Slow-growth approach: Difference between revisions

Latest revision as of 10:27, 16 November 2023

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:

w_{1\rightarrow 2}^{irrev}=\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 $w_{1\rightarrow 2}^{irrev}$ corresponds to the free-energy difference between the the final and initial state. In the general case, $w_{1\rightarrow 2}^{irrev}$ is the irreversible work related to the free energy via Jarzynski's identity^[2]:

exp^{-{\frac {\Delta A_{1\rightarrow 2}}{k_{B}\,T}}}={\bigg \langle }exp^{-{\frac {w_{1\rightarrow 2}^{irrev}}{k_{B}\,T}}}{\bigg \rangle }.

Note that calculation of the free-energy via this equation requires averaging of the term ${\rm {exp}}\left\{-{\frac {w_{1\rightarrow 2}^{irrev}}{k_{B}\,T}}\right\}$ 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].

Anderson thermostat

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. For a slow-growth approach 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.

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

Nose-Hoover thermostat

For a slow-growth approach run with Nose-Hoover thermostat, one has to:

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

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

[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:
-<math>
+::<math>
 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 $\dot{\xi}$, 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:
+== Anderson thermostat ==
+* For a slow-growth simulation, one has to perform a calcualtion very similar to {{TAG|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. For a slow-growth approach 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}}
@@ Line 31: / Line 37: @@
 #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>
+== Nose-Hoover thermostat ==
+* For a slow-growth approach run with Nose-Hoover thermostat, one has to:
+#Set the standard MD-related tags: {{TAG|IBRION}}=0, {{TAG|TEBEG}}, {{TAG|POTIM}}, and {{TAG|NSW}}
+#Set {{TAG|MDALGO}}=2, and choose an appropriate setting for {{TAG|SMASS}}
+#Define geometric constraints in the {{FILE|ICONST}}-file, and set the <tt>STATUS</tt> parameter for the constrained coordinates to 0
+#When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE.
 <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>
 == 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="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:Molecular dynamics]][[Category:Slow-growth approach]][[Category:Theory]][[Category:Howto]]