Constrained molecular dynamics: Difference between revisions

From VASP Wiki
No edit summary
 
(14 intermediate revisions by 2 users not shown)
Line 1: Line 1:
In general, constrained molecular dynamics generates biased statistical averages.
It can be shown that the correct average for a quantity <math>a(\xi)</math> can be obtained using the formula:
:<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:
:<math>
{\Delta}A_{1 \rightarrow 2} = \int_{{\xi(1)}}^{{\xi(2)}}\Bigl( \frac{\partial {A}} {\partial \xi} \Bigr)_{\xi^*} \cdot d{\xi}.
</math>
Note that as the free-energy is a state quantity, the choice of path connecting (1) with (2) is irrelevant.
<div id="SHAKE"></div>
<div id="SHAKE"></div>
Constrained molecular dynamics is performed using the SHAKE algorithm.<ref name="Ryckaert77"/>.
Constrained molecular dynamics is performed using the SHAKE{{cite|ryckaertt:jcp:1977}} algorithm.
In this algorithm, the Lagrangian for the system <math>\mathcal{L}</math> is extended as follows:
In this algorithm, the Lagrangian for the system <math>\mathcal{L}</math> is extended as follows:
:<math>
:<math>
Line 58: Line 35:
<div id="Slowgro"></div>
<div id="Slowgro"></div>


== Anderson thermostat ==
== References ==
 
   
* For a constrained molecular dynamics run with Andersen thermostat, one has to:
[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]
#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 STATUS parameter for the constrained coordinates to 0.
#When the free-energy gradient is to be computed, set {{TAG|LBLUEOUT}}=.TRUE.
 
== References ==
<references>
<ref name="Ryckaert77">[http://dx.doi.org/10.1016/0021-9991(77)90098-5 J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).]</ref>
<ref name="Carter89">[http://dx.doi.org/10.1016/S0009-2614(89)87314-2 E. A. Carter, G. Ciccotti, J. T. Hynes, and R. Kapral, Chem. Phys. Lett. 156, 472 (1989).]</ref>
<ref name="Otter00">[http://dx.doi.org/10.1080/00268970009483348 W. K. Den Otter and W. J. Briels, Mol. Phys. 98, 773 (2000).]</ref>
<ref name="Darve02">[http://dx.doi.org/10.1080/08927020211975 E. Darve, M. A. Wilson, and A. Pohorille, Mol. Simul. 28, 113 (2002).]</ref>
<ref name="Fleurat05">[http://dx.doi.org/10.1063/1.1948367 P. Fleurat-Lessard and T. Ziegler, J. Chem. Phys. 123, 084101 (2005).]</ref>
<ref name="Ryckaert77">[http://dx.doi.org/10.1016/0021-9991(77)90098-5 J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).]</ref>
</references>
----
 
[[Category:Molecular dynamics]][[Category:Constrained molecular dynamics]][[Category:Theory]][[Category:Howto]]

Latest revision as of 09:59, 15 October 2024

Constrained molecular dynamics is performed using the SHAKE[1] algorithm. In this algorithm, the Lagrangian for the system is extended as follows:

where the summation is over r geometric constraints, is the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:

with ξi(q) being a geometric parameter and ξi is the value of ξi(q) fixed during the simulation.

In the SHAKE algorithm, the Lagrange multipliers λi are determined in the iterative procedure:

  1. Perform a standard MD step (leap-frog algorithm):
  2. Use the new positions q(tt) to compute Lagrange multipliers for all constraints:
  3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
  4. repeat steps 2-4 until either |σi(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.

References