Statistical independence, molecular similarity

Section 15.4 provides a discussion of similarity measures, which depend on three factors (1) the representation used to encode the desired molecular and chemical information, (2) whether and how much information is weighted, and (3) the similarity function (sometimes called the similarity coefficient) that maps the set of ordered pairs of representations onto the unit interval of the real line. Each of these factors is discussed in separate subsections. Section 15.5 presents a discussion of a number of questions that address significant issues associated with MSA Does asymmetric similarity have a role to play Do two-dimensional (2D) similarity methods perform better than three-dimensional (3D) methods Do data fusion and consensus similarity methods exhibit improved results Are different similarity measures statistically independent How do we compare similarity methods Can similarity measures be validated S ection 15.6 provides a discussion of activity landscapes... 

It is often possible to obtain similar or identical results from statistical mechanics and from thermodynamics, and the assumption that a system will be in a state of maximal probability in equilibrium is equivalent to the law of entropy. The major difference between the two approaches is that thermodynamics starts with macroscopic laws of great generality and its results are independent of any particular molecular model of the system, while statistical methods always depend on some such model. 

Finally, there is worthy to point out the fact that inside the molecular organic crystals the hyper-symmetry prevails, due to the frequency with which the independent molecules have similar conformations, as revealed from the analysis of the statistics of the Cambridge stmctural database (Sona Gautham, 1992). 

In molecular crystals or in crystals composed of complex ions it is necessary to take into account intramolecular vibrations in addition to the vibrations of the molecules with respect to each other. If both modes are approximately independent, the former can be treated using the Einstein model. In the case of covalent molecules specifically, it is necessary to pay attention to internal rotations. The behaviour is especially complicated in the case of the compounds discussed in Section 2.2.6. The pure lattice vibrations are also more complex than has been described so far . In addition to (transverse and longitudinal) acoustical phonons, i.e. vibrations by which the constituents are moved coherently in the same direction without charge separation, there are so-called optical phonons. The name is based on the fact that the latter lattice vibrations are — in polar compounds — now associated with a change in the dipole moment and, hence, with optical effects. The inset to Fig. 3.1 illustrates a real phonon spectrum for a very simple ionic crystal. A detailed treatment of the lattice dynamics lies outside the scope of this book. The formal treatment of phonons (cf. e(k), D(e)) is very similar to that of crystal electrons. (Observe the similarity of the vibration equation to the Schrodinger equation.) However, they obey Bose rather than Fermi statistics (cf. page 119). 

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