The molecular pair approach

In this Appendix, the equivalence of the diffusion equation treatment and the molecular pair analysis is proved (see Chap, 8, Sect. 3.2) for the situation where there is a potential energy E/(r) between the reactants and the diffusion coefficient is tensorial and position-dependent. This Appendix is effectively a generalisation of the analysis of Berg [278]. The diffusion equation has a Green s function G(r, f r0, t0) which satisfies eqn. (161) 

The density distribution corresponding to the source of density at a point r0 and time i0f is (see Appendix A) 

Furthermore, this density satisfied the normal initial and boundary conditions 

Finally, the homogeneous density (termed a here) for reaction of species with a vast excess of the other follows from Green s theorem (Appendix A, Sect 3) as 

The rate coefficient is defined as the current of reactants together for the homogeneous density case 

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The diffusion equation analysis is discussed in Sect. 2. It has been used very much more frequently in studies of diffusion-limited reactions rates than the analysis based on molecular pair behaviour, which is discussed in Sect. 3. This is probably because the diffusion equation approach is rather more direct, clear and versatile than the molecular pair analysis (furthermore, time-dependent Green s functions are required for the molecular pair approach). Besides, the probability that a molecular pair will reencounter one another is often derived from a diffusion equation analysis in any case and under these circumstances the two approaches are identical. 

As an example of the application this work, Kapral [285] and Pagistas and Kapral [37] have considered the reaction rate between iodine atoms (or some other similar species) effectively distributed uniformly in solution. They compared their calculations with those of the diffusion equation analysis and with the molecular pair approach rather than compare rate coefficients, Kapral [285] compared the rate kernels (which are approximately the time derivatives of rate coefficients). Over long times, these kinetic theory and molecular pair rate kernels both reduce to the typical form of the Smoluckowski rate kernel. However, with parameters such as R — 0.43 nm and D = 6 x 10 9m2s 1, the time beyond which the rate kernels of kinetic theory and the Smoluchowski theory are in reasonably close agreement is 20 ps, a time much longer than the velocity... 

There seems little need or point to discuss much of the work done on the molecular pair analysis using random walk forms of h(f). Other than the original work by Noyes [5, 265], relatively few authors have been interested in the molecular pair approach. Northrup and Hynes [103] have discussed the motion of a reactant pair in detail. They noted that the probability of recombination of a pair formed in encounter contact is... 

Northrup and Hynes [103] have developed these ideas further and provided a direct comparison with the molecular pair analysis of Noyes (see Chap. 6, Sect. 2.3). They showed that the various parameters introduced by Noyes, and used also by other authors, can be defined by functions familiar in the diffusion equation analysis. Consequently, there seems no specific reason to advocate using the molecular pair approach in future. Those interested in this subject are urged to consult this excellent and readable article by Northrup and Hynes [103]. 

Very much more effort on the subject of diffusion-limited reaction rates has been devoted to theoretical aspects, most of which has been with the aid of the diffusion equation. Indeed, so much has now been written that there are many articles which have not even been mentioned here. Yet it should be emphasised that much of what can be usefully said about the theoretical analysis of reaction rates with the diffusion equation has been said, sometimes several times, for which the author takes some share of responsibility Both the subjects of homogeneous reaction and pair recombination have been exhaustively analysed. Because the molecular pair approach is identical to the diffusion equation analysis, if the Noyes h(t) expression is approximated by a diffusive Green s function, no further effort on the molecular pair approach is really necessary. 

In Sect. 3, the Noyes approach to analysing reaction rates based on the molecular pair approach is discussed [5]. Both this and the diffusion equation analysis are identical under conditions where the diffusion equation is valid and when the appropriate recombination reaction rate for a molecular pair is based on the diffusion equation. Some comments by Naqvi et al. [38] and Stevens [455] have obscured this identity. The diffusion equation is a valid approximation to molecular motion when the details of motion in a cage are no longer of importance. This time is typically a few picoseconds in a mobile liquid. When extrapolating the diffusion equation back to such times, it should be recalled that the diffusion is a continuum form of random walk [271]. While random walks can be described with both a distribution of jump frequencies and distances, nevertheless, the diffusion equation would not describe a random walk satisfactorily over times less than about five jump periods (typically 10 ps in mobile liquids). Even with a distribution of jump distances and frequencies, the random walk model of molecular motion does not represent such motion adequately well as these times (nor will the telegrapher s or Fokker-Planck equation be much better). It is therefore inappropriate to compare either the diffusion equation or random walk analysis with that of the molecular pair over such times. Finally, because of the inherent complexity of molecular motion, it is doubtful whether it can be described adequately in terms of average jump distances and frequencies. These jump characteristics are only operational terms for very complex quantities which derive from the detailed molecular motion of the liquid. For this very reason, the identification of the diffusion coefficient with a specific jump formula (e.g. D = has been avoided. 

An approach, similar to that employed in the analysis of tartrate mixtures, has been used for the chiral discrimination of amino acid (M/j/s) mixtures, using an amino acid of defined configuration as reference (S). The proton-bound trimers [S2-M H]+ form [S M H]+ and [S2H]+ fragments upon CID or MIKE decay (equations (9)-(12)). With two independent measurements of the fragmentation ratio [S-M-H] /[S2H] from either [S2-M -H] and [52-M5-H]" , the differences in binding energies can be determined. The relative gas phase basicities (GB) of the molecular pairs [S-M] and [S2] can be derived from equations (13) and (14). 

In Sect. 2.1, the timescale over which the diffusion equation is not strictly valid was discussed. When using the molecular pair analysis with an expression for h(f) derived from a diffusion equation analysis or random walk approach, the same reservations must be borne in mind. These difficulties with the diffusion equation have been commented upon by Naqvi et al. [38], though their comments are largely within the framework of a random walk analysis and tend to miss the importance the solvent cage and velocity relaxation effects. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

We now turn from the use of quantum mechanics and its description of the atom to an elementary description of molecules. Although most of the discussion of bonding in this book uses the molecular orbital approach to chemical bonding, simpler methods that provide approximate pictures of the overall shapes and polarities of molecules are also very useful. This chapter provides an overview of Lewis dot structures, valence shell electron pair repulsion (VSEPR), and related topics. The molecular orbital descriptions of some of the same molecules are presented in Chapter 5 and later chapters, but the ideas of this chapter provide a starting point for that more modem treatment. General chemistry texts include discussions of most of these topics this chapter provides a review for those who have not used them recently. 

The free energy T of the molecular fluid consists of the ideal and non-ideal contributions, J [p = T lp] + J "[p]. First the molecular DFT approach replaces the whole intermolecular interaction by the sum of all pair interactions which appear via the pair potential 4>. That procedure fixes the free energy of the system as functional of both the pair potential < (X, X2) and the pair density p< (xi, X2),... 

The molecular DFT approach derives an expression J-c[p] expanding the anisotropic excess parts of the perturbative pair interaction potentials into spherical harmonics. That way the inclusion of the anisotropic dispersion and quadrupolar interactions succeeds [7]. 

The molecular DFT approach describes the attractive interaction separating the isotropic part as the orientation average from the interaction between pairs of molecules. Regarding the anisotropic part of an attractive pair interaction as an excess to the isotropic part, it contributes the essentially smaller share to the energy of the system in comparison with the isotropic part. [7] shows that the anisotropic parts of the dispersive and quadrupolar interaction only very slightly affect the calculated isotherms. 

So far we have discussed chemical bonding only in terms of electron pairs. However, the properties of a molecule cannot always be explained accurately by a single structure. A case in point is the O3 molecule, discussed in Section 9.8. There we overcame the dilemma by introducing the concept of resonance. In this section we will tackle the problem in another way—by applying the molecular orbital approach. As in Section 9.8, we will use the benzene molecule and the carbonate ion as examples. Note that in discussing the bonding of polyatomic molecules or ions, it is convenient to determine fust the hybridization state of the atoms present (a valence bond approach), followed by the formation of appropriate molecular orbitals. 

Robert S. Mulliken (United States) for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method. Mulliken received the Nobel Prize in recognition of his work studying that nature of how electrons behaving in molecules, in particular for the molecular orbital approach that he developed. Molecular orbitals are formed by the overlap of the orbitals on individual atoms, and these can be used to rationalize whether bonds will exist between pairs of atoms, how strongly the pairs will be bonded, and what type of reactivity the molecule may be expected to undergo. 

More recently a molecule with a pair of planar aromatic moieties (Figure 6.4) was synthesized and used to exaggerate the difference between metallic and semiconducting SWNTs in the non-covalent functionalization and solubilization. The molecule representing essentially the molecular tweezers approach (Figure 6.4) exhibited significant selectivity toward... 

The molecular orbital approach to describing hydrogen also starts with two hydrogen nuclei (a and b) and two electrons (1 and 2), but we make no initial assumption about the location of the two electrons. We solve (at least in principle) the Schrodinger equation for the molecular orbitals around the pair of nuclei, and we then write a wave equation for one electron in a resulting MO ... 

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