Algorithms for Molecular Simulation Using Constraints

Because of their importance in the molecular simulation process, a wide variety of methods for handling holonomic constraints have been proposed in the past few decades. We summarize a few of these here. 

The term SHAKE normally refers to both the discretization mentioned above and the component-wise solution of the nonlinear equations (these methods were suggested together in the paper [322]). Using the traditional SHAKE approach we solve the Eqs. (4.27)-(4.29) using the iterative procedure of Sect. 4.3.3. If RATTLE is used for discretization, the iterative method of Sect. 4.3.3 can again be used to solve (4.30)-(4.32), followed by a subsequent solve of Eqs. (4.33)-(4.34). 

While SHAKE/RATTLE is a robust procedure the sequential, iterative character of nonlinear solvers needed for its implementation can lead inefficiencies when it is implemented on a parallel computer. Therefore, a number of the alternative methods avoid iteration. We briefly summarize a few of the popular alternative schemes here. 

M-Shake This is Newton-iteration-based implementation of SHAKE, using (4.18)-(4.19) to solve Eqs. (4.27)-(4.29). Methods like this were first proposed by Ciccotti and Ryckaert [84] in the context of rigid body molecular dynamics. An extended discussion of such methods with reference to their convergence, implementation, in particular linear system solvers, and variants such as SHAKE-SOR (which uses the successive over-relaxation method) can be found in [25]. A conjugate gradient method can also be used [392]. 

It is also possible to use internal coordinate representations to treat constraints, i.e. parameterization schemes, which are typically based on partitioning the coordinates of the system into a set of independent variables and a complementary collection of dependent variables (which are expressed in terms of the first set). These approaches result in systems of slightly smaller dimension, but typically greater complexity. Moreover, the parameterization is typically local which leads to additional implementation issues. For some details on such schemes and discussion of their use in molecular simulation, the reader is referred to [20, 190, 255]. 

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