Exponential integrals

Hence, as the second class of techniques, we discuss adaptive methods for accurate short-term integration (Sec. 4). For this class, it is the major requirement that the discretization allows for the stepsize to adapt to the classical motion and the coupling between the classical and the quantum mechanical subsystem. This means, that we are interested in discretization schemes which avoid stepsize restrictions due to the fast oscillations in the quantum part. We can meet this requirement by applying techniques recently developed for evaluating matrix exponentials iteratively [12]. This approach yields an adaptive Verlet-based exponential integrator for QCMD. 

Fig. 3. Stepsize r used in the simulation of the collinear photo dissociation of ArHCl the adaptive Verlet-baaed exponential integrator using the Lanczos iteration (dash-dotted line) for the quantum propagation, and a stepsize controlling scheme based on PICKABACK (solid line). For a better understanding we have added horizontal lines marking the collisions (same tolerance TOL). We observe that the quantal H-Cl collision does not lead to any significant stepsize restrictions.

M. Hochbruck, Ch. Lubich, and H. Selhofer Exponential integrators for large systems of differential equations. SIAM J. Sci. Comp. (1998) (to appear)... 

M. Hochbruck and Ch. Lubich. Exponential integrators for quantum-classical molecular dynamics. Tech. Rep., Universitat Tubingen, 1998. In preparation. 

Equation 9-15 gives the conversion expression for the second order reaction of a macrofluid in a mixed flow. An exponential integral, ei(a), which is a function of a, and its value can be found from tables of integrals. However, the conversion from Equation 9-15 is different from that of a perfectly mixed reactor without reference to RTD. An earlier analysis in Chapter 5 gives... 

The resulting integrals are expressed in terms of exponential integrals... 

Methods using tabulated values of the exponential integral Putting U = E/RT, eqn. (23) becomes... 

Methods using a simple approximation for the exponential integral... 

Alternative simple approaches to approximations for the exponential integral exploit the linearity of the log P(t/) against U relation over short ranges [533,542,543]... 

For the integral being evaluated here, b = 2/a0 and n = 3, so that the exponential integral can be written as... 

From equation 14.3-19, with MA defined by equation 4.3-4. From equation 13.52, with 3.4-10 or 14.3-20 and 13.4-2. dE is an exponential integral defined by E x) = y le y dy, where y is a dummy variable the integral must be evaluated numerically (e.g., using E-Z Solve) tabulated values also exist. 

This integral is related to the exponential integral (see Table 14.1). It cannot be solved in closed analytical form, but it can be evaluated numerically using the E-Z Solve software the upper limit may be set equal to 10f. 

For reaction A -> products constant-density, isothermal, steady-state operation. Second term is related to exponential integral see Table 14.1. 

Closely related to the gamma function are the exponential-integral ei(a) defined by the equation... 

Equation (41) involves the exponential integral which can be evaluated numerically or else is available as a standard tabulated function (see, for instance, ref. 33). Therefore... 

The exponential integral in eqn. (56) is a standard tabulated function [33], so predictions of conversion can be made and are plainly a function of Tk alone. Such calculations have been performed by various authors [43—46]. Hilder [47], when repeating these calculations, found eqn. (57) to be a simple and adequate approximation to eqn. (56). 

This is the conversion expression for second-order reaction of a macrofluid in a mixed flow reactor. The integral, represented by ei(a) is called an exponential integral. It is a function alone of a, and its value is tabulated in a number of tables of integrals. Table 16.1 presents a very abbreviated set of values for both ei(jc) and Ei(jc). We will refer to this table later in the book. 

A better estimate of the time ArDA, and hence of the period ATp, can be obtained using an integration of eqn (5.78). This involves the exponential integral, Ei(x ) where xt = (M/K) — 0h a function freely available in standard tabulations. We then find... 

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