Mathematical Description of the Molecules

There are many cases in polymer science where a simple mathematical description of the polymer suffices to cover its physical properties and those of networks formed from it. The general theoretical situation will be reviewed in Sect. 5, but it is appropriate at this point to see if there is a reasonably concise way to describe the polymer. Clearly if a polymer varies all along its length and has properties dependent on that variation there is no easy way e.g., it is useless to describe DNA or a protein in a simple way if their biological interactions are being studied. But the molecules described in the previous and discussed in this and the next Sections do have repeated properties and can be reasonably described. 

A different kind of mathematics then applies via the Wiener integral where the probability over large distances of finding a curve R(s) is 

For stiff or stiffish molecules this approach is not appropriate (more of this in Sect. 5). 

In differential geometry it is shown that there is a set of three vectors, t tangent, n normal, b binormal which emerge from differentiating R(s) according to the Serret-Frenet formulae  

A curve is completely described by k and t and this description k(s), t(s) is independent of any coordinate system, i.e., is intrinsic. The simple law of resistance to bending takes the form of an energy H = JeR ds and therefore a weight exp( — jeR ds/kT). 

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Thus molecular modeling can be defined as the generation, manipulation, calculation, and prediction of realistic molecular structures and associated physicochemical as well as biochemical properties by the use of a computer. It is primarily a mean of communication between scientist and computer, the imperative interface between human-comprehensive symbolism, and the mathematical description of the molecule. The endeavor is made to perceive and recognize a molecular structure from its symbolic representations with a computer. Thus functions of the molecular modeling include ... 

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. 

Quantum mechanics (QM) is the correct mathematical description of the behavior of electrons and thus of chemistry. In theory, QM can predict any property of an individual atom or molecule exactly. In practice, the QM equations have only been solved exactly for one electron systems. A myriad collection of methods has been developed for approximating the solution for multiple electron systems. These approximations can be very useful, but this requires an amount of sophistication on the part of the researcher to know when each approximation is valid and how accurate the results are likely to be. A significant portion of this book addresses these questions. 

Since diversity is a collective property, its precise quantification requires a mathematical description of the distribution of the molecular collection in a chemical space. When a set of molecules are considered to be more diverse than another, the molecules in this set cover more chemical space and/or the molecules distribute more evenly in chemical space. Historically, diversity analysis is closely linked to compound selection and combinatorial library design. In reality, library design is also a selection process, selecting compounds from a virtual library before synthesis. There are three main categories of selection procedures for building a diverse set of compounds cluster-based selection, partition-based selection, and dissimilarity-based selection. 

Each reflection is the result of diffraction from complicated objects, the molecules in the unit cell, so the resulting wave is complicated also. Before considering how the computer represents such an intricate wave, let us consider mathematical descriptions of the simplest waves. 

To calculate the static thermodynamic and molecular ordering properties of a system of molecules, the configurational partition function Qc of the system must be derived. Qc does not contain the kinetic energy, intramolecular and intermolecular vibrations, and very small rotations about molecular bonds. Qc does contain terms which deal with significant changes in the shapes of the molecules due to rotations about semiflexible bonds (such as about carbon-carbon bonds in n-alkyl [i.e., (-CH2-)X] sections) in a molecule. For mathematical tractability in deriving Qc, the description of the molecules in continuum space is mapped onto a... 

Polymers, unlike metals, find wide application as completely amorphous solids. The two amorphous forms of greatest interest are the glasses and the elastomers or rubbers. Polymers found their earliest technical application as elastomers over 100 years ago, and it is in this form that they make a unique contribution to society. In examining the structure of the amorphous state we shall confine attention to the really significant point, which is the conformation of the molecule. It will be shown that the conformation resembles that of the molecule in dilute solution sketched at the end of the previous chapter. By means of a simple model a mathematical description of the conformation will be obtained which has fundamental and practical consequences for the mechanical properties of elastomers, the stress-strain curve in particular. 

Mathematical description of the ODMR experiment on single molecules... 

An important advantage of using colloidal crystals as nanoporous membranes is their highly ordered nature, which allows using accurate mathematical descriptions of the transport rate [21-25]. The effective diffusivity of molecules in the fee lattice Dfcc, can be expressed as (8/t)Dsoi, where D oi is the diffusivity of molecules in... 

It is not possible to promote an electron in a real atom of carbon to give a real organic molecule. A diamond does not promote an electron from one orbital to another to transform itself into something else. It is possible, however, to start with a mathematical description of the atomic orbitals in carbon and have the program generate a solution based on the electrons being moved into four identical orbitals in molecular carbon. 

The current practical limit for ab-initio codes is around 50 first row atoms. With new algorithms and direct methods this should be extended to 100-200, or possibly more, first row atoms within the next few years. At the present time, the study of larger molecules requires some simplifications to be made. These can take the form of approximations to the Fock Matrix using simple mathematical descriptions of the physics or, alternatively, experimental data can be used to calibrate a parameterised form of the Fock matrix. This is the basis of the widely used MNDO, MINDO/3 and AMI semi-empirical methods (Ref 15). 

The mathematical description of the model is out of the scope of this paper. Briefly, in this model, each reactant beam density is fitted to gaussian radial and temporal distribution functions, the spread in relative translational energy is neglected and the densities are assumed to be constant within the probed volume, which is smaller than the reaction zone. These assumptions result in a simple analytic expression of the overlap integral. Calculations are carried out for each rovibrational state of the outcoming molecule and for extreme velocity vector orientations, i.e, forwards and backwards. An example of the correction function, F, obtained for the A1 + O2 reaction at = 0.49 eV is displayed on Fig. 1, together with the... 

This completes the mathematical description of the theory which, as we have already seen, provides a good account of the orientational order and thermodynamic properties of nematogens composed of flexible molecules of low molecular weight. 

Although intrinsic reaction coordinates like minima, maxima, and saddle points comprise geometrical or mathematical features of energy surfaces, considerable care must be exercised not to attribute chemical or physical significance to them. Real molecules have more than infinitesimal kinetic energy, and will not follow the intrinsic reaction path. Nevertheless, the intrinsic reaction coordinate provides a convenient description of the progress of a reaction, and also plays a central role in the calculation of reaction rates by variational state theory and reaction path Hamiltonians. 

Adenine as an isolated molecule has no symmetry elements and therefore might mathematically be considered chiral however, as in the case of glycine (Section 1.2.1), this description is not useful in chemistry since the enantiomers differ only by inversion through the weakly pyramidal nitrogen atom of the amine functionality, the main body of the molecule being planar. The inversion corresponds to a low-frequency vibration and a low-energy barrier such that single enantiomers... 

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