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Riemannian Optimization and the Hartree-Fock Method

Caio O. da Silva

TL;DR

Some of the main algorithms described in the literature are implemented to solve an optimization problem known as Hartree-Fock, a problem somewhat hard to solve and, as a consequence of this, it requires many tools from Riemannian Optimization.

Abstract

In the present work we studied a subfield of Applied Mathematics called Riemannian Optimization. The main goal of this subfield is to generalize algorithms, theorems and tools from Mathematical Optimization to the case in which the optimization problem is defined on a Riemannian manifold. As a case study, we implemented some of the main algorithms described in the literature (Gradient Descent, Newton-Raphson and Conjugate Gradient) to solve an optimization problem known as Hartree-Fock. This method is extremely important in the field of Computational Quantum Chemistry and it is a good case study because it is a problem somewhat hard to solve and, as a consequence of this, it requires many tools from Riemannian Optimization. Besides, it is also a good example to see how these algorithms perform in practice.

Riemannian Optimization and the Hartree-Fock Method

TL;DR

Some of the main algorithms described in the literature are implemented to solve an optimization problem known as Hartree-Fock, a problem somewhat hard to solve and, as a consequence of this, it requires many tools from Riemannian Optimization.

Abstract

In the present work we studied a subfield of Applied Mathematics called Riemannian Optimization. The main goal of this subfield is to generalize algorithms, theorems and tools from Mathematical Optimization to the case in which the optimization problem is defined on a Riemannian manifold. As a case study, we implemented some of the main algorithms described in the literature (Gradient Descent, Newton-Raphson and Conjugate Gradient) to solve an optimization problem known as Hartree-Fock. This method is extremely important in the field of Computational Quantum Chemistry and it is a good case study because it is a problem somewhat hard to solve and, as a consequence of this, it requires many tools from Riemannian Optimization. Besides, it is also a good example to see how these algorithms perform in practice.
Paper Structure (33 sections, 67 theorems, 275 equations, 1 figure, 3 tables)

This paper contains 33 sections, 67 theorems, 275 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. Riemannian Geometry
  3. Introduction
  4. Differential Topology
  5. Riemannian Geometry
  6. Quantum Mechanics
  7. Introduction
  8. History
  9. Theory
  10. Axioms
  11. Molecules (Electronic Structure)
  12. Spin
  13. Practice (Hartree--Fock)
  14. Computing the Energy
  15. Riemannian Optimization
  16. ...and 18 more sections

Key Result

Theorem 2.22

If $f:X \to Y$ and $g:Y \to Z$ are smooth functions, then

Figures (1)

  • Figure 1: Graph illustrating chapters dependencies.

Theorems & Definitions (279)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • Example 2.14: Open sets
  • ...and 269 more