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 $1 \rightarrow 2$ can be computed as: \begin{equation}\label{irev_work} w^{irrev}_{1 \rightarrow 2}=\int_Template:\xi(1)^Template:\xi(2) \left ( \frac{\partial {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt. \end{equation} 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~\cite{Jarzynski:97}: \begin{equation}\label{eq_jarzynski} {\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. \end{equation} Note that calculation of the free-energy via eq.(\ref{eq_jarzynski}) 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 Ref.~\cite{Oberhofer:05}.

• For a constrained molecular dynamics run with Andersen thermostat, one has to:

1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
2. Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB
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.

For a slow-growth simulation, one has to additionally:

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

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

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