Forces of constraint

To determine the free-energy profile (rather then the potential energy profile as above), one performs a (typically short) constant temperature (canonical ensemble) FPMD simulation of the system (while constraining the reaction coordinate to the prescribed value) and records the mean force of constraint along the chosen reaction coordinate. This process is then repeated for a number of points along the reaction coordinate connecting the reactant and... 

If forces of constraint are present, as in the case of the monomer-dimer study where the distance between the dimer subunits is fixed, displacement correction vectors must be added to Eq. (5) in... 

The definition of the atomic contribution to the virial of the external forces of constraint is nontrivial. Keith has developed the procedure for the atomic partitioning of null properties properties such as the sum of the Feynman forces on the nuclei, that sum to zero for the entire molecule [51]. Thus in analogy with the expression for a total system, the energy of atom A is given by... 

The first approach of Lagrangian dynamics consists of transforming to a set of independent generalized coordinates and making use of Lagrange s equations of the first kind, which do not involve the forces of constraint. The equations of constraint are implicit in the transformation to independent gen-... 

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. 

The integration of the equations of motion In this step, the forces of constraint and their time derivatives up to order s nax obtained from step 1, are used as input to the selected integration scheme, to generate the constrained coordinates. Note that the particular choice of numerical integration algorithm in step 2 determines the parameter of step 1. 

Having evaluated the forces of constraint and their derivatives up to or-der Sj gjj in step 1, we can integrate numerically the constrained equations of motion Eq. [2j ... 

With the Lagrange multipliers (kltg) available, the forces of constraint can now be computed using... 

The (7), obtained by solving Eq. [39], are substituted into Eq. [37] to provide the displacements necessary to satisfy the constraints. Subsequently, the constrained position vectors r(tQ -I- 5f) are obtained from Eq. [38] by adding these constraint corrections to the partially constrained position vectors. The actual derivatives of order of the forces of constraint must be computed (a priori) in the analytical method, whereas in the method of undetermined parameters, the approximate derivatives of order of the forces of constraint can be computed (a posteriori) if desired, by replacing the X (fo)) by the (7). 

Inserting Eq. [49] into Eq. [46] shows that the coordinates given by the method of undetermined parameters are accurate to 0(ht ). The error in the coordinates (local error) is of the 0(bt ) present in the basic Verlet scheme. Therefore, consistent with the preceding error analysis, no additional error is introduced in the method of undetermined parameters, and the constraints are exactly satisfied at every time step. If the approximate forces of constraints are desired, they can be computed a posteriori as... 

Inserting Eq. [49] into Eq. [50] shows that the forces of constraint are accurate to the same order as the integration algorithm adopted. Hence, with the Verlet algorithm the computation of the approximate forces of constraint at t reduces to the computation of the (y). Equivalently, by inserting Eq. [48] into Eq. [47], one obtains... 

Comparing Eq. [51] with Eq. [52] leads to the identical expression, Eq. [50], for the approximate forces of constraints. Note the factor of in front of the last term in Eq. [52]. A common error in the literature is to extract the constraint forces Gjitg) from the basic Verier expression for the unconstrained coordinates, Eq. [48], and to write the constrained coordinates as follows ... 

Not surprisingly, comparing Eq. [53] with Eq. [51] leads to the forces of constraint in Eq. [50], but with a factor of V2 discrepancy. This discrepancy is due to unwarranted attempts to apply Eq. [48], which should be used only in the computation of the unconstrained coordinates [r (t(j + 8t)J, to the constrained coordinates [r(rQ + 8, (7))). Unfortunately, a factor of Va is often artificially introduced into the equations to mask this inconsistency. For convenience and conformity with the most widely adopted convention, the rest of this chapter redefines the undetermined parameters 7) such that their new values are equal to half their previously defined values. With this new definition, Eq. [46] takes the form... 

Solution by Relaxation. The method which will next be described was first applied extensively by SouthwelB in certain engineering problems dealing with static equilibrium. The displacements of a structure subjected to known loads are determined by assuming an arbitrary displacement, computing the forces of constraint necessary to maintain the structure in its arbitrary displacement, and then varying the assumed displacement so as to relax all the forces of constraint to negligible values. 

The method is applicable to any set of simultaneous equations in the present example, the At are assigned arbitrary initial values, and X is also assigned an initial trial value, Xf The quantities analogous to the forces of constraint are the e s defined by... 

For many situations of interest elementary Newtonian mechanics is not directly applicable, since the system might be subject to a set of constraints. These constraints and the corresponding a priori unknown forces of constraint hamper both the setting up and the solution of the Newtonian equations of motion. For example, consider a planar pendulum of mass m with fixed length I oscillating in the two-dimensional xy-plane as sketched in Figure 2.2. Experimentally the fixed length of the pendulum may be realized by an iron rod... 

Figure 2.2 Planar pendulum of mass m and fixed length I In the xi/-plane. The motion of the mass is restricted to the dashed one-dimensional circle by action of an a priori unknown force of constraint Fc, and therefore completely described by a single generalized coordinate i9.

Equating the average force to the force of constraint is not precisely correct [36]. However, the assumptions involved in this approximation hold fairly well for the case of a small solute penetrating a membrane. The main advantage of this approach is that it allows simultaneous calculation of the average force and local diffusion coefficient. In combination, these two quantities provide an estimate of the membrane permeability to the solute of interest. 

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