Hexadecapole-induced dipole

The far i r spectra of simple molecular gases are now extremely well characterised For H2 and D2 the individual rotational lines are easily resolved (18) for larger species such as N2 ) or C02 the individual lines are merged by the translational broadening As a representative example we may consider the work of Poll and Huntv ) on N2 They fit the spectroscopic lineshape(21,22) vith the QID model and with a more sophisticated model in which the hexadecapole-induced dipole (HID) and overlap terms were included The simple QID model gave a remarkably good description of the lineshape and intensity... 

Similar rototranslational spectra (which may be roughly approximated by the envelope of their stick spectra) are observed in other gases as well. Figures 3.22 and 3.23 show the binary absorption spectra of pure methane and carbon dioxide. The smooth curves drawn through the data points represent line shape computations based on the multipole-induction model of the induced dipoles involved [75, 56, 141, 186]. A detailed analysis indicates that for the CH4-X system, CH4 octopole and hexadecapole induction both contribute roughly in comparable amounts to the observable spectra. Rototranslational spectra of several other systems are known see, for example, a review [58]. 

Here a designates the trace of the polarizability tensor of one molecule (l/47i o) times the factor of a represents the electric fieldstrength of the quadrupole moment q2. Other non-vanishing multipole moments, for example, octopoles (e.g., of tetrahedral molecules), hexadecapoles (of linear molecules), etc., will similarly interact with the trace or anisotropy of the polarizability of the collisional partner and give rise to further multipole-induced dipole components. 

If, for example, the induced dipole model is truncated at the order R 6 in the separation R between the molecular centers, account may be made of the dipoles induced by multipoles up to order — 4 (hexadecapole). Moreover, dipoles induced by derivatives of the local field at their center... 

Figure 4.1 shows the four significant induced dipole components for the rototranslational bands (left panel). The isotropic and anisotropic overlap components, B01 and — B21, dominate at near range (dotted). These fall off roughly exponentially with separation R so that at more distant range, the quadrupole-induction, B23, dominates it falls off more slowly, like R 4. A weak hexadecapole component, B45, is also present. The dashed lines show the classical (i.e., overlap-free) multipole induction contributions. These differ only at near range from the computed B23 and B45 components,... 

Fig. 10. The hexadecapole chaige distribution of the dxy dx y> component of the Ai -> Ti cobalt(III) transition with the primary alignment produced of the induced dipoles in the ligands B1, Aj, Ci and C2 (full arrows). Representative secondary alignments of induced dipoles with a common orientation, due to the particular primary correlation, (B1), are, gy(2) (Aj) and gy (B2) (broken arrows), and the orthogonal tertiary correlations, g/ (A2) and g l (Cl) (dotted arrows), arising from the particular secondary orientation, gy (Ap...

Fig. 11. The charge distribution of the dxy dx -y component of the Ai Ti cobalt(III) transition with the correlation produced of the induced dipole in the ligands, Bj, A2, Ciand C2 (solid arrows), and the corresponding dipole aligiunent in the ligands, B2, Aj, A2, and Ci (broken arrows) due to the field of the hexadecapole moment of the dyz dy component...

Since the molecules are moving past each other, each rotational line is translationally broadened Depending on the ratio of the mass to the moment of inertia, separated rotational lines or a broad featureless spectrum may be observed A model such as QID therefore makes specific predictions about the relative intensity of different rotational lines (or regions of the spectrum), it is possible to check the model in a way which is relatively insensitive to the form of the intermolecular potential The quadrupole-induced dipole gives rise to AJ = 2 transitions via the isotropic part of the polarizability of its collision partner in general a 2 -pole-induced dipole gives rise to AJ transitions (eg the hexadecapole ( 4) -induced dipole has AJ = 4) ( )... 

Both photon-assisted collisions and collision-induced absorption deal with transitions which occur because a dipole moment is induced in a collisional pair. The induction proceeds, for example, via the polarization of B in the electric multipole field of A. A variety of photon-assisted collisions exist for example, the above mentioned LICET or pair absorption process, or the induction of a transition which is forbidden in the isolated atom [427], All of these photon-assisted collision processes are characterized by long-range transition dipoles which vary with separation, R, as R n with n — 3 or 4, depending on the symmetry of the states involved. Collision-induced spectra, on the other hand, frequently arise from quadrupole (n = 4), octopole (n = 5) and hexadecapole (n = 6) induction, as we have seen. At near range, a modification of the inverse power law due to electron exchange is often quite noticeable. The importance of such overlap terms has been demonstrated for the forbidden oxygen —> lD emission induced by collision with rare gases [206] and... 

Molecules of still higher, e.g. octahedral, symmetry possess a Oowest) hexadecapole moment the electric fields of these hexadecapoles now induce a dipole moment in neighbouring molecules. Here, too, 9 0, and... 

In the ligand polarization mechanism for optical activity, the potential of the electric hexadecapole component, Hxy(x>-y>), produces a determinate correlation of the induced electric dipole moment in each ligand group which does not lie in an octahedral symmetry plane of the [Co Ng] chromophore (Fig. 8). The resultant first-order electric dipole transition moment has a non-vanishing component collinear with the zero-order magnetic moment of the dxy dxj yj transition in chiral complexes, and the scalar product of these two moments affords the z-component of the rotational strength, RJg, of the Aj -> Ti octahedral excitation. 

The dynamic coupling between the electric hexadecapole moment of a d—d transition in the metal ion and an induced electric dipole in each ligand is forbidden in a four-coordinate complex containing a tetrahedral chromophore, although the mechanism becomes formally allowed on reduction to D2(i or lower chromophoric symmetry. In tetrahedral complexes the dynamic coupling between a d—d quadrupolar... 

No transformation is needed for which is always real. The notation reflects the fact that 2/fc transforms like cos k(j) and like sin k(j). The factors of /T/2 ensure that a rotation of axes induces an orthogonal transformation of the moments. The first few of these moments coincide precisely with the Cartesian charge and dipole moment, and later ones describe the quadrupole, octopole and so on. A complete list is given in Table 1 for moments up to hexadecapole. 

Moon and Oxtoby presented a general theory for collision-induced absorption, which occurs in the near- and far-infrared region of the spectrum, in molecules. Speeific results were presented for the case of symmetric linear Dooh) and tetrahedral (Tj) molecules. The authors subsequently applied their nonasymptotic theory of the pair dipole moment to the eal-culation of binary spectral integrals and the far-infrared speetrum for dinitrogen.The authors also evaluate the eontributions to the seeond-order multipole model (including the anisotropy of the polarizability, the hexadecapole moment and the dipole-oetopole polarizability). 

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