Cluster properties¶

The same properties are computed for clusters and nuclei (any group of molecules), in the tables and the JSON report. All of them are computed on the group made whole across the periodic boundaries: its molecules are placed next to each other along the cluster's connections (or, for a nucleus, a minimum spanning tree of its molecules' centers), so a cluster wider than half the box stays whole. A cluster connected to its own periodic image (e.g. a slab spanning the box) cannot be made whole; the log warns once per such cluster.

Property	Report field	Table column	Unit
number of molecules	`size`	`Size`	—
mass	`mass`	—	amu (g/mol)
equivalent radius	`radius`	`Radius`	Å
radius of gyration	`radius_of_gyration`	—	Å
equivalent diameter	`diameter`	—	Å
equivalent volume	`volume`	—	Å³
density	`density`	`Density`	g/cm³
total charge	`charge`	`Charge`	e
dipole moment	`dipole_moment`	`Dipole`	D (debye)
sphericity	`sphericity`	`Spher`	— (0 to 1)
shape parameter	`shape_parameter`	`Shape`	— (−0.25 to 2)

Mass and charge¶

The mass is the sum of the atoms' masses, and the charge the sum of their partial charges, both from the topology.

Size¶

A cluster is not a sphere, but its size is summarised by the equivalent sphere: the uniform sphere with the same mass-weighted radius of gyration \(R_g\), whose radius is \(\sqrt{5/3}\,R_g\). The radius of gyration only sees the atoms' centers, so the sphere would end at the outer atoms' centers; half the atoms' mean van der Waals radius \(\bar r_\text{vdW}\) (by element, from MDAnalysis' table¹) is added to account for the atoms' own size:

\[ R = \sqrt{\tfrac{5}{3}}\,R_g + \tfrac{1}{2}\,\bar r_\text{vdW}, \qquad d = 2R, \qquad V = \tfrac{4}{3}\pi R^3, \qquad \rho = \frac{m}{V}. \]

Half, not the full van der Waals radius, because bonded atoms' van der Waals spheres overlap. On methanol-malic acid mixtures the full radius puts a lone malic acid at 0.70 g/cm³ and half at 1.21 g/cm³ (the solid is 1.61 g/cm³), and the choice barely changes large clusters. The density of small clusters is an estimate; compare it between clusters rather than with bulk densities.

The report also gives \(R_g\) itself, as radius_of_gyration, for analyses that need it without the equivalent sphere's assumptions.

This needs the atoms' elements: the command line guesses them from the atom names when the topology has none (e.g. a .tpr).

Dipole moment¶

The magnitude of the dipole vector \(\boldsymbol\mu = \sum_i q_i(\mathbf r_i - \mathbf r_\text{COM})\) about the group's center of mass, in debye (1 D = 0.2081943 e·Å). For a neutral group the dipole does not depend on the reference point; for a charged group it does, so dipoles of charged clusters are only comparable with that convention in mind.

The report's Nucleus.combined_dipole_moment is the dipole of all a cluster's nuclei taken together, as one group.

Shape¶

Both shape descriptors come from the eigenvalues \(\lambda_1, \lambda_2, \lambda_3\) of the mass-weighted gyration tensor, with \(\bar\lambda\) their mean, as defined by Dima and Thirumalai.²

Sphericity is one minus the asphericity \(\Delta\):

\[ \text{sphericity} = 1 - \Delta, \qquad \Delta = \frac{3}{2}\,\frac{\sum_i (\lambda_i - \bar\lambda)^2}{\left(\sum_i \lambda_i\right)^2}. \]

It is 1 when the three moments are equal (a sphere, but also e.g. a cube), 0 for a rod (all mass on a line), and 0.75 for a flat disk or ring. It is not Wadell's surface-area sphericity.

Shape parameter:

\[ S = 27\,\frac{\prod_i (\lambda_i - \bar\lambda)}{\left(\sum_i \lambda_i\right)^3}, \]

0 for a sphere, negative for oblate (flattened) shapes down to −0.25 for a disk, and positive for prolate (elongated) ones up to 2 for a rod.

Connections¶

For each connection, the report records the distance between the two molecules: their centers of mass' distance for cm rules, and the donor-acceptor distance of their shortest hydrogen bond for hb rules, with its donor-hydrogen-acceptor angle and the number of hydrogen bonds between them.

MDAnalysis takes its van der Waals radii from A. Bondi, "van der Waals Volumes and Radii", J. Phys. Chem. 68, 441–451 (1964), doi:10.1021/j100785a001; R. S. Rowland and R. Taylor, "Intermolecular Nonbonded Contact Distances in Organic Crystal Structures", J. Phys. Chem. 100, 7384–7391 (1996), doi:10.1021/jp953141+; and M. Mantina, A. C. Chamberlin, R. Valero, C. J. Cramer and D. G. Truhlar, "Consistent van der Waals Radii for the Whole Main Group", J. Phys. Chem. A 113, 5806–5812 (2009), doi:10.1021/jp8111556. ↩
R. I. Dima and D. Thirumalai, "Asymmetry in the Shapes of Folded and Denatured States of Proteins", J. Phys. Chem. B 108(21), 6564–6570 (2004). doi:10.1021/jp037128y ↩