functionals.rst 5.27 KB
Newer Older
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's avatar
jensj committed
37
===============
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'}).



98
PAW correction
jensj's avatar
jensj committed
99
==============
100

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's avatar
jensj committed
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's avatar
jensj committed
208
    4 \sigma_{\alpha \beta g} =
jensj's avatar
jensj committed
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's avatar
jensj committed
225
    \frac{Y_L \delta_{gg'}}{m_g} (
jensj's avatar
jensj committed
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'}).