Statistically derived potentials

Einally, structural properties that depend directly neither on the data nor on the energy parameters can be checked by comparing the structures to statistics derived from a database of solved protein structures. PROCHECK-NMR and WHAT IE [94] use, e.g., statistics on backbone and side chain dihedral angles and on hydrogen bonds. PROSA [95] uses potentials of mean force derived from distributions of amino acid-amino acid distances. 

The AGf, AGAa AGAs. and AGss values, and, correspondingly, In fi and a values depend on the electric state of the surface, i.e., on the electrode potential or charge. This isotherm was deduced by Frumkin [i] (and named after him soon) as a general case of the -> Langmuir isotherm, which corresponds to a = 0. A statistical derivation of the Frumkin isotherm is available [ii] various model considerations and relations to other types of isotherms are discussed in [iii]. Another typical form of the Frumkin isotherm is... 

Many catalyst researchers are unaware of the potential benefits of statistical design of experiments. Others have had unfortunate experiences with so-called designed experiments because they underestimated the influence of experimental uncertainties on the reliability of the conclusions. In both cases familiarity with the fundamentals of statistical inference in the experimentation strategy is beneficial (4). Statistically derived strategies can certainly offer many important benefits, although they obviously cannot replace creativity or sound technical judgment. 

In the mid-1990s the six sigma approach was developed and can be seen as one of the most recent approaches to center all statistic-based information of control mechanism with the aim of deriving potentials and target for quality assurance and improvement (Fig. 3). 

The validity of (3.76) actually goes far beyond mixtures of perfect gases. Systems in which the chemical potentials of the components can be expressed by (3.76) are called ideal systems. A special case of ideal systems are dilute solutions. A statistical derivation of (3.76) for dilute solutions may be found in LANDAU-LIFSHITZ, Theoretical Physics, Vol. V (1968). In dilute solutions, the mole fraction of a solute is approximately given as x. = N. /N where is the number of moles of the solvent. This enables us to rewrite the chemical potential of a solute approximately in terms of its concentration c. as... 

A potential energy function Emin is computed for each size bin based on statistics derived from the minimized structures. 

An impressive number of studies has recently been devoted to constructing different versions of these potential terms. Among them two main philosophies have prevailed. Some authors work in the context of standard statistical models and derive scores expressed as logCfrequencies). " Others work in the context of statistical mechanics and derive potential terms expressed as —kT log(frequencies), where k is Boltzmann s constant and T is the temperature. " The two quantities differ only in units and sign, and are hence essentially equivalent. 

As discussed above, to identify states of the system as those for the reactant A, a dividing surface is placed at the potential energy barrier region of the potential energy surface. This is a classical mechanical construct and classical statistical mechanics is used to derive the RRKM k(E) [4]. 

The concept of corresponding states was based on kinetic molecular theory, which describes molecules as discrete, rapidly moving particles that together constitute a fluid or soHd. Therefore, the theory of corresponding states was a macroscopic concept based on empirical observations. In 1939, the theory of corresponding states was derived from an inverse sixth power molecular potential model (74). Four basic assumptions were made (/) classical statistical mechanics apply, (2) the molecules must be spherical either by actual shape or by virtue of rapid and free rotation, (3) the intramolecular vibrations are considered identical for molecules in either the gas or Hquid phases, and (4) the potential energy of a coUection of molecules is a function of only the various intermolecular distances. 

Formula for the chemical potentials have been derived in terms of the formation energy of the four point defects. In the process the conceptual basis for calculating point defect energies in ordered alloys and the dependence of point defect concentrations on them has been clarified. The statistical physics of point defects in ordered alloys has been well described before [13], but the present work represents a generalisation in the sense that it is not dependent on any particular model, such as the Bragg-Williams approach with nearest neighbour bond energies. It is hoped that the results will be of use to theoreticians as well as... 

With this database in hand, a simple question is asked [29] How different is a knowledge-based potential derived from this lattice database compared to the actual energy function used to construct the database If statistical errors are negligible and the knowledge-based method is perfect, the answer is expected to be They are exactly the same. ... 

---