The results are presented in Table 3.3. From left to right the different columns indicate, respectively, the number of atoms included in the core/environment regions, the number of iterations made in each zone, the final energy of the located transition state structure, the total CPU time of the location, the CPU time devoted to calculate the initial Hessian and its percentage over the total CPU time.
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The first test just includes 3 atoms in the core: the shifting proton and both the proton donor and acceptor oxygen atoms. The successive tests progressively increase the number of atoms around the first three that are incorporated in the core region. The third column shows that the final energy has always the same value, what indicates that, in this case, whatever core / environment partition which includes the three atoms directly involved in the proton transfer leads to the right transition state structure. However, each test behaves in a different way as for the number of iterations and the CPU time. When the core is small a lot of environment iterations are required due to both the huge coupling between the two regions and the great size of the environment region. Conversely, the coupling is low when the core is big (what in addition implies a small environment region), what reduces significantly the number of environment iterations.
It seems that it exists a range of intermediate core sizes which involves a reduced number of core iterations. This probably comes from a compromise between a lower core/environment coupling and the efficiency of the RFO method in handling a progressively increasing number of core atoms as the size of the core grows.
These last results show clearly that when the core/environment partition is selected in an adequate way, the part of the Hessian matrix of the entire system corresponding to the core-core and environment-environment diagonal blocks are numerically much more important than the core-environment non-diagonal ones. A Hessian of this type is the most suitable to be used for any type of optimization[253]. And this fact justifies the efficiency of the micro-iterative method.
As for the CPU time, there is a good correlation between the total number of iterations and the difference between the total CPU time and the time devoted to the calculation of the initial Hessian. So, for instance, the core/environment partition 80/1218 converges with the smallest number of total iterations (653) lasting the most reduced CPU time (5774 s), the Hessian calculation excluded. On the other hand, as expected, the CPU time corresponding to the calculation of the initial Hessian increases monotonically very fast with the core size, requiring the 1% of the total CPU time when just 3 atoms are included in the core, but as much as the 73% for a core with 399 atoms.
Joining all the factors described above, it can be easily understood why it exists an interval of medium core sizes that minimizes the total CPU time, as seen in the fourth column, the optimal partition being 23/1275 (within the discrete, limited series studied here).
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Carbon configuration inversion reaction:
The results of the series of tests corresponding to the carbon configuration inversion step
when a full TS search in the core until convergence is performed before minimizing again the
environment are exhibited in Table 3.4. The first test has 7 atoms in the core, including the
stereogenic carbon atom and several atoms of His297 and Lys166. As for the number of
iterations and the CPU time these results follow the same trends as the proton transfer step
(Table 3.3). The difference lies on the final energies. It can be seen that three distinct sets of
energies are obtained, with a difference of roughly 3 kcal/mol between the test with 7 core
atoms and the tests of 138 or more core atoms.
This indicates that three different progressive approximations to the transition state structure
are located,
a number of 138 or more core atoms (in this discrete, limited
series) being required to converge to the right transition state structure as the core size
increases. (We consider that the right transition state structure of the corresponding potential energy
valley is the one that would be obtained including all the 1298 moving atoms in the core).
This fact highly contrasts with the proton transfer step where a core of three atoms is enough. This is a consequence of the global character of the changes involved in the carbon configuration step (in front of the local character of the proton transfer step), whose TS search requires a core including all the atoms that participate significantly in the reaction coordinate to be carried out successfully. Then, at first glance, it would seem that the best core/environment partition could be 34/1264 which requires 15453 s of total CPU time. However, this partition does not lead to a good enough approximation to the right transition state structure yet. Then, the optimal partition for this step is rather 138/1160 which lasts more total CPU time (17809 s) due to the bigger CPU time required to calculate the initial Hessian, although it involves clearly fewer iterations and spends less time in the location of the transition state structure once the initial Hessian has been calculated. Indeed this partition leads to the right TS. That is, as the different atoms which define the reaction coordinate are incorporated in the core, the description of the TS of the corresponding potential energy valley is progressively improved, until a good enough approximation to the right TS is reached.
Finally, it has to be mentioned that the high total CPU time of the test corresponding to the partition 643/655 is due not only to the CPU time required to calculate the initial Hessian, but also to the time employed to partially diagonalize this big matrix Hessian at each iteration4.6.
