Riemannian gradient descent for Hartree-Fock theory
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866915868199354368 |
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| author | Dinvay, Evgueni |
| author_facet | Dinvay, Evgueni |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Riemannian gradient descent for Hartree-Fock theory Dinvay, Evgueni Quantum Physics 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. |
| title | Riemannian gradient descent for Hartree-Fock theory |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2603.15870 |