Density-to-Potential Inversions in Density Functional Theory

Density functional theory and many of its extensions are formally exact quantum many-body theories. In practice, however, implementations of these theories use approximations for all but the most trivial systems. We present a set of inversion methods to numerically compute the exact potentials corresponding to given input densities. The results of these inversions may then be used to evaluate the quality of different density functional approximations and guide the design of new approximations. The inversion methods use classical gradient-based optimization routines that are constrained to satisfy the governing partial differential equations. Numerous examples are given to illustrate the strengths and weaknesses of the different inversion methods