Constraint RATTLE

The concept of a symplectic method is easily extended to systems subject to holonomic constraints [22]. For example the RATTLE discretization is found to be a symplectic discretization. Since SHAKE is algebraically equiva lent to RATTLE, it, too, has the long-term stability of a symplectic method. 

Rotation matrices may be viewed as an alternative to particles. This approach is based directly on the orientational Lagrangian (1). Viewing the elements of the rotation matrix as the coordinates of the body, we directly enforce the constraint Q Q = E. Introducing the canonical momenta P in the usual manner, there results a constrained Hamiltonian formulation which is again treatable by SHAKE/RATTLE [25, 27, 20]. For a single rigid body we arrive at equations for the orientation of the form[25, 27]... 

Phase-space displacements obeying the linear constraints of vanishing total linear momentum and total angular momentum, and of fixed bond lengths can be also generated using an iterative procedure such as RATTLE [47]. 

The solution for the 7 and It)] using SHAKE was termed RATTLE by Andersen. The material here is new in the sense that the original formulation of RATTLE was for bond-stretch constraints only. The treatment is general-... 

As with SHAKE, iterations over the constraints continue until all are satisfied, within a certain tolerance. When all constraints have been satisfied and the coordinates at (tQ + bt) are available, the forces + 80) can be computed for use in the second stage of RATTLE. In this second stage, the new velocities of the particles [r" " ( Q + 8f)) obtained in the current iteration are... 

As with SHAKE, RATTLE is illustrated in this chapter for the internal coordinate constraints of bond stretch, angle bend, and torsion. For bond-stretch constraints and the first stage of RATTLE, inserting the constraint a ( rj) of Eq. [95] into Eq. [131] yields... 

Lynch and Pettitt (36) employed a pure Nose thermostat and RATTLE-like algorithm for bond constraint, as well as a single fraction particle. Our approach (40) used quaternions (41) to allow separate Nose-Hoover thermostats for translational and rotational modes. We found that the system s tendency to freeze in metastable states could be overcome by introduction of multiple fractional particles (four were used). Curve (b) of Fig. 3 is an extreme example of runaway insertions at high chemical potential as compared to curve (a) where the actual chemical potential of water resulted in a stable density virtually undistin-guishable from the experimental one. 

A method that automatically preserves both the constraint and hidden constraint is the RATTLE method [12] which consists of the following steps ... 

Kutteh R (1998) RATTLE recipe for general holonomic constraints angle and brtsion constraints. CCP5 Newslett, 46 9... 

In an MD simulation, the eonstraints are treated in succession during one cycle of the iteration and the process is repeated until all constraints have converged to the desired aecuracy. An improvement of the SHAKE algorithm is the RATTLE algorithm, which was proposed by Andersen [19]. In RATTLE, the veloeity-Verlet algorithm is employed to integrate the dynamical equations. 

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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