============================= Planewaves and exact exchange ============================= 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: .. math:: \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: .. math:: c_{\bk n}(\bG) = \sum_\br e^{-i(\bG+\bk)\cdot\br}\tilde\psi_{\bk n}(\br) **Exact exchange** From the pair densities: .. math:: \tilde\rho_{\bk_1n_1 \bk_2n_2}(\br) = \tilde\psi_{\bk_1n_1}(\br)^* \tilde\psi_{\bk_2n_2}(\br) + ... = \\ \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), we get the exact exchange energy: .. math:: 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 .. math:: F(\bG)=\frac{e^{-\alpha G^2}}{G^2}, we get (see [#Sorouri]_): .. math:: 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 .. math:: \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: .. math:: 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 .. math:: \gamma = \frac{\Omega}{(2\pi)^2}\sqrt{\pi/\alpha}- \sum_{\bk}\sum_{\bG\neq\bk}F(\bk-\bG). The gradient is: .. math:: \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. .. [#Sorouri] *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)