Slow-growth approach

The free-energy profile along a geometric parameter $\xi$ can be scanned by an approximate slow-growth approach^[1]. 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 $1 \rightarrow 2$ can be computed as:

${\displaystyle w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}} \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. }$

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

${\displaystyle {\rm exp}\left\{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T} \right \}= \bigg \langle {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} \bigg\rangle. }$

Note that calculation of the free-energy via this equation requires averaging of the term ${\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}$ over many realizations of the $1 \rightarrow 2$ transformation. Detailed description of the simulation protocol that employs Jarzynski's identity can be found in reference ^[3].

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.

VASP can handle multiple (even redundant) constraints. Note, however, that a too large number of constraints can cause problems with the stability of the SHAKE algorithm. In problematic cases, it is recommended to use a looser convergence criterion (see SHAKETOL) and to allow a larger number of iterations (see SHAKEMAXITER) in the SHAKE algorithm. Hard constraints may also be used in metadynamics simulations (see MDALGO=11 | 21). Information about the constraints is written onto the REPORT-file: check the lines following the string: Const_coord

References

Contents

[woo1997-1] T. K. Woo, P. M. Margl, P. E. Blochl, and T. Ziegler, J. Phys. Chem. B 101, 7877 (1997).

[jarzynski1997-2] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).

[oberhofer2005-3] . Oberhofer, C. Dellago, and P. L. Geissler, J. Phys. Chem. B 109, 6902 (2005).

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