Moment quadrupole

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

Because of difficulties in calculating the non-adiabatic conpling terms, this method did not become very popular. Nevertheless, this approach, was employed extensively in particular to simulate spectroscopic measurements, with a modification introduced by Macias and Riera [47,48]. They suggested looking for a symmetric operator that behaves violently at the vicinity of the conical intersection and use it, instead of the non-adiabatic coupling term, as the integrand to calculate the adiabatic-to-diabatic transformation. Consequently, a series of operators such as the electronic dipole moment operator, the transition dipole moment operator, the quadrupole moment operator, and so on, were employed for this purpose [49,52,53,105]. However, it has to be emphasized that immaterial to the success of this approach, it is still an ad hoc procedure. 

Figure 4.18 from Buckingham A D 1959. Molecular Quadrupole Moments. Quarterly/ Reviews of the Chemical Society 13 183-214. Reproduced by permission of The Royal Society of Chemistry. 

Fig. 2.8 A quadrupole moment can he obtained from various arrangements of two positive and two negative chargt...

The quadrupole is the next electric moment. A molecule has a non-zero electric quadrupole moment when there is a non-spherically symmetrical distribution of charge. A quadrupole can be considered to arise from four charges that sum to zero which are arranged so that they do not lead to a net dipole. Three such arrangements are shown in Figure 2.8. Whereas the dipole moment has components in the x, y and z directions, the quadrupole has nine components from all pairwise combinations of x and y and is represented by a 3 x 3 matrix as follows ... 

Only if the total charge on the system (q) equals zero will the dipole moment be unchanged Similar arguments can be used to show that if both the charge and the dipole moment ar zero then the quadrupole moment is independent of the choice of origin. For convenience the origin is often taken to be the centre of mass of the charge distribution. 

This definition of the quadrupole is obviously dependent upon the orientation of the chargi distribution within the coordinate frame. Transformation of the axes can lead to alternativi definitions that may be more informative. Thus the quadrupole moment is commonl defined as follows ... 

I is the bond length. The experimental quadrupole moment is consistent with a charge, q, of approximately 0.5e. In fact, a better representation of the electrostatic potential around the nitrogen molecule is obtained using the five-charge model shown in Figure 4.20. 

The operator Zi zi xj + Za Za ZaXa that appears above is the z,x element of the eleetrie quadrupole moment operator Qz x it is for this reason that this partieular eomponent is labeled E2 and denoted the eleetrie quadrupole eontribution. 

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. 

Nuclide Natural abundance, % Spin I Sensitivity at constant field relative to NMR frequency for a 1-kG field, Mffz Magnetic moment J-T-1 Electric quadrupole moment Q, 10 m ... 

Finally, if the gas molecule possesses a quadrupole moment Q—examples are CO, COj and Nj—this will interact strongly with the field gradient F to produce a further contribution fQ to the energy. ... 

With nitrogen, the departure from spherical symmetry combined with the relatively strong quadrupole moment, leads to a blurring of the step-like character of the isotherm in the multilayer region (cf. Fig. 2.29(b)). 

All of the heteroatoms possess at least one naturally occurring isotope with a magnetic moment (Table 15). The nuclei and also possess an electric quadrupole moment... 

Isotope Natural abundance (%) Nuclear spin Electric quadrupole moment NMR frequency fora 23.5 kO field (MHz) Relative sensitivity... 

Forces Molecules are attracted to surfaces as the result of two types of forces dispersion-repulsion forces (also called London or van der Waals forces) such as described by the Lennard-Jones potential for molecule-molecule interactions and electrostatic forces, which exist as the result of a molecule or surface group having a permanent electric dipole or quadrupole moment or net electric charge. 

Dispersion forces are always present and in the absence of any stronger force will determine equihbrium behavior, as with adsorption of molecules with no dipole or quadrupole moment on nonoxidized carbons and silicahte. 

For a polar surface and molecules with permanent dipole moments, attraction is strong, as for water adsorption on a hydrophilic adsorbent. Similarly, for a polar surface, a molecule with a permanent quadrupole moment vidll be attracted more strongly than a similar molecule with a weaker moment for example, nitrogen is adsorbed more strongly than oxygen on zeolites (Sherman and Yon, gen. refs.). 

Now, the quadrupole moment can next be calculated by differentiating the potential to get the electric field due to the dipole moment. The reader can now see that an infinite series can be thus generated. The total electric field is simply the sum of all the individual multipole contributions, given by... 

In order to caleulate the force of attraction between the particle and surface. Hays had to estimate the dipole and quadrupole moments on the particle. He did this by assuming that the eharge distribution a 0) around the surface of the particle was given by... 

Hays further assumed that the dipole and quadrupole moments are given by a AnR ... 

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