Constrained molecular dynamics

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

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

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

[math]\displaystyle{ \sigma_i(q) = \xi_i({q})-\xi_i \; }[/math]

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):

   [math]\displaystyle{ v^{t+{\Delta}t/2}_i = v^{t-{\Delta}t/2}_i + \frac{a^{t}_i}{m_i} {\Delta}t }[/math]

   [math]\displaystyle{ q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }[/math]

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

   [math]\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})} }[/math]

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

   [math]\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 }[/math]

   [math]\displaystyle{ q^{t+{\Delta}t}_i = q^{t}_i + v^{t+{\Delta}t/2}_i{\Delta}t }[/math]

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.