# [Pw_forum] cell size dependence of Wannier spreads

Nicola Marzari marzari at MIT.EDU
Tue Mar 20 21:06:28 CET 2007

```Konstantin Kudin wrote:
>  Hi all,
>
>  Can anyone briefly summarize how Wannier function spreads converge
> with the size of the unit cell? The spreads somehow seem to be not so
> well defined ...
>
>  Thanks!
>  Kostya
>

Hi Kostya,

the thins to keep in mind is that the r and r^2 operators
in the Wannier algorithm are calculated using a reciprocal
space representation, and that since we have only a
discrete k-point mesh, we are actually *approximating* these
operators.

This is to say that as you increase the fineness of the k-point
sampling, not only your electronic structure becomes more accurate,
but that your approximate representation of the operators becomes
also more accurate. The "operators" converge more slowly than the
"electronic structure".

So, spreads can only be compared either for fully converged
calculations with respect to k-points, or when compatible meshes
are used (e.g. if you double the cell in one direction,
you halve the k-points).

If you are using CP, or any Gamma-only code, changing the unit cell
size also makes your spread operator different. So if you calculate
spreads in liquid water for a 64-molecule unit cell, they will be
larger (I think) than for a 32-molecule unit cell, but for the only
reason that the operator is changing.

One fix that might work (if needed) would be to calculate the spread
explicitly in real space - i.e. the unitary matrices and the electronic
structure converge rapdily with k-point sampling, and than with the
U_mn^k you construct the MLWFs in real space, and you calculate
expliclty <r> and <r^2>. Or, you could use the extrapolation technique
used by Yudong Wu in his PhD (was a student of Roberto that worked with
me on WFs, he graduated in 2003 or so, and a copy should be around in
Princeton), in which <psi|exp(iGr)|psi> was calculated for very small G,
smaller than the smallest in your Bravais lattice.

nicola

--
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Prof Nicola Marzari   Department of Materials Science and Engineering
13-5066   MIT   77 Massachusetts Avenue   Cambridge MA 02139-4307 USA
tel 617.4522758 fax 2586534 marzari at mit.edu http://quasiamore.mit.edu

```

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