[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.
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: The density,
is given by that of the Slater determinant formed by the spin-orbitals of the occupied states: where the coefficients
are the occupation numbers given by the Fermi–Dirac distribution at the temperature of the system with the restriction
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,
, substitute in the Kohn–Sham equation, extract a new density
is reached, the energy of the system is expressed as: Assume that we have an approximate electron density
, which is different from the exact electron density
based on the approximate electron density
Kohn–Sham equations are then solved with the XC and Hartree potentials and eigenvalues are then obtained; that is, we perform one single iteration of the self-consistency calculation.
The sum of eigenvalues is often called the band structure energy: where
and the exact total energy is to the second order of the error of the approximate electron density, i.e.,
Therefore, for many systems the accuracy of Harris energy functional may be sufficient.
The Harris functional was originally developed for such calculations rather than self-consistent convergence, although it can be applied in a self-consistent manner in which the density is changed.
Many density-functional tight-binding methods, such as CP2K, DFTB+, Fireball,[2] and Hotbit, are built based on the Harris energy functional.
In these methods, one often does not perform self-consistent Kohn–Sham DFT calculations and the total energy is estimated using the Harris energy functional, although a version of the Harris functional where one does perform self-consistency calculations has been used.