Local molecules

Figure 4.6 Schematic representation of a portion of the spectrum of linear XYZ local molecule. The scale is that appropriate to HCN. The energy levels are obtained using Eq. (4.54) with Ni = 144, N2 = 47, A, = -1.208 cm-1, A2 = -10.070 cm 1, A12 = -1.841...

An important problem of molecular spectroscopy is the assignment of quantum numbers. Quantum numbers are related to conserved quantities, and a full set of such numbers is possible only in the case of dynamical symmetries. For the problem at hand this means that three vibrational quantum numbers can be strictly assigned only for local molecules (f = 0) and normal molecules ( , = 1). Most molecules have locality parameters that are in between. Near the two limits the use of local and normal quantum numbers is still meaningful. The most difficult molecules to describe are those in the intermediate regime. For these molecules the only conserved vibrational quantum number is the multiplet number n of Eq. (4.71). A possible notation is thus that in which the quantum number n and the order of the level within each multiplet are given. Thus levels of a linear triatomic molecules can be characterized by... 

This Hamiltonian describes a generic linear triatomic molecule, as discussed in Chapter 4. When Xl2 = 0 we have a local molecule, and, as X]2 increases, we move from the local to the normal limit. 

We now turn to the case of identical sites. Actually, we require that the sites be identical in a strict sense, as we explain below. We use again the three-site case, but instead of three different sites a, b, and c we assume that the sites are identical. Since we are dealing with localized molecules, the sites are still distinguishable. The canonical PFs listed in Eqs. (2.2.2)-(2.2.5) now reduce to... 

Note that in this section we have a thermodynamic system of molecules possessing translational and rotational degrees of freedom. In the previous sections we treated a system of localized molecules. Therefore, the GPFs of the two systems are different but their BI is the same, provided the approximations made in Appendix B are valid. 

For simplicity, we assume that the internal PF of the ligand does not change upon binding, and that all the molecules involved in the process (C.14) are devoid of translational degrees of freedom (localized molecules). 

Now we consider the same Hamiltonian, but taken for the local molecule (T = 0). From the outset we include the additional variational mixing parameter A into the Ansatz. The understanding of the importance of introducing the mixing parameter in the local model in the contrast to the lattice one can be gained if one examines the expressions for Hamiltonians, in particular, those terms, which contain the mixing of two phonon modes. The mixing is contained mostly in the local term with /3,... 

Adsorbent-adsorbate potential energy calculations have been made for the adsorption of argon in the channels and intersections of Silicalite-I (Muller et al., 1989). The most favourable sites for localized adsorption are within the straight and sinusoidal channels, which together should be able to accommodate 20 molec uc-1. At a loading of 24 molec uc 1 all the available sites in the channels and intersections are probably occupied by localized molecules. 

There are several processes that can lead to the delocalization of an adsorbate molecule that prefers to adsorb with localization. One such process is illustrated in Fig. 2c, for the in-between molecule ii discussed above. This behavior of the molecule ii is referred to as restricted-access delocalization of the solvent molecule C (or M). Figure 2c is shown as an overhead view in Fig. 2d. Delocalized molecules C (shown as dashed circles) arise either from steric hindrance by surrounding localized molecules iib) or from the inability of C to center itself over a site as in iia and Fig. 2c. 

Less-polar solvent molecules B (CHCI3, CH2CI2, benzene, etc.) that do not localize nevertheless interact with adsorption sites. This is illustrated in Fig. Id for the binary mixtures A/B (A is nonpolar), and is contrasted in Fig. le for adsorption of a mobile phase A/C, where C is localizing. When nonlocalizing molecules of a polar mobile phase M are adjacent to localizing molecules of solute X (Fig. Ic) or solvent C (Fig, If), these noncova-lent interactions of M with surface sites can interfere with or displace corresponding interactions between the localized molecule and its site. This effect is referred to as site-competition delocalization. 

Alternatively, it can be argued that these lateral (noncovalent) interactions of a nonlocalizing solvent M with the surface are simply preempted by a localizing molecule C or X. This then represents a loss in energy of the final system (adsorption of C or X) which is equivalent to that proposed above and in Figs. lc,f. See the further discussion of Ref. (28b). 

Because of steric constraints on the hydrogen bonding of localized molecules XH and C, as well as possible involvement of the electron pair on C with the adsorbent site C ), the equilibrium constant for reaction (20b) should be smaller than for reaction (20a). 

An additional contribution to x, Em , or Eya due to interactions of species i, mobile phase M, or solute X with mobile-phase molecules in the adsorbed phase [Eqs. (17c), (17d)j A localization function (Table II) for localizing molecules C, X, or functional group k (Eqs. (13), (15), (37)] recognizes site-competition delocalization... 

The molecular partition function has been factorized into contributions g vib > resulting from the vibrations of the localized molecule as a whole around its equilibrium position, q ot, resulting from rotations, and gint, resulting from intramolecular vibrations. The configurational entropy Scant results from the degeneracy due to the distribution of the... 

In the Wilson equation, the effects of difference both in molecular size and intermolecular forces are incorporated by an extension of the Flory-Huggins relation (5-8). Overall solution volume fractions ( i = XiViJvi) are replaced by local volume fractions, d> which are related to local molecule segregations caused by differing energies of interaction between pairs of molecules. The concept of local compositions that differ from overall compositions is shown schematically for an overall equimolar binary solution in Fig. 5.6, which is taken... 

Fig. 9.5 Localizing O and non-localizing [ ] molecules and their interaction with an adsorbent. The silica surface is drawn in an overhead view and its adsorptive centres (silanol groups) are symbolized by asterisks.

A modification of the method for calculation of the energy of the hypothetical localized molecules results in a better correlation between the delocalization energy calculated by HMO theory and experimentally observed stability. A series of... 

One far-reaching perspective is to perform experiments on translationally localized molecules prepared in specific quantum states. Several methods were proposed to cool the internal degrees of freedom of the molecules, using lasers or lamps [80],... 

It is apparent that the 7 take the place in this formulation of the interaction tensors T of the conventional Cartesian formulation, but it should be emphasized once again that all the formulae given here refer to multipole moment components in the local, molecule-fixed frame of each molecule, whereas the corresponding Cartesian formulae deal in space-fixed components throughout and require a separate transformation between molecule-fixed and space-fixed frames. ( Space-fixed is perhaps a misleading term here, since the calculation is commonly carried out in a coordinate system with one of its axes along the intermolecular vector. However, the point is that in the Cartesian tensor notation there has to be a common set of axes for the system as a whole, and this can be the molecule-fixed frame for at most one of the molecules involved.)... 

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