.. _Ch:Algorithm:

Algorithm
*********

In RESPACK, the :math:`i`-th Wannier function is defined as 

.. math::

    w_{i, 0}({\bf r}) = \frac{1}{\sqrt{N_k}} \sum_{{\bf k}} \sum_{m=1}^{N_w} \sum_{n=N_s}^{N_s+N_b} U_{im}^{{\bf k}} U_{mn}^{{\bf k}, opt} \psi_{nk}(\bf{r}) 

Here, :math:`U` and :math:`U^{opt}` are unitary matricies, :math:`\psi_{n{\bf k}}(r)` is the :math:`n`-th Bloch wave function, and :math:`N_s` and :math:`N_b` are determined from the energy-window information.
The Bloch wave function is defined as

.. math::
    \psi_{n{\bf k}} =\frac{1}{\sqrt{N_k}} \frac{1}{\sqrt{\Omega}} \sum_{{\bf G}} C_{{\bf G}n}({\bf k}) e^{i({\bf k}+{\bf G})\cdot {\bf r}}

Here, :math:`C_{{\bf G}n}` is the expansion coefficient of the plane wave.
By inserting this coefficient, the Wannier function is written as 

.. math::

    w_{i, 0}({\bf r}) = \frac{1}{N_k} \sum_{{\bf k}} \sum_{{\bf G}} \tilde{C_{{\bf G}i}}({\bf k})\frac{1}{\sqrt{\Omega}}e^{i({\bf k}+{\bf G})\cdot r}

Here, :math:`\tilde{C_{\bf{G}i}}(\bf{k})` is the expansion coefficient of the plane wave for the Wannier function:

.. math::
    
    \tilde{C_{{\bf G} i}}({\bf k}) = \sum_{m=1}^{N_w} \sum_{n=N_s}^{N_s+N_b} U_{im}^{{\bf k}} U_{mn}^{{\bf k}, opt} C_{{\bf G}n}


Here, :math:`U^{{\bf k},opt}` is required at dielectric function calculation, and :math:`\tilde{C_{{\bf G}i}}({\bf k})` is required at Coulomb calculation.
This program generates :math:`\tilde{C_{{\bf G}i}}(\bf{k})` using :math:`C_{{\bf G}n}` made by QE and :math:`U^{{\bf k}}, U^{{\bf k}, opt}` made by Wannier90.

**K-points Order**

RESPACK performs calculations based on the information at irreducible k-points (:math:`\psi_{k\in irr}`). 
This is because the information at reducible k-points (:math:`\psi_{k \in reducible}`) can be generated by symmetry.
On the other hand, Wannier90 calculates :math:`U^{{\bf k}}`, and so on, based on :math:`\psi_{k \in reducible}`.
The :math:`{\bf k}`-order of :math:`\psi_{k \in reducible}` used in Wannier90 must be the same as the :math:`{\bf k}`-order of :math:`\psi_{k \in reducible}` generated by the symmetry in RESPACK.
In pre-process mode, an operation is performed to align the :math:`{\bf k}`-order.


.. _file_expression-label:

Relations between expressions and file names
--------------------------------------------

This program finally outputs the four files in the ``dir-wan`` directory.
The expressions and the corresponding files are shown below:

- :math:`\tilde{C_{{\bf G}i}}(\bf{k})` --- ``dat.wan``

- :math:`N_s, N_b`  --- ``dat.ns-nb``

- :math:`U^{{\bf k}}U^{{\bf k},opt}` --- ``dat.umat``

- :math:`<w_{i0}|r|w_{i0}>` --- ``dat.wan-center``


References
----------------

- K. Nakamura, Y. Yoshimoto, Y. Nomura, T. Tadano, 
  M. Kawamura, T. Kosugi, K. Yoshimi, T. Misawa, and Y. Motoyama,
  `Comput. Phys. Commun. 261, 107781 (2021) <https://doi.org/10.1016/j.cpc.2020.107781>`_

- N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt,
  `Rev. Mod. Phys. 84, 1419 (2012) <https://doi.org/10.1103/RevModPhys.84.1419>`_