Statistical mechanics formulations

It is possible to go beyond the SASA/PB approximation and develop better approximations to current implicit solvent representations with sophisticated statistical mechanical models based on distribution functions or integral equations (see Section V.A). An alternative intermediate approach consists in including a small number of explicit solvent molecules near the solute while the influence of the remain bulk solvent molecules is taken into account implicitly (see Section V.B). On the other hand, in some cases it is necessary to use a treatment that is markedly simpler than SASA/PB to carry out extensive conformational searches. In such situations, it possible to use empirical models that describe the entire solvation free energy on the basis of the SASA (see Section V.C). An even simpler class of approximations consists in using infonnation-based potentials constructed to mimic and reproduce the statistical trends observed in macromolecular structures (see Section V.D). Although the microscopic basis of these approximations is not yet formally linked to a statistical mechanical formulation of implicit solvent, full SASA models and empirical information-based potentials may be very effective for particular problems. 

The molecular approach, adopted throughout this book, starts from the statistical mechanical formulation of the problem. The interaction free energies are identified as correlation functions in the probability sense. As such, there is no reason to assume that these correlations are either short-range or additive. The main difference between direct and indirect correlations is that the former depend only on the interactions between the ligands. The latter depend on the maimer in which ligands affect the partition function of the adsorbent molecule (and, in general, of the solvent as well). The argument is essentially the same as that for the difference between the intermolecular potential and the potential of the mean force in liquids. 

Chapter B outlines a typical statistical-mechanical formulation of polypeptide conformations in terms of these three parameters and describes its use for the evaluation of s and tr from observed helix-coil transition curves. Then the reported values of AH and a for selected polypeptide-solvent pairs are given and their implications are briefly discussed from a molecular standpoint. Here AH denotes the transition enthalpy derived from s by a thermodynamic relation. 

Experimentally, fN is determined as a function of temperature T, solvent composition x, and degree of polymerization N fN = F xp(T, x, N) here Fexp stands for the experimentally obtained functional form. On the other hand, statistical-mechanical formulations allow fN to be expressed in terms of s, a, and N fN = Flhcor(s, a, N), where Fth denotes a theoretical function. Then it should be possible from a comparison of F p and Flheor to determine s and a as functions of T and x. How can this be achieved Since the pioneering work of Zimm et al. (17) in 1959 various methods have been proposed. Typical approaches are outlined below for the experimental situation in which a thermally induced helix-coil transition is observed. For most of the proposed methods such transition curves must be available for a series of samples of different N. Preferably, these samples ought to be sharp in molecular weight distribution and cover as wide a range of N as possible. 

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. 

A general scheme, based on a rigorous statistical mechanical formulation, for obtaining the interaction between two colloidal particles in a fluid has been outlined. The implementation of the theory is in its early stages. In the DLVO theory and the theory of HLC, it is assumed that the various contributions can be added together. In the MSA, the hard core and electrostatic terms will be additive. However, it is only at low electrolyte concentration that the effect of dipole orientation and the repulsive contribution of the double layer overlap will be additive. There is no reason to believe (or disbelieve) that the van der Waals term should also be additive. 

Other lattice models are noteworthy as well. Roe (1974), for instance, developed a statistical mechanical formulation for an adsorbed layer capable of exchanging polymer and solvent with the bulk solution. The grand canonical ensemble, first introduced by DiMarzio and Rubin (1971),... 

The next four chapters address several applications of MD to water and aqueous solutions. Floris and Tani describe tiie development of force fields for water-water and water-ion interactions in Chapter 10. Balbuena et al. analyze force fields for cation-water systems introducing new descriptions of short-range interactions. Li and Tomkinson assess the estimation of neutron scattering spectra of ice by MD and lattice dynamics simulations in Chapter 12. Tanaka in Chapter 13 discusses the stability and dynamics of ice and clathrate hydrate using Monte Carlo, MD, lattice dynamics simulations, and a statistical mechanical formulation. 

Huang K (1963) Statistical Mechanics. John Wiley Sons, Inc., New York Hulburt HM, Katz S (1964) Some Problems in Particle TechnologyiA statistical mechanical formulation. Chem Eng Sci 19 555-574... 

Hulburt, H. M. Katz, S. 1964 Some problems in particle technology. A statistical mechanical formulation. Chemical Engineering Science 19, 555-574. 

The rate of elementary reactions is statistical-mechanically formulated without the limitation mentioned in the introduction and the rate expression thus obtained is discussed and developed in this section. 

Mitchell (1993) indicates that there is no rigorous proof of the statistical mechanics formulation of the rate process theory, but it does tend to describe the behavior of many real systems. The method although interesting is not normally used in practice. 

The degree to which explanation depends on comparison between the collision, thermodynamic, and quantum mechanical/statistical mechanical formulations of molecular quantities. 

An important area of application for QM methods has been determining and describing reaction pathways, energetics, and transition states for reaction processes between small species. QM-derived first and second derivatives of energy calculated at stable and saddle points on PES can be used under statistical mechanics formulations [33, 34] to yield enthalpies and free energies of structures in order to determine their reactivity. Transition state theory and idealized thermodynamic relationships (e.g., AG[Po—>P] = kTln[P/Po]) allow temperature and pressure regimes to be spanned when addressing simple gas phase and gas-surface interactions. 

In the last 1930 s Kirkwood (16) and FrHhlich (17) independently developed statistical mechanical formulations which avoided introduction of a molecular cavity and instead introduced the resultant moment of representative molecules... 

Hulburt, H.M. and S. Katz. Some Problems in Particle Technology A Statistical Mechanical Formulation, Chem. Eng. Sci., 19 (1964), 555-574. 

The quantum statistical mechanical theory of relaxation phenomena (QSM theory) is a maximum entropy approach to nonequilibrium processes that is in the same spirit as the statistical mechanical formulations of equilibrium and nonequilibrium thermodynamics given in Sections 3.3 and 5.4. As in the statistical mechanical formulation of nonequilibrium thermodynamics, the maximum entropy principle is assumed to apply to systems out of equilibrium. 

The constant term may be identified with So we note that So — k log go, where is the statistical weight of the ground state, that is, the number of eigenfunctions corresponding to this state. This is the statistical-mechanical formulation of the third law of thermodynamics. If, as is usually true, the ground state of a system is non-... 

The statistical mechanical formulation of the transition-state theory (TST) can also be applied to the kinetics of reactions on surfaces. Let us consider the following reaction ... 

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