where E[ρ] is the exact ground-state energy, Ĥ is the corresponding (exact) Hamiltonian, and Φ is the noninteracting KS wave function (a minimizer of the kinetic energy constrained to yield ρ(r)). Finally, e c (r) is the correlation energy density and we strictly adhere to a specific definition (gauge), that of the electrostatic potential of the exchange-correlation hole transparently arising from the many-body wave functions (see Refs. 67 and 68 for the derivations). This specific gauge is chosen because of its physical soundness and proven advantages over other gauges when constructing DFAs . Computing the exact e c (r) is extremely expensive, requiring a series of wave functions adiabatically connecting the non-interacting and true wave functions, limiting the implementation to very small systems and basis sets .