Localized basis sets in electronic structure theory: How do you solve a large (yet sparse) generalized eigenvalue problem?

Kristopher Andersen, University of California, Davis

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Electronic structure calculations are a primary tool used in condensed matter physics to understand and predict the electronic properties of materials. Using density-functional theory, these calculations incorporate quantum mechanical, many-particle exchange and correlation effects to solve for the electronic ground state. The usual limitation of using density-functional theory in electronic structure calculations is the size of the system, simply how many atoms can be included in the calculation. Topical interest in defects, doping, and nanoscale materials all lead to prohibitively large calculations in terms of both storage and cost. Within these calculations, the dominant computational problem is the need to repeatedly solve a standard or (often) generalized eigenvalue problem resulting from the discretization of the continuous problem. Recent efforts have focused on using either localized basis set or finite-difference (no basis set) approaches. The common aim of these approaches is to introduce sparse, structured matrices into the resulting eigenvalue problem, such that the cost of a matrix-vector product is O(N) as the size of the system N gets large. The goal is then to find a computational algorithm to solve the eigenvalue problem that uses as few matrix-vector multiplications as possible. Here, we discuss a finite-element approach to density-functional theory. Finite-elements are nonorthogonal basis functions constructed from a linear combination of strictly local, piecewise polynomials. Using finite-elements as the basis provides a natural way to deal with the trade-off between cost and accuracy while still retaining the advantages of a basis set. The structure and sparsity of the resulting generalized eigenvalue problem will be discussed, as will a comparison of algorithms to solve it. These algorithms vary from Krylov subspace and Jacobi-Davidson approaches, to more recent minimization techniques of a generalized Rayleigh quotient using the steepest descent or conjugate gradient methods. A comparison to a recently proposed extension of the conjugate gradient method will be made.

Abstract Author(s): K. E. Andersen, J. E. Pask, W. E. Pickett