Slow-growth approach: Difference between revisions

Latest revision as of 13:54, 16 October 2024

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 exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}= \bigg \langle exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \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].

References

[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).

[1]

[2]

[3]

@@ Line 7: / Line 7: @@
 The resulting work needed to perform a transformation <math>1 \rightarrow 2</math>
 can be computed as:
-\begin{equation}\label{irev_work}
+::<math>
 w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}}  \left ( \frac{\partial                                      {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.
-\end{equation}
+</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}
-{\rm exp}\left\{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T} \right \}=
+::<math>
-\bigg \langle {\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \} \bigg\rangle.
+exp^{-\frac{\Delta A_{1 \rightarrow 2}}{k_B\,T}}=
-\end{equation}
+\bigg \langle  exp^{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T}} \bigg\rangle.
-Note that calculation of the free-energy via eq.(\ref{eq_jarzynski}) requires
+</math>
-averaging of  the term ${\rm exp} \left \{-\frac{w^{irrev}_{1 \rightarrow 2}}{k_B\,T} \right \}$
-over many realizations of the $1 \rightarrow 2$
+Note that calculation of the free-energy via this equation requires
+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
-can be found in Ref.~\cite{Oberhofer:05}.
+can be found in reference <ref name="oberhofer2005"/>.
-* For a constrained molecular dynamics run with Andersen thermostat, one has to:
-#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
-#Set {{TAG|MDALGO}}=1, and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
-#Define geometric constraints in the {{FILE|ICONST}}-file, and set the {{TAG|STATUS}} parameter for the constrained coordinates to 0
-#When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE.
-For a slow-growth simulation, one has to additionally:
-<ol start="5">
-<li>Specify the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt></li>
-</ol>
-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|SHAKE algorithm]]. In problematic cases, it is recommended to use a looser convergence criterion (see {{TAG|SHAKETOL}}) and to allow a larger number of iterations (see {{TAG|SHAKEMAXITER}}) in the [[#SHAKE|SHAKE algorithm]]. Hard constraints may also be used in [[#Metadynamics|metadynamics simulations]] (see {{TAG|MDALGO}}=11 {{!}} 21). Information about the constraints is written onto the {{FILE|REPORT}}-file: check the lines following the string: <tt>Const_coord</tt>
@@ Line 42: / Line 33: @@
 <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>
+<ref name="oberhofer2005">[https://pubs.acs.org/doi/abs/10.1021/jp044556a . Oberhofer, C. Dellago, and P. L. Geissler, J. Phys. Chem. B 109, 6902 (2005).]</ref>
 </references>
 ----
-[[The_VASP_Manual|Contents]]
-[[Category:Molecular Dynamics]][[Category:Slow-growth approach]][[Category:Theory]][[Category:Howto]]
+[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]