Semiclassical origins of DFT

Contrary to current belief, there is a systematic expansion that lies behind the successes of modern DFT calculations. This is a formal expansion in powers of $\hbar$ and was first identified four decades ago by Lieb and Simon in the case of Thomas-Fermi (TF) theory. In this semiclassical limit, the exact Hohenberg- Kohn functional $F [n]$, where $n(r)$ is the one-particle density, tends to its TF counterpart (local approximation for the kinetic energy and Hartree approximation for Coulomb repul- sion). In the same limit, the exchange-correlation energy,$E_{\xc} [n]$, used in the Kohn-Sham (KS) method tends to its local density approximation (LDA),

The focus of this project is to find the leading corrections to the local approximation in this expansion (as has been tried in nuclear physics, in terms of path integrals and density matrices, and more recently for quantum dots). For slowly-varying densities, these are the well-known gradient corrections which form the starting point of Perdewâ€™s generalized gradient approximations (GGAs). But for all other cases, including all atoms, molecules, and most solids, the dominant correction is an oscillation generated by the classical caustic structure of the KS potential, $v_s(r)$. These corrections have been derived in several specific situations. They are universal functionals of the potential, rather than of the density, and so have remained elusive in DFT.