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Shape

Textbook guide to molecular shape descriptors: gyration tensor, radius of gyration, asphericity, and the inertia tensor.

Conventions

  • Length Å; \(R_g\) in Å; tensors \(3\times 3\).
  • Descriptors are per cluster — find aggregates first (Cluster).

1. Gyration tensor and \(R_g\)

\[ S = \frac{1}{N}\sum_{i=1}^N (\mathbf{r}_i-\mathbf{r}_c)\otimes(\mathbf{r}_i-\mathbf{r}_c), \qquad R_g^2 = \operatorname{tr} S. \]

Mass-weighted form uses COM and masses \(\{m_i\}\). For polymers \(R_g\sim N^\nu\) with Flory exponent \(\nu\) (swollen \(\approx 3/5\), \(\theta=1/2\), globule \(1/3\)).

Eigenvalues \(\lambda_1\le\lambda_2\le\lambda_3\) define asphericity \(b=\lambda_3-\tfrac12(\lambda_1+\lambda_2)\), acylindricity \(c=\lambda_2-\lambda_1\), and relative shape anisotropy

\[ \kappa^2 = (b^2 + \tfrac34 c^2)/R_g^4 \]

(\(0\) sphere → \(1\) rod).

The inertia tensor is the mass-weighted cousin; its eigenvectors are principal axes for molecular frames and orientational analyses.

2. Usage

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, Cluster, ClusterCenters, CenterOfMass,
    RadiusOfGyration, GyrationTensor, InertiaTensor,
)

masses = np.full(len(xyz), 12.011)
nlist = NeighborList(cutoff=1.6)(frame)
clusters = Cluster(min_cluster_size=10)([frame], [nlist])
centers = ClusterCenters()([frame], clusters)
com = CenterOfMass(masses)([frame], clusters)
rg = RadiusOfGyration(masses)([frame], clusters, com)
S = GyrationTensor()([frame], clusters, centers)
I = InertiaTensor(masses)([frame], clusters, com)

Pass masses=None for geometric (unit-mass) descriptors.

3. Pitfalls

  1. Molecule split across PBC → unwrap relative to cluster center.
  2. Mass convention not reported.
  3. Cluster cutoff wrong → garbage aggregates.

See also