Slow-growth approach: Difference between revisions

Revision as of 15:35, 13 March 2019

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 Ref.~\cite{Oberhofer:05}.

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

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

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

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

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).

[1]

[2]

@@ Line 12: / Line 12: @@
 </math>
-In the limit of infinitesimally small $\dot{\xi}$, the work $w^{irrev}_{1 \rightarrow 2}$
+In the limit of infinitesimally small <math>\dot{\xi}</math>, the work <math>w^{irrev}_{1 \rightarrow 2}</math>
 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
+In the general case, <math>w^{irrev}_{1 \rightarrow 2}$ </math>is the irreversible work related
-to the free energy via Jarzynski's identity~\cite{Jarzynski:97}:
+to the free energy via Jarzynski's identity<ref name="jarzynski1997"/>:
-\begin{equation}\label{eq_jarzynski}
+<math>
 {\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}
+</math>
-Note that calculation of the free-energy via eq.(\ref{eq_jarzynski}) requires
-averaging of  the term ${\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}$
+Note that calculation of the free-energy via this equation requires
-over many realizations of the $1 \rightarrow 2$
+averaging of  the term <math>{\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}</math>
+over many realizations of the <math>1 \rightarrow 2</math>
 transformation.
 Detailed description of the simulation protocol that employs Jarzynski's identity
@@ Line 44: / Line 46: @@
 <references>
 <ref name="woo1997">[https://pubs.acs.org/doi/abs/10.1021/jp9717296 T. K. Woo, P. M. Margl, P. E. Blochl, and T. Ziegler, J. Phys. Chem. B 101, 7877 (1997).]</ref>
+<ref name="jarzynski1997">[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.78.2690 C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).]</ref>
 </references>
 ----