Abstract
We propose a limited-memory quasi-Newton method using the bad Broyden update and apply it to the nonlinear equations that must be solved to determine the effective Fermi momentum in the weighted density approximation for the exchange energy density functional. This algorithm has advantages for nonlinear systems of equations with diagonally dominant Jacobians, because it is easy to generalize the method to allow for periodic updates of the diagonal of the Jacobian. Systematic tests of the method for atoms show that one can determine the effective Fermi momentum at thousands of points in less than fifteen iterations.
Highlights
The accuracy and scope of density functional theory calculations is limited by the need to approximate the density functionals for the exchange-correlation energy [1–4]
The goal of this work is to present a new diagonally updated limited-memory quasi-Newton method we developed for solving a system of nonlinear equations associated with the weighted density approximation
In the remainder of this section, we present background material on the weighted density approximation, so that the nature of the system of nonlinear equations we are solving is clear. (In particular, the system of nonlinear equations is very large, but the Jacobian is dominated by contributions from its diagonal.) Section 2 presents the new method we developed
Summary
The accuracy and scope of density functional theory calculations is limited by the need to approximate the density functionals for the exchange-correlation energy [1–4]. This motivates the robust stream of research into exchange-correlation functionals and the development of new strategies for approximating the exchange-correlation energy. Based on the realization that local and semilocal forms for the exchange-correlation energy functional will always give qualitatively incorrect results for some systems [5–13], there has been significant recent work on developing nonlocal density functionals. The goal of this work is to present a new diagonally updated limited-memory quasi-Newton method we developed for solving a system of nonlinear equations associated with the weighted density approximation. In the remainder of this section, we present background material on the weighted density approximation, so that the nature of the system of nonlinear equations we are solving is clear. (In particular, the system of nonlinear equations is very large, but the Jacobian is dominated by contributions from its diagonal.) Section 2 presents the new method we developed
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