xtalforge

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Local environments and chemistry ​

Tasks 9.3.1-9.3.4. References: pymatgen analysis.local_env, analysis.bond_valence; O'Keefe and Brese, JACS 113 (1991) 3226; ICSD bond-valence data mining as shipped by pymatgen (MIT).

Periodic Voronoi ​

The Voronoi cell of an atom is the intersection of the bisector half-spaces

2 (r_j - r_i) . x <= |r_j - r_i|^2

for all neighbors within the cutoff, in coordinates relative to the central atom. The half-spaces are clipped from a large cube face by face (xtalforge_core::polyhedron); every new cap face is ordered around the plane normal. The volume follows from the signed tetrahedra of the outward faces, and the solid angle of a face is area * cos(theta) / d^2. Face planes are matched back to the neighbor that produced them by the normal cosine, so each neighbor carries its area and solid angle. The cutoff must include every face of the cell; at least twice the longest nearest-neighbor distance is safe.

Coordination and shape measures ​

Three strategies are available:

  • cutoff: every neighbor within a distance.
  • voronoi: faces with a solid angle above a threshold.
  • crystalnn: a CrystalNN-like weight exp(-((d - d_min)/0.4)^2) * sqrt(Omega/pi).

Continuous shape measures compare the neighbor vectors with ideal polyhedra of the same size. Both sets are centred and normalized to unit RMS radius; the best rotation comes from Kabsch alignment and the vertex correspondence from the Hungarian algorithm on the squared-distance cost matrix, iterated four times. The reported value is

CShM = 100 * sum |q - p|^2 / sum |p|^2

which is zero for the ideal shape. The ideal sets are linear, trigonal planar, tetrahedral, square planar, trigonal bipyramid, square pyramid, octahedral, pentagonal bipyramid, cube, square antiprism and icosahedral.

Bond valence and oxidation states ​

The bond valence of a pair is

v_ij = exp((R_ij - d_ij) / 0.31)
R_ij = r_i + r_j - r_i r_j (sqrt(c_i) - sqrt(c_j))^2 / (c_i r_i + c_j r_j)

with the 1991 r and c parameters. Only pairs where at least one element is in the O'Keefe electronegative set contribute, and the sign follows the electronegativity order. The site sum is signed.

Oxidation states use a maximum-a-posteriori search: for every element the mean bond-valence sum is compared with the ICSD Gaussian for each candidate state, multiplied by its occurrence prior. A depth-first search over the element states keeps the combination with the smallest negative log posterior plus 10 * |total charge|. The state is then assigned to every atom of that element. Cells that are not neutral are reported, never hidden.

RDF, PDF and fingerprints ​

The RDF is a pair histogram with the normalization g(r) = n(r) / (N rho 4 pi r^2 dr); partial functions use the concentration of the second element. A moving average can smooth every curve. The running coordination number integrates rho g(r) from zero. Bond angles use all neighbor pairs of every atom. The fingerprint stores the element fractions, the six cell parameters and the smoothed total RDF; its similarity is 0.4 * composition + 0.2 * cell + 0.4 * RDF cosine terms.