Diamond Version 5 User Manual: Polyhedra

Voronoi Polyhedra

Creation of Voronoi polyhedra

In this article:
- Voronoi polyhedra can be created for selected atoms or atoms of selected sites basing upon their Dirichlet domains.
- There are several options for the style of the polyhedron as well as if and how to show and connect the neighbouring atoms.
- Infos about a selected Voronoi polyhedron (neighbouring atoms, distances, bond strengths, solid angles, etc.) are available from the Properties view.

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Next article: Destroying polyhedra

You should also read or have read this closely related article: Dirichlet Domains

Sample #1: Magnetite

The first sample of this article will use magnetite from COD:9013529, so open the sample file "COD-9013529.diamdoc". The data sheet shows us three atom sites in the atomic parameter list: "FeT", "FeM", and "O".

Screenshot of sample "COD-9013529.diamdoc" showing unit cell with bonds and one coordination polyhedra for the three atom sites "FeT", "FeM", and "O" each. (Note: Screenshots on this page are made with Diamond version 4.)

"FeM" has a conventional (regular) octahedral environment (six Fe--O distances of 2.060 Angstroms). To study the Voronoi polyhedron of the site "FeM", we do the following with the structure picture of COD-9013529.diamdoc:
- Destroy all (Shift+Ctrl+D or Build/Destroy/All).
- Build/Add Atoms -> select site "FeM" -> OK. (This creates one single "FeM" atom directly at (1/2, 1/2, 1/2).
- Select this "FeM" atom (click on atom).
- Open the "Voronoi Polyhedra" dialog with Build/Polyhedra/Voronoi Polyhedra.

The first page of the "Voronoi Polyhedra" dialog shows a list of all atom sites, which you can use to choose the atoms in the structure picture to become center(s) of Voronoi polyhedron/a. Or you choose "Selected atoms in structure picture", provided at least one atom in the structure picture has been selected before calling the dialog. The middle part of the dialog page refers to the polyhedron shape and is similar to the shape and design of (conventional) coordination polyhedra that you already know from older Diamond versions. In the section concerning the atoms in the neighbourhood you define if and how far to create neighbouring atoms (that means all or certain atoms of the Dirichlet domain) and if and how to connect the central atom with the neighbouring atoms.

The second page shows the list of neighbouring atoms for the site that is selected or checked in the atom site list of the first page or (has been selected in the structure picture). Here the Dirichlet domain looks straightforward simple, six equidistant Fe--O distances of 2.06 Angstroms, leading to a regular cube with four vertices for the six faces each and a solid angle of 1/6 (or 16 2/3 percent).

Confirming the dialog with OK for the (selected) single "FeM" (central) atom, leads to the following Voronoi polyhedron picture. (To show the table of polyhedra, use the corresponding command from the "View" menu, "Table" sub-menu. Choose the option "Dirichlet domain (Voronoi polyhedron)" in the Properties view and adjust the column widths in the properties view.) The Voronoi polyhedron is the expected cube, formed by the intersecting planes of the atomic environment, which forms a regular octahedron.

Repeating this procedure for the other two sites "FeT" and O show that these domains/environments are somewhat complicated. The Dirichlet domain of "FeT" is not the tetrahedron -- we might expect -- but has 20 neighbouring atoms, four near O atoms 1.887 Angstroms away from the center, and followed by twelve more O atoms after a big distance gap at 3.494 Angstroms and four "FeT" atoms at 3.636 Angstroms each. The Voronoi polyhedron has 36 vertices and 156 faces:

And finally the situation for the O site. Here we also have a lot of neighbouring atoms (19) and a big distance gap after the four Fe neighbours, one "FeT" 1.887 and three "FeM" 2.060 Angstroms away from the center. A series of twelve O-O distances between 2.856 and 3.082 Angstroms follows, most of the O neighbours are "indirect neighbours" (meaning that the centerpoint of the vector center--neighbour lies out of the Voronoi polyhedron's surface). This is indicated by the asterisk before the sequential neighbour number in the list of neighbours in the properties view. Finally there are three "FeT" neighbours 3.494 Angstroms away. The Voronoi polyhedron has 34 vertices and 53 faces:

In summary, only for the "FeM" site all atoms of the Voronoi polyhedron (Dirichlet domain) do contribute to the atomic environment, whereas for "FeT" and "O" it is only a fraction (four of 20 or 19, rsp.) of the neighbouring atoms. The reason is that for the evaluation of the contributing (that means bonding) neighbouring atoms considers if and how far the "spherical domains" of the Voronoi polyhedra overlap. The "spherical domain" is a sphere having the same volume as the Voronoi polyhedron, and Rsd is this sphere's radius. The evaluation also considers the element-specific "Slater radii" (Rs) -- empirically defined by J. C. Slater (1964).

Sample #2: antimony trifluoride/pentafluoride adduct

We take up the sample from the related article about the Dirichlet domains, "PCD-1251073-unit-cell-with-Sb-polyhedra.diamdoc", and extend the unit cell with bonds and one coordination polyhedra for the four Sb sites each:
- "Build" -> "Fill" -> "Super Cell": Use the option "Double cell Z" and press OK. (This fills up the picture at the left side, since -z is on the left.)
- Press Shift+Ctrl+N to connect atoms directly.
- Now one by one, click on the same four (central) atoms like in the right neighbouring cell, that means "Sb1", "Sb2", "Sb3", and "Sb4" and run command "Build" -> "Polyhedra" -> "Voronoi Polyhedra" each. (This opens the "Voronoi Polyhedra" dialog each, confirm with OK.) The result should look like this. (To show the table of polyhedra, run the corresponding command from the "Table" sub-menu of the "View" menu.)

Again, like in the sample #1 (magnetite), we have the situation that not all atoms of the Voronoi polyhedron do contribute as (strongly connected) neighbouring atoms -- after evaluation of the overlap of spherical domains and Slater radii spheres.

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Next article: Destroying polyhedra

Reference for COD:9013529:
Fe3O4 -- magnetite -- Fd-3m (227), a = 8.3967 A
Bosi, F.; Halenius, U.; Skogby, H., "Crystal chemistry of the magnetite-ulvospinel series Locality: synthetic Note: Mgt100", American Mineralogist, 94, 181-189 (2009).

Reference for PCD:1251073:
Sb4F16 -- P 1 21/c 1 (14)
Gillespie R.J., Slim D.R., Vekris J.E., "The Crystal Structure of the 1:1 Adduct of Antimony Trifluoride and Antimony Pentafluoride", Journal of the Chemical Society. Dalton Transactions, Inorganic Chemistry (1972-), ?, 971-974 (1977).

Literature about Voronoi polyhedra and Dirichlet domains:
- "VoroWiki": http://www.voronoi.com.
- G.L. Dirichlet, Z. Reine Angew. Math. 40, 216 (1850).
- G.F. Voronoi, Z. Reine Angew. Math. 134, 198 (1908).
- Blatov V.A., Shevchenko A.P., Serezhkin V.N. Crystal space analysis by means of Voronoi-Dirichlet polyhedra. Acta Cryst. 1995, A51, 909-916.
- Blatov V.A. Multipurpose crystallochemical analysis with the program package TOPOS. IUCr CompComm Newsletter. 2006, 4-38.
- TOPOS web page: http://www.topos.ssu.samara.ru.

Consideration of Rsd and Slater radii:
- Serezhkin, V. N., Mikhailov, Yu. N. & Buslaev, Yu. A. (1997). Russ. J. Inorg. Chem. 42, 1871-1910.