Quantum theory, theme

This is a recurrent theme of quantum theory. Many quantum systems can be formulated exactly in terms of a wave equation and the behaviour of the system will be described exactly by the wavefunction, the solution to the wave equation. What is not always appreciated is that this is a mathematical description only, which does not ensure understanding of the event in terms of a comprehensible physical model. The problem lies therein that the description is only possible in terms of a wave formalism. Understanding of the physical behaviour however, requires reduction to a particle model. The wave description is no more than a statistically averaged picture of the behaviour of many particles, none of which follows the actual statistically predicted course. The wave description is non-classical, and the particle model is classical. Mechanistic understanding is possible only in terms of the classical approach, and a mathematically precise description only in terms of the wave formalism. The challenge of quantum theory is to reconcile the two points of view. 

The quantum theory of the previous chapter may well appear to be of limited relevance to chemistry. As a matter of fact, nothing that pertains to either chemical reactivity or interaction has emerged. Only background material has been developed and the quantum behaviour of real chemical systems remains to be explored. If quantum theory is to elucidate chemical effects it should go beyond an analysis of atomic hydrogen. It should deal with all types of atom, molecules and ions, explain their interaction with each other and predict the course of chemical reactions as a function of environmental factors. It is not the same as providing the classical models of chemistry with a quantum-mechanical gloss a theme not without some common-sense appeal, but destined to obscure the non-classical features of molecular systems. 

Given the plurality of the aforementioned discussions in both volumes, we hope that both senior and young quantum chemists and physicists with an interest in the specific theme of "unstable states in the continuous spectra" and in quantum theory, in general, will find the present set of two volumes resourceful, innovative, and helpful. 

The dominant theme of quantum theory is that its causal statements about a system are probabilistic. In other words, the epistemic rule of correspondence, which relates experience to quantum-theoretical states, involves probabilistic concepts in an essential way. In particular, an essential premise of quantum theory is that the physical condition or state of a system at a given time cannot be fully disclosed experimentally unless many measurements are made on replicas of the system prepared in a specified manner. Conversely, an inherent prerequisite of quantum theory is that a preparation of a system be specified and uniquely associated with a state prior to any attempt to reveal experimentally the characteristics of the state. It is this prerequisite that clearly distinguishes quantum mechanics from classical mechanics. It has been discussed extensively in the literature. 

One of the central themes in chemistry is the study of structure-property relationships. In its early days quantum chemistry was mostly concerned, preoccupied one could say, with questions relating to the nature of the chemical bond. Before the arrival of quantum theory, chemists believed in special cbemical forces that had short range, directional properties, and exhibited saturation—all three characteristics so different from well-known forces of physics. 

The answers to the above questions, not all of which need he presented here, were formulated between 1925 and 1926, in the revolution of modern quantum theory, which shook the foundations of physics and philosophy. Remarkably, the central theme of quantum theory was the nature of light, and what came to be called the wave-particle duality. But other broader implications of the new theory existed, and the first inkling of this was given in 1924 by Louis de Broglie (Fig. 3.26) in his doctoral dissertation. He postulated that particles may also possess wavelike properties and that these wavelike properties would manifest themselves only in phenomena occurring on an atomic scale, as dictated by Planck s constant. He also postulated that the wavelength of these matter waves, for a given particle such as an electron or proton, would be inversely proportional to the particle s momentum p, which is a product of its mass m and speed... 

In this chapter we give a brief review of some of the basic concepts of quantum mechanics with emphasis on salient points of this theory relevant to the central theme of the book. We focus particularly on the electron density because it is the basis of the theory of atoms in molecules (AIM), which is discussed in Chapter 6. The Pauli exclusion principle is also given special attention in view of its role in the VSEPR and LCP models (Chapters 4 and 5). We first revisit the perhaps most characteristic feature of quantum mechanics, which differentiates it from classical mechanics its probabilistic character. For that purpose we go back to the origins of quantum mechanics, a theory that has its roots in attempts to explain the nature of light and its interactions with atoms and molecules. References to more complete and more advanced treatments of quantum mechanics are given at the end of the chapter. 

Thermodynamic treatments in physical chemistry were effectively identical with the theory of the subjectin the nineteenth century. No oneunderstoodelectron transfer at interfaces at that time (J. J. Thompson did not discover the electron until 1897). But whereas the molecular kinetic approach gradually seeped into many parts of chemistry by the 1930s, the chemistry of electrode processes remained reluctantly bound up with the older thermodynamic viewpoint. The Faraday Society meeting in Manchester, U.K. in 1947 was a turning point in the application of a molecular-level concepts and even of quantum mechanics. By the mid-1950s, research papers in electrode process chemistry (except for those dealing with electroanalytica] themes)10 were fully kinetic. 

Nowadays, the basic framework of our understanding of elementary processes is the transition state or activated complex theory. Formulations of this theory may be found in refs. 1—13. Recent achievements have been the Rice—Ramsperger—Kassel—Marcus (RRKM) theory of unimol-ecular reactions (see, for example, ref. 14 and Chap. 4 of this volume) and the so-called thermochemical kinetics developed by Benson and co-workers [15] for estimating thermodynamic and kinetic parameters of gas phase reactions. Computers are used in the theory of elementary processes for quantum mechanical and statistical mechanical computations. However, this theme will not be discussed further here. 

Regarding TDDFT benchmark studies of chiroptical properties prior to 2005, the reader is referred to some of the initial reports of TDDFT implementations and early benchmark studies for OR [15,42,47,53,98-100], ECD [92,101-103], ROA [81-84], and (where applicable) older work mainly employing Hartree-Fock theory [52,55, 85,104-111], Often, implementations of a new quantum chemistry method are verified by comparing computations to experimental data for relatively small molecules, and papers reporting new implementations typically also feature comparisons between different functionals and basis sets. The papers on TDDFT methods for chiroptical properties cited above are no exception in this regard. In the following, we discuss some of the more recent benchmark studies. One of the central themes will be the performance of TDDFT computations when compared to wavefunction based correlated ab initio methods. Various acronyms will be used throughout this section and the remainder of this chapter. Some of the most frequently used acronyms are collected in Table 1. 

In Papers II and III, the centroid-based theory was significantly extended to treat perhaps one of the most challenging problems in condensed matter theory—the computation of general real-time quantum correlation functions (A(f)5(0)). Consistent with the general theme of this research, the properties of dynamical correlation functions were explored using the centroid-based perspective of quantum statistical mechanics. To be more specific, in one approach, real-time dynamical information was extracted with the help of the centroid-constrained formalism for imaginary-time... 

But many others believe the question of fundamentalism and reduction can stiU be studied within the context of science. One can still consider the more modest question of whether chemistry reduces to its sister science of physics.This question can be approached in a scientific manner by examining the extent to which chemical models or, indeed, the periodic system, can be explained by the most basic theory of physics, namely, quantum mechanics. It is this question that forms the underlying theme for this entire book, and it is a question that is addressed more and more exphcitly in later chapters as the story reaches the impact of modem physical theories on our understanding of the periodic system. 

The next important fact of lanthanide and actinide chemistry is going to be one of the central themes of this chapter, namely, relativistic effects. Relativistic effects arise from the differences in the assumed infinite speed of light in non-relativistic mechanics (whether classical or quantum mechanical) and the true speed of light. Although the special theory of relativity was proposed by Einstein in the very early part of the century, its impact in chemistry came about much later. In fact, it is not an exaggeration... 

The basic principle of VB theory is that a covalent bond forms when orbitals of two atoms overlap and a pair of electrons occupy the overlap region. In the terminology of quantum mechanics (Chapter 7), overlap of the two orbitals means their wave functions are in phase, so the amplitude between the nuclei increases (see Figure 7.5). The central themes of VB theory derive from this principle ... 

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