Some thoughts for blue moon method and slow growth in cp2k

[xuan Garrett]

Dear cp2k community,

Based on some of the topics in this google group and the official pages in cp2k.org, I notice that the blue moon method and slow growth method are not clearly displayed . So, here, I want to start a topic about this. 

You can see that the page is not available on official page from cp2k.org. And after I check from Prof. Hutter that actually, cp2k only outputs the lamda value (the constraint force) rather the whole blue moon ensemble for thermodynamic integration (TI). 

And the official example that uses the PMF to calculate the free energy itself provides the one example, where one should note that the if the COLVAR (CV) only refers to the single distance, then the mean force can be treated by <lamda>. 

If the CV is defined as the gap between two distances, such as d(A---B) - d(B---C) as in substitution reaction: A + B---C → A---B + C. Then, the mean force does not always follow the simple lamda, but <Z^(-1/2)*lambda> / <Z^(-1/2)>. 

Comparatively, for VASP users, because the LBLUEOUT tag supports the whole blue moon ensemble correction, we just need to follow its official steps with TI. But for cp2k users, the strict blue moon ensemble correction should be done yourself as suggested by Prof. Hutter. 

I don't know if developers will attempt to add the corrections to some simple CV definitions for blue moon ensemble.

And for the slow growth, in terms of cAIMD (not the slow growth in metadynamics of cp2k), maybe the tag in &CONSTRAINT/&COLLECTIVE as of TARGET_GROWTH states for this function. 

So, here, I attach one article with the correction formula for these two conditions, single distance and gap of two distances.

Best regards,

Xuan