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Conformational searches using molecular mechanics can be either systematic or 'random' using Monte Carlo methods (or other non-systematic methods) to sample conformational space.

In general, my understanding is that Monte Carlo methods perform just as well as systematic searches # assuming that the right parameters are used. This is often described in (organic) computational papers along the lines of :

A sufficient number of conformations were explored such that all low-energy conformers within a X kj/mol window of the global minima were found at least X times.

There is a lot of ambiguity over how many times a low energy conformer needs to be found in order to 'guarantee' that the conformational space has been fully explored. Certainly, in my own experience I can generate an ensemble that fulfils this criteria but is clearly non-complete (if I run the same search from a slightly different starting geometry I generate another ensemble with some different low-energy conformers).

My question is two-fold:

  • Is there any mathematical basis for the statement about finding a low energy conformation x times, or is it largely a case of convenience that works well empirically in the majority of situations ?
  • Is there really any way to know that a search is comprehensive without systematically exploring the entire PES ?

#: There was a report that Ken Houk lost a bet due to his belief that systematic sampling was the only true way to explore a potential energy surface, though I cant find the paper where this was discussed.

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    $\begingroup$ "A sufficient number of conformations were explored such that all low-energy conformers within a X kj/mol window of the global minima were found at least X times." Do you mean for the first X to be different from the second X, meaning that the second one should be called something else, such as Y? $\endgroup$ – user1271772 May 13 '18 at 23:58
  • $\begingroup$ Note that both systematic and random search can only quarantine finding global minimum for a certain probability. The question is not if random sampling is guaranteed correct, the question is which of the two methods are better in finding the answer with high probability in the shorter time. $\endgroup$ – Greg May 14 '18 at 1:13
  • $\begingroup$ We've been interested in this question - and are currently generating sets of exhaustively-sampled minima. $\endgroup$ – Geoff Hutchison May 15 '18 at 2:55
  • $\begingroup$ I'd guess that the answer is "it seems to work well" based on an estimate of statistics - if you're seeing the same conformer X times, the probability of missing minima can be hoped to be small. $\endgroup$ – Geoff Hutchison May 15 '18 at 2:56
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We have published² a probabilistic treatment of this question, but for the case of MD simulations (and not MC). The principal idea is to use Good-Turing statistics to estimate the probability of unobserved species (ie conformations that have not -as yet- been observed). I have no idea whether this treatment can be transferred to MC, but there is an open source program³ -based on R- that you could try to use.

² https://pubs.acs.org/doi/10.1021/ci4005817

³ https://github.com/pkoukos/GoodTuringMD

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  • $\begingroup$ That's an interesting approach $\endgroup$ – Geoff Hutchison May 15 '18 at 2:54

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