How to convert absolute cartesian coordinates to monoclinic coordinates?

rowlesmr

I'm having a bit of a brain fart, and just can't get this to work.

How can I convert absolute cartesian coordinates to monoclinic coordinates?

I've put together a structure using regular cartesian coordinates, and upon inspection, the symmetry is monoclinic, with be~=111 deg.

What is the transformation to go from (x,y,z) -> (x', y', z')? (and thence to fractional coordinates)

I've derived a couple of methods, and I'm getting some ridiculous results, and I don't know if that's because my derivation is wrong, or my arithmetic.

One matrix I've got is this:

[ 1   0  -cot(be) ] [x]    [x']
[ 0   1       0   ] [y] =  [y']
[ 0   0   csc(be) ] [z]    [z']

rowlesmr

Had some insight today while cleaning the house (does anyone want to buy a house in Perth?)

This is the transformation:

[ 1   0  -cot(be) ] [x]    [x']
[ 0   1      0    ] [y] =  [y']
[ 0   0      1    ] [z]    [z']

You just need to slide all the atoms along the x-axis to line them up with the new z-axis.

You then make fractional coordinates by dividing the transformed absolute coordinates by the size of the unit cell in the original cartesian coordinates.

The cell prms then become a, b = size of cell in cartesian, c=size of the sloped line that would become the new c-axis from the cartesian coordinates. be = be.

Whack it all in FINDSYM, and my 36 atom P1 unit cell becomes a 5 atom C2/c unit cell, with transformed axes, because why not?