Mathematical Methods for Physicists
- B Arfken, H J Weber
- Mathematical Methods for Physicists
Orbital-free density functional theory OF-DFT is a firstprinciples quantum mechanics method that can be formulated to scale linearly with system size.1 By contrast, KohnSham density functional theory KS-DFT Ref. 2 scales cubically with size initially and can be made to scale linearly in the asymptotic limit. The linear scaling within OF-DFT is achieved by eliminating the fictitious orbitals employed within KS-DFT, which are invoked to obtain an accurate estimate of the kinetic energy of the electrons. Instead, OFDFT relies upon explicit functionals of the electron density to compute all energy terms, including the kinetic energy. OF-DFT is significantly faster than KS-DFT. Currently, OF-DFT can be used to study samples consisting of tens of thousands of atoms on a single processor. Unfortunately, this increase in speed currently comes at a cost in accuracy. The kinetic energy within KS-DFT is exact in the limit of noninteracting electrons, whereas the form of the OF kinetic energy density functional KEDF is known exactly only for the uniform electron gas and for a single orbital, and approximations must be used for all other cases.3 Because of this limitation, OF-DFT is currently only as accurate as KSDFT for main group metals 10 meV /atom difference ,4 and for some properties of semiconductors.5 In most cases, the most accurate KEDF currently available is the Wang-Govind-Carter WGC KEDF with the density-dependent response kernel.4 The WGC KEDF is derived from a class of KEDFs that explicitly account for the linear response of the density of a uniform electron gas subject to small perturbations in the potential. As a result, these KEDFs work best for main group, nearly-free-electron-like metals. This class of KEDF was pioneered by Wang and Teter WT ,6 modified by Perrot,7 and Madden and co-workers,8–10 and generalized by WGC.11 These KEDFs all rely upon a linear-response kernel derived from a single fixed reference density. However, WGC introduced an important advance by accounting for a nonlocal density dependence in the linear-response kernel.4 This density dependence significantly improves upon the WT KEDF in many cases, particularly for describing vacancies, surfaces, and equations of state. Some properties of silicon, a representative covalent semiconductor, are also well described using two slightly different parameters in the 1999 WGC KEDF.5 However, sufficiently accurate KEDFs have yet to be developed for localized electron densities present in, e.g., transition metals,12 or molecules.3 Perhaps unfortunately, the WGC KEDF is still the most accurate KEDF for condensed matter to date.4 The kernel of the WGC KEDF is obtained by solving a second-order differential equation described below. Previous implementations of the WGC KEDF used a numerical solution to this differential equation. Here we present an analytic solution, which offers several advantages over the numerical solution, in our new software PROFESS PRinceton OrbitalFree Electronic Structure Software .1 First, the kernel can be computed on the fly for any given system using the analytic solution and stored on a grid with the appropriate mesh spacing. A numerical solution inevitably requires interpolation. Second, parameters in the kernel are simple to adjust in the analytic solution, requiring no extra effort. In a numerical implementation, one has to compute and store a different solution for every set of parameters. Third, knowledge of the analytic solution, which has not been reported previously, makes it possible to derive the WGC KEDF in real space, in order to perform accurate OF-DFT calculations under Dirichlet fixed boundary conditions as opposed to periodic boundary conditions PBCs . Dirichlet boundary conditions allow the study of aperiodic or isolated systems, such as a dislocation, grain boundary, or crack in a bulk material or a nanostructure e.g., quantum dot or wire , without periodic image artifacts inherent with periodic boundary conditions. In what follows, we first derive the analytic form of the kernel used in the WGC KEDF. With this in hand, we present our implementation of the WGC KEDF under Dirichlet boundary conditions, which maintains near-linear O(N lnN) scaling of the computation time with respect to system size. We validate our approach on the following test cases: a single aluminum atom in a vacuum, a cluster of 14 aluminum atoms in vacuum, and a cluster of 108 aluminum atoms arranged in a face-centered-cubic orientation without vacuum, but with the values of the electron density at the boundaries specified to be the same as in bulk fcc aluminum.