planewaves.rst 2.58 KB
 jensj committed Nov 05, 2010 1 2 3 4 5 6 ========== 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  Jens Jørgen Mortensen committed Feb 23, 2016 7 function as:  jensj committed Nov 05, 2010 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47  .. 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  jensj committed Feb 02, 2011 48 E_x' be defined as the sum above excluding the divergent terms  jensj committed Nov 05, 2010 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 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)  jensj committed Nov 23, 2010 99  \frac1N\sum_\bG\frac{C_{\bk_1n_1\bk_2n_2}(G)^*}{|\bk_1-\bk_2-\bG|^2}  jensj committed Nov 05, 2010 100 101 102 103 104 105 106 107 108 109  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)