tau-calc

A Python 3 script that calculates geometry indices, including τ₄, τ₅, O (octahedricity), and CShM (Continuous Shape Measures) values for various shapes of selected atoms from a crystallographic information file (CIF). The script saves you the tedious task of checking the two largest angles and calculating the τ-values. It also computes octahedricity and polyhedral volume and can print or save XYZ coordinates of the central atom and its neighboring atoms. Tables in Markdown format containing coordination number, geometry index, CShM, and polyhedral volume can optionally be generated and saved.

Introduction:

The geometry indices τ₄ and τ₅ helps in assigning a coordination geometry for four-coordinated (tetrahedral, trigonal pyramidal, square planar or seesaw geometry) or five-coordinates compounds (square pyramidal or trigonal bipyramidal geometry). Only the two largest angles enclosing the central atom are needed for the assignment. Please check the concise Wikipedia article or the papers above for more information. The octahedricity O is calculated with the following equation (see paper for more information):

$$ O = \sqrt{\frac{1}{15}\sum_{i=1}^{15}(\hat{\theta_i} - \theta)^2} $$

$\hat{\theta_i}$ = 180° for trans X-M-X angles and 90° for cis X-M-X angles
$\theta$ = experimental X-M-X angles

And is close to zero for an almost ideal octahedron

The CShM (Continuous Shape Measures) value approaches zero when a shape closely resembles the chosen ideal shape. For further details, refer to the paper and related articles on CShM.

The polyhedral volume is calculated using the Convex Hull Algorithm (implemented via SciPy and the Qhull Library).

External modules

gemmi, numpy, scipy, tabulate

Quick start

Start the script with:

python3 tau-calc.py filename.cif atom

The input is case sensitive.

The following output will be printed:

The predicted coordination number for atom is N.

The two largest angles are β = value and α = value.
Note: Angles for the calculation of τ₄, τ₄' and τ₅.

Number of cis angles ~ 90° (< 135°)    = value
Number of trans angles ~ 180° (> 135°) = value
Note: First value should be 12, second 3 for an octahedron.

--------------------------------------------------------------------------------
atom geometry index  ("<--" indicates the parameter for coordination number N):
--------------------------------------------------------------------------------
τ₄  =   value
τ₄' =   value 
τ₅  =   value <--
O   =   value 

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Continuous shape measure (CShM) for coordination number N:
--------------------------------------------------------------------------------
CShM S(Shape 1) = value ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░
CShM S(Shape 2) = value ░░░░
CShM S(Shape 3) = value ░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░

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Polyhedral volume (coordination number N) = value A³
--------------------------------------------------------------------------------

--------------------------------------------------------------------------------
Table of typical geometries and their corresponding τ₄ or τ₅ values: 
--------------------------------------------------------------------------------
Coordination number 4:
Tetrahedral          : τ₄ = 1.00       τ₄' = 1.00
Trigonal pyramidal   : τ₄ = 0.85       τ₄' = 0.85
Seesaw               : τ₄ = 0.43       τ₄' = 0.24
Square planar        : τ₄ = 0.00       τ₄' = 0.00

Coordination number 5:
Trigonal bipyramidal : τ₅ = 1.00                     
Square pyramidal     : τ₅ = 0.00                     

The likely structural parameter is marked with an arrow (<--).

Command-line options

• filename , required: the CIF (crystallographic information file)
• atom_name, required: the central atom, input is case sensitive, e.g. Co1 calculates τ for Co1
• -e atom(s), optional: exclude atoms, e.g. -e N1 N3 excludes bonds and angles to N1 and N3 from calculation
• -d N, optional: excludes atoms outside d = N Å from calculation, e.g. -d 2.1 excludes atoms with bond lengths larger than 2.1 Å from the central atom from calculation
• -v optional: verbose output, prints all bond lengths and angles of the central atom to the neighboring atoms and the XYZ coordinates
• -sxyz optional: save the XYZ coordinates of the central atom and its neighboring atoms (filename: cif_name-atom_name.xyz)
• -smd optional: save tables in Markdown format containing coordination number, geometry index, CShM, and polyhedral volume (filename: cif_name-atom_name.md)
• -ssh optional: save input file for the SHAPE program (filename: cif_name-atom_name.dat)

Remarks

• If the predicted coordination number is larger than 2, τ will be calculated independently of the real coordination geometry.
• The suggestion τ₄, τ₅ or O (<--) is based on the number of angles (6 for τ₄, 10 for τ₅, 15 for O).
• In octahedral coordination, all angles less than 135° are considered "90°" cis angles, while angles greater than 135° are considered "180°" trans angles.
• The XYZ coordinates of the neighboring atoms are given relative to the central atom, which is positioned at [0, 0, 0].
• The XYZ file (option: -sxyz) is useful for further studies of coordination geometry, such as with the Continuous Shape Measures (CShM) method.
• The polyhedral volume should match the value calculated by Olex2.
• More reference shapes for CShM can be defined in the code. You can find definitions of several ideal shapes
  here (cosymlib) or here (shape). Ideal shapes that include '-' are from the first source, while other shapes come from the second.
• The script's results should match those of the online calculator or the Shape program when using default options.
• Although a strictly defined τ₃ value does not exist, shape measures for 3-coordinated atoms are still calculated.
• If the calculation is not fast enough for your needs, you can uncomment the optimization process using the Hungarian algorithm. Ensure that the number of trials (num_trials) is set high enough to avoid missing the global minimum.

Known Issues

• The script is not very well tested with symmetry generated atom positions. However, this is rarely the case with small molecule structures.
• All flavors of τ and O are calculated as soon as two angles are present. So you have to check if it makes sense.
• The script can only remove atoms from the coordination sphere, not add atoms. Therefore, make sure that the connectivity list is appropriate.
• For the generation of the XYZ file, hydrogen atoms are generally ignored. As a result, in metal hydrides, hydrogen atoms will not be included in the XYZ file.
• The CShM method is rewritten from the C++ code and may still contain errors.
• As the number of vertices increases, both computation time and memory usage grow rapidly ($n!$). To address this issue, enable the optimization process using the Hungarian algorithm. Ensure that the number of trials (num_trials) is set sufficiently high to avoid missing the global minimum.

Polyhedra and Shape Reference

Examples

Example 1:

python3 tau-calc.py test.cif Hg1 -smd

The predicted coordination number for Hg1 is 4.

The two largest angles are β = 135.94° and α = 124.44°.
Note: Angles for the calculation of τ₄, τ₄' and τ₅.
 
Number of cis angles ~ 90° (< 135°)    = 5
Number of trans angles ~ 180° (> 135°) = 1
Note: First value should be 12, second 3 for an octahedron.
 
--------------------------------------------------------------------------------
Hg1 geometry index ("<--" indicates the parameter for coordination number 4):
--------------------------------------------------------------------------------
τ₄  =   0.71 <--
τ₄' =   0.67 <--
τ₅  =   0.19 
O   =  23.49 
 
--------------------------------------------------------------------------------
Continuous shape measure (CShM) for coordination number 4:
--------------------------------------------------------------------------------
S(SP-4, Square)                               =  32.0294 ░░░░░░░░░░░░░░░░░░░░░░░
S(T-4, Tetrahedron)                           =   2.8099 ░
S(SS-4, Seesaw or sawhorse)                   =   5.8584 ░░░
S(vTBPY-4, Axially vacant trigonal bipyramid) =   2.8078 ░
 
--------------------------------------------------------------------------------
Polyhedral volume (coordination number 4) = 9.7478 A³
--------------------------------------------------------------------------------
 
--------------------------------------------------------------------------------
Table of typical geometries and their corresponding τ₄ or τ₅ values: 
--------------------------------------------------------------------------------
Coordination number 4:
Tetrahedral          : τ₄ = 1.00       τ₄' = 1.00
Trigonal pyramidal   : τ₄ = 0.85       τ₄' = 0.85
Seesaw               : τ₄ = 0.43       τ₄' = 0.24
Square planar        : τ₄ = 0.00       τ₄' = 0.00

Coordination number 5:
Trigonal bipyramidal : τ₅ = 1.00                     
Square pyramidal     : τ₅ = 0.00

Results saved to test-Hg1.md

The saved tables are formatted as follows:

Table 1. Coordination number (C.N.), Geometry index, Polyhedral volume (in ų) of Hg1

| **Property**      |   **Value** |
|-------------------|-------------|
| C.N.              |      4      |
| τ₄                |      0.71   |
| τ₄'               |      0.67   |
| Polyhedral volume |      9.7478 |


Table 2. Continuous Shape Measures (CShM) of Hg1

| **Shape**   | **Description**                   |   **CShM** |
|-------------|-----------------------------------|------------|
| SP-4        | Square                            |    32.0294 |
| T-4         | Tetrahedron                       |     2.8099 |
| SS-4        | Seesaw or sawhorse                |     5.8584 |
| vTBPY-4     | Axially vacant trigonal bipyramid |     2.8078 |

Example 2:

python3 tau-calc.py test2.cif Ru1 -v

--------------------------------------------------------------------------------
Ru1 binds to:
--------------------------------------------------------------------------------
 N1-Ru1 2.065(3) Å      .
Ru1-N1  2.065(3) Å  3_665
Ru1-N1  2.065(3) Å  9_655
Ru1-N1  2.065(3) Å  6_655
Ru1-N1  2.065(3) Å 12_665
Ru1-N1  2.065(3) Å      7
 
The predicted coordination number for Ru1 is 6.

--------------------------------------------------------------------------------
Ru1 angles are:
--------------------------------------------------------------------------------
N1-Ru1-N1  78.98(19)°      .  3_665
N1-Ru1-N1  94.66(13)°      .  9_655
N1-Ru1-N1  92.37(18)°  3_665  9_655
N1-Ru1-N1  92.37(18)°      .  6_655
N1-Ru1-N1  94.66(13)°  3_665  6_655
N1-Ru1-N1 170.89(18)°  9_655  6_655
N1-Ru1-N1  94.66(13)°      . 12_665
N1-Ru1-N1 170.89(18)°  3_665 12_665
N1-Ru1-N1  94.66(13)°  9_655 12_665
N1-Ru1-N1  78.98(19)°  6_655 12_665
N1-Ru1-N1 170.89(18)°      .      7
N1-Ru1-N1  94.66(13)°  3_665      7
N1-Ru1-N1  78.98(19)°  9_655      7
N1-Ru1-N1  94.66(13)°  6_655      7
N1-Ru1-N1  92.37(18)° 12_665      7
 
The two largest angles are β = 170.89° and α = 170.89°.
Note: Angles for the calculation of τ₄, τ₄' and τ₅.
 
Number of cis angles ~ 90° (< 135°)    = 12
Number of trans angles ~ 180° (> 135°) = 3
Note: First value should be 12, second 3 for an octahedron.
 
--------------------------------------------------------------------------------
Ru1 geometry index ("<--" indicates the parameter for coordination number 6):
--------------------------------------------------------------------------------
τ₄  =   0.13 
τ₄' =   0.13 
τ₅  =   0.00 
O   =   7.12 <--
 
--------------------------------------------------------------------------------
Continuous shape measure (CShM) for coordination number 6:
--------------------------------------------------------------------------------
S(HP-6, Hexagon)                           =  28.5605 ░░░░░░░░░░░░░░░░░░░░░░░░
S(PPY-6, Pentagonal pyramid)               =  27.0546 ░░░░░░░░░░░░░░░░░░░░░░
S(OC-6, Octahedron)                        =   0.9516 ░
S(TPR-6, Trigonal prism)                   =  13.6312 ░░░░░░░░░░░
S(JPPY-6, Johnson pentagonal pyramid (J2)) =  30.8553 ░░░░░░░░░░░░░░░░░░░░░░░░░░
 
--------------------------------------------------------------------------------
Polyhedral volume (coordination number 6) = 11.5171 A³
--------------------------------------------------------------------------------
 
--------------------------------------------------------------------------------
Table of typical geometries and their corresponding τ₄ or τ₅ values: 
--------------------------------------------------------------------------------
Coordination number 4:
Tetrahedral          : τ₄ = 1.00       τ₄' = 1.00
Trigonal pyramidal   : τ₄ = 0.85       τ₄' = 0.85
Seesaw               : τ₄ = 0.43       τ₄' = 0.24
Square planar        : τ₄ = 0.00       τ₄' = 0.00

Coordination number 5:
Trigonal bipyramidal : τ₅ = 1.00                     
Square pyramidal     : τ₅ = 0.00                    

 
--------------------------------------------------------------------------------
XYZ coordinates of the central atom and its neighbors: 
--------------------------------------------------------------------------------
7
879418.cif Ru1
Ru  0.00000000  0.00000000  0.00000000
N   1.01427676 -1.42908956 -1.09132010
N   0.72983256  1.59407203 -1.09132010
N  -1.74608007 -0.16384466 -1.09132010
N   1.74476624 -0.16384466  1.09132010
N  -1.01559059 -1.42908956  1.09132010
N  -0.73114639  1.59407203  1.09132010

Example 3:

python3 tau-calc.py test3.cif Co1 -e N12

Excluded atoms: ['N12']
 
The predicted coordination number for Co1 is 5.

The two largest angles are β = 176.77° and α = 173.52°.
Note: Angles for the calculation of τ₄, τ₄' and τ₅.
 
Number of cis angles ~ 90° (< 135°)    = 8
Number of trans angles ~ 180° (> 135°) = 2
Note: First value should be 12, second 3 for an octahedron.
 
--------------------------------------------------------------------------------
Co1 geometry index ("<--" indicates the parameter for coordination number 5):
--------------------------------------------------------------------------------
τ₄  =   0.07 
τ₄' =   0.06 
τ₅  =   0.05 <--
O   =   4.65 
 
--------------------------------------------------------------------------------
Continuous shape measure (CShM) for coordination number 5:
--------------------------------------------------------------------------------
S(PP-5, Pentagon)                            =  30.9225 ░░░░░░░░░░░░░░░░░░░░░░░░
S(vOC-5, Vacant octahedron (J1))             =   0.3515 ░
S(TBPY-5, Trigonal bipyramid)                =   6.6922 ░░░░░
S(SPY-5, Square pyramid)                     =   2.2513 ░░
S(JTBPY-5, Johnson trigonal bipyramid (J12)) =   7.9883 ░░░░░░
 
--------------------------------------------------------------------------------
Polyhedral volume (coordination number 5) = 4.8918 A³
--------------------------------------------------------------------------------
 
--------------------------------------------------------------------------------
Table of typical geometries and their corresponding τ₄ or τ₅ values: 
--------------------------------------------------------------------------------
Coordination number 4:
Tetrahedral          : τ₄ = 1.00       τ₄' = 1.00
Trigonal pyramidal   : τ₄ = 0.85       τ₄' = 0.85
Seesaw               : τ₄ = 0.43       τ₄' = 0.24
Square planar        : τ₄ = 0.00       τ₄' = 0.00

Coordination number 5:
Trigonal bipyramidal : τ₅ = 1.00                     
Square pyramidal     : τ₅ = 0.00                      

Example 4

References

If you use τ₄, τ₅, the O index or CShM to describe the coordination geometry of your compounds, please cite one or more of the following articles:

τ₄:

"Structural variation in copper(i) complexes with pyridylmethylamide ligands: structural analysis with a new four-coordinate geometry index, τ₄"

Lei Yang, Douglas R. Powell, Robert P. Houser, Dalton Trans. 2007, 955-964.

DOI: https://doi.org/10.1039/B617136B

τ₄' (τ₄ improved):

"Coordination polymers and molecular structures among complexes of mercury(II) halides with selected 1-benzoylthioureas"

Andrzej Okuniewski, Damian Rosiak, Jarosław Chojnacki, Barbara Becker, Polyhedron 2015, 90, 47–57.

DOI: https://doi.org/10.1016/j.poly.2015.01.035

τ₅:

"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"

Anthony W. Addison, T. Nageswara Rao, Jan Reedijk, Jacobus van Rijn, Gerrit C. Verschoor, J. Chem. Soc., Dalton Trans. 1984, 1349-1356.

DOI: https://doi.org/10.1039/DT9840001349

O:

"Structural, electrochemical and photophysical behavior of Ru(II) complexes with large bite angle sulfur-bridged terpyridyl ligands"

Christopher M. Brown, Nicole E. Arsenault, Trevor N. K. Cross, Duane Hean, Zhen Xu, Michael O. Wolf, Inorg. Chem. Front. 2020, 7, 117-127.

DOI: https://doi.org/10.1039/C9QI01009B

CShM:

"Continuous Symmetry Measures. 5. The Classical Polyhedra"

Mark Pinsky, David Avnir, Inorg. Chem. 1998, 37, 5575–5582.

DOI: https://doi.org/10.1021/ic9804925

"Shape maps and polyhedral interconversion paths in transition metal chemistry"

Santiago Alvarez, Pere Alemany, David Casanova, Jordi Cirera, Miquel Llunell, David Avnir, Coord. Chem. Rev., 2005, 249, 1693–1708.

DOI: https://doi.org/10.1016/j.ccr.2005.03.031

The script uses the Gemmi library for CIF processing:

"GEMMI: A library for structural biology"

Marcin Wojdyr, Journal of Open Source Software 2022, 7 (73), 4200.

DOI: https://doi.org/10.1021/ic9804925

https://gemmi.readthedocs.io/en/latest/

https://github.com/project-gemmi/gemmi