# 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 ${\displaystyle \rho ({\vec {r}})}$, which is different from the exact electron density ${\displaystyle \rho _{0}({\vec {r}})}$. We construct exchange-correlation potential ${\displaystyle v_{xc}({\vec {r}})}$ and the Hartree potential ${\displaystyle v_{H}({\vec {r}})}$ based on the approximate electron density ${\displaystyle \rho ({\vec {r}})}$. 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:

${\displaystyle E_{band}=\sum _{i}\epsilon _{i},}$

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

${\displaystyle E_{Harris}=\sum _{i}\epsilon _{i}-\int dr^{3}v_{xc}({\vec {r}})\rho ({\vec {r}})-{\frac {1}{2}}\int dr^{3}v_{H}({\vec {r}})\rho ({\vec {r}})+E_{xc}[\rho ]}$

It was discovered by Harris that the difference between the Harris energy ${\displaystyle E_{Harris}}$ and the exact total energy is to the second order of the error of the approximate electron density, i.e., ${\displaystyle O((\rho -\rho _{0})^{2})}$. 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.