[QE-developers] Sum rule not valid for the second order IFCs given by QE?

[Ville Härkönen]

Hi!

The following sum rule for the second order IFCs should hold

\sum_{l,k,k'} \Phi_{\alpha\beta}(lk,k') = 0.           (1)

Eq. (1) should hold when one uses the IFCs from the 'flfrc'-file and I
checked this to be true (the largest contributions are of the order 10^-6
in the same units as the IFCs are written in the file) for the Si diamond
(Si-d) structure. Also the following sum rule (this and the previous
condition can be found from Born-Huang, Dynamical Theory of Crystal
Lattices, p. 221, 1954.)

\sum_{l,k,k'} \Phi_{\alpha\beta}(lk,k') [ x_{\gamma}(l)  + x_{\gamma}(k) -
x_{\gamma}(k') ] = 0,          (2)

should hold for all \alpha, \beta, \gamma. However, if the IFCs from the
'flfrc'-file are used, Eq. (2) does not hold (the largest terms are of the
order 10^-1), at least for the Si-d case in which Eq. (1) holds. I have
used the lattice translational vectors

\vec{x}(l) = \sum _{i} l_{i} \vec{a}_{i},     l_{i} = 0, 1,\ldots,L-1,

where \vec{a}_{i} are naturally chosen to be the same as in the QE input.
When I use the IFCs of the 'flfrc'-file to calculate the phonon
frequencies, the same result is obtained as given by QE.

Could someone please confirm whether or not Eq. (2) holds, when the IFCs
from the  'flfrc'-file are used? It's always possible that there is an
error in my own calculations, but I have checked many times and haven't
found any. If it turns out that Eq. (2) does not hold for the IFCs from the
'flfrc'-file, what might be the reason for that?

All the best,
Ville Härkönen
University of Jyväskylä
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