Method saddle-point

The calculation of the integrals in Eq. (55) in the classical limit in the improved Condon approximation (for the nuclear subsystem) using the saddle point method leads to two coupled equations for the electron wave functions of the donor and the acceptor in the transitional configuration ... 

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

Saddle point method, 68 Self-focusing, 83, 84, 86 Self-injection, 150... 

The integrals in the numerator and denominator can be estimated by using the saddle point method again. By expanding g (x) around the maximum contribution at x , we get, up to second order. 

We first note that Eq. (29), which we derived by taking the limit x — 0 of the result (26) for general x, can also be obtained by a more direct route. In the limit x —> 0, the sizes of particles in the smaller system become independent random variables drawn from n (a) the second phase can be viewed as a reservoir to which the small phase is connected. One writes the moment generating function for V(m) in the small phase as a product of xN independent moment generating functions of n (o) and then evaluates the integral over V m) by a saddle point method [36]. 

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... 

Taking into account relationships (75) and (76), over the long time interval z> 1 the integration of (73) may be performed asymptotically by the saddle-point method [135]. The main term of the asymptotic expansion can be obtained as the product of the power and stretched exponential universal relaxation laws [38] ... 

The integrals (XI.4.14) and (XI.4.15) can be evaluated with good accuracy by the method appropriate to exponentially peaked functions, namely the saddle-point method (Marcus, loc. cit). 

Finally, let us try to formulate the scattering matrix within the present semiclassical treatment. Using the adiabatic wave functions obtained by the saddle point method,... 

AG (T) have been derived a particularly useful one is obtained using the saddle-point method ". This expression is ... 

Kinetic coefficients. The kinetic adsorption and desorption coefficients can be estimated [82, 219, 412] if the form of the potential (Z) of interaction between a particle (a surfactant molecule) and the solution surface is known here Z is the coordinate measured from the surface into the bulk of liquid. If the function (Z) has the form of a potential barrier with a potential well, then the saddle-point method [261] implies... 

This observation is most helpful for our asymptotical treatment of the atomic systems suggesting that the saddle point approximation (Mathews Walker, 1970), is suitable to fairly analytical perform the involved integrals. According with the saddle point method, to evaluate an integral of type (4.325) the intermediate form (4.326) is approximated by the saddle-point recipe (3.154) specialized here as (see also the Appendix of the present volume) ... 

In this case, the saddle point method is inapplicable for the calculation of the integral over a in Eq. (27). The major contribution to the integral comes rather from the large values of o-. This leads to the following expression for the transition probability... 

In the domain of long times this integral can be evaluated with the use of the saddle-point method. Finally, one has [142] ... 

We now turn to methods for first-order saddle points. As already noted, saddle points present no problems in the local region provided the exact Hessian is calculated at each step. The problem with saddle point optimizations is that in the global region of the search, there are no simple criteria that allow us to select the step unambiguously. Thus, whereas for minimization methods it is often possible to give a proof of convergence with no significant restrictions on the function to be minimized, no such proofs are known for saddle-point methods, except, of course, for quadratic surfaces. Nevertheless, over the years several useful techniques have been developed for the determination of saddle points. We here discuss some of these techniques with no pretence at completeness. 

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