Tutorial on nuclear quantum effects

[peastman]

This adds a tutorial on using RPMD and adQTB to simulate nuclear quantum effects. We shouldn't merge this until 8.4 is released, since it uses a new feature.

[epretti]

This is a nice tutorial! I was curious, and initially slightly confused, by the remark

Notice that we are using a larger friction coefficient than we did for the other integrators. This is usually required for adQTB. If the friction is too low, it may be impossible to fully compensate for zero point energy leakage. Even reducing the noise magnitude to zero would leave too much energy in some modes. This can slow down motions and cause sampling to take longer, but it still ends up being much faster than RPMD.

It took me a bit to figure out what the tutorial was talking about in terms of noise spectra/magnitudes/friction until I looked at the user guide. Is this saying that setting $\gamma_f$ too low means that $\gamma_r(\omega)$ might not be able to be decreased enough for certain $\omega$ to get the correct energy distribution? And then, is "This" in the last sentence ("This can slow down...") referring to having to set $\gamma_f$ higher in general, not to the problem pointed out by sentence immediately preceding it?

I'm also curious if the tutorial can make any comment about what to look for in the converged $\gamma_r(\omega)$ to know if you need a higher $\gamma_f$. The user guide suggests that one should watch for $\gamma_r(\omega)$ "close to zero". I tried changing $\gamma_f$ from 20/ps to 5/ps, and the resulting minimum value of $\gamma_r(\omega)$ was around 3/ps, but the RDF still differed noticeable from RPMD, so perhaps there is more to the story. Alternatively, I may have just misunderstood the tutorial and user guide, in which case a comment guarding against whatever my misconception is might be useful to have in the tutorial.

[peastman]

Is this saying that setting γ f too low means that γ r ( ω ) might not be able to be decreased enough for certain ω to get the correct energy distribution?

Correct.

"This" in the last sentence ("This can slow down...") referring to having to set γ f higher in general

Right.

I don't entirely have a good answer on how to pick the friction coefficient. If it gets close to zero at any frequency, it's clearly too low. But increasing it further does seem to improve the results. Possibly this just means the adaptation doesn't do a perfect job of correcting for zero point energy leakage, but I'm not sure.

In some ways, parahydrogen isn't a great example for this because it's so simple: just one type of particle interacting through one isotropic interaction. It doesn't have a wide range of frequencies, so there isn't that much for the adaptation to do. Aside from the very low frequencies, the spectrum is mostly flat.

Here's the corresponding graph for a box of flexible water molecules. That's a more complex system with several distinct motions at different frequencies, so there's more for the adaptation to do. In some ways that makes it a better example, except of course that it's a classical water model designed to produce correct results without NQEs, so it would actually be a bad example to use.