Explicit techniques

Explicit techniques calculate the state of a system at a later time by using information from the state of the system at the present time. These methods are also known as step-by-step methods, which start from the initial condition given in Equation 11.1 and continue stepwise by computing approximate values of the solution at points that increase by a fixed number, h, which is known as the step size. Some of the expficit techniques used to obtain numerical solutions of Equation 11.1 are summarized as follows. 

Euler s method is first order and executes the int ration through the following formulation  

A more accurate technique used for numerical integration is the classical fourth-order Runge-Kutta (RK) method. In this 

Multiphase Catalytic Reactors Theory, Desi, Manufacturing, and Applications, First Edition. Edited by Zeynep Ilsen Onsan and Ahmet Kerim Avci. 2016 John Wiley Sons, Inc. Published 2016 by John Wiley Sons, Inc. 

An alternative fourth-order formulation is the Runge-Kutta-Gill method  

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In the finite difference method an explicit technique would evaluate the right-hand side at the /ith time level. 

The Field of Numerical Analysis.—As used here, numerical analysis will be taken to represent the art and science of digital computation. The art is learned mainly by experience hence, this chapter will be concerned with explicit techniques and the mathematical principles that justify them. Digital computation is to be contrasted with analog computation, which is the use of slide rules, differential analyzers, model basins, and other devices in which such physical magnitudes as lengths, voltages, etc., represent the quantities under consideration. 

Initially (time zero) the values of c are known and therefore the values of cj (concentrations at t = At) can be calculated by direct application of (25.93). This approach can be repeated and the values of c"+1 can be calculated from the previously calculated values of c . This is an example of an explicit finite difference method, where, if approximate solution values are known at time t = nAt, then approximate values at time tn+1 = (n+ 1) At may be explicitly and immediately calculated using (25.93). Typically, explicit techniques require that constraints be placed on the size of Af that may be used to avoid significant numerical errors and for stable operation. In a stable method, unavoidable small errors in the solution are suppressed with time in an unstable method a small initial error may increase significandy, leading to erroneous results or a complete failure of the method. Equation (25.93) is stable only if At < (Ax)2/2, and therefore one is obliged to use small integration timesteps. 

This solution can be obtained explicitly either by matrix diagonalization or by other techniques (see chapter A3.4 and [42, 43]). In many cases the discrete quantum level labels in equation (A3.13.24) can be replaced by a continuous energy variable and the populations by a population density p(E), with replacement of the sum by appropriate integrals [Hj. This approach can be made the starting point of usefiil analytical solutions for certain simple model systems [H, 19, 44, 45 and 46]. 

In this chapter, we look at the techniques known as direct, or on-the-fly, molecular dynamics and their application to non-adiabatic processes in photochemistry. In contrast to standard techniques that require a predefined potential energy surface (PES) over which the nuclei move, the PES is provided here by explicit evaluation of the electronic wave function for the states of interest. This makes the method very general and powerful, particularly for the study of polyatomic systems where the calculation of a multidimensional potential function is an impossible task. For a recent review of standard non-adiabatic dynamics methods using analytical PES functions see [1]. 

The solution Xh(t) of the linearized equations of motion can be solved by standard NM techniques or, alternatively, by explicit integration. We have experimented with both and found the second approach to be far more efficient and to work equally well. Its handling of the random force discretization is also more straightforward (see below). For completeness, we describe both approaches here. 

These various techniques were recently applied to molecular simulations [11, 20]. Both of these articles used the rotation matrix formulation, together with either the explicit reduction-based integrator or the SHAKE method to preserve orthogonality directly. In numerical experiments with realistic model problems, both of these symplectic schemes were shown to exhibit vastly superior long term stability and accuracy (measured in terms of energy error) compared to quaternionic schemes. 

The Car-Parrinello quantum molecular dynamics technique, introduced by Car and Parrinello in 1985 [1], has been applied to a variety of problems, mainly in physics. The apparent efficiency of the technique, and the fact that it combines a description at the quantum mechanical level with explicit molecular dynamics, suggests that this technique might be ideally suited to study chemical reactions. The bond breaking and formation phenomena characteristic of chemical reactions require a quantum mechanical description, and these phenomena inherently involve molecular dynamics. In 1994 it was shown for the first time that this technique may indeed be applied efficiently to the study of, in that particular application catalytic, chemical reactions [2]. We will discuss the results from this and related studies we have performed. 

The problem with most quantum mechanical methods is that they scale badly. This means that, for instance, a calculation for twice as large a molecule does not require twice as much computer time and resources (this would be linear scaling), but rather 2" times as much, where n varies between about 3 for DFT calculations to 4 for Hartree-Fock and very large numbers for ab-initio techniques with explicit treatment of electron correlation. Thus, the size of the molecules that we can treat with conventional methods is limited. Linear scaling methods have been developed for ab-initio, DFT and semi-empirical methods, but only the latter are currently able to treat complete enzymes. There are two different approaches available. 

In this chapter we shall consider four important problems in molecular n iudelling. First, v discuss the problem of calculating free energies. We then consider continuum solve models, which enable the effects of the solvent to be incorporated into a calculation witho requiring the solvent molecules to be represented explicitly. Third, we shall consider the simi lation of chemical reactions, including the important technique of ab initio molecular dynamic Finally, we consider how to study the nature of defects in solid-state materials. 

PW91 (Perdew, Wang 1991) a gradient corrected DFT method QCI (quadratic conhguration interaction) a correlated ah initio method QMC (quantum Monte Carlo) an explicitly correlated ah initio method QM/MM a technique in which orbital-based calculations and molecular mechanics calculations are combined into one calculation QSAR (quantitative structure-activity relationship) a technique for computing chemical properties, particularly as applied to biological activity QSPR (quantitative structure-property relationship) a technique for computing chemical properties... 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

This study is particularly noteworthy in the evolution of QM-MM studies of enzyme reactions in that a number of technical features have enhanced the accuracy of the technique. First, the authors explicitly optimized the semiempirical parameters for this specific reaction based on extensive studies of model reactions. This approach had also been used with considerable success in QM-MM simultation of the proton transfer between methanol and imidazole in solution. 

Also, refinery operations and particularly chemical plant processes tend to be unique, so that techniques described in this book do not necessarily apply in all cases. The presentation of material is therefore not meant to be an endorsement, either by the author or publisher, nor are any guarantees, either explicit or implicit made for designs based on the information provided in this book. 

Error probabilities that are used in decomposition approaches are all derived in basically the same manner. Some explicit or implicit form of task classification is used to derive categories of tasks in the domain addressed by the technique. For example, typical THERP categories are selections of switches from control panels, walk-around inspections, responding to alarms and operating valves. 

If an analytical solution is available, the method of nonlinear regression analysis can be applied this approach is described in Chapter 2 and is not treated further here. The remainder of the present section deals with the analysis of kinetic schemes for which explicit solutions are either unavailable or unhelpful. First, the technique of numerical integration is introduced. 

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