Kohn–Sham equations

In physics and quantum chemistry, specifically density functional theory, the Kohn – Sham equation is the one electron Schrödinger equation (more clearly, Schrödinger-like 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.

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} \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, 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

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

↑ 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.
↑ Parr, Robert G.; Yang, Weitao (1994). Density-Functional Theory of Atoms and Molecules. Oxford University Press. ISBN 978-0-19-509276-9.
↑ http://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html

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[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] ttp://muchomas.lassp.cornell.edu/P480/Notes/dft/node11.html