[MMTK] Estimating error in the calculated vibrational density of states

Konrad Hinsen research at khinsen.fastmail.net
Mon Apr 23 11:04:01 UTC 2012

Nitin writes:

 > I have been calculating the vibrational density of states (VDOS) of
 > water systems. Since the MD simulations are not infinitely long
 > there must be some kind of noise that gets introduced in
 > calculating the VDOS. I was wondering if there is some way (any
 > method/ reference) to estimate this noise/error so that when one
 > reports the VDOS then an error bar can be placed.

The finite length of an MD trajectory has two consequences for the
vibrational density of states:

1) The dynamics on time scales longer than the trajectory is not
   captured in the simulation and is therefore missing. The DOS
   is systematically under-estimated at the low-frequency end.

2) The statistical average in the calculation of the velocity
   autocorrelation function (from which the DOS is calculated by
   Fourier transform) is imperfect, leading to statistical error.

Neither of the two sources of error can be estimated without further
assumptions, because both are a consequence of missing information.

A way to eliminate the systematic error (1) is to assume an analytical
model for the VACF or the DOS and fit it to the simulation data. The
quality of the result then depends on the accuracy of the model.

A popular way to handle problems of type (2) is to repeat the
calculation for shorter subsets of the trajectory. Typically one would
divide the trajectory into N pieces and calculate the DOS for each
piece of reduced length (watching out for problem (1) of course).  The
variance among the N results can be used as an estimate for the
statistical error. However, this gives reasonable results only if your
trajectory is representative of the true dynamics. If your system has
slow processes that are not captured by the trajectory, any such
analysis results in a signification under-estimation of statistical

Konrad Hinsen
Centre de Biophysique Moléculaire, CNRS Orléans
Synchrotron Soleil - Division Expériences
Saint Aubin - BP 48
91192 Gif sur Yvette Cedex, France
Tel. +33-1 69 35 97 15
E-Mail: research AT khinsen DOT fastmail DOT net

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