Steepest Descent Saddle Point Method

Applying the method of steepest descent (or saddle-point method) to Eq. (3.49) yields... 

In this subsection, the rate constants of IC and ISC are introduced and their structures are detailed. In particular, applying the harmonic potential model to these rate constants provides with practical and applicable theoretical formulas. A numerical evaluation method based on steepest-descent (or saddle-point method) is then demonstrated for the computation. 

It should be noted that the expressions for IC and ISC cases Eqs. (71) and (72) are quite similar except for the electronic matrix elements and energy gaps. Although the Fourier integral involved in Wl fb given above can easily be carried out numerically, analytical expressions are often desired for this purpose, the method of steepest-descent [45-51] (saddle-point method) is commonly used. Take Eq. (73) as an example. Wl b will first be written as... 

As a consequence of proposition 3 we obtain In the vicinity of a minimizer of E the Newton vectors and the steepest descent vectors always point to the minimizer. In the vicinity of a saddle point the Newton vectors always point to the saddle point whereas a steepest descent vector point to the saddle point only if E is convex along that vector. This observation forms the basis for a modified Newton-like method which looks for stationary points of prescribed type (see Sect.2.4.3). 

Steadiness of vacuum and one-particle states, 657 Steck B.,5Z8 Steck operator, 538 Steepest descents, method of, 62 Stochastic processes, 102,269 Strangeness quantum number, 516 Strategic saddle point, 309 Strategy, 308 mixed, 309... 

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],... 

Steepest descent can terminate at any type of stationary point, that is, at any point where the elements of the gradient of /(x) are zero. Thus you must ascertain if the presumed minimum is indeed a local minimum (i.e., a solution) or a saddle point. If it is a saddle point, it is necessary to employ a nongradient method to move away from the point, after which the minimization may continue as before. The stationary point may be tested by examining the Hessian matrix of the objective function as described in Chapter 4. If the Hessian matrix is not positive-definite, the stationary point is a saddle point. Perturbation from the stationary point followed by optimization should lead to a local minimum x. ... 

Evaluating Eq. (51) by the method of steepest descents for large R, 8, n, we find that the saddle point n is given by... 

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... 

For arbitrary potentials, given the low frequencies and high intensities employed in current experiments, for the numerical evaluation of the amplitude (4.1) in the form (4.4) the method of steepest descent [also known as the saddle-point approximation (SPA)] is the method of choice. Thus, we must determine the values of fc, //, and t for which the action Sp(t,t, k) is stationary, so that its partial derivatives with respect to these variables vanish. This condition gives the equations... 

Substitution of this into Eq. (3.118) gives convergence at y = — 1. Application of the method of steepest descents at the saddle point y = h gives the semi-classical result... 

The most common assumption is one of a reaction path in hyperspace (Miller et al. 1980). A saddle point on the PES is found and the steepest descent path (in mass-weighted coordinates) from this saddle point to reactants and products is defined as the reaction path. The information needed, except for the path and the energies along it, is the local quadratic PES for motion perpendicular to the path. The reaction-path Hamiltonian is only a weakly local method since it can be viewed as an approximation to the full PES and since it is possible to use any of the previously defined global-dynamical methods with this potential. However, it is local because the approximate PES restricts motion to lie around the reaction path. The utility of a reaction-path formalism involves convenient approximations to the dynamics which can be made with the formalism as a starting point. 

This expression can be evaluated by using the steepest descent method. Let cx be the value of x at the saddle point we obtain... 

As we are interested only in large values of r/ , we can evaluate the integral directly by using the steepest descent method. At the saddle point, r = r/2 . We thus obtain... 

The integral is to be calculated over a path in the complex plane, cut along the negative half-axis (see Fig. 13.5). We must study I(n, S, u) for large values of vS and for this purpose we shall use the steepest descent method. The relevant saddle point corresponds to the minimum of the function... 

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. ... 

Of course, the surface is not quadratic and the Hessian is not constant from step to step. However, near a critical point, the Newton-Raphson method (Eq. (2)) will converge rapidly. The main difficulty is that the convergence of the Newton-Raphson method is local. Thus the method will converge to the nearest critical point to the starting point. Consequently, one must start the optimization with a Hessian that contains no or one negative eigenvalue according to whether a minimum or a saddle point structure is required. For a minimum, the steepest descent direction... 

Fig. 6. Failure of coordinate driving method due to a long valley not leading to the transition structure. The dotted curve is the steepest-descent path from the saddle point.

A correct description of the reaction coordinate (IRC) has been obtained by applying the method of the steepest descent from the saddle point to the neighboring minimum of the PES [19,27,32] (see Sect. 1.3.4.2) for the reaction H +CH4 (X = Y = H). The calculations confirm that along all the path of Eq. (5.2) the 3 symmetry of the reacting system is retained. Therefore the PES in the reaction zone is defined by only four independent parameters (Fig. 5.1). In earlier calculations of this reaction, for example in Ref. [22], where an extended basis of orbitals was employed and polarization functions were included, the reaction path and the transition state structure characteristics were computed in the reaction coordinate regime. The distance r2 (Fig. 5.1) was chosen as such a coordinate. The calculation led to the erroneous conclusion... 

---