GRADIENT TYPE OPTIMIZATION METHODS FOR ELECTRONIC STRUCTURE CALCULATIONS

Xin Zhang

doi:10.1137/130932934

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GRADIENT TYPE OPTIMIZATION METHODS FOR ELECTRONIC STRUCTURE CALCULATIONS

2014-01-01

DOI：10.1137/130932934

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Abstract

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The density functional theory (DFT) in electronic structure calculations can be formulated as either a nonlinear eigenvalue or a 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 guarantees for solving the latter are often not competitive with 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 rate (i.e., convergence to a stationary point from any initial solution) as well as local convergence rate can be obtained from the standard theory for gradient methods 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 on the manifold and the projection onto the manifold. These tasks are cheaper than eigenvalue computation and are often more suitable for parallelization as long as the evaluation of the total energy functional and its partial derivative is efficient. Numerical results show that they can outperform SCF consistently on many practical large systems.

Keywords:

density functional model

electronic structure calculation

gradient type methods

nonlinear eigenvalue problem

total energy minimization

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