Constrained molecular dynamics

Constrained molecular dynamics is performed using the SHAKE algorithm.. 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:

1. Perform a standard MD step (leap-frog algorithm):

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

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

2. Use the new positions q(t+Δt) to compute Lagrange multipliers for all constraints:

   ${\lambda }_{k}={\frac {1}{{\Delta }t^{2}}}{\frac {\sigma _{k}(q^{t+{\Delta }t})}{\sum _{i=1}^{N}m_{i}^{-1}\bigtriangledown _{i}{\sigma }_{k}(q^{t})\bigtriangledown _{i}{\sigma }_{k}(q^{t+{\Delta }t})}}$

3. Update the velocities and positions by adding a contribution due to restoring forces (proportional to λ_k):

   $v_{i}^{t+{\Delta }t/2}=v_{i}^{t-{\Delta }t/2}+\left(a_{i}^{t}-\sum _{k}{\frac {{\lambda }_{k}}{m_{i}}}\bigtriangledown _{i}{\sigma }_{k}(q^{t})\right){\Delta }t$

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

4. 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.

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