Mathematical descriptions, molecular similarity

For the mathematical description and understanding of transport processes, it is advantageous for their descriptions to have several common characteristics, regardless of the nature of the transport quantity, to allow them to be treated in a similar manner. Without knowledge of their fundamental causes at the molecular level, which corresponds to their historical development, transport processes can be described with help from quantities that can be quantitatively measured on a macroscopic level. One such quantity is that of flux. 

We learned in Chapter 5 that each solution to the Schrodinger equation, called a wave function, represents an atomic orbital. The mathematical pictures of hybrid orbitals in valence bond theory can be generated by combining the wave functions that describe two or more atomic orbitals on a single atom. Similarly, combining wave functions that describe atomic orbitals on separate atoms generates mathematical descriptions of molecular orbitals. 

For an in-depth description of the concept of molecular similarity and the mathematical methods that have been applied in assessing similarity see, Doucet Weber (1996, ch. 11). 

Heat and mass transfer are analogous processes. Molecular diffusion in homogeneous materials or phases is similar to heat transfer. Convective diffusion or convection in homogeneous materials or phases corresponds to heat transfer by convection. Mass transfer at the phase boundary corresponds to heat conduction. Mass transfer between phases occurs like heat transfer in several chronological steps. The slowest step controls the rate of the entire process. Thus the mathematical descriptions of heat and mass transfer operations are analogous. Calculation methods and approaches to calculate the heat transfer coefficients may similarly be used to calculate mass transfer coefficients. (See Table 1-18 in Chapter 1.7.2 for the analogy of heat and mass transfer.)... 

Application of valence bond theory to more complex molecules usually proceeds by writing as many plausible Lewis structures as possible which correspond to the correct molecular connectivity. Valence bond theory assumes that the actual molecule is a hybrid of these canonical forms. A mathematical description of the molecule, the molecular wave function, is given by the sum of the products of the individual wave functions and weighting factors proportional to the contribution of the canonical forms to the overall structure. As a simple example, the hydrogen chloride molecule would be considered to be a hybrid of the limiting canonical forms H—Cl, H Cr, and H C1. The mathematical treatment of molecular structure in terms of valence bond theory can be expanded to encompass more complex molecules. However, as the number of atoms and electrons increases, the mathematical expression of the structure, the wave function, rapidly becomes complex. For this reason, qualitative concepts which arise from the valence bond treatment of simple molecules have been applied to larger molecules. The key ideas that are used to adapt the concepts of valence bond theory to complex molecules are hybridization and resonance. In this qualitative form, valence bond theory describes molecules in terms of orbitals which are mainly localized between two atoms. The shapes of these orbitals are assumed to be similar to those of orbitals described by more quantitative treatment of simpler molecules. 

One of the key parameters for correlating molecular structure and chemical properties with bioavailability has been transcorneal flux or, alternatively, the corneal permeability coefficient. The epithelium has been modeled as a lipid barrier (possibly with a limited number of aqueous pores that, for this physical model, serve as the equivalent of the extracellular space in a more physiological description) and the stroma as an aqueous barrier (Fig. 11). The endothelium is very thin and porous compared with the epithelium [189] and often has been ignored in the analysis, although mathematically it can be included as part of the lipid barrier. Diffusion through bilayer membranes of various structures has been modeled for some time [202] and adapted to ophthalmic applications more recently [203,204]. For a series of molecules of similar size, it was shown that the permeability increases with octa-nol/water distribution (or partition) coefficient until a plateau is reached. Modeling of this type of data has led to the earlier statement that drugs need to be both... 

In the latter way of looking at the build-up of molecules, the successive addition of electrons to a positively charged system is reminiscent of the manner in which the atoms of the Periodic Table were considered in Chapter 1. Here again there are certain configurations permitted the electron clouds, and these cloud shapes (or probability density functions) can be described using quantum numbers. Such probability density descriptions are called molecular orbitals in analogy to the much simpler atomic orbitals. Although the initial setup and subsequent mathematical treatment for molecules are much more complicated than for atoms, there arise certain similarities between the two types of orbitals. 

These equations are all the more attractive in that similar mathematical forms can also be obtained from molecular considerations for the description of... 

Practical problems in the estimation of the lipophilicity of araliphatic and aliphatic compoimds led to the / hydrophobicity scales of Rekker and Leo/Hansch. However, all such descriptor scales depend on experimental determinations. New molecular descriptors were developed from scratch, starting with the work of Randic, Kier and Hall, i.e. the various molecular connectivity parameters %. Later the electrotopological state parameters and the Todeschini WHIM parameters were added. Whereas topological descriptors are mathematical constructs that have no unique chemical meaning, they are clearly related to some physicochemical properties and are suited to the description of compound similarities in a quantitative manner. Thus, despite several critical comments in the past, they are now relatively widely used in QSAR studies. Only a meaningless and excessive application in quantitative models, as far as the number of tested and included variables is concerned, still deserves criticism. 

Similar concerns apply to molecular orbitals. One constructs molecular orbitals and populates them with electrons is a manner analogous to an individual atom by adopting the linear combination of atomic orbitals (LCAO) approximation. While this might lend the impression that molecular orbitals are merely an extension of atomic orbitals, they are conceptually distinct. An atomic orbital is a description of the state of motion of an electron subject to the influence of a single nucleus plus other electrons. But molecular orbitals describe electron motions in the field of two or more nuclei plus the other electrons and the use of the LCAO method is merely a matter of mathematical convenience (Gavroglu and Simoes 2012, p. 83). The delocalized character of molecular orbitals is conceptually quite distinct from the idea of atomic orbitals, and Mulliken - one of the originators of the molecular orbital approach - was at pains to distinguish his conceptual scheme from the methods employed to compute them (ibid, pp. 84—85). 

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