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
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
|
.. _electrostatic potential:
===============================
Note on electrostatic potential
===============================
In the PAW formalism, the electrostatic potential from the
pseudo charge `\tilde{\rho}(\mathbf{r})` is obtained by solving a Poisson
equation:
.. math::
\nabla^2 \tilde{v}_H(\mathbf{r})=-4\pi\tilde{\rho}(\mathbf{r}).
To get the *real* all-electron electrostatic potential, we need the
all-electron charge density:
.. math::
\rho(\mathbf{r}) = \tilde{\rho}(\mathbf{r}) +
\sum_a \Delta\tilde{\rho}^a(\mathbf{r} - \mathbf{R}^a),
where `\Delta\tilde{\rho}^a` is an atomic PAW correction to the pseudo
charge density:
.. math::
\Delta\tilde{\rho}^a(\mathbf{r}) =
n_c^a(r) - \tilde{n}_c^a(r) -
\mathbb{Z}^a\delta(\mathbf{r}) -
\sum_{\ell=0}^{\ell_{\text{max}}} \sum_{m=-\ell}^\ell
Q_{\ell m}^a \hat{g}_{\ell m}^a(\mathbf{r}) +
\sum_{\sigma i_1 i_2} D_{\sigma i_1 i_2}^a
(\phi_{i_1}^a(\mathbf{r})\phi_{i_2}^a(\mathbf{r}) -
\tilde{\phi}_{i_1}^a(\mathbf{r})\tilde{\phi}_{i_2}^a(\mathbf{r})).
See :ref:`here <density>` for details.
So, the all-electron potential is:
.. math::
v_H(\mathbf{r}) = \tilde{v}_H(\mathbf{r}) +
\sum_a \Delta\tilde{v}_H^a(\mathbf{r} - \mathbf{R}^a)
and
.. math::
\Delta\tilde{v}_H^a(\mathbf{r}) =
\int d\mathbf{r}'
\frac{\Delta\tilde{\rho}^a(\mathbf{r}')}
{|\mathbf{r}-\mathbf{r}'|}.
Notice that the `Q_{\ell m}^a` have been chosen so that all multipole
moments of `\Delta\tilde{\rho}^a` are zero and therefore, the
potential from these correction charges (`\Delta\tilde{v}_H^a`) will
be non-zero only inside the atomic augmentation spheres.
The :meth:`~gpaw.calculator.GPAW.get_electrostatic_corrections`
method will return an array of integrated corrections:
.. math::
\int d\mathbf{r} \Delta\tilde{v}_H^a(\mathbf{r})
in units of eV Å\ :sup:`3`.
|
