Hexadecapole

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

Many molecules, such as carbon monoxide, have unique dipole moments. Molecules with a center of inversion, such as carbon dioxide, will have a dipole moment that is zero by symmetry and a unique quadrupole moment. Molecules of Td symmetry, such as methane, have a zero dipole and quadrupole moment and a unique octupole moment. Likewise, molecules of octahedral symmetry will have a unique hexadecapole moment. 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Once an approximation to the wavefunction of a molecule has been found, it can be used to calculate the probable result of many physical measurements and hence to predict properties such as a molecular hexadecapole moment or the electric field gradient at a quadrupolar nucleus. For many workers in the field, this is the primary objective for performing quantum-mechanical calculations. But from... 

Hexadecapole moment (Debye-Ang 3) XXXX= -1574.8336 YYYY= -1160.7386 ZZZZ=... 

The spherical harmonic density functions are referred to as multipoles, since the functions with 1 = 0, 1, 2, 3, 4, etc., correspond to components of the charge distribution p r) which give nonzero contributions to the monopole (/ = 0), dipole (/ = 1), quadrupole (/ = 2), octupole (/ = 3), hexadecapole (/ = 4), etc., moments of the atomic charge distribution. 

Similarly, octupoles and hexadecapoles can be thought of as arising from 2p3d and 3d3d atomic orbital products, which leads to n, = 3 and 4, respectively. The... 

We demonstrate the use of local coordinate systems with the molecule of tetrasulfur tetranitride, S4N4, (Fig. 4.2) as an example. It occupies a general position in its crystal s space group, with one molecule in the asymmetric unit. Thus, there are eight crystallographically independent atoms. If multipoles up to and including the hexadecapoles are included, the number of population parameters... 

A systematic analysis of the electrostatic interactions in the crystals of 40 rigid organic molecules was undertaken by Price and coworkers (D. S. Coombes et al. 1996). In this work, distributed (i.e., local) multipoles up to hexadecapoles, obtained from SCF calculations with 6-31G basis sets, scaled by a factor of 0.9 to allow for the omission of electron correlation, are used in the evaluation of the electrostatic interactions. The experimental lattice constants and structures are reproduced successfully, the former to within a few percent of the experimental... 

Figure 1. Photoabsorption cross section for the dipole plasmon in axially deformed sodium clusters, normalized to the number of valence electrons N - The parameters of quadrupole and hexadecapole deformations are given in boxes. The experimental data [39] (triangles) are compared with SRPA results given as bars for RPA states and as the strength function (49) smoothed by the Lorentz weight with A = 0.25 eV. Contribntions to the strength function from p =0 and 1 dipole modes (the latter has twice larger strength) are exhibited by dashed curves. The bars are given in eVA. ...

Fig. 3.17. The rototranslational spectrum of H2-CH4 at 195 K experimental points big dots H2 quadrupole-induced component dotted CH4 octopole-induced component dashed CH4 hexadecapole-induced component dot-dashed total heavy curve. Reproduced with permission from the National Research Council of Canada from [46].

Fig. 3.22. Rototranslational absorption spectra of CH4-CH4 pairs [141] at 296 K in the frequency range 50-400 cm-1. A simple decomposition of the measurement ( ) is attempted in terms of an octopole-induced component (a) a hexadecapole-induced component (b) and double transitions (c) the superposition (heavy) reproduces the measurement closely.

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,... 

For the quadrupole-induction matrix element 23 and even for the weak hexadecapole-induction term 45, for R — 00, the coefficients of R x 2 are in close agreement with the asymptotic values (X + ) /2a.(vj qx v f). Figure 4.1 (center panel) illustrates this agreement solid and dashed lines merge for R —> 00. 

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... 

It is interesting to note that in high-resolution studies of the spectra of solid hydrogen transitions were seen with a change of the rotational quantum numbers A J of 6 and 8 [102]. The suggestion was made that these could be caused by H2 multipole moments of higher order than the hexadecapole (or 24) moment, e.g., by the H2 26 and 28 multipole moments. Such transitions are weak and have hitherto not been included in any treatments of collision-induced absorption in gases. 

In formulas (22.12) and (22.13) k acquires only even values for k = 2 we have the usual electric quadrupole interaction, whereas for k = 4 we have the electric hexadecapole interaction, already observed in [145]. The expressions for the matrix elements of the hyperfine structure operators considered above for the closed shells follow straightforwardly from the... 

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