Real space correlations

Real space correlation functions are typically of the form

$$\langle A({\bf r}) A({\bf 0}) \rangle$$

and are straightforward to obtain by molecular dynamics: one has to compute the quantity of interest $A({\bf r})$ starting from the atomic positions and velocities for several configurations, construct the correlation function for each configuration, and average over the available configurations.

The simplest example is the pair correlation function g(r), which is essentially a density-density correlation. g(r) is the probability to find a pair a distance r apart, relative to what expected for a uniform random distribution of particles at the same density:  

$$\rho g(r) = \frac{1}{N} \left\langle \sum_{i=1}^N \sum_{\sta... ...scriptstyle j=1}{(j\not= i)}}^N \delta (r-r_{ij}) \right\rangle$$ (18)

This function carries information on the structure of the system. For a crystal, it exhibits a sequence of peaks at positions corresponding to shells around a given atom. The positions and magnitude of the peaks are a ``signature'' of the crystal structure (fcc, hcp, bcc, ...) of the system. For a liquid, g(r) exhibits its major peak close to the average atomic separation of neighboring atoms, and oscillates with less pronounced peaks at larger distances. The magnitude of the peaks decays exponentially with distance as g(r) approaches 1. In all cases, g(r) vanishes below a certain distance, where atomic repulsion is strong enough to prevent pairs of atoms from getting so close.

One quantity often computed from g(r) is the average number of atoms located between r₁ and r₂ from a given atom,

$$\rho \int_{r_1}^{r_2} g(r) 4\pi r^2\,dr .$$

This allows to define coordination numbers also in situations where disorder is present.

The calculation of g(r) is intrinsically a O(N²) operation, and therefore it can slow down considerably an optimized molecular dynamics program. If the behavior at large r is not important, it might be convenient to define a cutoff distance, and use techniques borrowed from fast force calculations using short-ranged potentials to decrease the computational burden. It should also be noted that periodic boundary conditions impose a natural cutoff at L/2, where L is the minimum between the box sizes L_x, L_y, L_z in the three directions. For larger distance the results are spoiled by size effects.