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Kohn-Sham equations

The Kohn-Sham equations are a set of eigenvalue equations within density functional theory (DFT). DFT attempts to reduce a many-body problem for the N particle wavefunction $\Psi(\mathbf{r}_1,s_1;\ldots;\mathbf{r}_N,s_N)$ (which depends on 4N variables) to one in terms of the charge density $\rho(\mathbf{r})$ (which depends on 3 variables), using the Hohenberg-Kohn theorems. Thus, one writes the total energy $E$ of the system as a functional of the charge density:

$E[\rho] = T[\rho] + \int V_{ext}(\mathbf{r}) \rho(\mathbf{r}) d\mathbf{r} + V_{H}[\rho] + E_{xc}[\rho] \,\;$ ,

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where $T$ is the kinetic energy of the system, $V e x t$ is an external potential acting on the system,

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

is the Hartree energy and $E x c$ is the exchange-correlation energy.

The straightforward application of this formula has two obstacles: first, the exchange-correlation energy is not known exactly (see DFT Approximations for the workaround), and second, the kinetic term must be formulated in terms of the charge density. As was first proposed by Kohn and Sham, the charge density can be written as the sum of the squares of a set of orthonormal wave functions $\psi_i(\mathbf{r},s)$ :

$\rho(\mathbf{r})=\sum_i^N\sum_s |\psi_i(\mathbf{r},s)|^2,$

which are solutions to the Schrödinger equation for $N$ noninteracting electrons moving in an effective potential $v_{eff}(\mathbf{r})$

$-{\hbar^2\over2m}\nabla^2\psi_i(\mathbf{r},s) + v_{eff}(\mathbf{r})\psi_i(\mathbf{r},s) = \varepsilon_i \psi_i(\mathbf{r},s),$

where the effective potential is defined to be

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

These three equations form the Kohn-Sham orbital equations in their canonical form. This system is then solved iteratively, until self-consistency is reached. Note that the eigenvalues $ε i$ have no physical meaning, only the total sum, which corresponds to the energy of the entire system $E$ through the equation

$E = \sum_{i}^N \varepsilon_i - V_{H}[\rho] + E_{xc}[\rho] - \int {\delta E_{xc}[\rho]\over\delta \rho(\mathbf{r})} \rho(\mathbf{r}) d\mathbf{r}$ .

References

W. Kohn and L. J. Sham, Phys. Rev. 140 (1965) A1133
R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press, New York, 1989).

Category: Density functional theory