Quantum mechanics phenomenological modeling

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. 

The reader already familiar with some aspects of electrochemical promotion may want to jump directly to Chapters 4 and 5 which are the heart of this book. Chapter 4 epitomizes the phenomenology of NEMCA, Chapter 5 discusses its origin on the basis of a plethora of surface science and electrochemical techniques including ab initio quantum mechanical calculations. In Chapter 6 rigorous rules and a rigorous model are introduced for the first time both for electrochemical and for classical promotion. The kinetic model, which provides an excellent qualitative fit to the promotional rules and to the electrochemical and classical promotion data, is based on a simple concept Electrochemical and classical promotion is catalysis in presence of a controllable double layer. 

Figure 38. Absorption spectrum of pyrazine in the energy region of the S1-S2 conical intersection. Shown are (a) quantum mechanical (full line) and semiclassical (dotted line) results for the four-mode model (including a phenomenological dephasing constant of T2 = 30 fs) and (b) the experimental data [271],...

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. 

The quantum mechanical mechanisms that underlie exchange coupling are complex, but can be modeled by a phenomenological Hamiltonian that involves the coupling of local spin operators Sa and SB, the so-called Heisenberg-Dirac-Van... 

In this respect, chemistry does not differ from other sciences. Contemporary chemical research is organized around a hierarchy of models that aid its practitioners in their everyday quest for the understanding of natural phenomena. The building blocks of the language of chemistry, including the representations of molecules in terms of structural formulae [1], occupy the very bottom of this hierarchy. Various phenomenological models, such as reaction types and mechanisms, thermodynamics and chemical kinetics, etc. [2], come next. Quantum chemistry, which at present is the supreme theory of electronic structures of atoms and molecules, and thus of the entire realm of chemical phenomena, resides at the very top. 

This, in turn, accounts for the deviation from Curie-law behavior in many paramagnetic materials, as described in the last section. Before a discussion on the nature of exchange interactions, a phenomenological model of ferromagnetic behavior by Weiss is presented, which precedes the understanding of its quantum mechanical origin. 

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. 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

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]. 

Studies of the interfacial structure often quote early models from the first half of the twentieth century, and although these models have led to useful insights about the nature of the interface, it is important to note that they are based exclusively on phenomenological characteristics and macroscopic observations. To establish a rigorous understanding of the interfacial structure, microscopic data from experimental and theoretical studies (e.g., statistical mechanical and quantum mechanical) of the metal-water interface are needed. 

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. 

Mesoscopic parameters, such as the full-diffusion tensor and potential V, are usually determined phenomenologically or from complementary approaches. For instance, dissipative properties described by the diffusion tensor can be obtained on the basis of hydrodynamic modeling (see below). The internal potential can be evaluated as a potential energy surface scan over the torsional angle 6. For small molecules this operation can be easily conducted at the DFT level, while for big molecules such as proteins, mixed quantum mechanical/molecular mechanics (QM/MM) methodologies can be employed. 

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