Subject quantum molecular

A simple graphical method of formula ting the independent valence-bond structures for a molecule was discovered by Rumer.1 This method has been extended to permit the secular equation for a set of resonating valence-bond structures to be written without difficulty. Quantum-mechanical treatments of aromatic and conjugated molecules have been carried out by many investigators. The subject of molecular quantum mechanics is too extensive to be reviewed in this book. 

This overview has also considered the advantages and disadvantages of a description using first principles dynamics. Its application to many-atom systems undergoing electronic transitions is a very active and challenging subject of molecular quantum dynamics. 

Molecular systems exist in discrete quantum states, the study of which lies in the realm of molecular structure and wave mechanics. Transitions between quantum states occur either by absorption or emission of radiation (spectroscopy) or by collisional processes. There are two main types of collisional transitions which are important in chemical physics these are first, reactive processes in which chemical rearrangement takes place (reaction kinetics), and secondly collisions in which the energy distribution is changed without overall chemical reaction. It may therefore be concluded that the energy transfer processes discussed here are of fundamental importance in all molecular systems, and that the subject, like molecular structure, is enormously varied and complex. 

The next section, entitled Basic Aspects of Molecular Similarity, gives a general overview of molecular similarity and the usual vocabulary used by chemists in this field. In the section entitled The Electron Density as Molecular Descriptor, some elements of quantum chemistry will be presented. These sections should not be expected by the reader to offer a rigorous discussion of all aspects of such broad fields, and therefore, only the most important definitions and concepts will be introduced. The following sections will then address extensively the subject of molecular quantum similarity in both theoretical and practical aspects. An ample references list will help the interested reader to look up the more specialized literature. 

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

Like the geometry of Euclid and the mechanics of Newton, quantum mechanics is an axiomatic subject. By making several assertions, or postulates, about the mathematical properties of and physical interpretation associated with solutions to the Scluodinger equation, the subject of quantum mechanics can be applied to understand behaviour in atomic and molecular systems. The fust of these postulates is ... 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Sclirodinger equation is still fomiidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions fonn the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

This section attempts a brief review of several areas of research on the significance of phases, mainly for quantum phenomena in molecular systems. Evidently, due to limitation of space, one cannot do justice to the breadth of the subject and numerous important works will go unmentioned. It is hoped that the several cited papers (some of which have been chosen from quite recent publications) will lead the reader to other, related and earlier, publications. It is essential to state at the outset that the overall phase of the wave function is arbitrary and only the relative phases of its components are observable in any meaningful sense. Throughout, we concentrate on the relative phases of the components. (In a coordinate representation of the state function, the phases of the components are none other than the coordinate-dependent parts of the phase, so it is also true that this part is susceptible to measurement. Similar statements can be made in momentum, energy, etc., representations.)... 

Whereas oxaziridine and diaziridine were partial subjects of comprehensive theoretical studies on cyclic compounds (73MI50800), diazirine and some of its simple derivatives were the special target of quantum chemical investigations. Since diazirine, the lowest molecular weight heterocycle, has only five atoms and is of high symmetry, there was a chance for ab initio calculations, which followed some semiempirical studies. 

Actually, the first attempts to use the electron density rather than the wave function for obtaining information about atomic and molecular systems are almost as old as is quantum mechanics itself and date back to the early work of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of only historical interest. We therefore refrain from an in-depth discussion of the Thomas-Fermi model and restrict ourselves to a brief summary of the conclusions important to the general discussion of DFT. The reader interested in learning more about this approach is encouraged to consult the rich review literature on this subject, for example by March, 1975, 1992 or by Parr and Yang, 1989. 

It is the objective of the present chapter to define matrices and their algebra - and finally to illustrate their direct relationship to certain operators. The operators in question are those which form the basis of the subject of quantum mechanics, as well as those employed in the application of group theory to the analysis of molecular vibrations and the structure of crystals. 

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