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.. _introduction_to_paw:
===================
Introduction to PAW
===================
A simple example
================
We look at the `2\sigma`\ * orbital of a CO molecule: |ts|
.. |ts| image:: 2sigma.png
The main quantity in the PAW method is the pseudo wave-function (blue
crosses) defined in all of the simulation box:
.. math::
\tilde{\psi}(\mathbf{r}) = \tilde{\psi}(ih, jh, kh),
where `h` is the grid spacing and `(i, j, k)` are the indices of the
grid points.
.. figure:: co_wavefunctions.png
In order to get the all-electron wave function, we add and subtract
one-center expansions of the all-electron (thick lines) and pseudo
wave-functions (thin lines):
.. math::
\tilde{\psi}^a(\mathbf{r}) = \sum_i C_i^a \tilde{\phi}_i^a(\mathbf{r})
.. math::
\psi^a(\mathbf{r}) = \sum_i C_i^a \phi_i^a(\mathbf{r}),
where `a` is C or O and `\phi_i` and `\tilde{\phi}_i` are atom
centered basis functions formed as radial functions on logarithmic
radial grid multiplied by spherical harmonics.
The expansion coefficients are given as:
.. math::
C_i^a = \int d\mathbf{r} \tilde{p}^a_i(\mathbf{r} - \mathbf{R}^a)
\tilde{\psi}(\mathbf{r}).
Approximations
==============
* Frozen core orbitals.
* Truncated angular momentum expansion of compensation charges.
* Finite number of basis functions and projector functions.
More information on PAW
=======================
You can find additional information on the :ref:`literature` page, or
by reading the :download:`paw note <paw_note.pdf>`.
Script
======
.. literalinclude:: co_wavefunctions.py
|
