Constrained and multiple time step dynamics

In complex molecular systems, there are motions with disparate characteristic time scales fast degrees of freedom, such as bond and angle vibrations, and slow degrees of freedom, such as molecular translation and rotation. The integration time step size is determined by fast degrees 

If the characteristic time scales of the important motions are spread over a wide spectrum, the system is called stiff. Because of the requirement of small time steps, long simulation times are computationally hard to reach. 

Over the last three decades, scientists have proposed a number of ingenious algorithmic solutions for stiff molecular systems. Two are worth mentioning here (albeit briefly, since presentation of the details is beyond the purview of this book)  

The bond length vibrations are fast and on occasion unimportant for calculation of accurate structural and thermodynamic properties. They can then be constrained, fixing the atomic distances at the equilibrium value. Bond angle vibrations can also be constrained, using pseudo-bonds to keep triplets of atoms at correct angles. 

In algorithms, each constraint is translated in a force acting along the bond or pseudo-bond. These forces are taken into account in the equations of motion with additional terms. Perhaps the two most widely used algorithmic implementations of constraints are the SHAKE (developed by Ryckaert, Ciccotti, and Berendsen in 1977) and RATTLE (developed by Andersen in 1983) algorithms (see Eurther reading). 

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