Coupled or micro-iterative method

The methods presented so far for big systems do not permit the explicit control on the number of positive eigenvalues of the Hessian. This feature can be avoided for minima but it is crucial when locating first-order saddle points. In addition, even in the case of minima, when energy and gradient evaluations are expensive an efficient method is required. In this case we need a Hessian matrix provided that its size permits an easy manipulation. The first option would be to freeze all the atoms that may not be important for our saddle point location, and then build up a Hessian for a core region not bigger than the size computationally affordable. The Newton-like search is performed only on this small core zone, while the environment is neither permitted to relax nor contribute to the reaction vector. This intuitive approximation has been applied in enzymatic reactions (see reference [158] for an example).

Since systems such as enzymes are rather flexible, a movement of an atom, group or a side chain provokes in turn a coupled movement of the interacting atoms. This means that the above approximation of a frozen environment may not be adequate as a definitive strategy. The logical solution would be to permit the environment atoms to relax during the search in the core. This is the so-called micro-iterative method.

The micro-iterative method is a strategy first used by Maseras and Morokuma [108] in the IMOMM scheme applied to organometallic systems. Few years later, the GRACE package [159,160] permitted the location of TS structures and IRC pathways in enzymatic systems of thousands of atoms. Several groups have applied micro-iterative method to enzymatic systems [81,161,84,95,120,162] and zeolites [65]. This method splits the system in two parts, a core zone where an accurate second order search is done, and an environment that is kept minimized with a cheap first order method. Both processes are carried out until consistency. This separation makes that the sum of the expenses of the two processes is considerably lower than a single global search.

Figure 1.5: Micro-iterative scheme. A quadratic search is performed in the core while a linear minimization keeps the environment relaxed

The computational requirements of a second order search are only needed to optimize the small core zone, while the big part of the system is moved according to a cheaper method.

This is maybe the only strategy that can locate real saddle points in big systems with the direct usage of second derivatives information. Obviously the control and information given by the eigenpair will be only referred to the core zone where the main relevant movements of the reaction are expected.

Since this method is a central part of this thesis, it will be explained in more detail in the third chapter where we develop, implement and test some crucial aspects of the micro-iterative strategy.