Method of Superposition

In Section 4.2.3 we described application of the method of superposition to infinite and semi-infinite systems. The method can also be applied, in principle, to finite systems, but it often becomes unwieldy (see Crank s discussion of the reflection method [2]). 

4 Method of Separation of Variables Diffusion on a Finite Domain 

A standard method to solve many partial-differential equations is to assume that the solution can be written as a product of functions, each a function of one of the independent variables. Table 5.2 provides several functional forms of such solutions. 

The following example illustrates the method. Consider a one-dimensional diffusion problem with the initial and boundary conditions for the domain 0 x L  

This situation may represent the diffusion of a high-vapor-pressure dopant out of a thin film (thickness L, initial dopant concentration c0) of silicon when placed in a vacuum. Assume that the variables are separable.7 Letting c(x,t) = X(x)T(t) and substituting into the diffusion equation gives 

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A fruitful approach for velocity computation in the first three zones of jets supplied from outlets with finite size was developed based on the hypothesis that momentum diffuses with distance from the source in the same manner as heat energy." 40 approach, developed by Elrod,Shepelev and Gelman, - and Regenscheit, utilizes the method of superposition of jet momentum from the multiple-jet system. These jets originate from the points with supply air veloc-... 

The discussion of the interaction of air jets supplied at some angle to each other shows that application of the method of superposition of the interacting jets momentums and surplus heat to predict velocity and temperatures in the combined flow results in inaccuracy when two unequal jets are supplied at a right angle. A different approach was undertaken in the studies of interaction of the main stream with vertical directing jets. Ti i... 

Mainly for considerations of space, it has seemed desirable to limit the framework of the present review to the standard methods for treating correlation effects, namely the method of superposition of configurations, the method with correlated wave functions containing rij and the method using different orbitals for different spins. Historically these methods were developed together as different branches of the same tree, and, as useful tools for actual applications, they can all be traced back to the pioneering work of Hylleraas carried out in 1928-30 in connection with his study of the ground state of the helium atom. 

The method of superposition of configurations described in the first paper (1928) implies that, after choosing a complete basic set of one-electron functions rpk rx) one can develop the space function in Eq. III. 1 in the form ... 

On the helium problem, the connection between the method of correlated wave function and the method of superposition of configurations has also been investigated in detail.8... 

The method of superposition of configurations as well as the method of different orbitals for different spins belong within the framework of the one-electron scheme, but, as soon as one introduces the interelectronic distance rijt a two-electron element has been accepted in the theory. In treating the covalent chemical bond and other properties related to electron pairs, it may actually seem more natural to consider two-electron functions as the fundamental building stones of the total wave function, and such a two-electron scheme has also been successfully developed (Hurley, Lennard-Jones, and Pople 1953, Schmid 1953). 

In the three following sections we will try to sketch the mathematical foundation for the three approaches which are most closely connected with the Hartree-Fock scheme, namely the methods of superposition of configurations (a), correlated wave functions (b), and different orbitals for different spins (c). We will also discuss their main physical implications. 

Arbitrary Complete Basic Set The mathematical background of the method of superposition of configurations is quite simple and straightforward, and we will... 

The method of superposition of configurations is essentially based on the assumption that the basic orbitals form a complete set. The most popular basis used so far in the literature is certainly formed by the hydrogen-like functions, which set contains a discrete and a continuous part. The discrete subset corresponds physically to the bound states of an electron around a proton, whereas the continuous part corresponds to a free electron scattered by a proton, or classically to the elliptic and hyperbolic orbits, respectively, in a central-field problem. 

To test the accuracy and convenience of the method of superposition of configurations, the problem of the ground state of the helium atom has recently been reexamined by several authors. According to Hylleraas (1928), the total wave function may be expressed in the form... 

The results in Table VI are of considerable interest also for atoms with more than two electrons, since they show the possibilities and limitations of the method of superposition of configurations/ when the latter are built up from one-electron functions... 

For atoms with more than two electrons, it is very difficult to obtain such a small absolute error in the energy as in the helium case, but, within an isoelectronic sequence, the relative error will, of course, go down rapidly with increasing atomic number Z. The method of superposition of configurations has been used successfully in a number of applications, particularly by Boys (1950-) and Jucys (1947-), and, for a more detailed survey of the work on atoms, we will refer to the special table on atomic calculations in the bibliography. This is a field of rapid development, where one can expect important new results within the next few years. 

Power Series Expansions and Formal Solutions (a) Helium Atom. If the method of superposition of configurations is based on the use of expansions in orthogonal sets, the method of correlated wave functions has so far been founded on power series expansions. The classical example is, of course, Hyl-leraas expansion (Eq. III.4) for the ground state of the He atom, which is a power series in the three variables... 

Lowdin, P.-O., and Redei, L., Combined use of the methods of superposition of configurations and correlation factor on the ground states of He-like ions."... 

Flow boiling of sodium Noyes and Lurie (1966) attempted to correlate experimental data of flowing sodium by the method of superposition,... 

The instantaneous local-source solutions in Table 5.1 can be used to build up solutions for general initial distributions of diffusant by using the method of superposition (see Section 4.2.3). 

Analyses of the climbing rates of many other dislocation configurations are of interest, and Hirth and Lothe point out that these problems can often be solved by using the method of superposition (Section 4.2.3) [2]. In such cases the dislocation line source or sink is replaced by a linear array of point sources for which the diffusion solutions are known, and the final solution is then found by integrating over the array. This method can be used to find the same solution of the loopannealing problem as obtained above. 

Let us also briefly discuss some developments in accounting for correlation effects while performing calculations of spectral properties of complex many-electron atoms and ions. The method of superposition of configurations based on the transformed radial orbitals, briefly described in Chapter 29, has demonstrated its effectiveness particularly for complex electronic configurations. Code for transformed radial orbitals is described in [11]. There the following transformed radial orbitals are recommended ... 

It should be pointed out that equation (8) was obtained by applying the method of superposition (Fisher et al., 1979), where two instantaneous plume, M/A, are initially released at x = 0 and travel at the speed of-w and u, respectively, to satisfy the no flux boundary at x = 0. Equation (8) can be simplified to equation (9) when there is no advective transport of the solute. 

The method of superposition can succeed when linearity exists and separability of the component concentrations can be assured. Unfortunately, many electrochemical situations do not satisfy this requirement, and in such instances other predictive methods, such as simulation, must be applied. 

This method of superposition of radial responses of two-coil induction probes does not practically allow us to find parameters of probes which are sensitive only to certain parts of medium, similar to special probes in electrical logging as micrologs. 

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