Procedures for the Equations of Motion

The variety of algorithms currently in use by molecular dynamics groups for the numerical integration of the equations of motion may be taken as evidence that no single procedure has been demonstrated to be superior to any other procedure under all conditions. As is generally the case in large-scale 

Before actually discussing the various algorithms, a few general observations are in order. Suppose that we know the values of the molecular coordinates and velocities (translational and rotational) at time t that is, we have specified the point in phase space. The object of any integration procedure is 

Having made these few remarks, we shall turn to some specific examples. A method especially attractive for its simplicity is due to Verlet. If jc is the x component of a particular molecule s center of mass at time r , and is the X component of the molecule s translational velocity, the Verlet algorithm is obtained by addition and subtraction of the Taylor series for Xn+i = x(t +M) andx -i = x(t -AO  

This last objection is overcome in a method recently used by Schofield  

Neither of the above procedures is suitable for rotational degrees of freedom. We shall illustrate with Eqs. (6). Suppose we replace x by one of the Euler angles a . Then v must be replaced by d . In Eq. (6a), f(T ) is now related to the time derivative of the angular velocity at t . From Eqs. (4), however, we see that this derivative is a function of the angular velocity itself, in addition to the angular coordinates. Because d cannot be determined until a . i is known [see Eq. (6b)], /(F ) cannot be calculated, and the method fails. A similar problem arises in Schofield s method, where /(F +i) in Eq. (7b) cannot be calculated for an angular degree of freedom. 

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