Quantum statistical mechanical models

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

The rich metric structure of macroscopic thermodynamics also presents unusually stringent tests of theoretical models. Attempts to understand thermodynamic phenomena at a molecular level seem to demand improved dynamical and quantum statistical thermodynamic models that adequately incorporate the subtleties of quantum-mechanical valency and bonding interactions. Development of such models is an active area of modem physical chemistry research, but a more complete survey of the current molecular theory of gases and liquids is beyond the scope of the present work. 

Davis and Oppenheim examined the validity of the IBC model from the point of view of quantum statistical mechanics using a generalized Wigner distribution function. They thus avoided the weak coupling approximation used by both Fixman and Zwanzig. They pointed out an inconsistency in Herzfeld s argument, since if a molecule undergoes a... 

There have been several other methods proposed for the statistical mechanical modeling of chemical reactions. We review these techniques and explain their relationship to RCMC in this section. These simulation efforts are distinct from the many quantum mechanical studies of chemical reactions. The goal of the statistical mechanical simulations is to find the equilibrium concentration of reactants and products for chemically reactive fluid systems, taking into account temperature, pressure, and solvent effects. The goals of the quantum mechanics computations are typically to find transition states, reaction barrier heights, and reaction pathways within chemical accuracy. The quantum studies are usually performed at absolute zero temperature in the gas phase. Quantum mechanical methods are confined to the study of very small systems, so are inappropriate for the assessment of solvent effects, for example. 

Since Hagg (1962) had aigued the importance of the quasi-local structures in the field theoretical models, they have been applied to the quantum statistical mechanics. 

Critics of first-principles models often point out that this AE corresponds to immobile atoms at zero Kelvin. For most catalytically interesting problems, the quantum mechanical AE is the largest contribution to a finite temperature reaction enthalpy or free energy. Statistical mechanical models can be used to determine and compute finite temperature and entropy contributions when necessary. In practice, the greater challenge is to solve the Schrodinger equation sufficiently accurately to give chemically useful values of AE. 

The matrix a , defined in Appendix B, is a contravariant metric tensor in the infinitely stiff coordinates. Each det (a ) term can be evaluated directly using a recursion relation [102,123] or via the Fixman relation, Eq. (82). In a quantum mechanics-based statistical mechanics model, the effect of infinite stiffness on each partition function is not ... 

The Quantum Statistical Mechanics of a Simple Model System... 

As reactants transfonn to products in a chemical reaction, reactant bonds are broken and refomied for the products. Different theoretical models are used to describe this process ranging from time-dependent classical or quantum dynamics [1,2], in which the motions of individual atoms are propagated, to models based on the postidates of statistical mechanics [3], The validity of the latter models depends on whether statistical mechanical treatments represent the actual nature of the atomic motions during the chemical reaction. Such a statistical mechanical description has been widely used in imimolecular kinetics [4] and appears to be an accurate model for many reactions. It is particularly instructive to discuss statistical models for unimolecular reactions, since the model may be fomuilated at the elementary microcanonical level and then averaged to obtain the canonical model. 

Thermal Properties at Low Temperatures For sohds, the Debye model developed with the aid of statistical mechanics and quantum theoiy gives a satisfactoiy representation of the specific heat with temperature. Procedures for calculating values of d, ihe Debye characteristic temperature, using either elastic constants, the compressibility, the melting point, or the temperature dependence of the expansion coefficient are outlined by Barron (Cryogenic Systems, 2d ed., Oxford University Press, 1985, pp 24-29). 

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