Moment electric hexadecapole

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... 

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... 

The non-zero tensor components of multipole moments have been determined specifically for the tetrahedraland octahedral symmetries, beside the axial symmetry for which we have the general formula (40a). Lately, Kielich and Zawodny, resorting to methods of group theory, have calculated and tabulated all non-zero and independent tensor components of electric dipole, quadrupole, octupole, and hexadecapole moments for 51 point groups (Tables 4—7). 

The preceding considerations lead to the condusion that, whereas numerical determinations of electric dipoles require, in accordance with equation (241a), the investigation of substances in the absence of molecular correlations, the quadrupole, octupole, and hexadecapole moments are accessible to determination only if at least pairwise radial correlations are present in the non-dipolar medium. Numerical values of electric moments of higher order are determined by the method of measuring second dielectric virial coeffidents worked out and successfully applied by Cole et al. It has yielded quadrupole, octupole, and hexadecapole moments in good numerical agreement with the values obtained by other methods (see Section 2). 

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... 

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... 

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. 

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. 

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... 

Electric hexadecapole. In microsystems of higher e.g. octahedral) symmetry, as we shall see further on, all the above defined moments vanish and it is necessary to define multipoles of higher orders, beginning by a hexadecapole system the permanent moment of which is defined by a symmetric tensor of rank 4 ... 

Specifically, octahedrally symmetric molecules like SF have, as their first electric moment, a hexadecapole, since all the lower moments vanish as a result of high symmetry (p = 0 = 2 = 0). 

Table 3 Number of mutually independent (7) and of non-zero (N) tensor elements of the electric dipole p quadrupole (6), octupole ( 2), and hexadecapole (4 ) moments, for all point groups... 

Table 7 Non-zero (N) and independent (7) elements of the hexadecapole electric moment tensor for point groups...

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. 

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. 

Only static and dynamic molecular properties involving electric dipole and quadrupole operators will be discussed below. However, electric properties related to higher-order electric multipole operators can also be determined in a similar manner to the properties described here, in terms of expectation values, linear and nonlinear response functions. Nevertheless, it should be kept in mind that although the same formalism is applied in the calculation of response functions involving octupole, hexadecapole, and higher moments, in practice it may... 

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