Semiclassics: The hidden theory behind the success of DFT

Abstract

It is argued that the success of DFT can be understood in terms of a semiclassical expansion around a very specific limit. This limit was identified long ago by Lieb and Simon for the total electronic energy of a system. This is a universal limit of all (non-relativistic) electronic structure: atoms, molecules, and solids. In the simple case of neutral atoms, this limit corresponds to an expansion of the total energy in powers of Z −1/3 . For the total energy, Thomas-Fermi theory becomes relatively exact in the limit. The limit can also be studied for much simpler model systems, including non-interacting fermions in a onedimensional well, where the WKB approximation applies for individual eigenvalues and eigenfunctions. Summation techniques lead to energies and densities that are functionals of the potential. We consider several examples in one dimension (fermions in a box, in a harmonic well, in a linear half-well, and in the Pöschl-Teller well. The effects of higher dimension are also illustrated with the three-dimensional harmonic well and the Bohr atom, non-interacting fermions in a Coulomb well. Modern density functional calculations use the Kohn-Sham scheme almost exclusively. The same semiclassical limit can be studied for the Kohn-Sham kinetic energy, for the exchange energy, and for the correlation energy. For all three, the local density approximation appears to become relatively exact in this limit. Recent work, both analytic and numerical, explores how this limit is approached, in an effort to deduce the leading corrections to the local approximation. A simple scheme, using the Euler-Maclaurin summation formula, is the result of many different attempts at this problem. In very simple cases, the correction formulas are much more accurate than standard density functionals. Several functionals are already in widespread use in both chemistry and materials that incorporate these limits, and prospects for the future are discussed.

Gradient expansions for the large-coupling strength limit of the Møller-Plesset adiabatic connection

“…As N → ∞ the densities ρ Bohr N (r) of eq (38) approach the Bohr atom TF profile 27,28,38 ρTFBohr that it grows as log(N ). For this case everything is analytic and it is easy to reach very large N , evaluating the GEA2 integral to high accuracy.…”

“…• p = − 2 3 : the Thomas-Fermi scaling of the Bohr atoms; 27,28,38,39 • p = 0 : the scaling used in For any density functional G[ρ] that under the uniform coordinate scaling of eq (22) behaves as…”

“…with ρTFna (r) the TF profile (integrating to 1) for neutral atoms, [25][26][27][28]37 led to the conclusion that their exchange energy as a function of N = Z should have, to leading orders, the large-N expansion a x N 5/3 +b x N . Extracting b x from exchange energies of neutral atoms allowed to fix the GEA2 coefficient for exchange in ref.…”

“…A case for which it is even simpler to make a detailed numerical analysis of I GEA2 is the Bohr atoms, 27,28,38 which have densities constructed by occupying hydrogenic orbitals…”

“…18 The purpose of this work is to derive the missing GEA2 for the strong-coupling functionals of the MP AC, in order to reduce empiricism and hopefully increase the accuracy of the interpolations along the MP AC. To this purpose, we use the ideas derived from the semiclassical limit of neutral atoms, which have been used in recent years in DFT for the analysis of the exchange and correlation functionals, [23][24][25][26][27][28] yielding new approximations such as PBEsol. 29 As we shall see, the functionals we need to approximate allow us to probe more extensively these ideas, revealing several interesting features that could be used more generally to build DFT approximations.…”