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Van Hove Correlations & Reorientational Dynamics

This page is a self-contained, textbook-style introduction to two time-resolved correlation functions MolPy ports from TRAVIS: the Van Hove correlation function \(G(r, t)\) — the time-dependent generalization of the radial distribution function — and the Legendre reorientational correlation functions \(C_1(t)\), \(C_2(t)\) that quantify how fast molecular vectors lose their orientation. Together they bridge the static structure of the structural guide and the transport coefficients of the transport guide: they show how structure decorrelates in time.

As elsewhere, the correlation kernels run in Rust (molrs); the MolPy layer unwraps trajectories and returns a typed result.

Conventions used throughout

  • Distances are in Å; time lags are given in frames (multiply by the frame spacing for ps).
  • \(\langle\cdots\rangle\) averages over particles and time origins.

1. The Van Hove function is g(r) resolved in time

The Van Hove correlation function counts particles at separation \(r\) after a delay \(t\), and splits naturally into a self and a distinct part:

\[ G(r,t) = \underbrace{\frac{1}{N}\Big\langle\sum_i \delta\big(r - |\mathbf r_i(t) - \mathbf r_i(0)|\big)\Big\rangle}_{G_s(r,t)\ \text{(same particle)}} + \underbrace{\frac{1}{N}\Big\langle\sum_{i\ne j}\delta\big(r - |\mathbf r_i(t) - \mathbf r_j(0)|\big)\Big\rangle}_{G_d(r,t)\ \text{(distinct particles)}}. \]
  • \(G_s(r,t)\) is the self-part: the distribution of single-particle displacements. At short time it is a sharp peak at \(r=0\); at long time it broadens into the Gaussian of Fickian diffusion, and its second moment is the mean-squared displacement. Non-Gaussian shapes flag hopping or caging.
  • \(G_d(r,t)\) is the distinct-part: at \(t=0\) it is exactly \(\rho\,g(r)\), and its decay shows how the structural shells around a particle wash out as neighbours diffuse away.

2. Computing the Van Hove function

from molpy.compute import VanHove

vh = VanHove(n_rbins=200, r_max=15.0, lags=[1, 5, 10, 50, 100])
result = vh(frames)

result.r_centers    # radial grid, Å
result.lags         # the time lags (frames)
result.g_self       # G_s(r, t): rows are lags, columns radial bins
result.g_distinct   # G_d(r, t) (present when result.has_distinct)

Choose lags to straddle the dynamics of interest — a few short lags to resolve the ballistic/caging regime and longer ones to reach the diffusive plateau.

3. Reorientation: how fast vectors forget their direction

For a unit vector \(\mathbf u(t)\) rigidly attached to a molecule (a bond, a dipole, a symmetry axis), the Legendre reorientational correlation functions are

\[ C_\ell(t) = \big\langle P_\ell\big(\mathbf u(0)\cdot\mathbf u(t)\big)\big\rangle, \qquad P_1(x)=x,\quad P_2(x)=\tfrac{1}{2}(3x^2-1). \]

Both decay from 1 (perfectly correlated) toward 0 (fully randomized) on the reorientational time scale. The order matters because different experiments probe different \(\ell\): dielectric relaxation measures \(C_1\), while NMR spin-relaxation, fluorescence depolarization, and IR/Raman lineshapes measure \(C_2\). Fitting an exponential tail \(C_\ell(t)\approx e^{-t/\tau_\ell}\) gives the reorientational correlation time \(\tau_\ell\); in the diffusive limit \(\tau_1/\tau_2 = 3\).

4. Computing reorientational correlations

import numpy as np
from molpy.compute import LegendreReorientation

pairs = np.array([[o, h1], [o, h2]], dtype=np.int64)   # O–H bond vectors
reor = LegendreReorientation(max_lag=500)
result = reor(frames, pairs)

result.lags   # lags (frames)
result.c1     # C_1(t)
result.c2     # C_2(t)

5. Pitfalls checklist

  1. r_max beyond half the box → the distinct part is corrupted by periodic images; keep r_max ≤ L/2.
  2. Lags longer than the trajectory supports → few time origins remain, so the long-lag tail is noisy; ensure many origins per lag.
  3. Reading \(\tau\) from a non-exponential head → fit \(C_\ell\) on its long-time exponential tail, not the librational sub-picosecond decay.
  4. Unnormalized vectors → supply genuine bond endpoints; the kernel forms unit vectors, but a degenerate pair (identical atoms) is undefined.
  5. Wrapped coordinates for \(G_s\) → single-particle displacements require unwrapped trajectories, or the self-part saturates at the box size.

6. References

  • L. Van Hove, Phys. Rev. 95, 249 (1954) — the correlation function G(r, t).
  • B. J. Berne, R. Pecora, Dynamic Light Scattering, Wiley (1976) — reorientational correlation functions and the \(C_1\)/\(C_2\) distinction.
  • M. Brehm, M. Thomas, S. Gehrke, B. Kirchner, J. Chem. Phys. 152, 164105 (2020) — TRAVIS.

See also