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Electronic Energy Functionals: Levy-Lieb principle within the Ground State Path Integral Quantum Monte Carlo

We propose a theoretical/computational protocol based on the use of the Ground State (GS) Path Integral (PI) Quantum Monte Carlo (QMC) for the calculation of the kinetic and Coulomb energy density for a system of $N$ interacting electrons in an external potential. The idea is based on the derivation of the energy densities via the $N-1$-conditional probability density within the framework of the Levy-Lieb constrained search principle. The consequences for the development of energy functionals within the context of Density Functional Theory (DFT) are discussed. We propose also the possibility of going beyond the energy densities and extend this idea to a computational procedure where the $N-1$-conditional probability is an implicit functional of the electron density, independently from the external potential. In principle, such a procedure paves the way for an {\it on-the-fly} determination of the energy functional for any system.