Constrained molecular dynamics: Difference between revisions

Latest revision as of 06:48, 19 July 2022

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

{\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:

\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):
$v_{i}^{t+{\Delta }t/2}=v_{i}^{t-{\Delta }t/2}+{\frac {a_{i}^{t}}{m_{i}}}{\Delta }t$

$q_{i}^{t+{\Delta }t}=q_{i}^{t}+v_{i}^{t+{\Delta }t/2}{\Delta }t$
Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
${\lambda }_{k}={\frac {1}{{\Delta }t^{2}}}{\frac {\sigma _{k}(q^{t+{\Delta }t})}{\sum _{i=1}^{N}m_{i}^{-1}\bigtriangledown _{i}{\sigma }_{k}(q^{t})\bigtriangledown _{i}{\sigma }_{k}(q^{t+{\Delta }t})}}$
Update the velocities and positions by adding a contribution due to restoring forces (proportional to λ_k):
$v_{i}^{t+{\Delta }t/2}=v_{i}^{t-{\Delta }t/2}+\left(a_{i}^{t}-\sum _{k}{\frac {{\lambda }_{k}}{m_{i}}}\bigtriangledown _{i}{\sigma }_{k}(q^{t})\right){\Delta }t$

$q_{i}^{t+{\Delta }t}=q_{i}^{t}+v_{i}^{t+{\Delta }t/2}{\Delta }t$
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.

How to

Geometric constraints are introduced by defining one or more entries with the STATUS parameter set to 0 in the ICONST-file. Constraints can be used within a standard NVT or NpT MD setting introduced by MDALGO=1|2|3. Note that fixing geometric parameters related to lattice vectors is not allowed within an NVT simulation (VASP would terminate with an error message). Constraints can be combined with restraints, time-dependent bias potentials (Metadynamics), monitored coordinates and other elements available within the context of MD.

References

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

[Ryckaert77-1] 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"/>.
@@ Line 58: / Line 35: @@
 <div id="Slowgro"></div>
-== Anderson thermostat ==
+== How to ==
+Geometric constraints are introduced by defining one or more entries with the STATUS parameter set to 0 in the {{FILE|ICONST}}-file. Constraints can be used within a standard NVT or NpT MD setting introduced by {{TAG|MDALGO}}=1|2|3. Note that fixing geometric parameters related to lattice vectors is not allowed within an NVT simulation (VASP would terminate with an error message). Constraints can be combined with restraints, time-dependent bias potentials ([[:Category:Metadynamics|Metadynamics]]), monitored coordinates and other elements available within the context of MD.
-* 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 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]]
+[[Category:Molecular dynamics]][[Category:Constrained molecular dynamics]][[Category:Theory]][[Category:Howto]]