Gradient type optimization methods for electronic structure calculations

TL;DR

This paper introduces gradient-based optimization methods on the Stiefel manifold for electronic structure calculations, offering guaranteed convergence and improved efficiency over traditional SCF methods, especially for large systems.

Contribution

The paper develops new gradient-type algorithms for direct minimization in DFT, with proven convergence and computational advantages over eigenvalue-based methods.

Findings

Methods outperform SCF on large systems

No eigenvalue computations needed, reducing costs

Suitable for parallelization and scalable implementations

Abstract

The density functional theory (DFT) in electronic structure calculations can be formulated as either a nonlinear eigenvalue or direct minimization problem. The most widely used approach for solving the former is the so-called self-consistent field (SCF) iteration. A common observation is that the convergence of SCF is not clear theoretically while approaches with convergence guarantee for solving the latter are often not competitive to SCF numerically. In this paper, we study gradient type methods for solving the direct minimization problem by constructing new iterations along the gradient on the Stiefel manifold. Global convergence (i.e., convergence to a stationary point from any initial solution) as well as local convergence rate follows from the standard theory for optimization on manifold directly. A major computational advantage is that the computation of linear eigenvalue problems is no longer needed. The main costs of our approaches arise from the assembling of the total energy functional and its gradient and the projection onto the manifold. These tasks are cheaper than eigenvalue computation and they are often more suitable for parallelization as long as the evaluation of the total energy functional and its gradient is efficient. Numerical results show that they can outperform SCF consistently on many practically large systems.