Monopole approximation

It should be pointed out that the Forster calculations are based on the point dipole assumption which may be inaccurate when the separation distance is similar to the molecular size, as is the case for LHCII. In this situation the transition monopole approximation should also be considered. For chla Chang [172] has estimated that this leads to a Forster correction factor of 0.6-2.0 depending on orientation. 

Let us now consider A tt atomic orbital is essentially a pure p orbital. If there is any polarization (as will be discussed below), this will involve a very small displacement Ar,r/ of the centroid of the orbital. Then a monopole approximation of the 77 term of Eq. (11.9) suffices, giving... 

The relevant bond distance Ra of the initial molecule and the geometry of the final product, namely. Fee and cp (see Fig. 11.1), are known from the problem. The actual calculation involves four steps (1) calculating Ar as indicated earlier (2) applying the theorem (11.11), calculate the distance between the centroid of the tt orbital on atom I and the nucleus fe (3) then using the monopole approximation (11.10) to obtain rp), the average inverse distance between the electrons on I and nucleus k and finally (4) using this rp) result in Eq. (11.12) and finding F u- is... 

For cr systems sp carbons and hydrogen), the charges are taken at their respective nuclear positions. This is part of the monopole approximation (11.10). For the aryl carbon atoms, we have Ar = 0 for obvious symmetry reasons. This leaves us with the olefinic double bonds. They are dealt with in the following manner. 

It may, however, be observed63,167 that because of the shortage of the intermolecular distances with respect to the molecular dimensions, this dipole approximation may be rather inaccurate in the particular case of the interactions between the purine and pyrimidine bases and that it is preferable to treat the problem in the monopole approximation, i.e., by considering all the negative and positive charges in the system as interacting in a simple Coulombic fashion. In this monopole approximation the total energy (2 M) may then be considered as the sum of three main contributions... 

It may perhaps be useful to remember that the dipole moments of the tautomers cannot and should not be considered as indicative of the relative values of such interactions in the first place, because appropriate calculations must be carried out in this case (as we have seen) in the monopole approximation and, second, because even in the dipole approximation, the mutual orientation of the dipoles of the interacting molecules is important. A glance at the data on the dipole moments of the different compounds mentioned here indicates that, in fact, there is no relation between the value of this moment in the different tautomers and the presence of such tautomers in the crystal. Thus, the dipole moments are predicted to be greater for the N(7)H form than for the N(9)H one in purine and adenine, but greater in the N(9)H form than in the N(7)H one in guanine, hypoxanthine, xanthine, 8-azaguanine, 8-azaxanthine, and 6-mercaptopurine. Also, no general relationship seems to exist between the relative values of the dipole moments and the stabilities of the different tautomers. 

This is because for j = k the diagrams in Figs. 18 c, k give identical contributions in the monopole approximation. The presence of an extra core hole will only give small higher order corrections to the relaxation shift. For j,k belonging to the same main shell, it is still a reasonable approximation to set... 

The energies and interactions in Eq. (66) can also be regarded as effective operators including the effects of all other configurations. Including relaxation and screening in the static monopole approximation one obtains... 

Very often it is not possible to obtain all basic scattering functions with the same accuracy. Even worse, in resonant scattering we are often left with the cross term I ,(h) only. This is still quite an acceptable situation, if the resolution to a monopole approximation of the structure is required, as A fh) and B fh) may be determined completely from the two remaining functions I (h) and I (h). However a straight-forward method for the evaluation of higher multipoles in the sense of the above calculation then does not exist any more. The analysis of resonant scattering in this case has to refer to models. 

Figure 2 Contour map of the negative of the molecular electrostatic potential for acetamide at the HF/S-ZIGI ) level calculated using the monopole approximation and the CHELP charges. Scale same as in Figure 1.

Figure 3 Contour map of the difference between the molecular electrostatic potential from the full wavefunction and that from the monopole approximation using the CHELP charges. Shading indicates approximate value of the potential in the region. Note scale change from Figures 1 and 2.

The last four terms depend only on the reference density, po, and represent the repulsive energy contribution, Flcp, discussed above. Thus, we just have to deal with the second-order terms. The second-order term in the charge density fluctuations dp(r), that is, the second term in Equation 5.51, is approximated by writing Ap as a superposition of atomic contributions, Ap0 = Apv. This approach decays quickly with the increasing distance from the corresponding center. To simplify the second term further, Elstner applied a monopole approximation ... 

The diagonal terms yvv model the dependence of the total energy on charge density fluctuations of the second order. The monopole approximation restricts the change of the electron density considered, and no spatial deformations are included. Only the change of energy with respect to... 

This transformation from the dipole to the monopole approximation in the calculations of the intermolecular forces has very significant consequences and represents an essential step in the improvement of the calculations. More recently further refinements have still been introduced in the calculations, prominent among which are the replacement of molecular polarizability by bond polarizabilities and the introduction of a supplementary short-range repulsion term generally in the form of a semiempirical function of the type proposed by Kitaygorodskii and used more extensively by Faidni and Simonetta. Details of these refinements should be looked for in the original papers. 

The final methods which will be discussed are those which consider the electrostatic potential or field within a zeolite cage or skeleton, and the effect that this may have upon possible catalytic sites and absorbed molecules. A number of these calculations have been carried out, and they can be divided into two categories those which use the monopole approximation and simply calculate the field or potential due to point charges placed at atom centres , or those which use a more rigorous approach and calculate the potential directly from the molecular wavefunction . ... 

As one can see, the potential at H-bonding distances from the molecule are not too badly represented. Although more sophisticated methods to represent the electrostatic potential surrounding the molecule are clearly feasible, neither STO nor 43IG wave functions reproduce experimental polarity especially well. Thus, it is not clear that it is worth the effort to go beyond the monopole approximation since even our derived monopole potentials begin to take a non-infinitesmal amount of time on the CDC 7600 if we wish a relatively fine grid. 

Polar interactions may be included either by using a monopole approximation, which replaces polar bonds by point charges on the bonded atoms and then uses a Coulombic potential, or by the more involved treatment of del Re The latter method uses a semi-... 

For the evaluation of interactions between atoms at distances of a few Angstroms, the monopole approximation gives, however, more accurate values of the potential than the dipole moment method. 

Some of these contributions are difficult to evaluate, since either the theory is uncertain or no quantitative formula is really available for the computation. The monopole approximation is preferred for the evaluation of the electrostatic potential. The computation of the energy of the hydrogen bond is most uncertain. The hydrophobic interactions, or more generally... 

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