Classical statistical mechanics, conformational

There now exist several methods for predicting the free energy associated with a compositional or conformational change.7 These can be crudely classified into two types "exact" and "approximate" free energy calculations. The former type, which we shall discuss in the following sections, is based directly on rigorous equations from classical statistical mechanics. The latter type, to be discussed later in this chapter, starts with statistical mechanics, but then combines these equations with assumptions and approximations to allow simulations to be carried out more rapidly. 

While free energies can be theoretically determined in a number of ways, free energy calculations is generally understood to refer to a class of simulations that relate, through equations of classical statistical mechanics, the free energy difference between two different molecular states or conformations to a thermodynamic ensemble average that depends on potential energy properties of those states or conformations. That is,... 

Classical statistical mechanics state that Boltzmann s distribution law applies to a single closed system held at a fixed temperature, and populating different energy levels. In contrast, the database of known protein structures is a collection of minimum-energy conformations belonging to distinct proteins, the structural components of which occur in quite different environments. 

A valuable reference on the subject of single chains is the work by Flory, described in his book Statistical Mechanics of Chain Molecules. A new book on the subject is of interest. The reader may also wish to study the 1974 paper by Flory in Macromolecules, which serves as an excellent introduction to the rotational isomeric state theory.Another classic book is Hopfinger s Conformational Properties of Macromolecules... 

Recently, semiempirical molecular-orbital methods have been combined with force-field-based molecular-dynamics techniques into hybrid schemes The interesting part of the system is described by quantum chemistry, while the surroundings are treated by a classical force field. These hybrid schemes allow calculation of the energy and gradients fast enough for molecular dynamics simulations of hundreds of picoseconds (10s time steps) duration to be feasible. This provides sufficient sampling for the calculation of many statistical-mechanical properties. A short synopsis is given of work carried out at ETH Zurich on conformational equilibria in solution, reactions in solution and enzyme reactions. 

The classical texts on rotational isomers and the calculation of chain conformations are Flory PJ (1966) Statistical Mechanics of Chain Molecules. Interscience, New York. Volkenstein, MV (1963) Configurational Statistics of Polymeric Chains. Engl translation. Interscience, New York. 

The first of these developments is perturbation theory. Its application to solution theory was perhaps first made by H. C. Longuet-Higgins in his conformal solution theory (Longuet-Higgins 1951). The formal theory of statistical mechanical perturbation theory is very simple in the canonical ensemble. If denotes the intermo-lecular potential energy of a classical A-body system (not necessarily the sum of pair potentials), the central problem is to evaluate the partition function. 

As mentioned previously, the definition of an empirical potential establishes its physical accuracy those most commonly used in chemistry embody a classical treatment of pairwise particle-particle and n-body bonded interactions that can reproduce structural and conformational changes. Potentials are useful for studying the molecular mechanics (MM), e.g., structure optimization, or dynamics (MD) of systems whereby, from the ergodic hypothesis from statistical mechanics, the statistical ensemble averages (or expectation values) are taken to be equal to time averages of the system being integrated via (7). 

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... 

A further complication associated with the application of molecular mechanics calculations to relative stabilities is that strain energy differences correspond to A (AH) between conformers with similar chromophores (electronic effects) and an innocent environment (counter ions and solvent molecules), whereas relative stabilities are based on A (AG). The entropy term, TAS, can be calculated by partition functions, and the individual terms of AS include vibrational (5vib), translational (5 trans) and rotational (Arot) components, and in addition to these classical terms, a statistical contribution (5stat). These terms can be calculated using Eqs. 3.40-3.43tl21]. 

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