Constrained molecular dynamics: Difference between revisions

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 $\mathcal{L}$ is extended as follows:

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

where the summation is over r geometric constraints, $\mathcal{L}^*$ is the Lagrangian for the extended system, and λ_i is a Lagrange multiplier associated with a geometric constraint σ_i:

{\displaystyle \sigma_i(q) = \xi_i({q})-\xi_i \; }

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

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

Perform a standard MD step (leap-frog algorithm):
${\displaystyle v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \frac{a^{t}_i}{m_i} {\Delta}t }$

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

${\displaystyle q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }$
repeat steps 2-4 until either |σ_i(q)| are smaller than a predefined tolerance (determined by SHAKETOL), or the number of iterations exceeds SHAKEMAXITER.

References

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

[ryckaertt:jcp:1977-1] J. P. Ryckaert, G. Ciccotti, and H. J. C. Berendsen, J. Comp. Phys. 23, 327 (1977).

[1]

@@ 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>
-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:
 :<math>
@@ Line 58: / Line 35: @@
 <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 {{TAG|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="Andersen80">[http://dx.doi.org/10.1063/1.439486 H. C. Andersen, J. Chem. Phys. 72, 2384 (1980).]</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>
-<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="Allen91">M. P. Allen and D. J. Tildesley, ''Computer simulation of liquids'', Oxford university press: New York, 1991.</ref>
-<ref name="Parrinello80">[http://dx.doi.org/10.1103/PhysRevLett.45.1196 M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980).]</ref>
-<ref name="Parrinello81">[http://dx.doi.org/10.1063/1.328693 M. Parrinello and A. Rahman, J. Appl. Phys. 52, 7182 (1981).]</ref>
-<ref name="Toton10">[http://dx.doi.org/10.1088/0953-8984/22/7/074205 D. Toton, C. D. Lorenz, N. Rompotis, N. Martsinovich, and L. Kantorovich, J. Phys.: Condens. Matter 22, 074205 (2010).]</ref>
-<ref name="Kantorovich08">[http://dx.doi.org/10.1103/PhysRevB.78.094305 L. Kantorovich and N. Rompotis, Phys. Rev. B 78, 094305 (2008).]</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]]