Harris functional

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.

Assuming that we have an approximate electron density $\rho(\vec r)$, which is different from the exact electron density $\rho_0(\vec r)$. We construct exchange-correlation potential $v_{xc}(\vec r)$ and the Hartree potential $v_{H}(\vec r)$ based on the approximate electron density $\rho(\vec r)$. 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:

$E_{band}=\sum_i \epsilon_i,$

where $i$ loops over all occupied Kohn-Sham orbitals. Harris energy functional is defined as

$E_{Harris} = \sum_i \epsilon_i - \int dr^3 v_{xc}(\vec r) \rho(\vec r) - \frac{1}{2} \int dr^3 v_{H}(\vec r) \rho(\vec r) + E_{xc}[\rho]$

It was discovered by Harris that the difference between the Harris energy $E_{Harris}$ and the exact total energy is to the second order of the error of the approximate electron density, i.e., $O((\rho-\rho_0)^2)$. 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,^[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. 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).^[3] This was however conclusively demonstrated to be incorrect.^[4][5]

References

1. Harris, J. (1985). "Simplified method for calculating the energy of weakly interacting fragments". Physical Review B 31 (4): 1770–1779. Bibcode:1985PhRvB..31.1770H. doi:10.1103/PhysRevB.31.1770.
2. Lewis, James P.; Glaesemann, Kurt R.; Voth, Gregory A.; Fritsch, Jürgen; Demkov, Alexander A.; Ortega, José; Sankey, Otto F. (2001). "Further developments in the local-orbital density-functional-theory tight-binding method". Physical Review B 64 (19): 195103. Bibcode:2001PhRvB..64s5103L. doi:10.1103/PhysRevB.64.195103.
3. Zaremba, E. (1990). "Extremal properties of the Harris energy functional". Journal of Physics: Condensed Matter 2 (10): 2479. Bibcode:1990JPCM....2.2479Z. doi:10.1088/0953-8984/2/10/018.
4. Robertson, I. J.; Farid, B. (1991). "Does the Harris energy functional possess a local maximum at the ground-state density?". Physical Review Letters 66 (25): 3265–3268. Bibcode:1991PhRvL..66.3265R. doi:10.1103/PhysRevLett.66.3265. PMID 10043743.
5. Farid, B.; Heine, V.; Engel, G. E.; Robertson, I. J. (1993). "Extremal properties of the Harris-Foulkes functional and an improved screening calculation for the electron gas". Physical Review B 48 (16): 11602–11621. Bibcode:1993PhRvB..4811602F. doi:10.1103/PhysRevB.48.11602.