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One strength of molecular dynamics with respect to other methods
such as Monte Carlo is the fact that the time evolution of the
system is followed and--therefore--information on the dynamics
of the system is fully present.
Experimental data (for instance, phonon dispersion curves
in the case of crystals) are usually available in the
wavevector-frequency space. Information obtained by MD in real space
and real time will therefore be Fourier-transformed with respect
to both space and time for useful comparisons with experimental results.
To perform dynamical analysis, one typically instructs the simulation
program to dump periodically on a mass storage medium
the positions and possibly the velocities of all the particles.
These can easily produce very large files: the calculation
of dynamical quantities requires plenty of room for data.
These data are then analyzed later by other programs to extract
the quantities of interest.
In some cases, dynamical analysis are directly performed by the
MD program while it is running, thus saving disk space.

One typical quantity is the Van Hove correlation function
, representing the probability to find an atom
at position at time *t* if there was an atom
at position at time *t*=0.
This quantity is connected by a double Fourier transform to
the *dynamic structure factor* reported by experiments:

| |
(21) |

As for static quantities, the most convenient way to proceed
in terms of computational effort is that of using the space
Fourier transform of the density as a building block:
| |
(22) |

This is a single-particle quantity which is very fast to compute.
From here, one constructs a time-dependent density-density
correlation function (also called *intermediate scattering function*)
by performing an average over a molecular dynamics trajectory:
| |
(23) |

The average is made for all the available choices for .The maximum *t* (connected with the frequency resolution) is
clearly limited by the length of the run, while the periodicity
with which data are collected--typically a few time steps--controls
the maximum frequency in the final spectra.
The dynamic structure factor is finally obtained by a time
Fourier transform:
| |
(24) |

Again, the allowed are quantized as a consequence of
periodic boundary conditions.
will exhibit peaks in the plane, corresponding to the dispersion of propagating modes.
The broadening of these peaks is related with anharmonicity.
By performing calculations at different temperatures, one can
for instance study the effect of temperature on the phonon spectra,
fully including anharmonic effects: in this sense, the approach
is clearly superior to standard lattice dynamics techniques.
The main inconvenience of MD is that it is not so straightforward
to extract eigenvectors.
When dealing with vibrational modes and different polarizations,
it is often convenient to follow the approach outlined above,
but using currents

| |
(25) |

in place of the density .

** Next:** Annealing and quenching: MD
** Up:** Running, measuring, analyzing
** Previous:** Reciprocal space correlations
*Furio Ercolessi*

*9/10/1997*