Quantum mechanics, phenomenological theory

The existing phenomenological theories of catalysis bear approximately the same relation to the electron theory as the theory of the chemical bond, which was prevalent in the last century and which made use of valence signs (and dealt only with these signs), bears to the modern quantum-mechanical theory of the chemical bond which has given the old valence signs physical content, thereby disclosing the physical nature of the chemical forces. 

The theory described so far is based on the Master Equation, which is a sort of intermediate level between the macroscopic, phenomenological equations and the microscopic equations of motion of all particles in the system. In particular, the transition from reversible equations to an irreversible description has been taken for granted. Attempts have been made to derive the properties of fluctuations in nonlinear systems directly from the microscopic equations, either from the classical Liouville equation 18 or the quantum-mechanical equation for the density matrix.19 We shall discuss the quantum-mechanical treatment, because the formalism used in that case is more familiar. 

One of the most important theoretical contributions of the 1970s was the work of Rudnick and Stern [26] which considered the microscopic sources of second harmonic production at metal surfaces and predicted sensitivity to surface effects. This work was a significant departure from previous theories which only considered quadrupole-type contributions from the rapid variation of the normal component of the electric field at the surface. Rudnick and Stern found that currents produced from the breaking of the inversion symmetry at the cubic metal surface were of equal magnitude and must be considered. Using a free electron model, they calculated the surface and bulk currents for second harmonic generation and introduced two phenomenological parameters, a and b , to describe the effects of the surface details on the perpendicular and parallel surface nonlinear currents. In related theoretical work, Bower [27] extended the early quantum mechanical calculation of Jha [23] to include interband transitions near their resonances as well as the effects of surface states. 

Because of the inadequacies of QED a fundamental theory of electrode processes is still lacking. The working theories are exclusively phenomenological and formulated entirely in terms of ionic distributions in the vicinity of electrode interfaces. An early, incomplete attempt [54] to develop a quantum mechanical theory of electrolysis based on electron tunnelling, is still invoked and extensively misunderstood as the basis of charge-transfer. It is clear from too many superficial statements about the nature of electrons that the symbol e is considered sufficient to summarize their important function. The size, spin and mass of the electron never feature in the dynamics of electrochemistry. 

Elementary electrostatic theory cannot account for the observed ionization equilibria in nonaqueous solvents. The functional approach provides a qualitative interpretation of all ionization phenomena, and this is in agreement with quantum mechanical results. This approach considers the coordinating properties of the solvents toward neutral solutes, cations, and anions and takes into account outer-sphere Coordination occurring both in the pure solvents and in the solutions. The application of the donicity and of other phenomenological properties allows a number of semiquantitative predictions. 

The remarkable situation in which we find ourselves in modem materials science is that physics has for some time been sufficiently developed, in terms of fundamental quantum mechanics and statistical mechanics, that complete and exact ab initio calculations of materials properties can, in principle, be performed for any property and any material. The term ab initio" in this context means without any adjustable or phenomenological or calibration parameters being required or provided. One simply puts the required nuclei and electrons in a box and one applies theory to obtain the outcome of a specified measurement. The recipe for doing this is known but the execution can be tedious to the point of being impossible. The name of the game, therefore, has been to devise approximations and methods that make the actual calculations doable with limited computer resources. Thanks to increased computer power, the various approximations can be tested and surpassed and more and more complex materials can be modelled. This section provides a brief overview of the theoretical methods of solid state magnetism and of nanomaterial magnetism in particular. 

Abstract Contribution of the Jahn-Teller system to the elastic moduli and ultrasonic wave attenuation of the diluted crystals is discussed in the frames of phenomenological approach and on the basis of quantum-mechanical theory. Both, resonant and relaxation processes are considered. The procedure of distinguishing the nature of the anomalies (either resonant or relaxation) in the elastic moduli and attenuation of ultrasound as well as generalized method for reconstruction of the relaxation time temperature dependence are described in detail. Particular attention is paid to the physical parameters of the Jahn-Teller complex that could be determined using the ultrasonic technique, namely, the potential barrier, the type of the vibronic modes and their frequency, the tunnelling splitting, the deformation potential and the energy of inevitable strain. The experimental results obtained in some zinc-blende crystals doped with 3d ions are presented. 

The Jahn-Teller effect requires quantum-mechanical description. Therefore, we will discuss the results obtained in a microscopic theory and later we will compare the expressions for ultrasonic absorption with one obtained in the phenomenological theory. 

A consistently quantum mechanical theory describing the coherent and stochastic dynamics of tetrahedral rotors has not been reported yet. Nevertheless, in the extreme situations where the facile re-orientations involve only one axis, the DQR theory will be rigorously valid also for such rotors. Specifically, if the unique axis is a three-fold axis, the stochastic term will have the same form as in Eq. (8) [or the equivalent form in Eq. (10)]. In the case of a two-fold axis, the AB term such as that in Eq. (5) will be obtained, but with the pair-permutation operator P replaced by the operator R defined above. One can thus reasonably expect that even in the cases where there are more than one facile re-orientation axes, applicability of the phenomenological AB approach will suffer similar restrictions as those specified in Subsection 4.4 for methyl-like rotors. 

Aspects of all these various topics were discussed at the fifth international Indaba workshop of the International Union of Crystallography at Berg-en-Dal, Kruger National Park, South Africa, 20-25 August 2006. In most instances the analysis terminated before a fundamental interpretation of the results, because there is no fundamental theory of chemistry. As all chemical interactions are mediated by electrons it is clear that such a theory must be quantum-mechanical. The only alternative is a phenomenological simulation of chemical processes with empirical classical models. There is no middle ground. 

On a modest level of detail, kinetic studies aim at determining overall phenomenological rate laws. These may serve to discriminate between different mechanistic models. However, to it prove a compound reaction mechanism, it is necessary to determine the rate constant of each elementary step individually. Many kinetic experiments are devoted to the investigations of the temperature dependence of reaction rates. In addition to the obvious practical aspects, the temperature dependence of rate constants is also of great theoretical importance. Many statistical theories of chemical reactions are based on thermal equilibrium assumptions. Non-equilibrium effects are not only important for theories going beyond the classical transition-state picture. Eventually they might even be exploited to control chemical reactions [24]. This has led to the increased importance of energy or even quantum-state-resolved kinetic studies, which can be directly compared with detailed quantum-mechanical models of chemical reaction dynamics [25,26]. 

Several different approaches have been utilized to develop molecular theories of chemical kinetics which can be used to interpret the phenomenological description of a reaction rate. A common element in all approaches is an explicit formulation of the potential energy of interaction between reacting molecules. Since exact quantum-mechanical calculations are not yet available for any system, this inevitably involves the postulation of specific models of molecules which only approximate the real situation. The ultimate test of the usefulness of such models is found in the number of independent macroscopic properties which can be correctly explained or predicted. Even so, it must be remembered that it is possible for incorrect models to predict reasonably correct macroscopic properties because of fortuitous cancellation of errors, insensitivity of the properties to the nature of the model, relatively large uncertainties in the magnitudes of the properties, or combinations of such effects. 

From the middle of the nineteenth century on, understanding of the phenomenology of interfaces became better at the molecular level, although the nature of the forces involved remained uncertain until the advent of quantum mechanical theory in the 1930s. The study of colloidal phenomena followed a similar track in that certain characteristics of colloidal systems were recognized and studied in the last century (and before), but a good quantitative understanding of the principles and processes involved remained elusive. 

To be more definite, the mass action law is a postulate in the phenomenological theory of chemical reaction kinetics. In the golden age of the quantum, chemistry seemed to be reducible to (micro)physics The underlying physical laws for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the diflBculty is only that exact application of these laws leads to equations much too complicated to be soluble (Dirac, 1929). As was clearly shown by Golden (1969) the treatment of chemical reactions needs additional requirements, even at the level of quantum statistical mechanics. The broad-minded book of Primas (1983), in which the author deeply analyses why chemistry cannot be reduced to quantum mechanics is strongly recommended. 

The transmission coefficient k which is also included in the expression for the preexponential factor, cannot be calculated using a purely phenomenological approach, but involves the quantum mechanical theory of an elementary act. For most simple electron transfer reactions, k = 1. 

Strictly speaking, the values of a = 1 or 0 are not necessarily associated with complete disappearance of the energy barrier between the two states. This barrier suffices to be potential independent. Such processes may be called quasibarrierless and quasiactivationless. An explanation of the reason why, even with a significant potential shift, there remains a permanent barrier, is beyond the scope of the phenomenological theory. This phenomenon was predicted on the basis of the quantum mechanical theory of an elementary act, and it was shown that certain reactions, particularly those of chlorine evolution, are, in fact, quasibarrierless. A detailed discussion of this problem can be found in Ref. 70. Here we shall only point out that the reason for this... 

Full understanding of modern chemistry would be impossible without quantum theory. Chemistry existed as its own, phenomenological science long before the year 1900, and has a number of features that defy explanation in terms of the classical laws of physics, for example, chemical bonds, reaction barriers in chemical reactions, and spectra. After the year 1900, a number of chemical phenomena have been described using quantum mechanics. Chemical bonds can now be accurately calculated with the help of a personal computer. Electron transfer can be understood, as can excitation energy transfer and other phenomena in photochemistry and photophysics. Chemistry has become a branch of physics chemical physics. 

One consequence of the discovery of the structure of atoms through the work of Rutherford was that the concept of chemical bonding became a subject of theoretical consideration. In Bohr s theory, the most loosely bound electrons, the valence electrons, are responsible for chemical bonding. Before quantum mechanics was completed, Kossel, Lewis, and Langmuir invented a phenomenological electronic concept, the electron pair bond, corresponding to the dash used previously to indicate a covalent bond between atoms. 

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