Constrained molecular dynamics: Difference between revisions

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== References ==
== References ==
<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>
<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>
</references>

Revision as of 14:23, 18 April 2022

Constrained molecular dynamics is performed using the SHAKE algorithm.[1]. In this algorithm, the Lagrangian for the system Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}} is extended as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\mathcal {L}}^{*} is the Lagrangian for the extended system, and λi is a Lagrange multiplier associated with a geometric constraint σi:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \sigma _{i}(q)=\xi _{i}({q})-\xi _{i}\;

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):
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): v_{i}^{{t+{\Delta }t/2}}=v_{i}^{{t-{\Delta }t/2}}+{\frac {a_{i}^{{t}}}{m_{i}}}{\Delta }t
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): q_{i}^{{t+{\Delta }t}}=q_{i}^{{t}}+v_{i}^{{t+{\Delta }t/2}}{\Delta }t
  2. Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {\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}})}}
  3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λk):
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): 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
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): q_{i}^{{t+{\Delta }t}}=q_{i}^{{t}}+v_{i}^{{t+{\Delta }t/2}}{\Delta }t
  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.

Anderson thermostat

  • 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. Set MDALGO=1, and choose an appropriate setting for ANDERSEN_PROB.
  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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