Symmetry elements from XYZ

Ian Clifton ian at melvin.dp.ox.ac.uk
Thu Apr 17 08:08:27 EST 1997

islam <islam at icrf.icnet.uk> writes:

> Given an orthogonal dataset of coordinates for 2 identical
> molecules is it possible to derive the symmetry operator relating
> them or determine if one exists ?
> Am I correct in thinking:
>  * the problem is equivalent to least squares fitting the
>    two molecules to derive a transformation matrix and then
>    decomposing/examining the derived matrix to its "simplest
>    form" e.g. rotation about a 3D axis

Yes, every rigid-body motion can be expressed as a rotation plus a translation.
I find this simple theorem remarkably helpful when thinking about this
sort of thing!
>  * in a sense as long as a transformation matrix can be obtained
>    to fit one molecule exactly onto another, then a symmetry op 
>    always exists defined by the matrix ?

I guess it depends on what you mean by "symmetry op" - do you mean a
transformation that's compatible with a given space group? This would
imply restrictions on the rotational part (1, 2, 3, 4, 6), and limit
the translational part to multiples of the unit cell edges and certain
fractions thereof.

Ian Clifton                   Phone: +44 1865 275631
Dyson Perrins Laboratory      Fax:   +44 1865 275674
Oxford University  OX1 3QY UK ian at biop.ox.ac.uk                          

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