Geometry Indices

The geometry indices ($\tau$) are structural parameters that indicate which geometry a coordination centre is taking, from the ligand–centre–ligand angles alone, so bond lengths play no part.

They are calculated by gidx. The coordination number is the number of ligands in the file (any other count is an error): four ligands give τ4 and τ4′, five give τ5, six give τ6 and τ6′, and eight give τ8 and τ8′. The angles each index is built from are reported too.

Note

The indices are only dependant on angles, so shapes that share their largest angles share their index.

Four and five ligands

Only the two largest of all the angles between pairs of ligands are used, the largest as $\beta$ and the second largest as $\alpha$ (so $\beta \ge \alpha$).

τ₄ ¹ is 0 for a square plane and 1 for a regular tetrahedron; the ideal seesaw (0.64) and trigonal pyramid (0.85) lie in between.

$${\tau }_4=\frac{360^{\circ }-(\alpha +\beta )}{360^{\circ }-2\theta }$$

τ₄′ ² has the same limits, but it also weighs the difference between the two angles, so geometries with the same $\alpha +\beta$ (the ideal seesaw and a tetrahedron flattened to 135° and 135°) no longer share a value.

$${\tau }_4^{\prime }=\frac{\beta -\alpha }{360^{\circ }-\theta }+\frac{180^{\circ }-\beta }{180^{\circ }-\theta }$$

τ₅ ³ is 0 for a square pyramid and 1 for a trigonal bipyramid.

$${\tau }_5=\frac{\beta -\alpha }{60^{\circ }}$$

Six ligands

τ₆ ⁴ uses the three principal (trans) angles ${\alpha }_1\ge {\alpha }_2\ge {\alpha }_3$. It is 0 for an octahedron and 0.75 for a trigonal prism with trans angles of 135°. The paper gives no rule for picking them in a distorted structure, so gidx pairs the six ligands into three disjoint pairs whose angles add up to the most (the three largest angles whenever those do not share a ligand).

$${\tau }_6=\frac{3\times 180^{\circ }-({\alpha }_1+{\alpha }_2+{\alpha }_3)}{180^{\circ }}$$

τ₆′ is Cosmochlore’s own complement to τ₆. It uses the five smallest of the 15 ligand–centre–ligand angles, ${\gamma }_1\le \cdots \le {\gamma }_5$, which may share a ligand. It is 0 for an octahedron (cis angles of 90°), 0.456 for the equal-edge trigonal prism and 1 for a pentagonal pyramid (base angles of 72°). It measures how tightly the ligands pack, so it also grows for chelates with small bite angles, and it can exceed 1 (a planar hexagon gives 1.667).

$${\tau }_6^{\prime }=\frac{5\times 90^{\circ }-{\sum }_{i=1}^5{\gamma }_i}{90^{\circ }}$$

Pentagonal pyramid

The paper quotes ${\tau }_6=1.00$ for a pentagonal pyramid, but it gets there by summing the five angles between adjacent base vectors (5 × 72° = 360°) instead of three trans angles. gidx takes three trans angles for ${\tau }_6$, so the ideal pentagonal pyramid, whose trans angles are 144°, 144° and 90°, reads 0.900 there and 1.000 in ${\tau }_6^{\prime }$, which uses those five base angles.

Note

The uniform trigonal prism (all edges equal) has trans angles of 135.58°, so its τ6 is 0.740. The 0.75 quoted in the paper is for trans angles of exactly 135°.

Eight ligands

τ₈ ⁵ is 0 for the square antiprism and the cube, and 1 for the trigonal dodecahedron. $\theta$ and $\varphi$ are the angles that the two diagonals of a “square face” of the eight ligands subtend at the centre. As in the paper, the face holding the first ligand is found from the ligand across it (the one at the third largest of its seven angles) and the two ligands closest to both; the other four ligands make the second face. Each face gives a τ₈ and both are reported: two faces that disagree (about 1 and 0) point to a bicapped trigonal prism.

$${\tau }_8=\frac{|\theta -\varphi |}{129.82^{\circ }-97.07^{\circ }}$$

τ₈′ ⁵ is 1 for the trigonal dodecahedron and 0 for the square antiprism. The cube also reads 1, so τ₈′ separates the antiprism from the other two and τ₈ the cube from the dodecahedron. ${\delta }_A$ and ${\delta }_B$ are the dihedral angles between the planes through the centre and each pair of A ligands, and of B ligands. The larger diagonal of a face joins its two A ligands and the smaller one its two B ligands; the first A ligand is paired with the A at the smallest angle to it, and the first B with the B at the largest.

$${\tau }_8^{\prime }=1-\frac{|180^{\circ }-({\delta }_A+{\delta }_B)|}{180^{\circ }-2\times 65.53^{\circ }}$$

The paper’s $\Delta ={\tau }_8^{\prime }-{\tau }_8$ (cube_delta) is output too, with the larger τ₈ of the two faces: it is 1 for a cube and 0 for the ideal dodecahedron and antiprism.

Warning

τ8 and τ8′ are defined for the dodecahedron, antiprism, cube and bicapped-prism family. For other eight-vertex shapes τ8′ can be negative (ETBPY-8: −0.533) or undefined: a plane through the centre and two exactly opposite ligands does not exist, and gidx reports an error (JGBF-8, HBPY-8).

Exact constants

gidx uses the exact angles calculated from the geometry of the reference shapes., so the ideal polyhedra give exactly 0 or 1. For τ₄ and τ₄′ these are $\theta$ and $360^{\circ }-2\theta$ (printed 109.5° and 141°).

For τ₈ and τ₈′, the paper’s 129.8°, 97.1° and 65.5° come from the hard-sphere polyhedra of Kepert ⁶. In his dodecahedron the A and B sites sit at 36.85° and 69.46° from the $S_4$ axis, so the face diagonals subtend $\arccos (-{\cos }^{2}36.85^{\circ })=129.8208^{\circ }$ and $\arccos (-{\cos }^{2}69.46^{\circ })=97.0714^{\circ }$. In his antiprism all edges are equal and the planes close to $\arccos (\sqrt{2}-1)=65.5302^{\circ }$.

Geometry indices of the ideal shapes

The values gidx gives for the built-in reference shapes with four, five and six vertices (centre at the origin), angles in degrees:

Symbol	Shape	Symmetry	α	β	τ4	τ4′
SP-4	Square	D4h	180.00	180.00	0.000	0.000
T-4	Tetrahedron	Td	109.47	109.47	1.000	1.000
SS-4	Seesaw	C2v	90.00	180.00	0.638	0.359
vTBPY-4	Vacant trigonal bipyramid	C3v	120.00	120.00	0.851	0.851

Symbol	Shape	Symmetry	α	β	τ5
PP-5	Pentagon	D5h	144.00	144.00	0.000
vOC-5	Vacant octahedron	C4v	180.00	180.00	0.000
TBPY-5	Trigonal bipyramid	D3h	120.00	180.00	1.000
SPY-5	Spherical square pyramid	C4v	151.04	151.04	0.000
JTBPY-5	Johnson trigonal bipyramid J12	D3h	120.00	180.00	1.000

Symbol	Shape	Symmetry	τ6	τ6′
OC-6	Octahedron	Oh	0.000	0.000
TPR-6	Trigonal prism	D3h	0.740	0.456
PPY-6	Pentagonal pyramid	C5v	0.900	1.000
JPPY-6	Johnson pentagonal pyramid J2	C5v	0.900	1.000
HP-6	Hexagon	D6h	0.000	1.667

Slightly distorted versions of the same shapes are found in the crystal structures below, one six-coordinate cadmium each (coordinates from their CIF files; “Paper τ6” is the value the paper gives for the site):

Structure	Closest ideal shape	Paper τ6	τ6	τ6′
IQATAY Cd1	Octahedron	0	0.000	0.120
IQATAY Cd2	Octahedron	0.06	0.059	0.188
IQATAY Cd3	Octahedron	0.16	0.159	0.305
IQATAY Cd4	Octahedron	0.17	0.173	0.293
Title compound Cd2	Octahedron	0.27	0.270	0.471
ULESAH Cd1	Trigonal prism	0.73	0.734	0.511
VAPSAG Cd1	Pentagonal pyramid	1.02	0.820	1.044
CIJBII Cd1	Pentagonal pyramid	1.0	0.682	0.995

For the two pyramids the paper’s value is $(1+{\tau }_6^{\prime })/2$ (1.022 and 0.998), see above.

Symbol	Shape	Symmetry	τ8	τ8′	Δ
SAPR-8	Square antiprism	D4d	0.000	0.000	0.000
TDD-8	Triangular dodecahedron	D2d	1.000	1.000	0.000
CU-8	Cube	Oh	0.000	1.000	1.000
BTPR-8	Biaugmented trigonal prism	C2v	1.167 / 0.000	0.408	−0.758
JBTPR-8	Johnson-Biaugmented trigonal prism (J50)	C2v	1.167 / 0.000	0.408	−0.758
JSD-8	Snub disphenoid (J84)	D2d	1.222	1.000	−0.222
[WF4(PH3)4]+	DFT structure from the paper 5	D2	0.356	0.406	0.049

$\Delta ={\tau }_8^{\prime }-{\tau }_8$ (the larger τ8 of the two faces). The two values of τ8 for the biaugmented trigonal prism are those of its two square faces. The last row is the DFT-optimised structure of the paper’s supporting information (donor atoms only, tests/tau8/WF4_PH3_4_D2.xyz), in between the dodecahedron and the antiprism; its two faces give the same τ8.

Chelate complexes in the supporting information

For some chelates, the τ₈ values listed in the paper’s supporting information ⁵ differ from this procedure: it lists τ₈ = 1.22 for [WF₄(dpb)₂]⁺ (D_2d), where gidx gives 1.10 from the coordinates in the same file, and 0.05 and 0.02 for the C₂ state of [WF₄(dpe)₂]⁺, where it gives 0.06 and 0.05. The source of the difference could not be traced.

References

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2.

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Addison, A. W., Rao, T. N., Reedijk, J., van Rijn, J. & Verschoor, G. C. Synthesis, structure, and spectroscopic properties of copper(II) compounds containing nitrogen–sulphur donor ligands; the crystal and molecular structure of aqua[1,7-bis(N-methylbenzimidazol-2′-yl)-2,6-dithiaheptane]copper(II) perchlorate. Journal of the Chemical Society, Dalton Transactions 13, 1349–1356 (1984).

4.

Stoeckli-Evans, H. et al. Di-μ3-chlorido-1:2:3κ3Cl;2:3:4κ3Cl-di-μ2-chlorido-1:2κ2Cl;3:4κ2Cl-tetra­kis­[(4-amino-1,5-dimethyl-2-phenyl-2,3-di­hydro-1H-pyrazol-3-one-κ2N4,O)chlorido­cadmium(II)] 1.7-hydrate: a new six-coordinate geometry index, τ6. Acta Crystallographica Section E: Crystallographic Communications 81, 393–400 (2025).

5.

Turnbull, D., Wetmore, S. D. & Gerken, M. Stabilisation of [W(V)F4]+ by N- and P-Donor Ligands: d1 Second-Order Jahn–Teller Effects in Octacoordinate Complexes. Chemistry – A European Journal 27, 11335–11343 (2021).

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