Riemannian gradient descent for Hartree-Fock theory

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.

Figures (7)

• Figure 1: Convergence of the Riemannian steepest gradient descent for H$_2$ Hartree-Fock model starting from a random Gaussian superposition. DIIS oscillates and converges in twice as many iterations.
• Figure 2: Convergence of the Riemannian conjugate gradient descent on the Stiefel manifold for Hartree-Fock model starting from a random Gaussian superposition.
• Figure 3: Convergence of the Riemannian conjugate gradient descent on the Stiefel manifold for Hartree-Fock model starting from a random Gaussian superposition, with an additional restart after each rejection of the conjugate direction.
• Figure 4: Convergence of the Riemannian conjugate gradient descent on the Stiefel manifold for Hartree-Fock and B3LYP energy models.
• Figure 5: Convergence of the SCF method accelerated by KAIN with one element in the history for Hartree-Fock and B3LYP energy models.