General Theory

Writing the full spin-dependent density as
$$\begin{align}\gamma({\bf x}) = \rho_{\alpha}({\bf r})\vert\alpha(s)\vert^{2} + \rho_{\beta}({\bf r})\vert\beta(s)\vert^{2} \end{align}$$
shows that electron density is readily separable into two parts. Integration over the spin coordinate s gives the regular density
$$\begin{align}\rho({\bf r}) = \int \gamma({\bf x}) \, ds = \rho_{\alpha}({\bf r}) + \rho_{\beta}({\bf r}). \end{align}$$
Remembering from Chapter 2 that the KS energy can be written as
 $$\begin{align} E[\gamma] = T_{s}[\gamma] + V[\gamma] + J[\gamma] + E_{xc}[\gamma], \end{align}$$
where an approximate functional is used for the exchange-correlation functional $E_{xc}[\gamma]$.

The KS treatment begins by writing $\gamma$ in terms of a set of orthonormal spin orbitals, $\chi_{i}$:
$$\begin{align}\gamma({\bf x}) = \sum^{occ}_{i=1} \vert \chi_{i} \vert^{2} \end{align}$$
leading to the KS equations. The great advantage of this is that it allows the Hartree-Fock procedure to be written as a special case of KS density functional theory, simply by defining the Fock exchange-only functional for $E_{xc}[\gamma]$ as
$$\begin{align}E_{x}[\gamma] = -\frac{1}{2} \sum^{occ}_{i,j} \iint \frac{\chi^{*}_... ... x}_{2})\chi_{i}({\bf x}_{2})}{r_{12}} \, d{\bf x}_{1} d{\bf x}_{2}. \end{align}$$
Note that this exchange energy is defined for any appropriately normalized spin orbitals $\chi_{i}$, and hence for any appropriate density $\gamma({\bf x})$. Thus, the accuracy of a density produced by an exchange-correlation functional under the KS formalism can be examined by comparing its value for E_x above with the true Hartree-Fock energy. Thus, we define
 $$\begin{align} E_{KS}[\gamma_{KS}] = T_{s}[\gamma_{KS}] + V[\gamma_{KS}] + J[\gamma_{KS}] + E_{x}[\gamma_{KS}]. \end{align}$$
Also, it should be apparent that
$$\begin{align}E_{KS} \ge E_{HF} \end{align}$$
as the HF energy is the lowest expectation energy that can be obtained from a single-determinant wavefunction. This also allows the correlation energy to be defined as the difference
$$\begin{align}E_{c}[\gamma_{KS}] = E_{xc}[\gamma_{KS}] - E_{x}[\gamma_{KS}]. \end{align}$$

For the conventional treatment of electron correlation a set of orthonormal virtual spin orbitals, $\chi_{a}^{KS}$ needs to be introduced. A Fock matrix for the KS determinant can then be constructed
$$\begin{align}F_{pq}^{KS} = T_{pq} + V_{pq} + \sum_{i=1}^{occ} (pi\vert\vert qi) \end{align}$$
where (pi||qi) is an antisymmetrized two-electron integral
$$\begin{align}(pq\vert\vert rs) = \iint \chi_{p}({\bf x}_{1})\chi_{r}({\bf x}_{2}... ...f x}_{1})\chi_{q}({\bf x}_{2}) \right] \, d{\bf x}_{1} d{\bf x}_{2}. \end{align}$$
Unless the Fock functional is used, the Fock matrix will not be diagonal. In particular, there will be nonzero elements F_ia connecting the occupied and virtual spin orbitals. To simply diagonalise this matrix would allow the occupied and virtual orbitals to mix, altering the density. This can be avoided by separating the matrix into two parts,
$$\begin{align}F^{KS} = F^{KS}(\mathrm{OO}+\mathrm{VV}) + F^{KS}(\mathrm{OV}) \end{align}$$
corresponding to nonzero occupied-occupied, virtual-virtual blocks in the first part and nonzero occupied-virtual blocks in the second. The matrix F^KS(OO+VV) can then be diagonalised, providing a new set of spin orbitals that could be described as the canonical Fock orbitals for the constrained KS determinant. These new occupied orbitals will be an orthogonal transformation of the KS occupied spin orbitals and will yield the same density, that is
$$\begin{align}\gamma = \sum_{i=1}^{occ}\vert\chi_{i}\vert^{2} = \sum_{i=1}^{occ}\vert\chi_{i}^{KS}\vert^{2} = \gamma^{KS}. \end{align}$$
The single determinant wavefunction formed from the occupied $\chi_{i}$, written here as $\Psi_{0}$, will be equal to $\Psi_{KS}$.

Second-order Møller-Plesset theory has been used to determine the conventional correlation energy, starting from the $\Psi_{0}$ wavefunction. At first order, the MP energy is simply E_KS and the second-order correction is
 $$\begin{align} E^{(2)} = -\sum_{ia}\frac{F_{ia}^{2}}{\epsilon_{a}-\epsilon_{i}} -... ...\vert ab)^{2}}{\epsilon_{a}+\epsilon_{b}-\epsilon_{i}-\epsilon_{j}}. \end{align}$$
Note that F_ia appears in equation (3.13) as the off-diagonal elements are part of the perturbation Hamiltonian. If the Fock functional is used to form the KS density all F_ia will vanish and equation (3.13) will reduce to the more familiar form of equation (1.75). This new term again essentially allows for a mixing of the occupied and virtual orbitals leading to a tendency for the KS orbitals to move toward HF, therefore modifying the density. Also, the F_ia contribution to E⁽²⁾ is much smaller than the second part of equation (3.13), and have therefore been omitted here.