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Is there a threshold? Could I say that my final geometries are the same with 'energy difference = 0.05 meV/atom' ?

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    $\begingroup$ @robert If it is a gas-phase molecule can you describe what exactly what you are trying to do? $\endgroup$
    – mykd
    Aug 5 '20 at 15:12
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    $\begingroup$ To check if your structure is optimized, you need to look at the forces on the atoms. If you set out to relax the cell and the atoms themselves, a good threshold is 0.0001 eV/Angstrom (This is the default value on Quantum ESPRESSO). $\endgroup$
    – Xivi76
    Aug 5 '20 at 17:09
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    $\begingroup$ +1. But there might be two energy-equivalent minima in the energy landscape, so the energy at one optimized geometry will be the same as the energy at a different geometry. In this case the energy difference between the two geometries will be 0.000... eV and 0.000... eV/atom. Am I understanding the question wrong? $\endgroup$ Aug 5 '20 at 23:24
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    $\begingroup$ @NikeDattani I agree: for a floppy molecule it's easy to have several geometries whose energy agrees to, say, 0.05 meV per atom, but the geometries are still quite different and lead to different properties. Dividing by the number of atoms makes no sense unless you're dealing with simple crystals. $\endgroup$ Aug 6 '20 at 8:51
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    $\begingroup$ In my view this is an ill-posed question - going by the strict definition, different final geometries are obviously not the same, and the question does not provide enough context to understand what @Alfred considers "the same". Unless more context can be provided, I vote to close this question. $\endgroup$ Aug 14 '20 at 11:44
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The question makes no sense, since a solid state system might have different phases, or a molecule might have different conformers. They may all be proper local minima of the energy functional, and with very small overall total energy differences.

For instance, J. Phys. Chem. A 117, 2269 (2013) is a benchmark study for the 52 different conformers of the melatonin molecule (C13H16N2O2). The molecule has thus 33 atoms, and an energy difference threshold of 0.05 meV/atom would yield 1.65 meV in total, which would mean 0.038 kcal/mol. In this case it thus turns out that provided if you work hard to find the minima you can differentiate between the conformers by an energy criterion. However, if you go to a larger system, the simple energy criterion might not work.

In the worst case you might have two conformers that are structurally different, but have the same energy; e.g. the Jahn-Teller effect can result in a double well potential.

The best way to distinguish whether you have indeed converged to the same structure also requires checking the minimal geometry itself, by both visual and computational means. You might establish the similarity of the structure by fingerprints, for example; e.g. show that $ R = \sum_{i}^{N_{\rm atoms}} ({\bf x}_i - {\bf x}_i')^2$, where ${\bf x}$ and ${\bf x}'$ are your two structures, is small. (Note that permutations of the nuclei is allowed so you should minimize $R$ with respect to swaps of identical atoms.)

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