Path of steepest descent

Vector quantities, such as a magnetic field or the gradient of electron density, can be plotted as a series of arrows. Another technique is to create an animation showing how the path is followed by a hypothetical test particle. A third technique is to show flow lines, which are the path of steepest descent starting from one point. The flow lines from the bond critical points are used to partition regions of the molecule in the AIM population analysis scheme. 

The MEP is defined as the path of steepest descent in mass-weighted Cartesian coordinates. This is also called intrinsic reaction coordinate (IRC). In reality, we know that many other paths close to the IRC path would also lead to a reaction and the percentage of the time each path is taken could be described by the Boltzmann distribution. 

Given the definition of the geometry of the transition states in TST as the highest energy point in the minimum energy pathway from reactants to products, the formal definition of MEP is as follows. The MEP is, in one direction, the path of steepest descents from the transition state to reactants while, in the other direction, it is the path of steepest descents from transition state to products. For reasons which will not be discussed here, the formal definition of MEP includes the statement that the pathway is expressed in mass scaled Cartesian coordinates of the position of the atoms (introduced in Chapter 3, e.g. x is replaced by x = ). This simplifies... 

For JT problems of higher dimensions such as that for the T (e t2) problem, the adiabatic potential V is complicated and cannot be written down in an analytical form. However, in such problems, the least action path can be approximated by the minimum energy path (or path of steepest descent) on the adiabatic potential surface. It is the path for which the tangent to it is parallel to the gradient of the APES. 

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... 

These conditions cause C to be the path of steepest descent from the saddle point. To achieve this result we need to establish a relation between the real and imaginary parts of / (2). 

In our most straightforward implementation of VTST for gas-phase reactions, rather than allow arbitrary orientations of the dividing surface, we consider a one-parameter sequence of dividing surfaces that are defined in terms of a reaction path [12,13]. This procedure is applicable to complex problems, and it immediately provides a practical improvement over the conventional choice of placing the dividing surface at the saddle point. A robust choice for the reaction path is the minimum energy path (MEP), that is, the path of steepest descent in the mass-scaled coordinates [14]. The coordinates on this path are denoted q (j ) as a function of a progress variable s, and the path is defined by... 

This is an equation for the path of steepest decent. There are two paths of steepest descent from the saddle point, one toward reactants and the other toward the products. The combined path from reactants to products is called the intrinsic reaction path (IRP) [33-39]. Friction has eliminated the oscillatory motion from this path, so that it resembles the dashed path shown in Fig. 3. 

Figure 3. Another expanded view of the PES of Fig. 1, sketching the projection of a classical trajectory onto the PES (solid line) and a similar projection of the path of steepest descent (dashed line).

Equation (3.7) is an equation for the path of steepest descent in the 3N mass-weighted Cartesian coordinates. In fact, we can write similar equations for the path of steepest descent in any set of coordinates, say some complete set of (3A - 6)-internal coordinates, such as interatomic distances, R ... 

The dashed curve indicates a path of steepest descent. 

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