Part II Theory
3.4 Molecular Dynamics (MD)
3.4.6 Data Analysis
As we mentioned earlier, all the information that a molecular dynamics simulation gives is contained in the trajectories of the particles. A fair number of time steps need to be discarded before the system is in an equilibrium state and all the information from the initial condition is lost by that point, due to the numerical errors.
To determine when the system reaches equilibrium, it is often useful to look at the changes of total energy (potential plus kinetic) versus time. For a system in an NVE ensemble, it is often necessary to achieve an energy fluctuation less than 0.1%.
For a system in an NVT ensemble, it is common to have a temperature fluctuation
of a large amplitude, but one must ensure that the frequency of fluctuation agrees well with the one predicted by the size and components of the system. A velocity distribution function is also helpful to determine if the system is in equilibrium (the distribution should be close to the Maxwell-Boltzmann distribution at the corresponding temperature).
While the phase of the system can be found from the instantaneous positions of the atoms in an equilibrium state, one has to remember that the atoms are vibrating around their equilibrium positions, so to get accurate lattice parameters, time averages of the positions must be calculated. Some thermodynamic properties of the system can be obtained by taking derivatives of the results from a set of different environments. For example, the bulk modulus can be obtained by running an NVE MD under a few different volumes, and comparing the calculated pressures. The same method also applies to the thermal expansion coefficient.
A radial distribution function (RDF), calculated by binning the distances be- tween atom-atom pairs, gives useful information about the distribution of atoms around each other. A RDF is particularly useful for amorphous or other disordered materials. Pair distribution functions (PDF) measured from X-ray scattering can be converted to RDFs, and directly compared to the ones from molecular dynamics, sometimes with very good agreement. The mean square displacement (MSD) over time t tells how far atoms travel from their original positions. It is useful to check for solid-liquid phase changes because in liquid systems, the MSD tends to increase linearly with time.
We are mostly interested in obtaining phonon dynamics information from these trajectories. In principle, with a system of sufficient size, small enough time step, and long enough run-time, it is possible to reproduce all the phonon properties to good resolution within the limit of the interatomic potentials. To calculate the vibrational spectrum from the trajectories, we need to define a velocity autocorrelation function (VAF) first:
C(t)= hv(0)·v(t)i
hv(0)·v(0)i, (3.17)
where v(t) is the velocity of atoms and hi denotes the average over all atoms, and is normalized to the velocities at t = 0. Partial VAFs for certain atoms (for example, one species in a multi-element compound), or velocities at certain symmetry directions can also be defined accordingly.
The VAF gives the information on the dynamics of an "average" atom. It always starts at unity and usually decays to zero in the long-time limit because eventually all the information about the initial state is lost and so is the "correlation".
In a solid material, the VAF contains all information on the vibrational spectrum.
[53] The square of the Fourier transform of a VAF yields the phonon density of states
g(ω)= 1 2π
∞
Z
−∞
C(t)exp(iωt)dt
2
. (3.18)
Because of the discrete nature of the MD simulations, it is convenient to calculate discrete VAFs
Cm= hvm·v0i
hv0·v0i, (3.19)
and use a fast Fourier transform to obtain the phonon DOS. One problem remains:
the VAF often goes to zero eventually and as a result, when using Eq. 3.17, late time steps in longer MD simulations will not contribute to the statistical quality of the result because the VAF has decayed to zero there. One way to solve this problem is breaking up long MD runs into shorter ones, and using an averaged VAF. A smarter way to do it is using an average of all possible VAFs that can be constructed from the trajectories [53]
Cm= 1 M −m
M−m−1
X
n=0
hvn+m·vni
hv0·v0i . (3.20)
For a MD simulation withm steps, there arem−1 different VAFs with different ranges. The smallert parts always have better statistical quality and they happen to be the parts that contain the most information. It is important to have the correct weighting factor here.
The resolution of the phonon DOS is mostly determined by the size and the
number of time steps, but the sizes of the system also matters. There are some rules on the choice of the size and the number of time steps for achieving a desired resolution. Using a partial VAF, partial phonon DOS functions for specific atoms and phonon dispersion relations can also be calculated similarly. Although the vibrational information at theΓ point is always missing because it corresponds to the correlations with atoms at infinite distances, it is possible to approximate it with aqpoint that is close enough to the center of the Brillouin zone.
In practice, any direct Fourier transform of the VAF gives noisy results. It is best to append the VAF with a trailing series of zeros several times larger in size.
Because a steep step function causes oscillations after Fourier transformation, it is also necessary to smooth the interface between the VAF and the zeros in case the VAF does not drop very close to zero there (as is usually the case).
An integration of the VAF over time gives the diffusion coefficientD
D= 1 3
∞
Z
0
hv(0)·v(t)idt, (3.21)
which is related to the MSD.