Transition monopole

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. 

If wavefunctions are calculated using semiempirical methods that assume zero overlap between atomic orbitals (AOs) on different atomic centers, then a MuUiken population analysis [84, 85] can be applied to the calculated TD to yield transition monopoles distributed over each atomic center. Such an approach has proven to be effective [82] an advantage being that the interaction between distributed monopoles can be computed considerably faster then that between TDCs. At the same time, the basic topology of the donor-acceptor... 

Monochromator for slow neutrons. Monochromator fur langsame Neutronen 375. Monopole transition, Monopol-tlbergang 509. Monte Carlo method, Monte Carlo-Methode 464f., 494, 499, 503. 

The way that transition dipole moments interact together in multichro-mophore assemblies was initially examined by Kasha in his pioneering work on excitonic coupling. The original point dipole treatment was later refined by Hunter and Sanders who have developed a transition monopole treatment that allows quantitative prediction of the effects of excitonic interaction. Figure 13.4 qualitatively summarizes the nature of the absorption band shifts depending on the... 

Figure 7.5 shows the values of //2i(eo calculated for a pair of fran -butadiene molecules by the point-dipole (Eq. 7.18) and transition-monopole (Eq. 7.15) expressions. In panel A, the orientations of both molecules are held fixed while the center-to-center distance is varied in B the second molecule is rotated at a fixed distance. As a rule of thumb, the point-dipole approximation is reasonably satisfactory if the intermolecular distance is more than 4 or 5 times the length of the chromophores, although the relative error can still be substantial in some situations. 

Fig. 7.5 Electronic interaction matrix elements for two trani-butadiaie molecules as calculated by the transition-monopole (Eq. (7.15), solid curves) ot point-dipole (Eq. 7.18, dashed curves) expressions. Molecule 1 is fixed in position in the xy plane, centered at the raigin, with its long axis parallel to the x axis. The transition dipole for its lowest-energy excitatirai lies in the xy plane at an angle of 169° from the positive x axis, f /n is taken to be 1. In (A), molecule 2 is centered at various points along either x or y, in the same orientation as molecule 1 the abscissa gives the center-to-center distance. In B, molecule 2 is centered at (10 A, 0,0) and is rotated in the xy plane the abscissa indicates the angle (6) between the two transition dipoles...

Transition-Monopole Treatments of Interaction Matrix Elements and Mixing... 367... 

To calculate the distances r, Vy and between the different transition monopoles, it is convenient to let the first molecule s center be the origin and its axes define the x, y and z direction of the dimer system. The second molecule may be related to the first by a translation and three rotations, the angles and axes of which are defined by the Euler transformations (21). Thus, the second molecule may be defined by the location of its center, (X, Y, Z), and its three Euler rotations (a, b, c). 

The transition monopoles of Bchla or BchlZ have been calculated using the configurations... 

We have shown that both Qy excitonic transitions of Bchl and Chi dimers in vivo are bathochromically shifted with respect to their monomeric Qy transition. Apparently, this observation cannot be explained by admixing with CR states whereas it can be accounted for if one considers electrostatic and van-der Waals interactions among the paired molecules. Assuming these interactions are 2.6-3.0 times the interactions among the Qy transition monopoles in strongly coupled molecules, we have simulated the positions and the intensities of the Qy excitonic transitions in P-860 and P-960. The same approach has led to the reproduction of P-680 s spectra (CD and optical absorption), however, here the Qy transition monopoles are loosely coupled to each other. 

The monopole interaction 8 , which yields the isomer shift, is easy to treat it is just additive to all transition energies. Thus, the recorded spectrum has uniform shifts of all resonance hues with no change in their relative separations. 

To calculate Mossbauer spectra, which consist of a finite number of discrete lines, the nuclear Hamiltonian, and thus also Hsu, has to be set up and solved independently for the nuclear ground and excited states. The electric monopole interaction, that is, the isomer shift, can be omitted here since it is additive and independent of Mj. It can subsequently be added as an increment 5 to the transition energies of each of the obtained Mossbauer lines. 

Energy diagrams can also be helpful to stress these concepts. Table 26.2 represents the four cases we described. In these diagrams, the intermediates a and b lead to the two products A and B. The difference in the transition state free energies AAG will determine the selectivity. In the first three quadrants, I, II and III, the major product formed is B. In particular, we discussed in this study the monopolizing case. We can see that the selectivity, in this case, is dictated by relative intermediate stability. 

The selection rules appropriate for a shake-up transition are of the monopole type2, 76. The intensity of a shake-up peak depends on the overlap integral between the lower state molecular orbital from which the electron is excited (in the neutral molecule) and the upper state molecular orbital to which the electron is excited (in the core-ionized molecule). Consequently one expects transitions of the type au au, ag " ag> 7T nu, and irg - ng with g u and u - g transitions forbidden. 

As in the previous case of infrared transitions, one wants to calculate the line strengths S(v,J —> v, J ) defined in Eq. (2.127). For Raman transitions there are two contributions, as discussed in Chapter 1. The so-called trace scattering is induced by the monopole operator... 

The aspherical features of the density are described by the summation added to the /c-expression. The summation includes an additional monopole, which may be omitted for first- and second-row atoms, but is necessary to describe the outer s-electron shell of transition metal atoms, which is much more diffuse than the outermost d shell. 

If the field has three components, then it is easy to see that the the objects that will be formed will be pointlike. Such objects are called monopoles. Although there are some (stable) monopole and anti-monopole configurations which can annihilate, this process is not very efficient as one does not expect these objects to have long range interactions (an annihilation between a monopole and an anti-monopole is in general extremely unlikely). The net result of this is that at some epoch such a phase transition will convert some amount of energy into these monopoles and anti-monopoles which then behave as non relativistic matter. 

Before that, let us note that it seems more appropriate that inflation does not end earlier than at GUT epoch because of the monopole problem (we prefer that inflation ends when the temperature is below that of any phase transition... 

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