a critical point of a molecule composed connecting several molecules
using critical points of the Hartree-Fock functional corresponding to the
molecules as the basis functions under several assumptions. By the error
estimate of Newton's method we can see that the electronic structures of the
molecular fragments are preserved.
