Riemannian Newton methods for energy minimization problems of Kohn-Sham type

TL;DR

This work develops Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds to solve constrained energy minimization problems of Kohn–Sham type, including the Gross–Pitaevskii model. By deriving explicit Riemannian gradients and Hessians and exploiting manifold geometry via retractions and horizontal lifts, the authors create an inexact Newton framework that is robust to discretization and outperforms traditional SCF and gradient-based schemes in representative quantum-chemistry and quantum-gas problems. Numerical experiments on Gross–Pitaevskii and Kohn–Sham systems (e.g., pentacene and graphene) demonstrate that a single or few Newton steps can achieve high accuracy with significantly fewer iterations, albeit with higher per-step cost, underscoring a favorable overall efficiency. The results indicate a promising, dimension-independent path for fast Newton-type solvers in KS-type PDE discretizations and motivate further development of scalable solvers and implementation strategies.

Abstract

This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross-Pitaevskii and Kohn-Sham models. In particular, we introduce the Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates its supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.

Figures (7)

• Figure 5.1: Ground state in the harmonic trap (potential in gray, properly rescaled) for $\kappa=10,100,1000$ (from left to right).
• Figure 5.2: Convergence history of the residuals for the ground state in the harmonic trap for $\kappa=10,100,1000$ (from left to right).
• Figure 5.3: Computing the ground state in the harmonic trap for $\kappa=1000$: Iteration count of the Riemannian Newton method to fall below the tolerance $10^{-10}$ (left) and the error in the minimal energy (right) versus mesh size $h$. The dashed line indicates order $h^4$.
• Figure 5.4: Piecewise constant disorder potential $\vartheta_\text{rand}$ (left, black elements refer to the value $\varepsilon^{-2}$, white elements refer to the value $0$, $\varepsilon=2^{-6}$) and the corresponding ground states for $\kappa=1$ (middle) and $\kappa =10$ (right).
• Figure 5.5: Convergence history of the residuals for the ground state in a disorder potential for $\kappa=0.1,1,10$ (from left to right).

Theorems & Definitions (11)

• Example 2.1: Gross--Pitaevskii model
• Example 2.2: Kohn--Sham model
• Remark 2.3
• Theorem 3.1
• proof
• Remark 3.2
• Theorem 4.1
• proof
• Theorem 4.3: Positive Hessian
• proof