# 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

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

$\frac{\partial E_x}{\partial\tilde\psi_{\bk_1n_1}(\br)}= -\pi\Omega\sum_{\bk_2n_2}f_{\bk_1n_1}f_{\bk_2n_2} e^{i(\bk_1-\bk_2)\cdot\br}\tilde\psi_{\bk_2n_2}(\br) \frac1N\sum_\bG\frac{C_{\bk_1n_1\bk_2n_2}(G)^*}{|\bk_1-\bk_2-\bG|^2} e^{-i\bG\cdot\br},$
where $$1/|\bk_1-\bk_2-\bG|^2$$ is replaced by $$\gamma$$ for the term where $$\bk_1=\bk_2$$ and $$\bG=0$$.