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).

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[2]

[3]

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+The free-energy profile along a geometric parameter <math>\xi</math> can be scanned by an approximate slow-growth
+approach<ref name="woo1997"/>.
+In this method, the value of <math>\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>\dot{\xi}</math>.
+The resulting work needed to perform a transformation <math>1 \rightarrow 2</math>
+can be computed as:
-In general, constrained molecular dynamics generates biased statistical averages.
+::<math>
-It can be shown that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
+w^{irrev}_{1 \rightarrow 2}=\int_{{\xi(1)}}^{{\xi(2)}}  \left ( \frac{\partial                                      {V(q)}} {\partial \xi} \right ) \cdot \dot{\xi}\, dt.
-:<math>
-a(\xi)=\frac{\langle |\mathbf{Z}|^{-1/2} a(\xi^*) \rangle_{\xi^*}}{\langle |\mathbf{Z}|^{-1/2}\rangle_{\xi^*}},
 </math>
-where <math>\langle ... \rangle_{\xi^*}</math> stands for the statistical average of the quantity enclosed in angular parentheses computed for a constrained ensemble and <math>Z</math> is a mass metric tensor defined as:
-:<math>
-Z_{\alpha,\beta}={\sum}_{i=1}^{3N} m_i^{-1} \nabla_i \xi_\alpha \cdot \nabla_i \xi_\beta, \, \alpha=1,...,r, \, \beta=1,...,r,
-</math>
-It can be shown that the free energy gradient can be computed using the equation:<ref name="Carter89"/><ref name="Otter00"/><ref name="Darve02"/><ref name="Fleurat05"/>
-:<math>
-\Bigl(\frac{\partial A}{\partial \xi_k}\Bigr)_{\xi^*}=\frac{1}{\langle|Z|^{-1/2}\rangle_{\xi^*}}\langle |Z|^{-1/2} [\lambda_k +\frac{k_B T}{2 |Z|} \sum_{j=1}^{r}(Z^{-1})_{kj} \sum_{i=1}^{3N} m_i^{-1}\nabla_i \xi_j \cdot \nabla_i |Z|]\rangle_{\xi^*},
-</math>
-where <math>\lambda_{\xi_k}</math> is the Lagrange multiplier associated with the parameter <math>{\xi_k}</math> used in the [[#SHAKE|SHAKE algorithm]].<ref name="Ryckaert77"/>
-The free-energy difference between states (1) and (2) can be computed by integrating the free-energy gradients over a connecting path:
+In the limit of infinitesimally small <math>\dot{\xi}</math>, the work <math>w^{irrev}_{1 \rightarrow 2}</math>
-:<math>
+corresponds to the free-energy difference between the the final and initial state.
-{\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}}\Bigl( \frac{\partial {A}} {\partial \xi} \Bigr)_{\xi^*} \cdot d{\xi}.
+In the general case, <math>w^{irrev}_{1 \rightarrow 2}</math> is the irreversible work related
+to the free energy via Jarzynski's identity<ref name="jarzynski1997"/>:
+::<math>
+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 as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.
+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 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:
+== References ==
-<ol start="5">
+<references>
-<li>Specify the transformation velocity-related {{TAG|INCREM}}-tag for each geometric parameter with <tt>STATUS=0</tt></li>
+<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>
-</ol>
+<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>
+----
-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>
+[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]