Moments of a charge distribution

The moments of a charge distribution provide a concise summary of the nature of that distribution. They are suitable for quantitative comparison of experimental charge densities with theoretical results. As many of the moments can be obtained by spectroscopic and dielectric methods, the comparison between techniques can serve as a calibration of experimental and theoretical charge densities. Conversely, since the full charge density is not accessible by the other experimental methods, the comparison provides an interpretation of the results of the complementary physical techniques. The electrostatic moments are of practical importance, as they occur in the expressions for intermolecular interactions and the lattice energies of crystals. 

The above moments of a charge distribution can be related to the statistical moments of random variables. The orbital centroid vector is namely identical to the central first moment M(Q of a random variable , with density function p(x) ... 

The multipolarity of a transition J - J is given by Eq. (19-f)- The fields of electric 2 -pole and magnetic 2 -pole radiation may both be derived from the time variation of the Z-th moment of a charge distribution. Although the magnetic radiation is weaker than the associated electric radiation by a factor of the order v/cY in intensity, the two types of transition have the same parity and show similar behaviour in internal conversion and pair production phenomena which depend on the form of the field near the origin of the radiation. It is therefore convenient to describe both electric 2 -pole and magnetic 2 -pole transitions as of order A = L... 

An important feature of the electric multipole moments, as defined in Eqs. (4.4)-(4.6) and (4.8), is that the first non-vanishing moment of a charge distribution is independent of the choice of the origin Rq- However, all the higher moments depend on this origin. Thus, the dipole moment of a neutral molecule or the quadrupole moment of a neutral and non-polar molecule are both independent of the origin Rq, whereas the dipole moment of an ion or the quadrupole moment of a neutral but polar molecule will depend on the origin Rq [see Exercise 4.2]. 

This suggests the definition of the moments of a charge distribution. The zeroth moment... 

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