Critical points of the discretized Hartree-Fock functional of connected molecules preserving structures of molecular fragments

TL;DR

This paper introduces a Newton's method-based approach on the Grassmann manifold to find critical points of the discretized Hartree-Fock functional, preserving molecular fragment structures during molecular assembly.

Contribution

It develops a differential calculus on the Grassmann manifold with a new retraction, enabling the construction of molecular critical points from fragment critical points.

Findings

Method preserves electronic structures of molecular fragments.

Provides error estimates for Newton's method in this context.

Constructs critical points for connected molecules from fragment data.

Abstract

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.