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Phonon frequencies
The central quantity in the calculation of the phonon frequencies is
the force-constant matrix
, since the
frequencies at wavevector
are the eigenvalues of the
dynamical matrix
, defined as:
![\begin{displaymath}
D_{s \alpha , t \beta} ( {\bf k} ) = \frac{1}{\sqrt{M_s M_t}...
...R}_j^0 + {\bf\tau}_t - {\bf R}_i^0 - {\bf\tau}_s) \right] \; .
\end{displaymath}](img6.png) |
(1) |
where
is a vector of the lattice connecting different
primitive cells and
is the position of the atom
in
the primitive cell. If we have the complete force-constant matrix,
then
and hence the frequencies
can be obtained at any
, so that
can be computed to any required precision. In principle, the elements
of
are non-zero for arbitrarily large
separations
, but in practice they decay rapidly with separation,
so that a key issue in achieving our target precision is the cut-off
distance beyond which the elements can be neglected.
Dario Alfe`
2012-02-20