Steepest descent mapping

Identification of inherent structures leads to a natural division of the multidimensional configuration space into nonoverlapping regions, one for each inherent structure, that cover the entire space. The most direct way to accomplish this is to use steepest descent mapping, defined by solutions to the equation set... 

Figure 5-3 The top part of the figure shows the isolines of the misfit functional map and the steepest descent path of the iterative solutions in the space of model parameters. The bottom part presents a magnified element of this map with just one iteration step shown, from iteration (n. — 1) to iteration number ti. According to the line search principle, the direction of the steepest ascent at iteration number n must be perpendicular to the misfit isoline at the minimum point along the previous direction of the steepest descent. Therefore, many steps may be required to reach the global minimum, because every subsequent steepest descent direction is perpendicular to the previous one, similar to the path of experienced slalom skiers.

The actual path mapped out by the MEP on the PES is dependent on coordinate system. However, changes in coordinate system do not alter the nature of the stationary points on the PES (i.e. minima, TSs, etc.). One coordinate system, mass-weighted Cartesian coordinates (see Section 10.2.3), is especially significant for reaction dynamics, and the MEP in this coordinate system is known as the intrinsic reaction coordinate (IRC) [162]. In this section, we use the terms MEP, IRC, steepest descent path, and reaction path synonymously. 

Here, we consider how we can approximately describe the size dependence of the potential energy in Fig. 23.3. As discussed above, molecular coordinates at the imaginary time t = 8/2 are distributed according to the exact ground state wavefunction, I o(/ )p. Each structure at t = /2 along the VPIMD trajectory is mapped onto a nearest local minimum structure in the configuration space. This can be realized by the steepest descent minimization technique ... 

If mass-weighted coordinates were not introduced, a displacement of the point representing the system on the potential energy map would not follow the direction of the negative gradient or the steepest descent (on a geographic map such a motion would look natural, because slow rivers flow this way). Indeed, the formula analogous to Eq. (14.24) would have the form... 

By following the steepest descent path we change the coordinates of all the N atoms of the system. If the atoms were moving infinitely slowly the path would be mapped out by the classical equations of motion. However, the displacement of the atoms should not be mass dependent. This can be avoided by introducing mass-weighted coordinates in a similar manner to what is done in the normal mode treatment of vibrational motion. We therefore introduce the socalled intrinsic reaction path (IRP) [19]. The IRP is defined as the steepest descent path in mass-weighted coordinates. Thus we have the mass weighted cartesian position coordinates and notation, such that... 

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