[MMTK] scaling the amplitudes of the atomic displacement vectors

Konrad Hinsen hinsen at cnrs-orleans.fr
Tue Aug 19 22:23:22 EDT 2003


> In module NormalModes the atomic displacement vectors are scaled acording
> to
>
> amplitude=Numeric.sqrt(2.*temperature*Units.k_B/Units.amu) /
> (2.*Units.pi*self.frequencies[index])
>
> Could you explain where factor 2 in
> Numeric.sqrt(2.*temperature*Units.k_B/Units.amu) is from?

Here's one possible derivation:

The average potential energy in each mode is 0.5*temperature*Units.k_B from 
the equipartition theorem. It is also equal to half of the potential energy 
at maximum amplitude (because the average of sin(x)**2 over a period is 0.5), 
i.e. 0.25*k*amplitude**2. This yields

   amplitude**2 = 2.*temperature*Units.k_B/k

which shows where the factor 2 comes from. Replacing k by mass*omega**2 yields 
the complete formula you quoted.

Konrad.
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