In density functional theory (DFT), the Harris energy functional is a non-self-consistent approximation to the Kohn-Sham density functional theory. 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.
Assuming that we have an approximate electron density ρ ( r → ) , which is different from the exact electron density ρ 0 ( r → ) . We construct exchange-correlation potential v x c ( r → ) and the Hartree potential v H ( r → ) based on the approximate electron density ρ ( r → ) . Kohn-Sham equations are then solved with the XC and Hartree potentials and eigenvalues are then obtained. The sum of eigenvalues is often called the band energy:
E b a n d = ∑ i ϵ i ,
where i loops over all occupied Kohn-Sham orbitals. Harris energy functional is defined as
E H a r r i s = ∑ i ϵ i − ∫ d r 3 v x c ( r → ) ρ ( r → ) − 1 2 ∫ d r 3 v H ( r → ) ρ ( r → ) + E x c [ ρ ]
It was discovered by Harris that the difference between the Harris energy E H a r r i s and the exact total energy is to the second order of the error of the approximate electron density, i.e., O ( ( ρ − ρ 0 ) 2 ) . 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 DFTB+, Fireball, 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. These codes are often much faster than conventional Kohn-Sham DFT codes that solve Kohn-Sham DFT in a self-consistent manner.
While the Kohn-Sham DFT energy is Variational method (never lower than the ground state energy), the Harris DFT energy was originally believed to be anti-variational (never higher than the ground state energy). This was however conclusively demonstrated to be incorrect.
