Constrained molecular dynamics: Difference between revisions

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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>
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<div id="Slowgro"></div>
<div id="Slowgro"></div>


== Anderson thermostat ==
=== Constrained molecular dynamics ===
For a description of constrained molecular dynamics see {{TAG|Constrained molecular dynamics}}.


* For a constrained molecular dynamics run with Andersen thermostat, one has to:
* 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 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}}
# Select the thermostat:
#Define geometric constraints in the {{FILE|ICONST}}-file, and set the {{TAG|STATUS}} parameter for the constrained coordinates to 0
## Set {{TAGDEF|MDALGO|1}}, and choose an appropriate setting for {{TAG|ANDERSEN_PROB}}.
## Set {{TAGDEF|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.


 
== References ==
== References ==
   
<references>
[[Category:Advanced molecular-dynamics sampling]][[Category:Theory]]
<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 13:06, 21 October 2025

Constrained molecular dynamics is performed using the SHAKE[1] algorithm. In this algorithm, the Lagrangian for the system [math]\displaystyle{ \mathcal{L} }[/math] is extended as follows:

[math]\displaystyle{ \mathcal{L}^*(\mathbf{q,\dot{q}}) = \mathcal{L}(\mathbf{q,\dot{q}}) + \sum_{i=1}^{r} \lambda_i \sigma_i(q), }[/math]

where the summation is over r geometric constraints, [math]\displaystyle{ \mathcal{L}^* }[/math] is the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:

[math]\displaystyle{ \sigma_i(q) = \xi_i({q})-\xi_i \; }[/math]

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):
    [math]\displaystyle{ v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \frac{a^{t}_i}{m_i} {\Delta}t }[/math]
    [math]\displaystyle{ q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }[/math]
  2. Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
    [math]\displaystyle{ {\lambda}_k= \frac{1}{{\Delta}t^2} \frac{\sigma_k(q^{t+{\Delta}t})}{\sum_{i=1}^N m_i^{-1} \bigtriangledown_i{\sigma}_k(q^{t}) \bigtriangledown_i{\sigma}_k(q^{t+{\Delta}t})} }[/math]
  3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
    [math]\displaystyle{ v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \left( a^{t}_i-\sum_k \frac{{\lambda}_k}{m_i} \bigtriangledown_i{\sigma}_k(q^{t}) \right ) {\Delta}t }[/math]
    [math]\displaystyle{ q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }[/math]
  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.

Constrained molecular dynamics

For a description of constrained molecular dynamics see Constrained molecular dynamics.

  • For a constrained molecular dynamics run with Andersen thermostat, one has to:
  1. Set the standard MD-related tags: IBRION=0, TEBEG, POTIM, and NSW
  2. Select the 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.

References

  1. ↑ J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).
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