Multipole moments quadrupole

A neutral molecule will, in general, have a charge distribution that is not spherical. In addition to a dipole moment it can have higher multipole moments, quadrupole, etc. Each such higher moment will generate its own electric field. These will decrease faster with distance and are therefore neglected in the first approximation. 

The three moments higher than the quadrupole are the octopole, hexapole and decapoli. Methane is an example of a molecule whose lowest non-zero multipole moment is the octopole. The entire set of electric moments is required to completely and exactly describe the distribution of charge in a molecule. However, the series expansion is often truncated after the dipole or quadrupole as these are often the most significant. 

The multipole moments (charge, dipole, quadrupole) of each cell are then calculated mining over the atoms contained within the cell. The interaction between all of the 3 in the cell and another atom outside the cell (or indeed another cell) can then be lated using an appropriate multipole expansion (see Section 4.9.1). 

The dipole, quadrupole etc. moments are in general not conserved, i.e. a set of population atomic charges does not reproduce the original multipole moments. 

Of course, the na, o a 11, and oah NBOs of the H-bonding region are important contributors to the dipole, quadrupole, and higher-multipole moments of the monomers. Thus, certain multipoles may appear to explain the geometry through their close connections to these NBOs, but this is not an incisive way to describe the physical situation. 

A rotation of the H2 molecule through 180° creates an identical electric field. In other words, for every full rotation of a H2 molecule, the dipole induced in the collisional partner X oscillates twice through the full cycle. Quadrupole induced lines occur, therefore, at twice the (classical) rotation frequencies, or with selection rules J — J + 2, like rotational Raman lines of linear molecules. Orientational transitions (J — J AM 0) occur at zero frequency and make up the translational line. Besides multipole induction of the lowest-order multipole moments consistent with... 

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. 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

We will return to the quadrupole interaction in following chapters, but we now re-examine the general expansion of the electrostatic interaction and, in particular, the possibility of other nuclear electrostatic multipole moments. Because our multipole expansion is performed in a coordinate system with origin at the centre of charge of the protons p in the nucleus, the nuclear electric dipole moment is zero. However, this result arises only from our choice of origin and we now show that there are much... 

It includes the interactions of distributed multipole moments Q (up to a quadrupole) labeled t and u. The T matrix provides the Coulomb energy appropriate for particular multipoles and includes the distance between sites a and b and their relative orientations. The short range (penetration) component of the electrostatic energy, in a manner similar to the Ar-CC>2 case, can be absorbed into the exchange repulsion term. 

Figure 2.4 Interactions between multipole moments. Dipole moments are indicated by arrows, and an element of the quadrupole tensor by the double lobe.

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). 

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 ... 

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