Constraints

The inclusion of constraints to the fastest movements that are not of great interest in themselves (e.g. bond vibration) permit increasing the time step of the MD simulation. As a consequence of a larger time step the simulation becomes computationally cheaper.

The most commonly used method for applying holonomic constraints is the SHAKE procedure [178]. The procedure is based on the determination of Lagrange multipliers ($l_k$) imposed as a constriction to the equations of motion. For $n$ constraints with $d_k$ as the corresponding constrain distance we have

$$m_i\frac{d^2q_i(t)}{dt^2}= -\frac{\delta}{\delta q_i}[V({\bf q})+\sum_k^n l_k(t) (q^2_k-d^2_k)]$$ (2.110)

Here we will only mention that for solving the above equations in SHAKE procedure the Lagrange multipliers are determined iteratively and therefore they depend on a threshold. In small systems the procedure can be carried out by a matrix inversion [179].

While SHAKE works in Cartesian coordinates Tobias and Brooks generalized this to an arbitrary internal coordinate[180].

In addition, constraints in Molecular Dynamics simulations can be applied to other interesting areas. It may be used, for example in Potential of Mean Force calculations or in Ligand Binding techniques [181,182].