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):

   $v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \frac{a^{t}_i}{m_i} {\Delta}t$

   $q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t$

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):

   $v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \left( a^{t}_i-\sum_k \frac{{\lambda}_k}{m_i} \bigtriangledown_i{\sigma}_k(q^{t}) \right ) {\Delta}t$

   $q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t$

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.