Riemannian gradient descent for Hartree-Fock theory

TL;DR

This paper introduces a Riemannian optimization approach for Hartree-Fock theory in Sobolev space, enabling robust and efficient convergence using geometric methods and preconditioning, with compatibility for adaptive discretizations.

Contribution

It develops a Riemannian framework for Hartree-Fock optimization directly in Sobolev space, including explicit gradient formulas and algorithms, advancing infinite-dimensional quantum chemistry methods.

Findings

Robust convergence demonstrated on small molecules.

Competitive performance against traditional SCF-DIIS schemes.

Effective from random initial guesses in small molecule cases.

Abstract

We present a Riemannian optimization framework for Hartree-Fock theory formulated directly in the Sobolev space $H^1$. The orthonormality constraints are interpreted geometrically via infinite-dimensional Stiefel and Grassmann manifolds endowed with the embedded $H^1$ metric. Explicit expressions for Euclidean and Riemannian gradients, tangent-space projections, and retractions are derived using resolvent operators, avoiding distributional formulations. The resulting algorithms include Riemannian steepest descent and a preconditioned nonlinear conjugate gradient method equipped with Armijo backtracking and Powell-type restarts. Particular attention is given to physically motivated preconditioning based on inversion of the kinetic energy operator. The framework is naturally compatible with adaptive multiwavelet discretizations, where Coulomb-type convolutions can be evaluated efficiently. Numerical experiments demonstrate robust convergence and competitive performance compared to conventional SCF-DIIS schemes. In addition, for small molecules the gradient descent method converges from random initial guesses. The proposed formulation provides a geometrically consistent and discretization-independent perspective on electronic structure optimization and offers a foundation for further developments in infinite-dimensional Riemannian methods for quantum chemistry.