Harris functional ( From Wikipedia )

In density functional theory (DFT), the Harris energy functional is a non-self-consistent approximation to the Kohn–Sham density functional theory.^[1] It gives the energy of a combined system as a function of the electronic densities of the isolated parts. The energy of the Harris functional varies much less than the energy of the Kohn–Sham functional as the density moves away from the converged density.

Background

Kohn–Sham equations are the one-electron equations that must be solved in a self-consistent fashion in order to find the ground state density of a system of interacting electrons:

${\displaystyle \left({\frac {-\hbar ^{2}}{2m}}\nabla ^{2}+v_{\rm {H}}[n]+v_{\rm {xc}}[n]+v_{\rm {ext}}(r)\right)\phi _{j}(r)=\epsilon _{j}\phi _{j}(r).}$

The density, ${\displaystyle n,}$ is given by that of the Slater determinant formed by the spin-orbitals of the occupied states:

${\displaystyle n(r)=\sum _{j}f_{j}\vert \phi _{j}(r)\vert ^{2},}$

where the coefficients ${\displaystyle f_{j}}$ are the occupation numbers given by the Fermi–Dirac distribution at the temperature of the system with the restriction ${\textstyle \sum _{j}f_{j}=N}$, where ${\displaystyle N}$ is the total number of electrons. In the equation above, ${\displaystyle v_{\rm {H}}[n]}$ is the Hartree potential and ${\displaystyle v_{\rm {xc}}[n]}$ is the exchange–correlation potential, which are expressed in terms of the electronic density. Formally, one must solve these equations self-consistently, for which the usual strategy is to pick an initial guess for the density, ${\displaystyle n_{0}(r)}$, substitute in the Kohn–Sham equation, extract a new density ${\displaystyle n_{1}(r)}$ and iterate the process until convergence is obtained. When the final self-consistent density ${\displaystyle n(r)}$ is reached, the energy of the system is expressed as:

${\displaystyle E[n]=\sum _{j\in {\text{occupied}}}\epsilon _{j}-{\tfrac {1}{2}}\int v_{\rm {H}}[n]n(r)\,\mathrm {d} r-\int v_{\rm {xc}}[n]n(r)\,\mathrm {d} r+E_{\rm {xc}}[n]}$.

https://en.wikipedia.org/wiki/Harris_functional