Slow-growth approach

The free-energy profile along a geometric parameter $\xi$ can be scanned by an approximate slow-growth approach. In this method, the value of $\xi$ is linearly changed from the value characteristic for the initial state (1) to that for the final state (2) with a velocity of transformation $\dot{\xi }$. The resulting work needed to perform a transformation $1\to 2$ can be computed as:

$w_{1\to 2}^{irrev}=\underset{\xi (1)}{\overset{\xi (2)}{\int }}\left(\frac{\partial V(q)}{\partial \xi }\right)\cdot \dot{\xi }dt.$

In the limit of infinitesimally small $\dot{\xi }$, the work $w_{1\to 2}^{irrev}$ corresponds to the free-energy difference between the the final and initial state. In the general case, $w_{1\to 2}^{irrev}$ is the irreversible work related to the free energy via Jarzynski's identity:

$ex{p}^{-\frac{\mathrm{\Delta }{A}_{1\to 2}}{k_{B}T}}=⟨ex{p}^{-\frac{w_{1\to 2}^{irrev}}{k_{B}T}}⟩.$

Note that calculation of the free-energy via this equation requires averaging of the term $\mathrm{e}\mathrm{x}\mathrm{p}\{-\frac{w_{1\to 2}^{irrev}}{k_{B}T}\}$ over many realizations of the $1\to 2$ transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in reference .

How to

• For a slow-growth simulation, one has to perform a calcualtion very similar to Constrained molecular dynamics but additionally the transformation velocity-related INCREM tag for each geometric parameter with STATUS=0 has to be specified:

1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Choose a thermostat:
   1. Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB
   2. Set MDALGO=2, and choose an appropriate setting for SMASS
3. Define geometric constraints in the ICONST file, and set the STATUS parameter for the constrained coordinates to 0
4. When the free-energy gradient is to be computed, set LBLUEOUT=.TRUE.

5. Specify the transformation velocity-related INCREM-tag for each geometric parameter with STATUS=0.

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