planewaves.rst 2.58 KB
Newer Older
jensj's avatar
jensj committed
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
7
function as:
jensj's avatar
jensj committed
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's avatar
jensj committed
48
`E_x'` be defined as the sum above excluding the divergent terms
jensj's avatar
jensj committed
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's avatar
jensj committed
99
   \frac1N\sum_\bG\frac{C_{\bk_1n_1\bk_2n_2}(G)^*}{|\bk_1-\bk_2-\bG|^2}
jensj's avatar
jensj committed
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)