Slow-growth approach: Difference between revisions

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can be computed as:
can be computed as:


<math>
::<math>
w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}}  \left ( \frac{\partial                                      {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.
w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}}  \left ( \frac{\partial                                      {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.
</math>
</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.
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.
transformation.
Detailed description of the simulation protocol that employs Jarzynski's identity
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:
== 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 {{TAG|INCREM}} tag for each geometric parameter with <tt>STATUS=0</tt> has to be specified:
#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
#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}}
#Choose a thermostat:
#Define geometric constraints in the {{FILE|ICONST}}-file, and set the {{TAG|STATUS}} parameter for the constrained coordinates to 0
## Set {{TAG|MDALGO}}=1, and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}
## Set {{TAG|MDALGO}}=2, and choose an appropriate setting for {{TAG|SMASS}}
#Define geometric constraints in the {{FILE|ICONST}} file, and set the STATUS parameter for the constrained coordinates to 0
#When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE.
#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">
<ol start="5">
<li>Specify the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt></li>
<li>Specify the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt>.</li>
</ol>
</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>




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<references>
<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="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>
</references>
----
----
[[The_VASP_Manual|Contents]]


[[Category:Molecular Dynamics]][[Category:Slow-growth approach]][[Category:Theory]][[Category:Howto]]
[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]

Latest revision as of 13:10, 21 October 2025

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

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

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

[math]\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. }[/math]

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

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


References

  1. ↑ T. K. Woo, P. M. Margl, P. E. Blochl, and T. Ziegler, J. Phys. Chem. B 101, 7877 (1997).
  2. ↑ C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
  3. ↑ . Oberhofer, C. Dellago, and P. L. Geissler, J. Phys. Chem. B 109, 6902 (2005).
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