Partially constrained coordinates

The partially constrained coordinates [r t + 8t, [X ° (to),. . . , (Sm, -i)(f )))) are computed as described for the analytical method. The La-grangian multipliers and their time derivatives up to order — 1 are computed using Eqs. [10] and [11] as in step 1 of the preceding section. The forces 

Having computed the forces of constraint and their time derivatives up to order we can integrate numerically the equations of motion, Eq. [2], for the partially constrained coordinates r (t + 8t,. ..,  

When the panially constrained coordinates r (t + 8f, X (tQ),. .., have been computed, the undetermined parameters 7) can be obtained by inserting Eq. [38] into Eq. [36], giving 

The most general form of holonomic constraint is nonlinear in the particle positions. Even the simple bond-stretch constraint is nonlinear. Consequently, Eq. [39] is in general a system of / coupled nonlinear equations, to be solved for the / unknowns (7). This nonlinear system of equations must be contrasted with the linear system of equations Eqs. [10] and [11] (which is also in general part of the method of undetermined parameters) used in the analytical method to solve for the Lagrangian multipliers and their derivatives. A solution of Eq. [39] can be achieved in two steps  

Taylor expansion of the holonomic constraints The Taylor expansion for a function of N position vectors, about (r), is  

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Equation [40] is used to Taylor-expand every holonomic constraint in the molecular model, in Eq. [39], about the partially constrained coordinates ]r (t + 8tj ]X<° (tg),. . . , ( " (tQ)])]. Standard tests must be used to establish... 

Because of its flexibility, macrocycle (44) (Black McLean, 1968) has the choice of three possible coordination modes (46)-(48) around an octahedral metal ion nevertheless, for a given metal, all isomers will not be of equal energy and hence all three may not be observed experimentally. In contrast, the macrocycle (45) is partially constrained since each set of three donor atoms (S-N-S) is incorporated in a fully conjugated section of the ligand s backbone (Lindoy Busch, 1969). These two conjugated sections will tend to remain planar such that only a configuration corresponding to (46) can occur around an octahedral metal ion if all donor atoms remain coordinated. 

The complex nitrate spectrum can be partially analysed by using oxygen-18 shifts [34], in the manner described in Section 2.2. But there is an additional useful feature. The vibronic structure includes internal modes of the nitrate ions. These modes have nitrogen-15 shifts which identify them unambiguously [35]. The frequencies of these modes in coordination compounds are well-established so the optical frequencies can be used to determine the energy of the pure electronic transition to which they are coupled, while the polarisation of the vibronic spectrum constrains the choice of the electronic symmetry. 

The Kramers result for k T) of Eq. (3.41) has been tested by Wilson and co-workers [14] in their MD simulations of model aqueous nucleophilic substitution reactions. Specifically, these authors determined by MD simulation both the exact Kmd(T) and Kramers Kkr(T) transmission coefficients for 12 Sjv2 systems [14a] and for one system [14b]. The coefficients Kmd(T) were determined from ensembles of reactive and nonreactive MD trajectories. The coefficients kkr(T) were found from Eq. (3.41), with the parameters copmf and being computed via an MD implementation of our partial clamping model [21]. Namely, (Opmf and J (S) are computed via constrained MD simulations in which the reaction coordinate X is held fixed at its transition state value x while the remaining degrees of freedom of the solution are allowed to move freely subject to this single constraint. 

This problem can be partially circumvented if part of the reaction coordinate V (preferably the slowest component ) is known a priori. Applying constrained MD techniques, the system can then be guided over the reaction barrier by performing a series of simulations at different fixed values of V in which all the other degrees of freedom of the system are allowed to relax freely. This procedure has the additional advantage that the corresponding fiee energy of activation can be determined... 

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