RDF¶

Textbook guide to the radial distribution function \(g(r)\) — the probability of finding a particle at distance \(r\) from a reference particle, normalized by an ideal gas of the same density.

Conventions

Length Å; \(g(r)\) dimensionless; keep \(r_\max \le L/2\).
Workflow: build a NeighborList, then histogram.

1. Definition¶

For \(N\) particles in volume \(V\) with density \(\rho = N/V\),

\[ g(r) = \frac{V}{N^2}\Big\langle\sum_i\sum_{j\ne i}\delta(r-r_{ij})\Big\rangle \Big/ 4\pi r^2. \]

Equivalently, if \(n(r)\,\mathrm{d}r\) is the mean neighbour count in \([r,r+\mathrm{d}r]\),

\[ g(r) = \frac{n(r)}{4\pi r^2\,\rho\,\mathrm{d}r}. \]

Limits: \(g\to 0\) as \(r\to 0\) (excluded volume); \(g\to 1\) as \(r\to\infty\); peaks are coordination shells. The first minimum after the first peak is the natural cutoff for Persist, Cluster, and Order.

1.1 Coordination number¶

\[ n(R) = 4\pi\rho \int_0^R r^2\, g(r)\,\mathrm{d}r. \]

Evaluate at the first minimum of \(g(r)\) for the first-shell coordination number.

2. Computing \(g(r)\)¶

import numpy as np
import molpy as mp

rng = np.random.default_rng(0)
xyz = rng.uniform(0.0, 20.0, size=(200, 3))
frame = mp.Frame()
frame["atoms"] = {"x": xyz[:, 0], "y": xyz[:, 1], "z": xyz[:, 2]}
frame.box = mp.Box.cubic(20.0)

from molpy.compute import NeighborList, RDF

nlist = NeighborList(cutoff=10.0)(frame)
result = RDF(n_bins=200, r_max=10.0)([frame], [nlist])
result.rdf, result.bin_centers

Average over a trajectory by passing parallel lists of frames and neighbor lists. Neighbor cutoff must be \(\ge\) r_max.

Figure 1. Schematic liquid \(g(r)\): core exclusion, first shell peak, decay to 1.

3. Pitfalls¶

r_max > L/2 → MIC contamination.
Neighbor cutoff < r_max → truncated \(g(r)\).
Too few bins → low first peak / coordination number.
Single frame → \(g(r)\) is an ensemble average.

4. References¶

M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids, 2^nd ed. (2017).
J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids, 4^th ed. (2013).