functionals.rst 5.27 KB
 dulak committed Mar 21, 2009 1 2 3 4 5 6 7 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 .. _xc_functionals: ==================================== Exchange and correlation functionals ==================================== .. index:: libxc Libxc ===== We used the functionals from libxc_. ... Calculation of GGA potential ============================ In libxc_ we have (see also "Standard subroutine calls" on ccg_dft_design_) \sigma_0=\sigma_{\uparrow\uparrow}, \sigma_1=\sigma_{\uparrow\downarrow} and \sigma_2=\sigma_{\downarrow\downarrow} with .. math:: \sigma_{ij} = \mathbf{\nabla}n_i \cdot \mathbf{\nabla}n_j .. _libxc: http://www.tddft.org/programs/octopus/wiki/index.php/Libxc .. _ccg_dft_design: http://www.cse.scitech.ac.uk/ccg/dft/design.html Uniform 3D grid  jensj committed Jan 14, 2011 37 ===============  dulak committed Mar 21, 2009 38 39 40 41 42 43 44 45 46 47 48 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  We use a finite-difference stencil to calculate the gradients: .. math:: \mathbf{\nabla}n_g = \sum_{g'} \mathbf{D}_{gg'} n_{g'}. The x-component of \mathbf{D}_{gg'} will be non-zero only when g and g' grid points are neighbors in the x-direction, where the values will be 1/(2h) when g' is to the right of g and -1/(2h) when g' is to the left of g. Similar story for the y and z components. Let's look at the spin-k XC potential from the energy expression \sum_g\epsilon(\sigma_{ijg}): .. math:: v_{kg} = \sum_{g'} \frac{\partial \epsilon(\sigma_{ijg'})}{\partial n_{kg}} = \sum_{g'} \frac{\partial \epsilon(\sigma_{ijg'})}{\partial \sigma_{ijg'}} \frac{\partial \sigma_{ijg'}}{\partial n_{kg}} Using v_{ijg}=\partial \epsilon(\sigma_{ijg})/\partial \sigma_{ijg}, \mathbf{D}_{gg'}=-\mathbf{D}_{g'g} and .. math:: \frac{\partial \sigma_{ijg'}}{\partial n_{kg}} = (\delta_{jk} \mathbf{D}_{g'g} \cdot \mathbf{\nabla}n_{ig'} + \delta_{ik} \mathbf{D}_{g'g} \cdot \mathbf{\nabla}n_{jg'}), we get: .. math:: v_{kg} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (v_{ijg'} [\delta_{jk} \mathbf{\nabla}n_{ig'} + \delta_{ik} \mathbf{\nabla}n_{jg'}]). The potentials from the general energy expression \sum_g\epsilon(\sigma_{0g}, \sigma_{1g}, \sigma_{2g}) will be: .. math:: v_{\uparrow g} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (2v_{\uparrow\uparrow g'} \mathbf{\nabla}n_{\uparrow g'} + v_{\uparrow\downarrow g'} \mathbf{\nabla}n_{\downarrow g'}) and .. math:: v_{\downarrow g} = -\sum_{g'} \mathbf{D}_{gg'} \cdot (2v_{\downarrow\downarrow g'} \mathbf{\nabla}n_{\downarrow g'} + v_{\uparrow\downarrow g'} \mathbf{\nabla}n_{\uparrow g'}).  jensj committed Jan 07, 2011 98 PAW correction  jensj committed Jan 14, 2011 99 ==============  dulak committed Mar 21, 2009 100   jensj committed Jan 07, 2011 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 Spin-paired case: .. math:: \Delta E = \sum_g 4 \pi w r_g^2 \Delta r_g [\epsilon(n_g, \sigma_g) - \epsilon(\tilde n_g, \tilde\sigma_g)], where w is the weight ... .. math:: n_g = \sum_{i_ii_2} D_{i_1i_2} \phi_{j_1g} Y_{L_1} \phi_{j_2g} Y_{L_2} + n_c(r_g) = \sum_L n_{Lg} Y_L, where .. math:: n_{Lg} = \sum_q D_{Lq} n_{qg} + \delta_{L,0} \sqrt{4 \pi} n_c(r_g) and .. math:: D_{Lq} = \sum_p D_p G_{L_1L_2}^L \delta_{q_p,q} = \sum_p D_p B_{Lpq}. .. math:: \mathbf{\nabla} n_g = \sum_L Y_L \sum_{g'} D_{gg'} n_{Lg'} \hat{\mathbf{r}} + \sum_L \frac{n_{Lg}}{r_g} r \mathbf{\nabla} Y_L = a_g \hat{\mathbf{r}} + \mathbf{b}_g / r_g. Notice that r \mathbf{\nabla} Y_L is independent of r - just as Y_L is. From the two contributions, which are orthogonal (\hat{\mathbf{r}} \cdot \mathbf{b}_g = 0), we get .. math:: \sigma_g = a_g^2 + \mathbf b_g \cdot \mathbf b_g / r_g^2. .. math:: \frac{\partial \Delta E}{\partial n_{Lg}} = 4 \pi w \sum_{g'} r_{g'}^2 \Delta r_{g'} \frac{\partial \epsilon}{\partial \sigma_{g'}} \frac{\partial \sigma_{g'}}{\partial n_{Lg}}. Inserting .. math:: \frac{\partial \sigma_{g'}}{\partial n_{Lg}} = 2 a_{g'} Y_L D_{g'g} + 2 \mathbf b_g \cdot (r \mathbf{\nabla} Y_L) \delta_{gg'} / r_g^2, we get .. math:: \frac{\partial \Delta E}{\partial n_{Lg}} = 8 \pi w \sum_{g'} r_{g'}^2 \Delta r_{g'} \frac{\partial \epsilon}{\partial \sigma_{g'}} a_{g'} Y_L D_{g'g} + 8 \pi w \Delta r_g \frac{\partial \epsilon}{\partial \sigma_g} \mathbf b_g \cdot (r \mathbf{\nabla} Y_L).  jensj committed Jan 14, 2011 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207  Non-collinear case ------------------ .. math:: \mathbf{m}_g = \sum_L \mathbf{M}_{Lg} Y_L. .. math:: n_{\alpha g} = (n_g + \alpha m_g) / 2. .. math:: 2 \mathbf{\nabla} n_{\alpha g} = \mathbf{\nabla} n_g + \alpha \sum_L ( Y_L \sum_{g'} D_{gg'} \frac{\mathbf{m}_g \cdot \mathbf{M}_{Lg'}}{m_g} \hat{\mathbf{r}} + \frac{\mathbf{m}_g \cdot \mathbf{M}_{Lg}}{m_g r_g} r \mathbf{\nabla} Y_L) .. math:: = (a_g + \alpha c_g) \hat{\mathbf{r}} + (\mathbf{b}_g + \alpha \mathbf{d}_g) / r_g. .. math::  jensj committed Jan 14, 2011 208  4 \sigma_{\alpha \beta g} =  jensj committed Jan 14, 2011 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224  (a_g + \alpha c_g) (a_g + \beta c_g) + (\mathbf{b}_g + \alpha \mathbf{d}_g) \cdot (\mathbf{b}_g + \beta \mathbf{d}_g) / r_g^2. .. math:: \frac{\partial c_g}{\partial \mathbf{M}_{Lg'}} = \frac{Y_L}{m_g} ( D_{gg'} \mathbf{m}_g + \delta_{gg'} \mathbf{m}_g' - \delta_{gg'} \frac{\mathbf{m}_g \cdot \mathbf{m}_g'}{m_g^2} \mathbf{m}_g). .. math:: \frac{\partial (\mathbf{d}_g)_\gamma}{\partial \mathbf{M}_{Lg'}} =  jensj committed Jan 14, 2011 225  \frac{Y_L \delta_{gg'}}{m_g} (  jensj committed Jan 14, 2011 226 227 228 229 230  \mathbf{m}_g r \nabla_\gamma Y_L + \sum_{L'} \mathbf{M}_{L'g} r \nabla_\gamma Y_{L'} - \frac{\mathbf{m}_g}{m_g^2} \sum_{L'} \mathbf{m}_g \cdot \mathbf{M}_{L'g} r \nabla_\gamma Y_{L'}).