Hexadecapole Interaction

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... 

Mentioned briefly here are some highlights of the most recent developments relating to the quadrupolar halogens. Cederberg has investigated the potential evidence for a non-zero nuclear electric hexadecapole interaction by analysirw... 

Indium Halides.— The radiofrequency spectrum of In F has been unsuccessfully examined with the aim of finding an electric hexadecapole interaction of the indium nucleus (the upper limit for any such interaction was 2 kHz). 7... 

It should be noted that the quadrupole interaction is only the first nonzero term in the mathematical expression governing the variation of the electric potential V over the nuclear volume the second is the nuclear hexadecapole interaction. " which depends on the coupling between the nuclear electric hexadecapole moment H and the fourth derivative of the electric potential The nuclear... 

From quantum mechanical considerations, it can be proven that the only non-vanishing terms in Equation 2 are the first, third, and other odd-ordered ones. The first term is simply a constant, and, therefore, the orientationally-dependent term in the energy is the quadrupolar energy (hexadecapole interactions are extremely small if they exist at all) ... 

There are small but noticeable differences in potential curves from model 2(a) and a point quadrupole interaction [12]. Simulations of these hamiltonians produce small differences in liquid structure and thermodynamic results which can be rationalized in terms of the additional quadrupole-hexadecapole interaction for the- charges [43]. Calculation of S(q, (o) for solid a-Na showed that the frequency of a particular librational phonon was insensitive to the method used to include the quadrupole-quadrupole interaction [45]. For the case where dipoles) quadrupoles and higher moments are all important comparison of the representations has not been made. A study of 2-d dipoles [44] reports different results for different representations at low temperatures. 

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... 

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. 

In non-dipolar dielectrics sufiSciently dense for molecular interaction, the temperature-dependent polarization (241) is generally non-zero. Such interaction will lead to an effect consisting in the induction, in any given molecule immersed in the dense medium, of a dipole moment M by the fluctuating electric field of the permanent quadrupoles, > octu-poles, >hexadecapoles, and in general multipoles" of its nei bours. 

A multipole series should strictly be truncated at a given power, w = -(/j + /z + 1), of the intersite separation R, rather than by only using the multipoles Qik up to a given value of / on each molecule. Hence, to include the important quadrupole-quadrupole interactions (the terms in Q k Q2k )i the series should be taken up to R and thus also should include the hexadecapole charge Q k QIq and Qlo Qik) and octopole-dipole (Ql Qik and Q k Qlk ) terms. 

To describe the hexadecapole resonances, a more complicated theory is needed, because the presence of the hexadecapole moment requires ground-state angular momentum > 2 and second-order interactions with the light. 

The pioneering finite temperature Monte Carlo study [282] of the orientational disordering of commensurate and incommensurate monolayers of CO on graphite was based on empirical potentials and 64 molecules in a rectangular periodic cell. The CO-surface interactions were modeled with the Fourier representation [324, 326], and the necessary Lennard-Jones parameters were obtained from a fit to the measured isosteric heats of adsorption [287]. The nonelectrostatic CO-CO intermolecular interactions were based on Buckingham-type potentials as parameterized in Refs. 238 and 287. The electrostatics was represented by a three-site point-charge distribution located on the molecular axis [282]. The chosen values yield a reasonable representation of the moments up to the hexadecapole the dipole moment, however, is larger than the experimental value. These interactions... 

This is the famous crystal-field operator for an octahedron, which splits the ai-shell into eg and t2g subshells. The crystal-field interaction is usually parameterized by the crystal-field parameter, lODq, which corresponds to the splitting of the eg and t2g orbitals. The term in brackets here is the octahedral invariant of rank 4. In the multipole expansion this corresponds to a hexadecapole operator. In normalized form it reads... 

Because of orientation-dependent terms in both the moments and the Boltzmann factor values of B are much siore sensitive to molecular anisotropies than the pressure virial coefficient or the gas shear viscosity as a function of temperature. For nonpolar molecules quadrupole moment effects are large in the case of CO2 for example demonstrating the importance of quadrupole moments Q s 4.2 X 10 esitcii)> inferred from B while octopole and even hexadecapole effects can be recognized for more symmetrical molecules e.g. CH and SFg. For polar molecules permanent dipole interactions also come into play and anisotropy of repulsive forces (shape) is also important. The result is a very wide range in magnitudes and sign of B even for relatively simple molecules and comparison of calculated values with experiment is a sensitive test of multipole moments and anisotropies of used in the calculation. All these matters are discussed in detail by Sutter (21). 

Briefly, the interaction of (18.107) is expressed in a multipole expansion in the same manner as described in section 1.3 for the electrostatic portion of the hyperfine interaction. Qualitatively, this expansion contains terms corresponding to the electric quadrupole, hexadecapole, etc., interactions of the ion with the CEF. These separate interactions will be characterized by products of parameters (r ), /), and (r ), averaged over the 4f" electron configuration of the ion, with lattice sums A representing the contributions of the surrounding ions to the various successive gradients of the CEF. 

The term of the expanded electrostatic energy involves interaction between dipole-hex-adecapole (i = 1, = 4). hexadecapole-dipole(i = 4, l = l), quadrupole-octupole... 

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