# How large an energy difference in Hartree can be used to distinguish two different structures?

For example, I use two different geometries of the same molecule to do optimisation in Gaussian, and the two resultant geometries look roughly the same. The energy difference between the two optimised geometries is only 0.0000001 a.u.. So can I view them as the same geometry? Or is the energy difference large enough to distinguish them? How large an energy difference is large enough for you to see two geometries as different geometries?

• When you do a geometry optimization, one of the convergence criteria is the change in energy between iterations. I think 0.0000001 a.u., is below the threshold for convergence in energy (if you use standard settings). This suggest that the two structures are the "same". – Erik Kjellgren Feb 9 '18 at 22:16

I wouldn't use energy as a general criteria to discriminate conformations; a root mean squared deviation (RMSD) of superimposed structures would be a better metric. This is because two very different conformations may have very similar energy. For example, if you perform a relaxed energy scan of one of the dihedrals of cyclopentane, you get nearly the same energy for most of the dihedral angles, bu very different ring conformations.

If you found an energy difference of 0.0000001 a.u. and the superimposition of the conformations is nearly perfect, I'd say they are in the same minimum of the energy surface.

• Your talking about confirmation is a good example. But in my case, the two geometries are not conformers. They are just slightly different in bond lengths (around 0.0002 angstrom). I'm thinking whether I can see them as the same point on the potential energy surface, or they are actually two different points... – OhLook Feb 9 '18 at 22:57
• I consider them to be in the same point in the energy surface - 0.0002 angstrom and 0.0000001 a.u. is pretty much nothing! Gaussian stops the minimization because it considers that both structures have optimized. If the minimization was continued, they would certainly land in the same exact conformation. – diogom Feb 9 '18 at 23:04
• By the way, what are the details of your criteria for superimposed structures? How small a deviation is acceptable to see two geometries as the same minimum on PES? – OhLook Feb 9 '18 at 23:53
• I doubt there is a need for a cutoff. If a molecule is fully optimized from multiple starting structures, there is either a very small deviation (e.g. 0.0002) or something much bigger (e.g. 0.5 angstroms). Did you ever find something between - like 0.01 angstroms? – diogom Feb 10 '18 at 1:27

That's a pretty incredible threshold. To put things in perspective, if $1 E_{\mathrm{h}} \approx \pu{627 kcal/mol}$, then $\pu{0.0000001 a.u.} = \pu{6.27e-5 kcal/mol}$. Energy differences between conformers are often only a few kcal/mol, but this is a full 5 orders of magnitude smaller. Even $k_{\mathrm{B}}T$, which is $\pu{207 cm^{-1}}$ at room temperature, is $\pu{0.59 kcal/mol}$. If two molecules are truly the same conformer, then small geometry differences (bond lengths, angles) will 1. computationally converge to identical structures by slightly tightening convergence criteria, and 2. exist experimentally as identical vibrationally-averaged structures. That is, any vibrational mode has enough energy to cover this difference. Using a similar argument to the above, $\pu{1 cm^{-1} = 0.0029 kcal/mol}$, which is 2 orders of magnitude larger than the criterion you gave.

Consider the default energy convergence criteria for a few popular packages:

$$\begin{array}{lc} \hline \text{Program} & \text{Max energy change}\,(E_{\mathrm{h}}) \\ \hline \text{Gaussian 16} & \pu{1.0e-7} \\ \text{Molpro 2015.1} & \pu{1.0e-6} \\ \text{ORCA 4.0} & \pu{5.0e-6} \\ \text{Psi4 1.1} & \pu{1.0e-6} \\ \text{Q-Chem 5.0} & \pu{1.0e-6} \\ \hline \end{array}$$

So can I view them as the same geometry? Or is the energy difference large enough to distinguish them?

For two molecules you know to be the same conformer, $\pu{0.0000001 a.u.}$ is indistinguishable and not far from numerical noise due to loss of precision.

How large an energy difference is large enough for you to see two geometries as different geometries?

I am not aware of an official reference. However, for the case of a typical organic molecule that doesn't have shallow basins in its potential energy surface, I would consider the default geometry optimization convergence criteria enough.