Kohn–Sham equations

In physics and quantum chemistry, specifically density functional theory, the Kohn–Sham equation is the Schrödinger equation of a fictitious system (the "Kohn–Sham system") of non-interacting particles (typically electrons) that generate the same density as any given system of interacting particles.^[1][2] The Kohn–Sham equation is defined by a local effective (fictitious) external potential in which the non-interacting particles move, typically denoted as v_s(r) or v_eff(r), called the Kohn–Sham potential. As the particles in the Kohn–Sham system are non-interacting fermions, the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals that are the lowest energy solutions to

$\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+v_{{{\rm {eff}}}}({\mathbf r})\right)\phi _{{i}}({\mathbf r})=\varepsilon _{{i}}\phi _{{i}}({\mathbf r})$

This eigenvalue equation is the typical representation of the Kohn–Sham equations. Here, ε_i is the orbital energy of the corresponding Kohn–Sham orbital, φ_i, and the density for an N-particle system is

$\rho ({\mathbf r})=\sum _{i}^{N}|\phi _{{i}}({\mathbf r})|^{2}.$

The Kohn–Sham equations are named after Walter Kohn and Lu Jeu Sham (沈呂九), who introduced the concept at the University of California, San Diego in 1965.

Kohn–Sham potential

In Kohn-Sham density functional theory, the total energy of a system is expressed as a functional of the charge density as

$E[\rho] = T_s[\rho] + \int d\mathbf r\ v_{\rm ext}(\mathbf r)\rho(\mathbf r) + E_{H}[\rho] + E_{\rm xc}[\rho]$

where T_s is the Kohn–Sham kinetic energy which is expressed in terms of the Kohn–Sham orbitals as

$T_{s}[\rho ]=\sum _{{i=1}}^{N}\int d{\mathbf r}\ \phi _{i}^{*}({\mathbf r})\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\right)\phi _{i}({\mathbf r}),$

v_ext is the external potential acting on the interacting system (at minimum, for a molecular system, the electron-nuclei interaction), E_H is the Hartree (or Coulomb) energy,

$E_{H}={e^2\over2}\int d\mathbf r\int d\mathbf{r}'\ {\rho(\mathbf r)\rho(\mathbf r')\over|\mathbf r-\mathbf r'|}.$

and E_xc is the exchange-correlation energy. The Kohn–Sham equations are found by varying the total energy expression with respect to a set of orbitals, subject to constraints on those orbitals,^[3] to yield the Kohn–Sham potential as

$v_{{{\rm {eff}}}}({\mathbf r})=v_{{{\rm {ext}}}}({\mathbf {r}})+e^{2}\int {\rho ({\mathbf {r}}') \over |{\mathbf r}-{\mathbf r}'|}d{\mathbf {r}}'+{\delta E_{{{\rm {xc}}}}[\rho ] \over \delta \rho ({\mathbf r})}.$

where the last term

$v_{{{\rm {xc}}}}({\mathbf r})\equiv {\delta E_{{{\rm {xc}}}}[\rho ] \over \delta \rho ({\mathbf r})}$

is the exchange-correlation potential. This term, and the corresponding energy expression, are the only unknowns in the Kohn–Sham approach to density functional theory. An approximation that does not vary the orbitals is Harris functional theory.

The Kohn–Sham orbital energies ε_i, in general, have little physical meaning (see Koopmans' theorem). The sum of the orbital energies is related to the total energy as

${\displaystyle E=\sum _{i}^{N}\varepsilon _{i}-E_{H}[\rho ]+E_{\rm {xc}}[\rho ]-\int {\delta E_{\rm {xc}}[\rho ] \over \delta \rho (\mathbf {r} )}\rho (\mathbf {r} )d\mathbf {r} }$

Because the orbital energies are non-unique in the more general restricted open-shell case, this equation only holds true for specific choices of orbital energies (see Koopmans' theorem).

References

1. Kohn, Walter; Sham, Lu Jeu (1965). "Self-Consistent Equations Including Exchange and Correlation Effects". Physical Review. 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
2. Parr, Robert G.; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford University Press. ISBN 978-0-19-509276-9.
3. http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html