Kohn-Sham Theory

The kinetic energy has a large contribution to the total energy. Therefore even the 1% error in the kinetic energy of the Thomas-Fermi-Weizsacker model prevented DFT from being used as a quantitative predictive tool. Thus DFT was largely ignored until 1965 when Kohn and Sham [93] introduced a method which treated the majority of the kinetic energy exactly.

The theory begins by considering the noninteracting reference system: N noninteracting electrons, each in one of N orbitals, $\psi_{i}$. Such a system will be defined by the Hamiltonian
$$\begin{align}\hat{H}_{s}=\sum_{i}^{N}(-\frac{1}{2}\nabla_{i}^{2})+\sum_{i}^{N}v_{s}({\bf r}_{i}) \end{align}$$
which has an exact eigenfunction that is the single determinant constructed from the N lowest eigenstates of the one-electron equations
$$\begin{align}\left[ -\frac{1}{2}\nabla^{2}+v_{s}({\bf r}) \right] \psi_{i} = \varepsilon_{i} \psi_{i}. \end{align}$$
The corresponding Euler-Lagrange equation is
$$\begin{align}\mu = v_{s}({\bf r}) + \frac{\delta T_{s}[\rho]}{\delta \rho({\bf r})}. \end{align}$$

For this system the kinetic energy and electron density are given exactly by

$$T_{s}[\rho] = \sum_{i}^{N} \langle \psi_{i} \vert -\frac{1}{2}\nabla_{i}^{2}\vert\psi_{i}\rangle$$ (7.1)

$$\rho({\bf r}) = \sum_{i}^{N}\vert\psi_{i}({\bf r})\vert^{2}$$ (7.2)

and the total energy is given by
$$\begin{align}E[\rho] = T_{s}[\rho] + \int v_{s}({\bf r})\rho({\bf r})\,d{\bf r} \end{align}$$
The quantity $T_{s}[\rho]$ is well-defined, but not the exact kinetic energy, $T[\rho]$, defined in equation (2.7). Kohn and Sham reformulated the interacting problem so that its kinetic component is defined to be $T_{s}[\rho]$ and rearranging equation (2.7) to give
$$\begin{align}E[\rho] = T_{s}[\rho ]+ J[\rho] + \int v({\bf r})\rho({\bf r}) + E_{xc}[\rho], \end{align}$$
where $E_{xc}[\rho]$ is the exchange-correlation energy, made up of the non-classical electron-electron repulsion and also the difference between the exact and noninteracting kinetic energy
$$\begin{align}E_{xc}[\rho] = T[\rho] - T_{s}[\rho] + V_{ee}[\rho] - J[\rho]. \end{align}$$
The Euler-Lagrange equation (2.12) now becomes
 $$\begin{align} \mu = v_{\mathrm{eff}}({\bf r}) + \frac{\delta T_{s}[\rho]}{\delta \rho({\bf r})} \end{align}$$
where the Kohn-Sham (KS) effective potential, v_eff is defined as
$$\begin{align}v_{\mathrm{eff}}({\bf r}) = v({\bf r}) + \int \frac{\rho({\bf r}')}{\vert{\bf r}-{\bf r}'\vert}\,d{\bf r}' + v_{xc}({\bf r}) \end{align}$$
and the exchange-correlation potential, v_xc is
 $$\begin{align} v_{xc}({\bf r}) = \frac{\delta E_{xc}[\rho]}{\delta \rho({\bf r})}. \end{align}$$

Kohn and Sham noticed that equation (2.48) is the same as that for a non-interacting system moving in the potential $v_{\mathrm{eff}}({\bf r})$. Thus, the exact density can be obtained by solving the N one-electron equations (the restricted KS equations)
 $$\begin{align} \left[ -\frac{1}{2}\nabla^{2}+v_{\mathrm{eff}}({\bf r}) \right] \psi_{i} = \varepsilon_{i} \psi_{i}. \end{align}$$
Notice that v_eff depends on $\rho({\bf r})$, via equation (2.50), hence the KS equations must be solved iteratively.

The KS equations are very similar to the Hartree-Fock equations. In fact, setting the exchange-correlation potential to the HF exchange potential,
$$\begin{align}v_{xc}({\bf r}) = - \int \sum_{j} \frac{\psi_{j}({\bf r}')\psi_{j}(... ...{\vert{\bf r}-{\bf r}'\vert}\hat{P}_{{\bf r}{\bf r}'} \, d {\bf r}', \end{align}$$
yields the HF equations. Drawing too many similarities to HF is dangerous, however. Firstly, the KS orbitals are simply a way of representing the density; they are not (as in HF) an approximation of the wavefunction. In particular, Koopmans' theorem [94] -- that the ionization potentials and electron affinities are approximated by the negative of the HF occupied and virtual orbital eigenvalues respectively -- is invalid for KS orbitals. The highest occupied KS eigenvalue has been shown to be the negative of the first ionization potential, though [95]. Also, HF theory is variational, providing an upper bound to the exact energy, yet DFT is only variational if the exact energy functional is used.

The above analysis is only appropriate for closed shell molecules. Because the KS equations so closely follow the restricted HF equations, both the restricted open shell and unrestricted methodologies are readily available. However, the KS equations are formally exact (given the exact $E_{xc}[\rho]$), so it must be able to produce an excess of $\beta$ electron density at points in the molecule [96], and therefore only the unrestricted formalism is appropriate. The unrestricted KS equations are
$$\begin{align}\left[-\frac{1}{2}\nabla^{2}+v_{\mathrm{eff}}^{\alpha}({\bf r}) \ri... ...\right] \psi_{i}^{\beta} &= \varepsilon_{i}^{\beta} \psi_{i}^{\beta} \end{align}$$
where
$$\begin{align}v_{\mathrm{eff}}^{\alpha}({\bf r}) &= v({\bf r}) + \frac{\delta J[\... ...a E_{xc}[\rho_{\alpha},\rho_{\beta}]}{\delta \rho_{\beta}({\bf r})}. \end{align}$$

One problem with the above derivation of the KS equations is that the density must be non-interacting v-representable. That is, there must exist a potential v_s that will produce the same density as the exact wavefunction. If the density is not non-interacting v-representable, the determinant formed from the KS orbitals will be an excited state [97]. The criteria that make a density non-interacting v-representable are unknown.

Just as in HF theory, the KS equations are solved by expanding the orbitals over a basis set. The major advantage of DFT is that the basis set requirements are far more modest than the more conventional correlated methods [98,99]. In DFT the basis set only needs to represent the one electron density -- the inter-electron cusp is accounted for by the effective potential, v_eff. In the more traditional methods the basis set describes the entire N-electron wavefunction, requiring an accurate description of the cusp which is sensitive to the basis set.