Harris functional

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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 , which is different from the exact electron density . We construct exchange-correlation potential and the Hartree potential based on the approximate electron density . 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:

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

It was discovered by Harris that the difference between the Harris energy 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 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]


  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.