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Radical Voronoi: Domains, Voids & Charge Integration

This page is a self-contained, textbook-style introduction to the radical Voronoi tessellation MolPy ports from TRAVIS, and the three analyses built on it: connected-domain detection, void analysis, and electron-density integration into per-molecule charges and dipoles. A Voronoi tessellation assigns every point in space to its nearest atom; the radical (Laguerre / power) variant weights the partition by atomic radii, which is the physically correct division for systems of differently-sized atoms.

The tessellation and reductions run in Rust (molrs); they operate directly on positions, radii, and the box rather than on Frame objects.

Conventions used throughout

  • Positions and radii are in Å; volumes in ų.
  • The tessellation is fully periodic (minimum image).
  • A cell is one atom's radical-Voronoi polyhedron; domains and voids are groupings of cells.

1. The radical tessellation partitions space by power distance

In an ordinary Voronoi tessellation, the boundary between two atoms is the perpendicular bisector of their separation — correct only when all atoms are the same size. The radical tessellation instead places the boundary where the power distances are equal,

\[ \pi_i(\mathbf r) = |\mathbf r - \mathbf r_i|^2 - R_i^2, \]

so a larger atom (bigger \(R_i\)) claims a proportionally larger cell. Each atom's cell volume is a meaningful local volume — the basis for local density, packing fraction, and the analyses below.

from molpy.compute import RadicalVoronoi

cells = RadicalVoronoi()(positions, radii, box)   # -> VoronoiCells
cells.neighbors(i)    # cells sharing a face with cell i

2. Domains group cells that belong together

Labelling each atom (by species, charge sign, polarity, …) and merging face-sharing cells with the same label yields domains — the connected mesoscopic regions that characterize nanostructured liquids, such as the polar and apolar networks of ionic liquids. voronoi_domains returns the per-label domain sizes and volumes:

from molpy.compute import voronoi_domains

domains = voronoi_domains(cells, labels)   # labels: per-atom integer label

3. Voids are the empty cells of the tessellation

Flagging cells that contain no "real" occupant and aggregating their connected clusters maps the void space — cavities and free volume relevant to diffusion, gas solubility, and porosity:

from molpy.compute import voronoi_voids

voids = voronoi_voids(cells, is_void, box_volume)   # is_void: per-cell bool

4. Voronoi integration yields molecular charges and dipoles

Integrating a volumetric electron density over each radical-Voronoi cell and summing per molecule partitions the total charge into atomic/molecular contributions — the Voronoi route to per-molecule charges and dipole moments. These molecular dipoles are precisely what the dipole-autocorrelation route to infrared spectra consumes, which is why this analysis is the bridge from an ab initio MD electron density (e.g. a cube trajectory) to a predicted IR spectrum.

from molpy.compute import VoronoiIntegration

moments = VoronoiIntegration()(
    positions, radii, atomic_numbers, atom_to_mol, n_mol, grid, box
)
# moments -> per-molecule charges and dipole vectors

5. Pitfalls checklist

  1. Radii choice → the partition depends on the atomic radii; use a consistent, physically motivated set (e.g. van der Waals or covalent) and report it.
  2. Non-periodic box → the builder is periodic; supply the simulation box, not a free frame, or surface cells are unbounded.
  3. Label/void arrays length → labels and is_void must have one entry per atom/cell, in tessellation order.
  4. Grid resolution for integration → too coarse a density grid biases the integrated charges; converge the grid spacing.
  5. Charge neutrality → integrated per-molecule charges should sum to the system charge; a large residual signals a grid or radius problem.

6. References

  • B. J. Gellatly, J. L. Finney, J. Non-Cryst. Solids 50, 313 (1982) — radical (power) Voronoi tessellation.
  • M. Thomas, M. Brehm, B. Kirchner, Phys. Chem. Chem. Phys. 17, 3207 (2015) — Voronoi integration of the electron density for molecular dipoles.
  • M. Brehm, M. Thomas, S. Gehrke, B. Kirchner, J. Chem. Phys. 152, 164105 (2020) — TRAVIS; domain and void analysis.

See also