Differential, complete exact

A precise calculation of AGd as a function of the cationic radius would be very difficult because it would involve a complete conformational analysis of a large and complicated ligand system (82). Nevertheless, the dependency of the cation selectivity on steric interactions is capable of illustration. The term AGd can be estimated very crudely by using Hooke s law. As is shown in Fig. 16, ligands that are differentiated only by the radius of their equilibrium cavities can easily discriminate between cations of different size. This may explain why valinomycin and antamanide, two antibiotics with similar coordination spheres (54, 66), do not prefer the same cation (82). As it is no easy task to predict the exact dimensions of the cavity for a proposed ligand, the tailored synthesis of such ligands is conceivable yet problematic. 

So far we have seen that a periodic function can be expanded in a discrete basis set of frequencies and a non-periodic function can be expanded in a continuous basis set of frequencies. The expansion process can be viewed as expressing a function in a different basis. These basis sets are the collections of solutions to a differential equation called the wave equation. These sets of solutions are useful because they are complete sets. Completeness means that any arbitrary function can be expressed exactly as a linear combination of these functions. Mathematically, completeness can be expressed as... 

The aim of molecular orbital theory is to provide a complete description of the energies of electrons and nuclei in molecules. The principles of the method are simple a partial differential equation is set up, the solutions to which are the allowed energy levels of the system. However, the practice is rather different, and, just as it is impossible (at present) to obtain exact solutions to the wave equations for polyelectronic atoms, so it is not possible to obtain exact solutions for molecular species. Accordingly, the application of molecular orbital theory to molecules is in a regime of successive approximations. Numerous rigorous mathematical methods have been utilised in the effort to obtain ever more accurate solutions to the wave equations. This book is not concerned with the details of the methods which have been used, but only with their results. 

Notice that this is an integral representation of Laplace s equation for temperature. We will need to specify the extra function so it can become a complete representation. It is important to point out here that we have not made any approximation when deriving this formulation, making it an exact solution of the differential equation, V2T = 0. 

Errors and confusion in modelling arise because the complex set of coupled, nonlinear, partial differential equations are not usually an exact representation of the physical system. As examples, first consider the input parameters, such as chemical rate constants or diffusion coefficients. These input quantities, used as submodels in the detailed model, must be derived from more fundamental theories, models or experiments. They are usually not known to any appreciable accuracy and often their values are simply guesses. Or consider the geometry used in a calculation. It is often one or two dimensions less than needed to completely describe the real system. Multidimensional effects which may be important are either crudely approximated or ignored. This lack of exact correspondence between the model adopted and the actual physical system constitutes the basic problem of detailed modelling. This problem, which must be overcome in order to accurately model transient combustion systems, can be analyzed in terms of the multiple time scales, multiple space scales, geometric complexity, and physical complexity of the systems to be modelled. 

The widespread application of MO theory to systems containing a bonds was sparked in large part by the development of extended Hiickel (EH) theory by Hoffmann (I) in 1963. At that time, 7r MO theory was practiced widely by chemists, but only a few treatments of a bonding had been undertaken. Hoffmann s theory changed this because of its conceptual simplicity and ease of applicability to almost any system. It has been criticized on various theoretical grounds but remains in widespread use today. A second approximate MO theory with which we are concerned was developed by Pople and co-workers (2) in 1965 who simplified the exact Hartree-Fock equations for a molecule. It has a variety of names, such as complete neglect of differential overlap (CNDO) or intermediate neglect of differential overlap (INDO). This theory is also widely used today. 

In principle, one can write down all of these forces and formulate the Newtonian equations of motion for the fluid this yields a complicated differential equation known as the Navier-Stokes equation [1-3]. A complete solution of the Navier-Stokes equation gives the exact trajectory and velocity of each fluid element. In practice, the calculations are often difficult because one must simultaneously account for all fluid elements and the interactions between these elements caused by the viscous drag forces. (The simultaneous motion of many interacting fluid elements is analogous to the simultaneous motion of many interacting mechanical objects, the latter being so complicated that it is described as the many body problem. ) However, in certain cases, the Navier-Stokes equation is reduced to a tractable form by the existence of steady low-velocity flow and high symmetry in the flow conduit (e.g., capillary tubes of circular cross section). We will examine such simple cases shortly. 

The Born Oppcnheimer electronic Hamiltonian was given previously in (6.77) and a method of obtaining exact solutions is described in appendix 6.1. If the Bom expansion (6.131) is substituted into the complete non-relativistic Schrodinger equation, using the Hamiltonian (6. 130), we obtain a set of coupled differential equations for the functions... 

In order to complete the closure, the various length scales in the models above must be prescribed or related to the other independent variables through a differential equation. Daly and Harlow use a dynamical equation for S>, derived exactly from the Navier-Stokes equations and then closed by assumptions. The S) equation will be discussed presently. Daly and Harlow are now considering the use of two length-scale equations for the dissipating and energy-containing eddies. 

To "solve" this system of simultaneous equations, we want to be able to calculate the value of [A], [B] and [C] for any value of t. For all but the simplest of these systems of equations, obtaining an exact or analytical expression is difficult or sometimes impossible. Such problems can always be solved by numerical methods, however. Numerical methods are completely general. They can be applied to systems of differential equations of any complexity, and they can be applied to any set of initial conditions. Numerical methods require extensive calculations but this is easily accomplished by spreadsheet methods. 

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