Dipole-Length Representation

In eq. (13-1) the first equality gives the definition of the oscillator strength in the dipole-length representation, while the second equality gives the definition in the momentum (or velocity) representation. As usual, and T, are the total molecular wavefunctions for the final and initial states, and Ef and Et are the energies of the final and initial states, respectively. 

It should be noted that in the RPA, the dipole oscillator strengths calculated in dipole velocity, dipole length, or mixed representation and all sum rules would be identical, and the TRK sum rule, Eq. (13), would be fulfilled exactly, that is, be equal to the number of electrons if the computational basis were complete [30,34,35]. Comparison of the oscillator strengths calculated in the different formulations thus gives a measure of the completeness of the computational basis in addition to the fulfillment of the Thomas-Reiche-Kuhn sum rule (vide infra). 

Due to the appearance of the position operator, this is called the dipole oscillator strength in the length representation. One can consider the oscillator strength as the trace of a tensor of cartesian components... 

If no microscopic model is available to calculate the dipole correlation length, one might consider A, as a phenomenological parameter of the polarization model. However, before investigating the applicability of the polarization model to the interactions between silica surfaces, a more microscopic representation is used in the next section to obtain an expression for and to derive a relation between the surface dipoles and the polarization of water near the interface. 

To the extent that the polarization of physical atoms results in dipole moments of finite length, it can be argued that the shell model is more physically realistic (the section on Applications will examine this argument in more detail). Of course, both models include additional approximations that may be even more severe than ignoring the finite electronic displacement upon polarization. Among these approximations are (1) the representation of the electronic charge density with point charges and/or dipoles, (2) the assumption of an isotropic electrostatic polarizability, and (3) the assumption that the electrostatic interactions can be terminated after the dipole-dipole term. 

The structural representation follows from the fact that the bond length between C-3 and C-4 is greater than that between C-2 and C-3 and between C-4 and C-5. The ionization potential is 8.89 eV, the electron being removed from the third r-MO (see Fig. 5.2b). The dipole moment is 0.71 D, with the negative end situated on the 0-atom. In contrast, the dipole moment of tetrahydrofiiran is 1.75 D. The small dipole moment of furan confirms that one electron pair of the 0-atom is included in the conjugated system and therefore delocalized. Furan has the following UV and NMR data ... 

The differences in the bond lengths, especially between the bonds N/C-2 and N/C-4 indicate that the delocalization of the r-electrons is affected by the heteroatoms. As in the case of furan, the structural formula with two r-bonds is a good representation of the electronic structure of the molecule. The ionization energy of oxazole is 9.83 eV and its dipole moment is 1.5 D. 

Figure 6. Schematic representation of the odd-even effect on the orientation of the CF3-CH2 surface dipoles in CF3-terminated SAMs on gold. (Note The depicted structure serves only to illustrate the relative alternation of the average terminal group orientation as a function of chain length.)...

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