Potential energy surfaces steepest descent paths

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. 

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... 

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. 

A reaction-path based method is described to obtain information from ab initio quantum chemistry calculations about the dynamics of energy disposal in exothermic unimolecular reactions important in the initiation of detonation in energetic materials. Such detailed information at the microscopic level may be used directly or as input for molecular dynamics simulations to gain insight relevant for the macroscopic processes. The semiclassical method, whieh uses potential energy surface information in the broad vicinity of the steepest descent reaction path, treats a reaction coordinate classically and the vibrational motions perpendicular to the reaction path quantum mechanically. Solution of the time-dependent Schroedinger equation leads to detailed predictions about the energy disposal in exothermic chemical reactions. The method is described and applied to the unimolecular decomposition of methylene nitramine. 

This paper reviews recent (and current) work in my research group which is aimed at developing practical methods for describing reaction dynamics in polyatomic systems in as ab initio a framework as possible. To overcome the dimensionality dilemma of polyatomic systems—i.e., the fact that the potential energy surface depends on 3N-6 internal coordinates for an N atom system—we have developed dynamical models based on the intrinsic reaction path", i.e., the steepest descent path which connects reactants and products through the transition state (i.e., saddle point) on the potential energy surface. ... 

Fig. 13. The computational methods used for constructing a photochemical reaction path. The full path is computed by joining different MEPs, each one providing information on a specific part of the excited- or ground-state potential-energy surface. The IRD method is used to compute the steepest relaxation directions departing from the FC point (excited-state relaxation) or Cl (ground-state relaxation). The IRC method is used to compute the steepest-descent line defined by the computed IRDs. The CIO method is used to compute the lowest-energy conical intersection point directly. With TSO we indicate the standard transition structure optimization procedure.

Figure 6.1. Potential energy surface within a given region showing the col, or saddle point, and the minimum-energy path, which is that of steepest descent in either direction from the col. The activated complex exists at the col.

Figure 2 Gradient extremal paths on the MUller-Brown mixlel potential energy surface. Solid lines contours dotted lines selected steepest descent lines bold lines gradient extremals. Reproduced with permission from J. Q. Sun and K. Ruedenberg, J. Chem. Phys., 1993, 98, 9707. Copyright (1993) American Institute of Physics...

If this relation is substituted into equations (2) and (3), it can be seen that the steepest descent path through any point on a gradient extremal has zero curvature. For a quadratic expansion of the energy about an arbitrary point on the potential surface, a step of Ax can be taken toward the gradient extremal path. 

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