The correction of the mean force to calculation of free energy in CP2K

[YE-FEI LI]

Dear CP2K users and developers:
    I am interested into calculation of free energy under a designed 
reaction pathway, and I have some puzzle about the basic idea for 
calculating the free energy by utilizing thermodynamic integration and 
Lagrangian multiplier. The mean force can be obtained from Lagrangian 
multiplier plus some specific correction according to different constrain. 
For one bond length constrain, the mean force is the average of Lagrangian 
multiplier. However, for other constrain, the mean force might not be the 
average of Lagrangian multiplier, and the correction is needed.
    For instance, if I use the distance difference of r12-r34 as constrain, 
it seems the mean force is still the average of Lagrangian multiplier. (Am 
I correct?) However, if I use distance difference of r12-r31 as constrain, 
it seems the correction is needed, because r12 and r31 is coupled. (Am I 
correct?) In this case, if I want to calculate the free energy along r12-r31, 
CP2K will not do the correction automatically, since CP2K only output the Lagrangian 
multiplier. Thus I must do the correction by myself by using the trajectory 
of MD? 
Best,
Yefei
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