The mapping, exact or approximate, of a many-body problem onto an effective single-body problem is one of the most widely used conceptual and computational tools of physics. Here, we propose and investigate the inverse map of effective approximate single-particle equations onto the corresponding many-particle system. This approach allows us to understand which interacting system a given single-particle approximation is actually describing, and how far this is from the original physical many-body system. We illustrate the resulting reverse engineering process by means of the Kohn-Sham equations of density-functional theory. In this application, our procedure sheds light on the nonlocality of the density-potential mapping of density-functional theory, and on the self-interaction error inherent in approximate density functionals.
Coe, J. P., Capelle, K., & D’amico, I. (2009). Reverse engineering in many-body quantum physics: Correspondence between many-body systems and effective single-particle equations. Physical Review A, 79(3), . https://doi.org/10.1103/PhysRevA.79.032504