Kohn–Sham equations

History

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

Kohn received a Nobel Prize in Chemistry in 1998 for the Kohn–Sham equations and other work related to density functional theory (DFT).

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 $${\displaystyle E[\rho ]=T_s[\rho ]+\int d\mathbf{r}{v}_{\text{ext}}(\mathbf{r})\rho (\mathbf{r})+E_{\text{H}}[\rho ]+E_{\text{xc}}[\rho ],}$$

where T_s is the Kohn–Sham kinetic energy, which is expressed in terms of the Kohn–Sham orbitals as $${\displaystyle T_s[\rho ]=\sum _{i=1}^N\int d\mathbf{r}{\phi }_i^{\ast }(\mathbf{r})\left(-\frac{{\hslash }^2}{2m}{\mathrm{\nabla }}^2\right){\phi }_i(\mathbf{r}),}$$

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 $${\displaystyle E_{\text{H}}[\rho ]=\frac{e^2}{2}\int d\mathbf{r}\int d{\mathbf{r}}^{\prime }\frac{\rho (\mathbf{r})\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|},}$$

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, to yield the Kohn–Sham potential as $${\displaystyle v_{\text{eff}}(\mathbf{r})=v_{\text{ext}}(\mathbf{r})+e^2\int \frac{\rho ({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\mathbf{r}}^{\prime }+\frac{\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf{r})},}$$ where the last term $${\displaystyle v_{\text{xc}}(\mathbf{r})\equiv \frac{\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf{r})}}$$ 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 $${\displaystyle E=\sum _i^N{\epsilon }_i-E_{\text{H}}[\rho ]+E_{\text{xc}}[\rho ]-\int \frac{\delta E_{\text{xc}}[\rho ]}{\delta \rho (\mathbf{r})}\rho (\mathbf{r})d\mathbf{r}.}$$

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).