Euler step

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

For the mathematics of this, consider the discrete equation resulting from the Euler method, as in (4.8). Note that the new point, yn+i is formed from the old point yn by the addition of a term, here St f(yn). With RK, these terms are given the symbols fcj, there are from one to several of them, and they are added in a weighted manner. The procedure is to generate a number of these k s. One begins with an Euler step,... 

Gonzalez and Schlegel developed an implicit second-order integrator for reaction path following (GS2) [177,178], which is shown in Fig. 10.6c. Each GS2 step consists of two components. First, an explicit Euler step of length A is taken from the current point, x,, to a pivot point, x. ... 

The distinction between taking an LQA step and taking an Euler step along the reaction path is illustrated in Figure 2 on the Muller-Brown func-... 

In practice, a relatively small step is taken along one of these curves, and then potential energy surface information is determined at the new point. A subsequent step is then taken based on this new information. Figure 3 shows a comparison of the Euler and LQA methods for a sequence of such steps. The step length. As, is fairly large to illustrate the distinction between the two methods. The Xs result from a series of Euler steps. They are seen in the region... 

The new target OPs are estimated by taking a forward Euler step on the string evolution equation ... 

The solution curve x-x(s) is parameterized by the curve length s (see also Sect. 3.4 below, regarding this and other in variants). An Euler step would be given by... 

The method of Ishida, Morokuma, and Komomicki (IMK) is shown in Figure 3(a). It is a modification of the explicit Euler method that adds a stabilization step hence it is also known as the Euler stabilization method (ES). An explicit Euler step of length a is taken from along the tangent... 

The Muller-Brown method (MB) (Figure 3b) is an implicit Euler method. A step of length a is taken from 3c, and the energy is minimized under the constraint of constant step length, 3c,+i - 3c, = a. At the minimum, the residual gradient is parallel to the step. Thus the implicit Euler step can also be written as 3c, +, = 3c, - agi+1 / g,+11 = x/ +, ... 

But, once we have computed we can use it to approximate the integrand of (4.114) more accurately. We take the mid-point of the fiiU explicit Euler step, x -I- lP /2, and evaluate the function at this point... 

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