Reorientation¶

A molecule in a liquid does two things: it moves, and it turns. MSD measures the moving. This page measures the turning.

Rotational relaxation is what NMR spin-lattice relaxation, dielectric spectroscopy, and fluorescence anisotropy all probe, so it is one of the few dynamical quantities you can compare against experiment fairly directly — but only if you use the same \(\ell\) the experiment does.

Two correlation functions, not one¶

Track a unit vector \(\mathbf{u}(t)\) fixed in the molecule — an O–H bond, a dipole, the long axis of a rod. Its reorientation is measured by the Legendre correlation functions

\[ C_\ell(t) = \big\langle P_\ell\big(\mathbf{u}(t)\cdot\mathbf{u}(0)\big)\big\rangle, \]

with \(P_1(x) = x\) and \(P_2(x) = \tfrac12(3x^2-1)\). Both start at 1 and decay to 0 as the molecule forgets its initial orientation.

Why two? Because different experiments couple to different ranks. \(C_1\) is what dielectric relaxation and infrared absorption see, since those depend on the dipole direction. \(C_2\) is what NMR and depolarized light scattering see, because those depend on the axis rather than its sign — \(P_2\) is unchanged by \(\mathbf{u} \to -\mathbf{u}\). Quoting "the rotational correlation time" without saying which \(\ell\) is a common and avoidable ambiguity.

For simple rotational diffusion with rotational diffusion constant \(D_r\) there is an exact answer:

\[ C_\ell(t) = e^{-\ell(\ell+1)D_r t}, \qquad \tau_\ell = \frac{1}{\ell(\ell+1)D_r}, \]

so

\[ \boxed{\;\frac{\tau_1}{\tau_2} = 3\;} \]

independently of \(D_r\), of temperature, and of the molecule. That ratio is the most useful diagnostic on this page: it holds whenever reorientation proceeds by many small random steps, and it breaks when it does not — jump reorientation, strong hydrogen-bond networks, and glassy systems all push it away from 3.

Testing it where the answer is known¶

Here are 300 rods undergoing pure rotational diffusion — small random rotations, nothing else — with \(C_1\) and \(C_2\) computed from the resulting trajectory.

Figure 1. \(C_1\) and \(C_2\) for rods undergoing rotational diffusion. Both decay exponentially; the fitted relaxation times are 294 and 97 frames, a ratio of 3.02 against the exact 3.

That agreement is the validation — of the compute, of the estimator, and of the sampling. On a real molecular liquid the same measurement becomes physics: a ratio near 3 says diffusive reorientation, and a ratio far from it says the molecule is turning by some other mechanism.

Computing it¶

LegendreReorientation reads the tracked vectors from each frame's bonds topology block — (atomi, atomj), giving the vector from atom i to atom j. There is no separate vector argument. Whatever you want to track, express it as a bond.

import numpy as np
import molpy as mp
from molpy.compute import LegendreReorientation

rng = np.random.default_rng(0)
n_rods, n_frames = 300, 400
axis = rng.normal(size=(n_rods, 3))
axis /= np.linalg.norm(axis, axis=1, keepdims=True)
centres = rng.uniform(0.0, 100.0, size=(n_rods, 3))
bonds = {
    "atomi": np.arange(n_rods, dtype=np.uint32),
    "atomj": np.arange(n_rods, 2 * n_rods, dtype=np.uint32),
}

frames = []
for _ in range(n_frames):
    kick = rng.normal(0.0, 0.06, size=(n_rods, 3))
    kick -= (kick * axis).sum(axis=1, keepdims=True) * axis   # rotation only
    axis = axis + kick
    axis /= np.linalg.norm(axis, axis=1, keepdims=True)
    pos = np.concatenate([centres - 0.5 * axis, centres + 0.5 * axis])
    frame = mp.Frame()
    frame["atoms"] = {"x": pos[:, 0], "y": pos[:, 1], "z": pos[:, 2]}
    frame.box = mp.Box.cubic(200.0)
    frame["bonds"] = bonds
    frames.append(frame)

Subtracting the component along axis before adding the kick is what keeps the motion a rotation: a displacement along the axis would change the rod's length instead of its direction.

result = LegendreReorientation(max_lag=200)(frames)
c1, c2 = np.asarray(result.c1), np.asarray(result.c2)
print(round(float(c1[0]), 3), round(float(c2[0]), 3))   # -> 1.0 1.0
print(c1.shape)                                          # -> (201,)

Both start at exactly 1, as any normalized correlation function must — the first thing to check. Extract relaxation times by fitting the logarithm over the range where the curve is neither saturated nor buried in noise:

lag = np.asarray(result.lags, dtype=float)
# Single-exponential τ from log-linear fit on the decaying part of C_ℓ(t).
keep1 = (c1 > 0.05) & (lag > 0)
keep2 = (c2 > 0.05) & (lag > 0)
tau1 = -1.0 / np.polyfit(lag[keep1], np.log(c1[keep1]), 1)[0]
tau2 = -1.0 / np.polyfit(lag[keep2], np.log(c2[keep2]), 1)[0]
print(round(tau1 / tau2, 2))                             # -> 3.02

max_lag is in frames, not time, and stride sets the spacing between time origins — raise it to trade statistics for speed on long trajectories.

When it goes wrong¶

\(C_\ell(0) \ne 1\). The bond block is wrong, or the vectors are not being normalized as you assume. Stop here.

\(C_2\) decays more slowly than \(C_1\). Impossible for real reorientation — \(\tau_1 > \tau_2\) always. Suspect a swapped assignment.

\(\tau_1/\tau_2\) is far from 3. Sometimes physics, sometimes not. Genuine causes: jump reorientation, anisotropic rotation, hydrogen-bond-mediated turning. Artefacts: too short a trajectory, or fitting a stretched exponential with a single time. Check the fit residuals before claiming the physics.

The curves are noisy at long lag. Same time-origin problem as MSD: few independent origins remain out there. Fit the early, well-sampled decay.

The correlation does not reach zero. Something is restricting rotation — a frozen system, a rod stuck in a crystalline environment, or a constraint you forgot. In a liquid it must decay.

Check yourself¶

Confirm both curves start at 1.00.
Fit \(\tau_1\) and \(\tau_2\) and check the ratio. On a diffusively reorienting system you should get 3 to within a few percent, as above.
Halve the angular step in the example. Both times should double and the ratio should not move.

References¶

P. Debye, Polar Molecules, Chemical Catalog Company (1929) — rotational diffusion and \(C_1\).
D. A. McQuarrie, Statistical Mechanics, Harper & Row (1976), ch. 21 — Legendre reorientational correlation functions.
A. Luzar, D. Chandler, J. Chem. Phys. 98, 8160 (1993) — reorientation in hydrogen-bonded liquids, where the ratio departs from 3.