Van Hove¶

This page is a self-contained, textbook-style introduction to two time-resolved correlation functions: the Van Hove function \(G(r,t)\) — the time-dependent generalization of the radial distribution function — and the Legendre reorientational correlations \(C_1(t)\), \(C_2(t)\) 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.

Correlation kernels run in the high-performance backend; the MolPy layer unwraps trajectories where needed and returns a typed result.

Conventions used throughout

Distances in Å; time lags in frames (multiply by the dump interval for ps or fs).
\(\langle\cdots\rangle\) averages over particles and time origins.
Self Van Hove needs unwrapped single-particle trajectories.

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

Van Hove (1954) asked: given a particle at the origin at time \(0\), what is the probability density of finding a particle (the same one, or another) at distance \(r\) after time \(t\)? The answer splits into self and distinct parts:

\[ G(r,t) = \underbrace{ \frac{1}{N}\Big\langle \sum_i \delta\big(r - |\mathbf{r}_i(t)-\mathbf{r}_i(0)|\big) \Big\rangle }_{G_s(r,t)\ \text{(same particle)} + \underbrace{ \frac{1}{N}\Big\langle \sum_{i\neq j}\delta\big(r - |\mathbf{r}_i(t)-\mathbf{r}_j(0)|\big) \Big\rangle }_{G_d(r,t)\ \text{(distinct particles)}. \]

1.1 Self-part \(G_s(r,t)\)¶

\(G_s\) is the distribution of single-particle displacements:

\(t\to 0\): a sharp peak at \(r=0\) (particles have not moved).
Ballistic regime: peak shifts as \(\langle r\rangle\sim\langle v\rangle t\).
Cage regime (dense liquids): probability piles up near the first-neighbour distance while the tagged particle rattles in its cage.
Diffusive regime: \(G_s\) broadens into a Gaussian whose second moment is the MSD.

Non-Gaussian shapes (shoulders, secondary peaks) flag hopping, dynamic heterogeneity, or jump diffusion — central diagnostics in glass physics.

1.2 Distinct-part \(G_d(r,t)\)¶

At zero lag the distinct part recovers the RDF:

\[ \boxed{\;G_d(r,0)=\rho\,g(r)\;} \]

As \(t\) grows, the coordination shells wash out: neighbours leave and are replaced. The time for the first peak of \(G_d\) to decay is a structural relaxation time complementary to density-density correlations in \(k\)-space.

1.3 Moments, MSD, and the non-Gaussian parameter¶

The second moment of the self-part is the mean-squared displacement:

\[ \big\langle r^2(t)\big\rangle = 4\pi\int_0^\infty r^4\, G_s(r,t)\,\mathrm{d}r = \mathrm{MSD}(t). \]

(In 3-D with the radial measure \(4\pi r^2\,\mathrm{d}r\) on the probability density convention used here; implementations store a histogram consistent with their binning.) The non-Gaussian parameter

\[ \alpha_2(t) = \frac{3}{5}\frac{\langle r^4(t)\rangle}{\langle r^2(t)\rangle^2}-1 \]

vanishes for pure Fickian (Gaussian) diffusion and peaks when particles hop between cages — a standard glass-physics diagnostic.

1.4 Intermediate scattering function (connection)¶

Scattering experiments measure the intermediate scattering function

\[ F_s(k,t) = \big\langle e^{-i\mathbf{k}\cdot[\mathbf{r}(t)-\mathbf{r}(0)]}\big\rangle, \]

which is the Fourier transform of \(G_s(r,t)\). The same physics appears as a peak broadening in \(G_s\) or as a decay of \(F_s(k,t)\) at the structure-factor peak wavenumber \(k^*\). MolPy’s Van Hove API works in \(r\)-space; use it when real-space shells and hopping shoulders matter.

Figure 1. Schematic self Van Hove \(G_s(r,t)\) at fixed lag: a peak that broadens and shifts outward as \(t\) increases (Fickian limit is Gaussian).

2. Computing the Van Hove function¶

import numpy as np
import molpy as mp

def _frame(step: int) -> mp.Frame:
    rng = np.random.default_rng(0)
    xyz = rng.uniform(0.0, 20.0, size=(200, 3)) + 0.1 * step
    frame = mp.Frame()
    frame["atoms"] = {"x": xyz[:, 0], "y": xyz[:, 1], "z": xyz[:, 2]}
    frame.box = mp.Box.cubic(20.0)
    return frame

frames = [_frame(step) for step in range(20)]

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        # time lags (frames)
result.g_self      # G_s(r, t): rows = lags, columns = radial bins
result.g_distinct  # G_d(r, t) when result.has_distinct

Choose lags to straddle the dynamics of interest — short lags for ballistic/caging, longer ones for the diffusive broadening. Keep r_max ≤ L/2 for a clean distinct part under periodic boundaries.

3. Reorientation: how fast vectors forget their direction¶

3.1 Legendre correlations¶

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

\[ \boxed{\; C_\ell(t) = \big\langle P_\ell\big(\mathbf{u}(0)\cdot\mathbf{u}(t)\big)\big\rangle, \qquad P_1(x)=x,\quad P_2(x)=\tfrac12(3x^2-1) \;} \]

Both decay from \(1\) (perfect memory) toward \(0\) (fully randomized orientation). Different experiments probe different \(\ell\):

Experiment / response	Order
Dielectric relaxation, IR	\(C_1\)
NMR spin relaxation, fluorescence anisotropy, Raman	\(C_2\)

3.2 Correlation times¶

Fit the long-time exponential tail (not the librational head):

\[ C_\ell(t)\approx e^{-t/\tau_\ell} \quad\Rightarrow\quad \tau_\ell = \int_0^\infty C_\ell(t)\,\mathrm{d}t \quad\text{(or from the fit)}. \]

In the Debye rotational diffusion limit,

\[ \tau_\ell = \frac{1}{\ell(\ell+1)D_R} \quad\Rightarrow\quad \frac{\tau_1}{\tau_2}=3. \]

Large deviations of \(\tau_1/\tau_2\) from 3 signal jump reorientation (e.g. water’s large-amplitude H-bond exchanges) rather than small-step diffusion.

3.3 Link to spectra and dielectrics¶

Collective \(C_1\) of the total dipole underlies dielectric spectroscopy.
Single-molecule \(C_1\)/\(C_2\) set vibrational lineshape envelopes and NMR correlation times — complementary to the velocity-based VDOS / IR route.

4. Computing reorientational correlations¶

import numpy as np
from molpy.compute import LegendreReorientation

# Bond endpoints come from the frame's `bonds` topology block.
for f in frames:
    f["bonds"] = {"atomi": np.array([0, 0]), "atomj": np.array([1, 2])}

reor = LegendreReorientation(max_lag=5)
result = reor(frames)

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

5. Pitfalls checklist¶

r_max beyond half the box → distinct part corrupted by periodic images.
Lags longer than the trajectory supports → few time origins; noisy tails.
Reading \(\tau\) from a non-exponential head → fit the long-time tail of \(C_\ell\), not the librational sub-picosecond decay.
Degenerate vectors → identical head/tail atoms make \(\mathbf{u}\) undefined.
Wrapped coordinates for \(G_s\) → self-part saturates at the box size; unwrap single-particle trajectories.
Comparing \(\tau_1\) and \(\tau_2\) → ensure both use the same vector definition and fit window before quoting \(\tau_1/\tau_2\).

6. References¶

L. Van Hove, Phys. Rev. 95, 249 (1954) — \(G(r,t)\).
B. J. Berne, R. Pecora, Dynamic Light Scattering, Wiley (1976) — reorientational correlations and the \(C_1\)/\(C_2\) distinction.
W. Kob, H. C. Andersen, Phys. Rev. E 51, 4626 (1995) — non-Gaussian parameter and dynamical heterogeneity.
J.-P. Hansen, I. R. McDonald, Theory of Simple Liquids, 4^th ed., ch. 7–8.
M. Brehm, M. Thomas, S. Gehrke, B. Kirchner, J. Chem. Phys. 152, 164105 (2020) — AIMD analysis stack.