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[Xavier Gonze <gonze@pcpm.ucl.ac.be>]

Dear Andriy Nevidomskyy,

Andriy Nevidomskyy wrote:
> Dear Prof Xavier Gonze,
>
> While calculating phonon band structure of *benzene* molecule using
> abinis_v3.3.3 I was suprized to find negative frequencies in the output
> file. What could it mean?
> Some elements of dynamical matrix are negative
> (all are real though), suggesting that the structure might be unstable -
> since second derivatives of the energy are negative...
>
> However, the geometry of the molecule has been optimized so that forces
> on all atoms are quite small as well as are the stresses. Maybe it's the
> fact that benzene molecule is effectively 2D, that is crucial?
>
> Even if we admit that dynamical matrix contains negative elements,
> wouldn't it lead to *complex*, rather than *negative*, phonon 
frequencies?
>
> The output of phonons in Gamma (q=0,0,0) is below:
> ! Note that acoustic frequencies are not small by modulus = 60 cm-1!!!
> *********************************************************
>  Phonon frequencies in cm-1    :
> - -0.737886D+02 -0.685974D+02 -0.617728D+02 -0.528370D+02 -0.378516D+02
> - -0.249379D+02  0.401239D+03  0.401978D+03  0.603561D+03  0.607614D+03
> -  0.646034D+03  0.714213D+03  0.824161D+03  0.824692D+03  0.956362D+03
> -  0.956485D+03  0.985671D+03  0.999253D+03  0.100249D+04  0.103988D+04
> -  0.104072D+04  0.114675D+04  0.116831D+04  0.116939D+04  0.134262D+04
> -  0.134521D+04  0.147872D+04  0.147936D+04  0.160277D+04  0.160339D+04
>
>
> -  0.304616D+04  0.305551D+04  0.305698D+04  0.307338D+04  0.307372D+04
> -  0.308502D+04
> ********************************************************
>

One should observe six modes with zero frequency, corresponding
to 3 pure translation modes and 3 pure rotation modes. Instead
of these zero-frequency modes, you observe a group of six
'negative' modes (which means unstability - actually these are
 imaginary frequencies, as you have noticed,
but that's the way ABINIT deliver them - easier for automatic
post-processing )
in the range -73 to -24 cm-1, clearly separated from other modes,
these starting at 401 cm-1 . I think that this behaviour is NOT the
consequence of a bug. But there are indeed a few things to
understand here !

The plane wave implementation has nice features, but it cannot
go to the point of being tranlationally invariant, because
of the exchange and correlation grid (also present in other
implementations, in any case). So, it is normal that the
translational and rotational modes are not found with
a zero frequency. The observed frequencies can be made
smaller and smaller, by using finer and finer XC grids,
but the convergence is extremely slow.

Then, comes explanations related to the range of
observed frequencies. Note that your highest frequency
is on the order of 3000 cm-1. The frequencies are
the SQUARE ROOTS of the eigenvalues of the dynamical
matrix. Thus, the relative XC "noise" that generates non-vanishing
frequencies might be smaller than one part per thousand, on
typical matrix elements of the dynamical matrix, which
is indeed perfectly sensible. Last important
note : Hydrogen is the lightest atom (12 times lighter than
Carbon).

Actually, Xavier Rocquefelte and Gian-Marco Rignanese
have observed that range of non-zero frequencies for
other H-containing molecules, but not for molecules without H
(the residual frequencies were less than 10 cm-1 usually).

At last, you mention that the molecule should be unstable,
with these frequencies, while you have done a perfectly
satisfactory optimisation of structure. However, note that
these frequencies are rather small, and will not generate
large forces, especially since they are varying fast in space
(being related to the XC grid).

It seems that we have to live with this problem in ABINIT.
Note that in IFC, the acoustic sum rule is imposed
automatically so that the frequencies of the
translational modes can be made much closer to zero.
A treatment of rotational modes should be possible,
but has not been implemented.

>
> Finally, one more conceptual question.
> When should one care about the LO-TO splitting?
> Is it present in all classes of solids?
> Is it present only in crystal solids or also in molecules?

It is associated with long-range electrostatic interaction.
It should be ignored for molecules and for metals.

>
> Do I understand correctly that in order to take LO-TO splitting into 
account

So, only in the case of insulators,


> within ABINIT one has to do the following:
> 1) calculate d/dk perturbations (only in q=Gamma!) with rfelfd=2.
> 2) calculate phonons in q=Gamma specifying rfelfd=3 (which would use the
> result of d/dk from previous calculation)

OK.

> 3) calculate phonons close to Gamma?

Not needed.

>
> If this sketch is correct, how would ABINIT know, while doing 3), that it
> has to take LO-TO splitting into account? ABINIT cannot perform d/dk
> perturbations away from Gamma!
> Thus, what is the mechanism of calculating LO-TO splitting at q 
close, but
> not equal to Gamma?


Precisely at Gamma, there are non-analyticites, that must be separated
from the other contributions to the dynamical matrix. Thus, inside
ABINIT, there are at a few places, tests on whether q is exactly
0 0 0 or not. If not, all the contributions can be treated at once.
If at Gamma, then the non-analyticities are removed. They must be 
reintroduced thanks to effective charges, and so on, giving LO-TO
splitting.
In any case, the limit of non-zero q wavevectors frequencies
is obtained by taking the analytical result at Gamma, and adding
to it the onon-analytical contribution, precisely given
by the LO-TO splitting.

Good continuation,
Xavier