Steepest descent paths

Figure B3.5.1. Contour line representation of a quadratic surface and part of a steepest descent path zigzagging toward the minimum.

As shown by Valtazanos and Ruedenberg [93], steepest descent paths (e.g., the Fiikui intrinsic reaction coordinate)... 

R. Olender and R. Elber, Yet another look at the steepest descent path , J. Mol. Struct. Theochem and the proceeding of the WATOC symposium, 398-399, 63-72 (1997)... 

As explained above, the QM/MM-FE method requires the calculation of the MEP. The MEP for a potential energy surface is the steepest descent path that connects a first order saddle point (transition state) with two minima (reactant and product). Several methods have been recently adapted by our lab to calculate MEPs in enzymes. These methods include coordinate driving (CD) [13,19], nudged elastic band (NEB) [20-25], a second order parallel path optimizer method [25, 26], a procedure that combines these last two methods in order to improve computational efficiency [27],... 

Fig. 16 (a) R (D + RX) and P (D,+ + R + X ) zero-order potential energy surfaces. Rc and Pc are the caged systems, (b) Projection of the steepest descent paths on the X-Y plane J, transition state of the photoinduced reaction j, transition state of the ground state reaction W, point where the photoinduced reaction path crosses the intersection between the R and P zero-order surfaces R ., caged reactant system, (c) Oscillatory descent from W to J on the upper first-order potential energy surface obtained from the R and P zero-order surfaces. 

The projection on the Xi Yi plane of the steepest descent path followed by the system during the photoinduced reaction (calculated as in Section 3) is shown in Fig. 16b. Past the saddle point corresponding to the transition state of the photoinduced reaction, f, the system follows the steepest descent path on the P surface en route to the caged product state, Xx = I, F, = 1, until it reaches... 

Fig. 20 Reaction pathways in the reduction of methyl (a) and /-butyl chloride (b) by NO". , reactant and products , transition states. In (a) and (b), the full line is the mass-weighted IRC path from the reactant to the product states the dashed line is a ridge separating the Sn2 and ET valleys and the dotted-dashed line is the mass-weighted IRC path from the Sn2 product state to the ET product state (homolytic dissociation). The dotted line in (a) represents the col separating the reactant and the SN2 product valleys. The dotted line in (b) represents the steepest descent path from the bifurcation point, B, to the Sn2 product. In (a), B is the point of the col separating the reactant and the SN2 product valleys where the ridge separating the SN2 and ET valleys starts.

The geometrical properties of the PES in the vicinity of a transition state mean that the steepest descent path down from the transition state (also generally calculated in mass-weighted coordinates, and called the intrinsic reaction coordinate or IRC) will usually lead only to a single reactant in one direction and a single product (or intermediate) in the other. However, a transition state can sometimes be shared by more than one reactant and/or product. One of these cases arises when the PES possesses a so-called valley-ridge inflection point (VRI). °... 

The Rearrangement of 1,2,6-Heptatriene. The experimental facts and basic mechanistic idea behind this reaction were outlined in section 1.2. The molecular dynamics study began with a single CASSCF(8,8)/6-31G(d) trajectory started from TSl (see Fig. 21.2) with no kinetic energy (not even ZPE) in any of the real-frequency normal modes. The purpose of such an unphysical trajectory calculation is to see what is the steepest descent path down from the transition state... 

The methods of constructing different reaction paths are described in numerous papers and reviews (see, for example, Truhlar and Garrett [1984, 1987], Garrett et al. [1988], Ischtwan and Collins [1988], and references therein). In the IRC method proposed by Fukui [1970], the steepest descent path from the saddle point of a multidimensional PES V(X) to the reactant and product valleys is found by numerically solving the equation... 

The choice of reaction path definition used as the reference for such a constrained dynamics is arbitrary any path may be used in practice. However, a natural choice in order to ensure that the simulation moves along the bottom of the potential energy valley connecting reactants/products with TS is the intrinsic reaction path (IRP) of Fukui.46,47 IRP by definition goes along the bottom of such a valley. IRP simply corresponds to a steepest descent path in a mass-weighted coordinates ... 

Starting at a saddle point, a path of steepest descent can be defined on the potential energy surface by using the gradient function 8W/8Qj the path of steepest descent is uniquely determined by extremal values of the gradient unless a stationary point is reached (55). Besides the minima corresponding to the reactant and product asymptotes, a potential energy surface may exhibit some additional minima due to, e.g., van der Waals (59) complexes or intermediates (see later). In such cases, the reactant and product asymptote can be interconnected by several steepest descent paths and the construction... 

There has been much debate in recent years concerning the invariance properties of the minimum energy path the steepest descent path (65-70a)... 

Carpenter and Borden made two important conclusions that raise significant concerns about the traditional physical organic notions of reaction mechanisms. First, nonstatistical dynamics can occur even when intermediates exist in relatively deep potential energy wells, not just on flat caldera-like surfaces. Second, multiple products can be formed from crossing a single TS. The steepest descent path from a TS can only link to a single product, but reactions can follow nonsteepest descent paths that reach different products. 

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.

This is an identical equation for the minimum energy path [9,10] or the so-called steepest descent path. The first implementation of an algorithm to compute minimum energy paths based on the above formula was the LUP (Locally Updated Planes) method [9] that did not include the constraint on the displacement size. This is formally correct since different parameterizations of the path are possible, but may lead to numerical problems in which the distances between the intermediates grow without control. This was adjusted to produce more stable algorithms by the Nudge Elastic Band approach [11] and later by the String method [10]. 

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