Real-Space Basics

This chapter discusses numerical methods for solving several important differential equations in computational chemistry. It does not cover the conceptually related problem of coarse-grained modeling of large amplitude motions in polymers and biological macromolecules.  

As outlined above, several means of representing partial differential equations in real space exist. Here, for the most part, we choose the simplest (the FD representation), restrict ourselves to second-order-accurate representations, and operate in one dimension so as to bring out all the important concepts and avoid getting wrapped up in details. A short introduction to FE representations is also included. 

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These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

Traditionally the performance of HRTEM is judged in terms of its ability to resolve two adjacent atom columns. Resolution is ruled by a few basic principles A position dependent image intensity g(r) is described as a convolution of the specimen function f(r) with a point spread function h(r). It is convenient to express this convolution in real space as a product in reciprocal space ... 

But the key to the theory of bond energy is in the description of real-space core and valence regions in atoms and molecules therein lies the basic idea that gives rise to the notion of molecular chemical binding, expressed as a sum of atomlike terms. That marks the beginning of our story. 

The basic equations of the nonadiabatic corrections to electronic energy are 1. Ground - state energy correction (real - space orbital representation),... 

The reader may find a careful discussion on these issues in Ballentine s article [10], see also Ref. [1]. At this point, observe that the system is basically taken as an object or more technically an entity existent in laboratory (real) space a res corporea, a substance that is known through anyone of its properties (attributes) in one word, an object in the sense of classical physics. Of course, observable asks for observer thus, an object/subject philosophy underlies this particular view of QM presentation that now it becomes a representation. Note that if knowledge is defined as the relation of given representations to well-defined objects, we will also have a problem with the theory of knowledge by endorsing our views on quantum states. 

The chemist s intuition is that the eclipsed and staggered polymers can t be very different—at least, not until the ligands start bumping into each other, and for such steric effects there can be, in turn, much further intuition. The band structures may look different, since one polymer has one, the other two basic electronic units in the cell. Chemically, however, they should be similar, and we can see this by returning from reciprocal space to real space. Figure 36, which compares the DOS of the staggered and eclipsed polymers, shows just how alike they are in their distribution of levels. 

This relation is the basic equation governing a renormalization group in real space. According to Eq. (159) we shall obtain the critical exponent v for correlation length ... 

A detailed description of the experimental set-up has been given by Block and Czanderna (ref.7). The basic principles are as follows. The catalyst is prepared in the form of a field emitter tip by electrochemically etching a thin wire (0 w 0.1 mm). At its apex this field emitter closely resembles a hemisphere and its surface can be imaged in real space with atomic lateral resolution by field ion microscopy (FIM). 

Approximation B lends itself to very convenient tests if we rewrite our basic formula as follows El = f il — ikhD k - If accuracy turns out to be virtually the same as that of approximation A. This can be understood because the latter obeys to practically the same formula, namely E — /7fc k -Still the fact remains that we cannot justify both our approximations, A and B, because their parameters, 7, 7 and 7, cannot be simultaneously constant. This is so because 1/7 is the weighted average of 1/7 and hk (Section 2) and Vk. at le2ist, is certainly not to be treated as a constant. This criticism illustrates the pitfalls of these formulas when used with approximate constant )k parameters, notwithstanding the acceptable numerical results. On the other hand, this definition of the average I/7 pin-points the place of Politzer s formula in the framework of core-valence theory in real space as a simple form of the basic relationship E). = El + EJP". 

It is remarkable that Vernadsky still speaks in this connection that there is no necessity to talk about a special geometrical space, which is connected with life (Vemadsky, 1988, p. 212). He changed his mind later. This shows that in 1927 Vernadsky s concept of space-time was still not elaborated. But he already proposes the basic notion of the real space of the naturalist, on which he later builds his understanding of the problem. 

Any position in real space is given by the real-space vector R which is a linear combination of the three basic vectors i, 2/ arid 3, namely... 

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