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On the Hartree-Fock Eigenvalue Problem

Richard A Zalik

TL;DR

The paper addresses the Hartree–Fock eigenvalue problem by introducing a convolution-based reformulation that removes the Laplacian, casting the nonlinear PDE as an algebraic system suitable for new approximation methods. It builds a rigorous framework around harmonic-convolution quantities $\mathfrak{s}_{ac}$, $\mathfrak{p}$, and $\mathfrak{q}$ and proves that, under suitable regularity, convolution commutes with $\nabla^2$, enabling a purely algebraic HF formulation. The authors demonstrate two concrete approaches: a general convolution with a kernel $w$ and a Poisson-kernel-based smoothing $u_{a,t}=\psi_a\ast P_t$, establishing convergence, norm bounds, and key PDE relations that preserve HF content while eliminating the Laplacian. The results point to practical numerical benefits for Roothaan–Hall-type methods and grid-based solvers, offering a mathematically solid pathway to alternative approximate electronic-structure solutions.

Abstract

Using properties of harmonic functions in multidimensional space, we transform the Hartree-Fock eigenvalue problem into a more tractable eigenvalue problem in which the Laplacian is eliminated. This new formulation may facilitate the development of novel ways of finding approximate solutions of the electronic problem.

On the Hartree-Fock Eigenvalue Problem

TL;DR

The paper addresses the Hartree–Fock eigenvalue problem by introducing a convolution-based reformulation that removes the Laplacian, casting the nonlinear PDE as an algebraic system suitable for new approximation methods. It builds a rigorous framework around harmonic-convolution quantities , , and and proves that, under suitable regularity, convolution commutes with , enabling a purely algebraic HF formulation. The authors demonstrate two concrete approaches: a general convolution with a kernel and a Poisson-kernel-based smoothing , establishing convergence, norm bounds, and key PDE relations that preserve HF content while eliminating the Laplacian. The results point to practical numerical benefits for Roothaan–Hall-type methods and grid-based solvers, offering a mathematically solid pathway to alternative approximate electronic-structure solutions.

Abstract

Using properties of harmonic functions in multidimensional space, we transform the Hartree-Fock eigenvalue problem into a more tractable eigenvalue problem in which the Laplacian is eliminated. This new formulation may facilitate the development of novel ways of finding approximate solutions of the electronic problem.
Paper Structure (6 sections, 6 theorems, 69 equations)

This paper contains 6 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminary Results
  4. Convolution
  5. The Poisson Kernel
  6. Conclusion

Key Result

Theorem 1

Let $V$ denote the set of points in $\text{${\mathbb{R}}$}^3$ other than the singular points $\xi_c$; then $\nabla^2\psi_a \in C(V) \cap L_2$. If $\psi_a$ is also in $L_1$ (or if $\lim_{x\rightarrow \infty}\psi_a(x)=0$), then $\nabla^2\psi_a$ is in $L_1$ as well.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • Theorem 6
  • ...and 1 more