Dynamic constraint

The earliest molecular dynamics simulations using realistic potentials were of atoms interacting under the Lennard-Jones potential. In such calculations the only forces on the atoms are those due to non-bonded interactions. It is rather more difficult to simulate molecules because the interaction between two non-spherical molecules depends upon their relative orientation as well as the distance between them If the molecules are flexible then there will also be intramolecular interactions, which give rise to changes in conformation. Clearly, the simplest model is to treat the species present as rigid bodies with no intramolecular conformational freedom. In such cases the dynamics of each molecule can often be considered in terms of translations of its centre of mass and rotations about its centre of mass. The force on the molecule equals the vector sum of all the forces acting at the 

The constraint force can be introduced into Newton s equations as a Lagrange multiplier (see Section 1.10.5). To achieve consistency with the usual Lagrangian notation, we write F(,j, as —A and so Fcx equals Am. Thus  

Equations (7.50) and (7.51) contain three unknowns tfx/d, cfy/d and A). A third equation that links x and y is the equation of the slope, which can be written in the following form  

This constraint equation is expressed in terms of x and y rather than their second derivatives. However, as cr(x, y) = 0 holds for all x, y, it follows that da = 0 and d a = 0 also. Consequently, the constraint equation can be written  

In the general case, the equations of motion for a constrained system involve two types of force the normal forces arising from the intra- and intermolecular interactions, and the forces due to the constraints. We are particularly interested in the case where the constraint requires the bond between atoms i and j to remain fixed. The constraint influences the Cartesian coordinates of atoms i and j. The force due to this constraint can be written as follows  

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The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

Algorithms for macromolecular dynamics and constraint dynamics. Mol Phys 34,1311-1327. 

These six equations are insufficient to give a closure of the EMMS model that involves eight variables. The closure is provided by the most unique part of the EMMS model, that is, the introduction of stability condition to constraint dynamics equations. It is expressed mathematically as Nst = min, which expresses the compromise between the tendency of the fluid to choose an upward path through the particle suspension with least resistance, characterized by Wst = min, and the tendency of the particle to maintain least gravitational potential, characterized by g = min (Li and Kwauk, 1994). 

Constraint dynamics is just what it appears to be the equations of motion of the molecules are altered so that their motions are constrained to follow trajectories modified to mclude a constraint or constraints such as constant (total) kinetic energy or constant pressure, where the pressure in a dense adsorbed phase is given by the virial theorem. In statistical mechanics where large numbers of particles are involved, constraints are added by using the method of undetermined multipliers. (This approach to constrained dynamics was presented many years ago for mechanical systems by Gauss.) Suppose one has a constraint g(R, V)=0 that depends upon all the coordinates R=rj,r2...rN and velocities V=Vi,V2,...vn of all N particles in the system. By differentiation with respect to time, this constraint can be rewritten as l dV/dt -i- s = 0 where I and s are functions of R and V only. Gauss principle states that the constrained equations of motion can be written as ... 

This simple example could, of course, have been solved by simply substituting the constraint equation into the original function, to give a function of just one of the variables. However, in many cases this is not possible. The Lagrange multiplier method provides a powerful approach which is widely applicable to problems involving constraints such as in constraint dynamics (Section 7.5) and in quantum mechanics. 

The requirement that the time step is approximately one order of magnitude smaller than the shortest motion is clearly a severe restriction, particularly as these high-frequency motions are usually of relatively little interest and have a mmimal effect on the overall behaviour of the system. One solution to this problem is to freeze out such vibrations by constraining the appropriate bonds to their equilibrium values while still permitting the rest of the degrees of freedom to vary under the intramolecular and inter molecular forces present. This enables a longer time step to be used. We will consider such constraint dynamics methods in Section 7.5. 

The physical effects of introducing constraints into a molecular model have been discussed by several authors. > This chapter is concerned mainly with the methods of constraint dynamics. In addition to descriptions of bond-stretch and angle-bend constraints, dihedral (or torsional) constraints are explicitly considered. Torsional modes generally have frequencies comparable to those of other modes, and the weak coupling condition is not satisfied in this case. Hence the constraint approximation is not justified for torsion. This fact is particularly important because torsional motions play a major role in conformation interconversion in small molecules as well as polymers, and constraining them can seriously alter the dynamics of the original, unconstrained system. 

The second approach, which uses the Lagrange multiplier technique, consists of retaining the set of constrained coordinates and making use instead of Lagrange s equations of the second kind, which involve the forces of constraints. The Lagrange equations of the second kind together with the equations of constraints are used to solve for both the coordinates and the forces of constraints. Use of this approach with Cartesian coordinates has come to be known as constraint dynamics. This chapter is concerned with the various methods of constraint dynamics. 

There is a regrettable lack in the literature of exposition of the fundamental theory behind constraint dynamics. In this chapter we develop the basic theory for general forms of holonomic constraints and arbitrary integration algorithms. The benefits of such a general exposition are threefold. 

Accordingly, in discussing the various algorithms of constraint dynamics, it is instructive to classify them and relate them to one another, as described next. 

The basic constraint dynamics method employed There are two distinct methods of constraint dynamics the analytical method and the method of undetermined parameters. The reason for the names will become apparent. 

Although this chapter is written as a review of the methods of constraint dynamics, a substantial part of the material is new. In the next section, the analytical method is described in detail in its most general form. The gradual divergence of the constraints and the need for a constraint correction scheme are discussed. Finally, the method of Edberg et al. is discussed in the context of the analytical method, as a special case with = 0 and holonomic bondstretching constraints, together with a constraint correction scheme. 

Finally, further refinements and developments, such as the SETTLE algorithm and the special treatment of planar molecules in constraint dynamics, are briefly covered. A number of areas of possible progress are also mentioned. 

Although this chapter is concerned primarily with holonomic constraints, we comment on the role of nonholonomic constraints in the present context. Because the analytical dynamics theory of a system of particles subject to holonomic constraints is well established, the issues in (holonomic) constraint dynamics are mainly algorithmic in nature, as seen in this chapter. The situation is more complex for molecular dynamics with nonholonomic constraints, where theoretical difficulties exist. " ... 

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