Equilibrium statistical mechanics quantum

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The coimection between statistical mechanics and thennodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a hannonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

Progress in the theoretical description of reaction rates in solution of course correlates strongly with that in other theoretical disciplines, in particular those which have profited most from the enonnous advances in computing power such as quantum chemistry and equilibrium as well as non-equilibrium statistical mechanics of liquid solutions where Monte Carlo and molecular dynamics simulations in many cases have taken on the traditional role of experunents, as they allow the detailed investigation of the influence of intra- and intemiolecular potential parameters on the microscopic dynamics not accessible to measurements in the laboratory. No attempt, however, will be made here to address these areas in more than a cursory way, and the interested reader is referred to the corresponding chapters of the encyclopedia. 

In equilibrium statistical mechanics involving quantum effects, we need to know the density matrix in order to calculate averages of the quantities of interest. This density matrix is the quantum analog of the classical Boltzmann factor. It can be obtained by solving a differential equation very similar to the time-dependent Schrodinger equation... 

Marcus uses the Born-Oppenheimer approximation to separate electronic and nuclear motions, the only exception being at S in the case of nonadiabatic reactions. Classical equilibrium statistical mechanics is used to calculate the probability of arriving at the activated complex only vibrational quantum effects are treated approximately. The result is... 

MSN.51.1. Prigogine and F. Mayne, On the application of non-equilibrium statistical mechanics to quantum held theory, Phys. Lett. 21, 42—44 (1966). 

Raymond Kapral carried out his doctoral studies on molecular quantum mechanics at Princeton University and received his Ph.D. in 1967. He pursued his postdoctoral work at the Massachusetts Institute of Technology where his interests switched to non-equilibrium statistical mechanics. He then took a faculty position at the University of Toronto in 1969 where he is currently Professor of Chemistry. He is a Fellow of the Royal Society of Canada and received the Palladium Medal of the Chemical Institute of Canada in 2003. European collaborations have not only provided an opportunity to mix quantum and classical mechanics but also to mix science with good food and wine. 

The monomer label u is now interpreted as the Trotter (imaginary) time, and M as the mass of the quantum particle. The density matrix is relevant to the equilibrium statistical mechanics of a quantum particle, such as an electron in a dirty metal. 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

John S. Dahler, b. 1930, B.S. (1951) and M.S. (1952) University of Wichita Ph.D. (1955), University of Wisconsin at Madison University of Minnesota, Departments of Chemistry and of Chemical Engineering and Materials Science, Asst. Prof. (1959-61), Assoc. Prof. (1961-64), Prof. (1964-). Research Interests Non-equilibrium statistical mechanics and quantum theory of collisions. 

The most fundamental description of processes, in the present context, would be based on molecular considerations. A molecular description is distinguished by the fact that it treats an arbitrary system as if it were composed of individual entities, each of which obeys certain rules. Consequently, the properties and state variables of the system are obtained by summing over all of the entities. Quantum mechanics, equihbrium and non-equilibrium statistical mechanics, and classical mechanics are typical methods of analysis, by which the properties and responses of the system can be calculated. 

In going from statics to dynamics we need new experimental tools and also theoretical machinery that allows for the dependence on time. This means that the stationary states that are usually the subject of an introductory quantum mechanics course have to be extended to non-stationary ones. Fairly often, classical dynamics is sufficient to describe the time evolution but there are a number of interesting exceptions. Non-equilibrium statistical mechanics is necessary to describe systems with many degrees of freedom and their far-from-equilibrium pattern formation. 

Cao, J., Voth, G.A. The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties. J. Chem. Phys. 100 (1994) 5093-5105 II Dynamical properties. J. Chem. Phys. 100 (1994) 5106-5117 III. Phase space formalism and nalysis of centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6157-6167 IV. Algorithms for centroid molecular dynamics. J. Chem. Phys. 101 (1994) 6168-6183 V. Quantum instantaneous normal mode theory of liquids. J. Chem. Phys. 101 (1994) 6184 6192. 

Chemistry can be divided (somewhat arbitrarily) into the study of structures, equilibria, and rates. Chemical structure is ultimately described by the methods of quantum mechanics equilibrium phenomena are studied by statistical mechanics and thermodynamics and the study of rates constitutes the subject of kinetics. Kinetics can be subdivided into physical kinetics, dealing with physical phenomena such as diffusion and viscosity, and chemical kinetics, which deals with the rates of chemical reactions (including both covalent and noncovalent bond changes). Students of thermodynamics learn that quantities such as changes in enthalpy and entropy depend only upon the initial and hnal states of a system consequently thermodynamics cannot yield any information about intervening states of the system. It is precisely these intermediate states that constitute the subject matter of chemical kinetics. A thorough study of any chemical reaction must therefore include structural, equilibrium, and kinetic investigations. 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

The development of theoretical chemistry ceased at about 1930. The last significant contributions came from the first of the modern theoretical physicists, who have long since lost interest in the subject. It is not uncommon today, to hear prominent chemists explain how chemistry is an experimental science, adequately practiced without any need of quantum mechanics or the theories of relativity. Chemical thermodynamics is routinely rehashed in the terminology and concepts of the late nineteenth century. The formulation of chemical reaction and kinetic theories take scant account of statistical mechanics and non-equilibrium thermodynamics. Theories of molecular structure are entirely classical and molecular cohesion is commonly analyzed in terms of isolated bonds. Holistic effects and emergent properties that could... 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

Giauque, whose name has already been mentioned in connection with the discovery of the oxygen isotopes, calculated Third Law entropies with the use of the low temperature heat capacities that he measured he also applied statistical mechanics to calculate entropies for comparison with Third Law entropies. Very soon after the discovery of deuterium Urey made statistical mechanical calculations of isotope effects on equilibrium constants, in principle quite similar to the calculations described in Chapter IV. J. Kirkwood s development showing that quantum mechanical statistical mechanics goes over into classical statistical mechanics in the limit of high temperature dates to the 1930s. Kirkwood also developed the quantum corrections to the classical mechanical approximation. 

Consequently, while I jump into continuous reactors in Chapter 3, I have tried to cover essentially aU of conventional chemical kinetics in this book. I have tried to include aU the kinetics material in any of the chemical kinetics texts designed for undergraduates, but these are placed within and at the end of chapters throughout the book. The descriptions of reactions and kinetics in Chapter 2 do not assume any previous exposure to chemical kinetics. The simplification of complex reactions (pseudosteady-state and equilibrium step approximations) are covered in Chapter 4, as are theories of unimolecular and bimolecular reactions. I mention the need for statistical mechanics and quantum mechanics in interpreting reaction rates but do not go into state-to-state dynamics of reactions. The kinetics with catalysts (Chapter 7), solids (Chapter 9), combustion (Chapter 10), polymerization (Chapter 11), and reactions between phases (Chapter 12) are all given sufficient treatment that their rate expressions can be justified and used in the appropriate reactor mass balances. 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

The virial isotherm equation, which can represent experimental isotherm contours well, gives Henry s law at low pressures and provides a basis for obtaining the fundamental constants of sorption equilibria. A further step is to employ statistical and quantum mechanical procedures to calculate equilibrium constants and standard energies and entropies for comparison with those measured. In this direction moderate success has already been achieved in other systems, such as the gas hydrates 25, 26) and several gas-zeolite systems 14, 17, 18, 27). In the present work AS6 for krypton has been interpreted in terms of statistical thermodynamic models. 

Ray Kapral came to Toronto from the United States in 1969. His research interests center on theories of rate processes both in systems close to equilibrium, where the goal is the development of a microscopic theory of condensed phase reaction rates,89 and in systems far from chemical equilibrium, where descriptions of the complex spatial and temporal reactive dynamics that these systems exhibit have been developed.90 He and his collaborators have carried out research on the dynamics of phase transitions and critical phenomena, the dynamics of colloidal suspensions, the kinetic theory of chemical reactions in liquids, nonequilibrium statistical mechanics of liquids and mode coupling theory, mechanisms for the onset of chaos in nonlinear dynamical systems, the stochastic theory of chemical rate processes, studies of pattern formation in chemically reacting systems, and the development of molecular dynamics simulation methods for activated chemical rate processes. His recent research activities center on the theory of quantum and classical rate processes in the condensed phase91 and in clusters, and studies of chemical waves and patterns in reacting systems at both the macroscopic and mesoscopic levels. 

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

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