Ideal bond length

Fig. 10.3. The BaO and Ti02 layers that compose BaTiOs. The ideal bond lengths and the corresponding cell lengths are shown in pm.

It must be possible to choose parameters for the unit cell and atomic coordinates that reproduce as closely as possible the ideal bond lengths calculated using the network equations (3.3) and (3.4), without bringing any atoms into too close contact. 

Note that R drops precipitously in the first stages of refinement after ALBP side chains replace those of P2. Note also that R and the deviations from ideal bond lengths, bond angles, and planarity of peptide bonds decline smoothly throughout the later stages of refinement. The small increase in R at the end is due to inclusion of weaker reflections in the final round of simulated annealing. 

Another way to demonstrate this effect is shown in three sample calculations using the MOMEC force fieldt52-57 58 96 99-124-140 155) jn Table 3.5 the experimental M-N distances (r(obs)), the calculated bond lengths (r(calc)), the corresponding force constants (k) and ideal bond lengths (r0) are shown for some sample calculations of some hexaamminemetal complexes. 

In the previous three lessons, variations in the parameters for the Co-N bond length were considered. Here we will examine the effects of changing the force constant and the ideal bond-length for the Co-N potential on the isomer distribution. Shown in Table 17.13.2 are the parameters used and the corresponding strain energies calculated for the three isomers. 

Ideal bond length r0 Force constant kb mer- Strain energies [kJ mol ] s-fac, mer-u-fac, s-fac-u-fac) ... 

Ideal bond length r0 Force constant kh Isomer proportions [%] ... 

Although conventional Verlet-type molecular dynamics places restraints on bond lengths and bond angles, one could conceivably want to implement these restrictions as holo-nomic constraints. This is supported by the observation that the deviations from ideal bond lengths and bond angles are usually small in X-ray crystal structures. There are essentially two possible approaches to solve Newton s equations (Eq. 12) with holonomic constraints. The first involves a switch from Cartesian coordinates f) to generalized internal ones ft. Having thus redefined the system, one would solve equations of motion for the... 

As an example of the application of the method we quote values for the refinement of protease A from Streptomyces griseus [97]. The R value for some 12 662 reflections in the resolution range 8.0-1.8 A was 0.139 for some 5912 variable parameters for 1250 protein atoms and 175 water molecules. The final structure differed from ideal bond lengths by an overall root mean square (rms) deviation of 0.02 A and the probable error in atomic co-ordinates was of the order of 0.15 A. 

A complete force field consists of a functional form, as exemplified in Figure 1, and a set of parameters. For example, for each type of bond in the example force field, two parameters are needed an ideal length k (corresponding to the bond length in a hypothetical unstrained molecule), and a stretching force constant fcj. The latter can be seen as the relative stiffness of the bond, and determines how much the energy increases upon a certain distortion. Some parameters, such as the ideal bond length, correspond closely to observables. However, the optimum set of parameters can rarely be identified by observation. 

The simplex optimizations only run for a specified number of cycles, typically lOA to 30N. If the same point is best for 3N cycles, the optimization is also terminated. The method is very robust, and boundary conditions for parameters are easily implemented (for example, ideal bond lengths should always be positive). However, convergence is slow when too many parameters are included. As a rule of thumb, no more than 10 parameters should be included in a standard simplex optimization, but a recently introduced biasing procedure where the inversion point is offset toward the best points can make the method competitive for up to 30-40 parameters (15). 

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