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In physics and quantum chemistry, the Kohn–Sham equations describe the mapping between a fictitious system (the "Kohn–Sham system") of non-interacting particles and a real system of interacting particles. Both systems reproduce the same charge density, which in Density Functional Theory (DFT) takes the role of the wavefunction in describing the system. They provide a closed set of equations used by most practical DFT calculations. An initial guess of the Kohn-Sham orbitals can be made and the equations can be iterated towards self-consistency. In principle, the Kohn–Sham equations represent an exact reformation of quantum mechanics. In practice, an unknown term in the equations has to be approximated and this hinders the accuracy of the approach.

The Kohn–Sham particles obey a modified schrodinger equation where the external potentail is replaced by a local fictitious potential called the Kohn–Sham potential. typically denoted as v_s(r) or v_eff(r). 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

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

Here, ε_i is the orbital energy of the corresponding Kohn–Sham orbital, φ_i. This eigenvalue equation is the most important of the Kohn–Sham equations and is sometimes referred to as the Kohn-Sham equation. The Kohn-Sham potential is defined such that the density of the Kohn–Sham system and the interacting system are identical.

The density for a non-interacting 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 density functional theory, the total energy of a system is expressed as a functional of the charge density as

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

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), V_H is the Hartree (or Coulomb) energy,

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

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 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. The advantage of splitting up the energies this way is that this unknown contribution is also the smallest. This means that even poor approximations to the exchange-correlation energy can produce surprisingly accurate results. 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

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

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

Category:Density functional theory