# Choice of coordinate system for geometry optimization

For geometry optimization, most QM codes tend to use internal coordinates. Most codes also support Cartesian coordinates. I have always heard the usual "use Cartesian as a last resort, always use internals" advice.

Recently I had to optimize small water clusters with Gaussian, and I found that in most cases, the optimization often stops when one of the dihedral angles goes close to $$180^\circ$$, with the error "Tors failed for dihedral". My supervisor's advice was to start from the last frame where the optimization stopped (which would build a new set of coordinates I imagine). However, I found that if I used Cartesian optimization from the start, then I actually saved (my) time, which is otherwise lost in restarting the optimization, often multiple times.

I also found that there were multiple types of internal coordinates, for example natural internal coordinates, Z-matrix supplied from outside, redundant internal coordinates and delocalized internal coordinates. I don't completely understand the difference between those.

Another thing I noticed was that GAMESS never ran into the dihedral problem with internal coordinates, and GAMESS uses delocalized internals instead of redundant internal coordinates of Gaussian.

I found only one paper that discusses the choice of Cartesian vs redundant internals but it is very old [1]. I was able to run some small molecules on my laptop, and delocalized internals seem to be the fastest for all of them.

So, my question — Is there any benchmark that compares the different coordinate systems for optimization, particularly the various types of internal coordinates, for small to large molecules? I would also be interested to read any benchmarks that you run, so feel free to post those.

Follow up question— Is it really that bad to use Cartesian coordinates? because I found that they generally take 1-2x the time. So calling them last resort seems a bit extreme.

Reference:

1. H. B. Schlegel, "A comparison of geometry optimization with internal, cartesian, and mixed coordinates", Int. J. Quantum Chem., 44, 243-252. https://doi.org/10.1002/qua.560440821