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.


  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

2 Answers 2


You may want to have a look at this paper: "Geometry optimization made simple with translation and rotation coordinates" https://aip.scitation.org/doi/10.1063/1.4952956

This is not a comprehensive benchmark but it does contain a few examples ranging from water clusters of 12 water molecules to small proteins. According to this paper Cartesian coordinates are on average a poor choice for small water clusters, but it seems to have a better worst-case performance than delocalized internal coordinates.

Internal coordinates have numerical instabilities when some of the angles approach 180 degrees. If these angles are already linear from the very start of the optimization, the program can construct the internal coordinates in such a way that they do not include the linear angles, or at least not in a way that causes numerical instabilities. Usually problems only arise when an angle that was originally not linear becomes linear during the optimization; this does not happen very often for covalently linked systems, as you usually draw linear angles (like alkynes or nitriles) as linear from the start, but it happens more frequently for noncovalent clusters - it's very common that you draw a nonlinear hydrogen bond in your initial guess structure but which turns out to become linear (by the way, if you restart the optimization the program generates a new set of internal coordinates that avoids that new linear angle, this is why restarting the geometry optimization works). Thus, people who work with large molecules may favor internal coordinates more than people who work with noncovalent clusters (like you) do. This may be the reason why they view Cartesian coordinates as a last resort but you do not fully agree with them.

  • 5
    $\begingroup$ Wow, I read the paper you cited and I'm really surprised to see that different optimization methods can also lead to different final structures. I never thought about this. I was convinced that the difference between the coordinate system can affect only the number of optimization steps. Thank you for pointing out this paper! $\endgroup$
    – NickZ
    Commented May 23, 2021 at 0:55

See our paper from 2006 and references therein: IJQC2006.

M. Swart and F.M. Bickelhaupt, "Optimization of strong and weak coordinates", Int. J. Quant. Chem. 2006, 106, 2536-2544

In this paper we took Baker's delocalized coordinates, combined it with ideas from Lindh for separation into weak and strong coordinates, and generalized this. These adapted delocalized coordinates have been included in the QUILD program, and showed a major improvement (in terms of geometry optimization/TS-search cycles needed).

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    $\begingroup$ In general we try to avoid link only answer, as the link could expire. Can you include a reference to the paper and maybe a short summary so the answer is self-contained? $\endgroup$
    – Tyberius
    Commented Jul 20, 2023 at 19:31
  • $\begingroup$ Done. The link will not expire. It's DOI.org. $\endgroup$
    – MSwart
    Commented Jul 21, 2023 at 20:48
  • $\begingroup$ @MSwart since you worked in this area, you might find it easy to answer this new question! $\endgroup$ Commented Oct 14, 2023 at 18:44

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