Rethinking the Kohn-Sham inverse problem

Density functional theory (DFT) is a cornerstone of modern electronic structure theory. In the Kohn-Sham scheme, the many-electron Schrödinger equation is replaced by a set of effective single-particle equations. Thus, the full complexity of the quantum mechanical many-particle effects is mapped to the exchange-correlation potential vxc(r). Almost all DFT calculations done in practice rely on approximations to vxc(r). However, numerical representations of the quasi-exact vxc(r) can be obtained from quasi-exact densities by inverting the Kohn-Sham procedure. This inverse Kohn-Sham scheme is an important source of insight into exact DFT. Here, we review the inverse Kohn-Sham problem and explain in detail several aspects of why Kohn-Sham inversion is intrinsically difficult. We then present several inversion schemes and discuss their pros and cons, specifically addressing the effects of statistical uncertainties that are inevitable in quantum Monte Carlo reference densities. We use these schemes to obtain representations of vxc(r) that correspond to the ground-state densities that have become available from accurate diffusion Monte Carlo calculations on real space grids for the Li2 and N2 molecules, and the C atom. In the latter, the highest occupied orbital has a nodal line and the exchange-correlation potential goes to a different asymptotic value in this direction. As an outlook, we discuss the possibility of interlacing quantum Monte Carlo and Kohn-Sham theory by using the quasi-exact Kohn-Sham determinant to fix the nodes in a diffusion Monte Carlo calculation.