would you do FMO-10 if you could?

The many-body expansion of the energy is a popular way to reduce the cost of computing the energy of an entire system too large to fit into your computer. The strategy is quite simple: chop your large system into several small pieces, do your quantum mechanics on each piece. In practice you can stop here (1-body), but often people to pair-corrections (2-body) to get good results and triples-corrections (3-body) if they want to be fancy. There are many implementations of this, including the eXplicit-POLarization method, the Fragment Molecular Orbtital method and the electrostatically-embedded many-body method, just to name a few.

My question is: would you do a 10-body calculation if you could?

Here is an argument against it, and it does not involve anything to do with accuracy, but rather computational cost.

The number of n-mers (1-mers, 2-mers, 3-mers and so on) increase quite drastically if one does not include approximations, but I can hear you ask: how bad is it?

For 16 water molecules, each water molecule is a 1-mer, the total number of calculations one would need to perform in order to calculate the n-body calculation is presented in the figure below


where we see that a 10-body calculation would require a total 58650 unique calculations whereas the 3-body calculation would require a "mere" 696.

Edit: For a discussion about timings, +Jan Jensen wrote a blog post on the subject.

That's at least an argument against dreaming of large n-body calculations unless you severely sit down and think about eliminating some of these calculations.

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Oxygen, you little rascal! Part 2.

Background
In the ongoing saga (read part I) with acrolein solvated in water, we were having problems with our high quality embedding potential just not converging nicely when increasing system sizes, QM-region sizes. We were using a functional to get good shielding constants (the KT3 functional) with a basis set designed to get good shielding constants (the apcS-1 basis set) but no matter what tried, the widely adopted TIP3P potential for water just did it better than we did. And we were just confused!

Solution
The problem turned out to be the basis set that with the diffuse functions leaked electron density out into the MM-region when using the PE potential and this was causing some disturbance with the induced dipoles. On the contrary, our belief is that TIP3P is a much too soft potential and there is a large degree of error cancellation because of that. Both are effects I would think to be very small and have no real impact but I had also forgotten that we were talking about nuclear shielding constants which are sensitive as f*ck the wave function.

I switched to the smaller pcS-1 basis set and all that leaking-trouble went away. Everything that we did not expect was gone. TIP3P was converging rather slowly and the PE-model was quickly converging as is shown below for an average of 120 snapshots.

The PE numbers (blue) are converged at QM size 3 but even size 1 and 2 show some good indications. For size 1, acrolein is solvated in purely classical water. As we would expect, the improved potential means that we are converging (fast) towards a value for the absolute shielding of acrolein in water valued -223.2 ppm.

Outlook
We will return to the use of diffuse basis functions and a proper quantum mechanical fix in a later paper as this work here sparked my co-worker +Jógvan Magnus Haugaard Olsen into actually picking up work he had done on repulsion during his Ph.D. (pdf of his thesis)

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