Riemannian Optimization and the Hartree-Fock Method

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.