Quantum-statistical treatments

The classical model predicts thermal motion to vanish at very low temperatures, in contradiction to the zero-point vibrations which follow from the quantum-mechanical treatment of oscillators. For temperatures at which hv % kBT, the spacing of the discrete energy levels cannot be neglected, so the classical model is no longer valid. 

For a harmonic oscillator with frequency v, the potential energy equals M7kJu2 where k. is the force constant. Using the virial theorem, which states 

According to statistical mechanical theory, the total energy of the oscillator is a function of the partition function z, through the relation 

Substitution into Eq. (2.48), and adding the zero-point energy hv/2, gives 

Combining this result with Eq. (2.47), and substitution of k — 47i2mv, where in is the reduced mass of the harmonic oscillator, gives the quantum-statistical temperature dependence of m2 as 

---

In [11] the alpha-particles were included into the EOS, and detailed comparisons of the outcome with respect to the alpha-particle contribution has been made. We will elaborate this item further, first by using a systematic quantum statistical treatment instead of the simplifying concept of excluded volume, second by including also other (two- and three-particle) correlations. 

Computational Techniques. For evaluation of kf by eqs. (7) and (8), the vibrational energy level sum at a given total energy must be found. It has been shown, both through experiment and computation,9 19 20 that for polyatomic molecules, even with energies above 100 kcal. mole-1, it is necessary to use a quantum statistical treatment to find this sum. Classical approximations are totally inadequate and drastically in error. High speed machine computational techniques and simplified approximation formulas have been developed, which allow this quantum-statistical summation to be done with relative ease these methods are described and summarized in Appendix I. 

It comes out, that polymer network is a very specific ensemble of entities [II] in terms of classification in the introductory chapter of this book, it has been possible to transform some nanoeffects to macroscale. At the same time, as we shall see, it is expected to be liable to statistical, mechanical, and even quantum statistics treatment. Recent experiments with networks of nanotube type, as quantum wires capable of ballistic transfer of charge [12,13], can be reported as the extension of a such approach to carbon-carbon network. 

Achieving the full potential of mechanistic studies will require the development of new simulation methodologies. For example, proton transport systems require truly quantum statistical treatments but efficient techniques for the incorporation of quantum effects into large-scale simulations are currently immature. In a similar vein, techniques for the inclusion of polarization effects need further development. Furthermore, improvements in sampling methods may greatly accelerate equilibration of slow degrees of freedom in the membrane and improve mixing in multicomponent systems. 

Object.—Quantum statistics was discussed briefly in Chapter 12 of The Mathematics of Physics and Chemistry, and as far as elementary treatments of quantum statistics are concerned,1 that introductory discussion remains adequate. In recent years, however, a spectacular development of quantum field theory has presented us with new mathematical tools of great power, applicable at once to the problems of quantum statistics. This chapter is devoted to an exposition of the mathematical formalism of quantum field theory as it has been adapted to the discussion of quantum statistics. The entire structure is based on the concepts of Hilbert space, and we shall devote a considerable fraction of the chapter to these concepts. 

What was the distinction between quantum chemistry and chemical physics After the Journal of Chemical Physics was established, it was easy to say that chemical physics was anything found in the new journal. This included molecular spectroscopy and molecular structures, the quantum mechanical treatment of electronic structure of molecules and crystals and the problem of chemical binding, the kinetics of chemical reactions from the standpoint of basic physical principles, the thermodynamic properties of substances and calculation by statistical mechanical methods, the structure of crystals, and surface phenomena. 

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 last decade has witnessed an intense interest in the theory of radiative association rate coefficients because of the possible importance of the reactions in the interstellar medium and because of the difficulty of measuring these reactions in the laboratory. Several theories have been proposed these are all directed toward systems of at least three or four atoms and utilize statistical approximations to the exact quantum mechanical treatment. The utility of these treatments can be partially gauged by using them to calculate three body rate coefficients which can be compared with laboratory measurements. In order to explain these theories briefly, it would be helpful to write down equations for the mechanism of association reactions. Consider two species A+ and B that come together with bimolecular rate coefficient kj to form a complex AB+ which can then be stabilized radiatively with rate coefficient kr, be stabilized collisionally with helium with rate coefficient kcoll, or redissociate with rate coefficient k j ... 

The above statement must not be taken in a pejorative sense. It is agreed that a knowledge of the component systems is a condition sine qua non. But only as long it is realized that the theoretical study of a macromolecule should be undertaken by a quantum statistical method, after the individual systems have been fully studied. In this connection, the theories developed in the quantum mechanical treatments of fluids and solids can be of use. The ultimate purpose is the development of a theory that will account for the specific reactivity of macromolecules. 

Essentially, the problem falls into two parts. Firstly, the calculation of the multi-body reaction potential surface and secondly, the determination of the properties of the reaction products from the knowledge of the potential surface. The reverse process of inverting experimental data to yield a potential energy surface is more complicated and has rarely been attempted. The calculation of potential surfaces and of product distributions may be carried out at various levels of sophistication using classical, semi-classical or full quantum mechanical treatments. Gross features of the reaction potential surfaces may be related to various product properties by simplistic model calculations. Statistical theories may also be used in cases where the lifetime of the collision is long enough to justify their use. 

For chemically simple molecules such as this, the relative intensities of a multiple are symmetric about their mid-point and are determined by statistical considerations, the intensities being (more or less) proportional to the coefficients of the binomial expansion. In other words, they go as Pascal s triangle, in a doublet 1 1, a triplet 1 2 1, a quartet 1 3 3 1, etc. The spacing between all the split resonances, the triplet for CH3 and the quartet for CH is the same and is independent of the strength of the applied field. This spacing depends on the interaction between the groups and is described in the quantum mechanical treatment by a spin-spin coupling constant, J. 

Quantum effects can be recovered by quantum simulations. Currently there are two main types of quantum simulation methods used. One is based on the time-dependent Schrodinger equation. The other is based on Feynman s path integral (PI) quantum statistical mechanics. [7,8] The former is usually complicated in mathematical treatment and needs also large computational resources. Currently, it can only be used to simulate some very limited systems. [77] MD simulations based on the latter have been used more than a decade and are gaining more and more popularity. The main reason is that in PIMD simulations, the quantum systems are mapped onto corresponding classical systems. In other words the quantum effects can be recovered by making a series of classical simulations with different effective potentials. 

I7e. Heat Capacities at High Temperatures.—Although the theoretical treatment of heat capacities requires the limiting high temperature value to be 3/2, i.e., 5.96 cal. deg. g. atom , experimental determinations have shown that with increasing temperature Cv increases still further. The increase is, however, gradual for example, tfie heat capacity of silver is 5.85 cal. deg. g. atom at 300° K and about 6.5 cal. deg. g. atom at 1300° K. This increase is attributed mainly to the relatively free electrons of the metal behaving as an electron gas. By the use of the special form of quantum statistics, viz., Fermi-Dirac statistics, applicable to electrons, the relationship... 

Quantitative models is a heterogeneous group of models expressed in mathematical language. This includes what can be called hard models of general applicability, e.g. thermodynamic models, quantum mechanical models, absolute rate theory, as well as soft models or local models, usually expressed in terms of analogy and similarity, e.g. linear free energy relationships (LFERs), correlations for spectroscopic structural determination, empirical determined kinetic models, and as we shall see, models obtained by statistical treatment of experimental data from properly designed experiments. 

At the present time, by far the most useful non-empirical alternatives to Cl are the methods based on density functional theory (DFT) . The development of DFT can be traced from its pre-quantum-mechanical roots in Drude s treatment of the electron gas" in metals and Sommerfeld s quantum-statistical version of this, through the Thomas-Fermi-Dirac model of the atom. Slater s Xa method, the laying of the formal foundations by... 

Until now, our formulation of statistical thermodynamics has been based on quantum mechanics. This is reflected by the definition of the canonical ensemble partition function Q, which turns out to be linked to matrix elements of the Hamiltonian operator H in Eq. (2.39). However, the systems treated below exist in a region of thermodjniamic state space where the exact quantum mechanical treatment may be abandoned in favor of a classic dc.scription. The transition from quantum to classic statistics was worked out by Kirkwood [22, 23] and Wigner [24] and is rarely discussed in standard texts on statistical physics. For the sake of completeness, self-containment, and as background information for the interested readers we summarize the key considerations in this section. 

By introducing the notion of various statistical physical ensembles in Section 2.2.1, we saw that wc can make the quantum iiurhaitical treatment consistent with several constraints imposed at the macroscopic level of description. That way we obtain an understanding of a thermal system at the microscopic level that is, we can interpret thermodynamic properties in terms of the interaction between the micro.scopic constituents forming a macroscopic system. 

---