Abstract
We improve on the Thomas–Fermi approximation for the single-particle density of fermions by introducing inhomogeneity corrections. Rather than invoking a gradient expansion, we relate the density to the unitary evolution operator for the given effective potential energy and approximate this operator by a Suzuki–Trotter factorization. This yields a hierarchy of approximations, one for each approximate factorization. For the purpose of a first benchmarking, we examine the approximate densities for a few cases with known exact densities and observe a very satisfactory, and encouraging, performance. As a bonus, we also obtain a simple fourth-order leapfrog algorithm for the symplectic integration of classical equations of motion.
Highlights
All practical applications of density-functional theory (DFT) to systems of interacting particles require trustworthy approximations to the functionals for the kinetic energy and the interaction energy or—more relevant for the set of equations that one needs to solve self-consistently—to their functional derivatives
While the Kohn–Sham (KS) scheme [1] avoids approximations for the kinetic-energy functional (KEF), this comes at the high price of a CPU-costly solution of the eigenvalue-eigenstate problem for the effective single-particle Hamiltonian
The popular alternative to KS-DFT proceeds from the KEF in Thomas–Fermi (TF) approximation [2,3] and improves on that by the inclusion of inhomogeneity corrections in the form of gradient terms, with the von Weizsacker term [4] as the leading correction. This gradient expansion is notorious for its lack of convergence, the wrong sign of the von Weizsacker term for one-dimensional systems [6], and the vanishing of all corrections for two-dimensional systems [6,7,8]—or so it seems [9]
Summary
All practical applications of density-functional theory (DFT) to systems of interacting particles require trustworthy approximations to the functionals for the kinetic energy and the interaction energy or—more relevant for the set of equations that one needs to solve self-consistently—to their functional derivatives. While the Kohn–Sham (KS) scheme [1] avoids approximations for the kinetic-energy functional (KEF), this comes at the high price of a CPU-costly solution of the eigenvalue-eigenstate problem for the effective single-particle Hamiltonian. The popular alternative to KS-DFT proceeds from the KEF in Thomas–Fermi (TF) approximation [2,3] and improves on that by the inclusion of inhomogeneity corrections in the form of gradient terms, with the von Weizsacker term [4] as the leading correction. There is, in particular, a relatively simple five-factor approximation that is correct to fourth order It promises a vast improvement over the TF approximation without the high costs of the KS method and without the dimensional limitation of the Ribeiro et al method. The here developed higher-order ST approximations have the potential to significantly improve upon standard second-order leapfrog algorithms [21,22] or even fourth-order Runge–Kutta methods
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
