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

[Paolo Giannozzi]

Interesting. I report here a few comments I received from other people
about the second sum rule:


   1. try to compute it on the dynamical matrix at Gamma for the supercell
   you can obtain with "matdyn.x" starting from interatomic the force constants
   2. it seems to be associated with a rotation or homogeneous deformation
   and should be zero only at equilibrium (with respect to both internal
   coordinates and volume): try to compute it as a function of the volume.
   3. it weights more heavily force constants that are far from the origin.
   These are truncated in a rather arbitrary way, so it might be tricky to get
   it with a good degree of accuracy

Paolo

On Mon, Jun 8, 2020 at 10:19 PM Ville Härkönen <ville.j.harkonen at gmail.com>
wrote:

> 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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-- 
Paolo Giannozzi, Dip. Scienze Matematiche Informatiche e Fisiche,
Univ. Udine, via delle Scienze 208, 33100 Udine, Italy
Phone +39-0432-558216, fax +39-0432-558222
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