Octupole moment, electric

Electric octupole (Figure 4). Three-dimensional space is defined in general by three distinct vectors ri, rs, rs and the electric octupole moment is consequently given by a tri-vector of the form qr r rs. The permanent... 

In analogy to the definition of electric dipole moment, electric multipole moments are also defined. In particular, the quadrupole moment Q and the octupole moment U are defined as ... 

A molecular electric potential exists because some types of atoms in a molecule attract electrons better than others. For instance, it is well known that fluorine attracts electrons more than hydrogen in HF, resulting in a nonzero dipole moment for this molecule. In the linear molecule carbon dioxide, electronic charge migrates toward the oxygens giving the molecule a measurable quadrupole moment. Even tetrahedral methane has a measurable octupole moment due to charge separation in its C —H bonds its dipole and quadrupole moments are zero by symmetry. 

Ab initio SCF-MO calculations have been performed on BFg, and alterations in predicted geometries with different gaussian-orbital basis sets noted. Further non-empirical SCF calculations on this molecule have been reported by Rothenberg and Schaeffer. Values of the dipole moment, quadrupole moment, octupole moment, second and third moments of the electronic charge distribution, diamagnetic susceptibility, and electric field gradient were calculated. 

Here q is the net charge (monopole), p is the (electric) dipole moment, Q is the quadrupole moment, and F and F are the field and field gradient (dF/dr), respectively. The dipole moment and electric field are vectors, and the pF term should be interpreted as the dot product (p F = HxFx + iiyFy + n F. The quadrupole moment and field gradient are 3 x 3 matrices, and QF is the sum of all product terms. For an external field it is rarely necessary to go beyond the quadrupole term, but for molecular interactions the octupole moment may also be important (it is for example the first nonvanishing moment for spherical molecules like CH4). 

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... 

As examples of molecules having, in a linear approximation, isotropic polarizability, we adduce ones of tetrahedral symmetry such as CH, CO, and the like. Here, neither a permanent dipole nor quadrupole is available, and the lowest non-zero moment is an octupole having the non-zero components Qxis The electric field of such octupoles, given by equation (48e), induces a dipole in another molecule, and we have by (204) in the approximation of pairwise correlations ... 

For tetrahedral molecules the linear polarizability is isotropic, and the first non-zero permanent electric moment is the octupole Q = TlJe electric fields of these octupoles induce a dipole moment in any given molecule, whence 0 and we have by equation (241) to a suflBciently good approximation ... 

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). 

TaUe 6 Non-zero (N) and independent (/) elements of the octupole electric moment tensor S2 for point groups... 

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 basic au for dipole, quadrupole and octupole electric moments are given as ... 

In 1996, Munn extended the microscopic theory of bulk second-harmonic generation from molecular crystals to encompass magnetic dipole and electric quadrupole effects [96] and included all contributions up to second order in the electric field or bilinear in the electric field and the electric field gradient or the magnetic field. This was accomplished by replacing the usual polarization of Refs. 72 and 84 by an effective polarization as well as by defining an effective quadrupole moment. Consequently, the self-consistently evaluated local electric field and electric field gradient were expressed in terms of various molecular response coefficients and lattice multipole tensor sums (up to octupole). In this... 

Explicit expressions for the traceless electric quadrupole, octupole and hexade-capole moments are... 

In a spherical nucleus we assume the charge distribution to be spherical and the nucleus acts as a monopole. In the deformed nuclei, the nuclear charge has a non-spherical distribution. The potoitial at a point x,y,z (Fig. 11.6) will be found to vary depending on the charge distribution and mode of rotation of the nucleus. The nuclear charge may be distributed to form a dipole, a quadrupole, etc. Nuclei are therefore divided into differoit classes dq>ending on their electrical moments monopoles, dipoles, quadrupoles, octupoles etc. 

Large ions normally feature a complex distribution of charges and thus have nonzero electric quadrupole, octupole, and higher multipole moments. However, those produce zero torque in homogeneous fields and thus are not relevant to ion alignment in high-field IMS. 

The experiments show that in the ground states of nuclei the electric moments of 2 order with odd I and all 2-pole magnetic moments with even I are missing. Only electric monopole, magnetic dipole, electric quadrupole, magnetic octupole, etc., moments are observed in experiments. This means that the 2-pole character unambiguously determines whether the moment is electric or magnetic, and these latter attributes may be omitted. This rule can be viewed as a consequence of a symmetry property (parity) of the nuclear states. 

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