Planewaves

With \(N=N_1N_2N_3\) grid points: \(\br^T=(g_1/N_1,g_2/N_2,g_3/N_3)\mathbf A\), where \(g_c=0,1,...,N_c-1\), we get a plane wave expansion of the wave function as:

\[\tilde\psi_{k n}(\br) = \frac{1}{N} \sum_\bG e^{i(\bG+\bk)\cdot \br}c_{\bk n}(\bG),\]

where the coefficients are given as:

\[c_{\bk n}(\bG) = \sum_\br e^{-i(\bG+\bk)\cdot\br}\tilde\psi_{\bk n}(\br)\]

Exact Exchange

From the pair densities:

\[ \begin{align}\begin{aligned}\begin{split}\tilde\rho_{\bk_1n_1 \bk_2n_2}(\br) = \tilde\psi_{\bk_1n_1}(\br)^* \tilde\psi_{\bk_2n_2}(\br) + ... = \\\end{split}\\\frac{1}{N^2} \sum_{\bG\bG'} e^{i(\bG-\bk_1+\bk_2)\cdot \br} c_{\bk_1n_1}(\bG)^* c_{\bk_2n_2}(\bG+\bG') = \sum_\bG e^{i(\bG-\bk_1+\bk_2)\cdot \br}C_{\bk_1n_1\bk_2n_2}(\bG),\end{aligned}\end{align} \]

we get the exact exchange energy:

\[E_x = -\pi\Omega \sum_{\bk_1n_1} \sum_{\bk_2n_2} f_{\bk_1n_1}f_{\bk_2n_2} \sum_\bG \frac{|C_{\bk_1n_1\bk_2n_2}(\bG)|^2}{|\bk_1-\bk_2-\bG|^2},\]

where the weight of a \(\bk\)-point is included in \(f_{\bk n}\). Let \(E_x'\) be defined as the sum above excluding the divergent terms for \(\bk_1=\bk_2\) and \(\bG=0\). With

\[F(\bG)=\frac{e^{-\alpha G^2}}{G^2},\]

we get (see [1]):

\[E_x = E_x' -\pi\Omega\sum_{\bk_1n_1n_2}f_{\bk_1n_1}f_{\bk_1n_2} |C_{\bk_1n_1\bk_1n_2}(0)|^2 \left(\sum_{\bk_2\bG}F(\bk_1-\bk_2-\bG)- \sum_{\bk_2}\sum_{\bG\neq\bk_1-\bk_2}F(\bk_1-\bk_2-\bG)\right).\]

In the limit of an infinitely dense sampling of the BZ and a not too small \(\alpha\), we get

\[\sum_{\bk_2\bG}F(\bk_1-\bk_2-\bG)= \frac{N_k\Omega}{(2\pi)^3}\int_{\text{BZ}}F(\bk)d\bk= \frac{N_k\Omega}{(2\pi)^2}\sqrt{\pi/\alpha},\]

where \(N_k\) is the number of \(\bk\)-points.

Finally:

\[E_x = E_x' -\pi\Omega\sum_{\bk_1n_1n_2}f_{\bk_1n_1}f_{\bk_1n_2} |C_{\bk_1n_1\bk_1n_2}(0)|^2\gamma,\]

where

\[\gamma = \frac{\Omega}{(2\pi)^2}\sqrt{\pi/\alpha}- \sum_{\bk}\sum_{\bG\neq\bk}F(\bk-\bG).\]

The gradient is:

\[\frac{\partial E_x}{\partial\tilde\psi_{\bk_1n_1}(\br)}= -\pi\Omega\sum_{\bk_2n_2}f_{\bk_1n_1}f_{\bk_2n_2} e^{i(\bk_1-\bk_2)\cdot\br}\tilde\psi_{\bk_2n_2}(\br) \frac1N\sum_\bG\frac{C_{\bk_1n_1\bk_2n_2}(G)^*}{|\bk_1-\bk_2-\bG|^2} e^{-i\bG\cdot\br},\]

where \(1/|\bk_1-\bk_2-\bG|^2\) is replaced by \(\gamma\) for the term where \(\bk_1=\bk_2\) and \(\bG=0\).

[1]Accurate and Efficient Method for the Treatment of Exchange in a Plane-Wave Basis, A. Sorouri, W.M.C. Foulkes, and N.D.M. Hine, J. Chem. Phys. 124, 064105-1 – 064105-7 (2006)