Multipole operators

The transition operator, or electric multipole operator, is a tensor of rank k and it is given the symbol We, thus, have... 

Motivated by these considerations, we have recently proposed a multi-chain fermionic dynamical symmetry model (FDSM) which was developed to specifically address the above raised questions. Our starting point is the Ginocchio SO(8) model 10(since from now on only fermion groups will be mentioned, we shall drop the use of the F superscript to denote them). In our opinion, Ginocchio was the first person to seriously pursue the concept of multi-chain dynamical symmetries from a fermionic viewpoint. The main ingredients of the Ginocchio model can be summarized as follows. If one were to take the fermion pair (i.e. a+a+ type of operators) with =0 S) and 2(D) and certain multipole operators (i.e. a+a type of operators), both types are constructed from... 

The multipole operators are here defined as spherical tensors ... 

It is also possible to make ab initio calculations of the A s, however, and at the same time to improve the Uns51d scheme by assuming that the A s are dependent on the indices 1, labelling the multipole operators (2 -poles) associated with the excitations. Such a non-empirical Unsold scheme has been proposed by Mulder et The average excitation energy is defined as the ratio ... 

The expansion gives the interaction operator as a series of multipole-multipole operators, which can be taken term by term. The quantities Q nt) are the components of multipole moments referred to axes fixed in the molecules, primes distinguishing quantities for centre b. The moments transform as spherical harmonics, and are defined in (II.5),... 

An electric multipole operator of order (151) is symmetric in all n indices (JJi Jn-i) and thereby constitute (n+2)(n+l)/2 linearly independent quantities. In the electric multipole expansion (148) the electric multipole of order n is multiplied with the (n — l)st derivative of the electric field. This... 

In the static case, electric multipoles are often redefined to give forms that are traceless on any two indices. In this manner, the above restrictions (154) can be incorporated [49]. The general form of such traceless multipole operators is... 

Before 1970 the multipole expansion (by which we mean the expansion in powers of MR) of the interaction operator was usually truncated after the R dipole-dipole term, so that the only dispersion interaction term was —C R. Around 1970 it became clear that this approximation was not sufficient and that more terms were needed. However, the straightforward application of the Taylor expansion, and its natural formulation in terms of Cartesian tensors [77], soon becomes cumbersome. Nineteenth century potential theory [78,79] came to the rescue. In this theory the multipole series is rephrased in terms of associated Legendre functions, which enables a closed form of it. Multipole operators are defined as... 

Traceless forms of the higher multipole operators have been given by Buckingham.12 Much of the work reviewed will refer to the case of a spatially uniform applied field, when the perturbed hamiltonian reduces to... 

Using the table of multipoles (p. el74), we may easily write down the multipole operators for the individual molecules. The lowest moment is the net charge (monopole) of the molecules ... 

Since we are dealing with point charges, the computation of the multipole moments are reduced to inserting into the multipole operator the values of the coordinates of the corresponding charges. 

An inspection of eq. (7.55) shows that the matrix elements ne eded for an evaluation of p Q) are the same as those appearing in the radiation problem. Thus p(Q) can be expressed in terms of the multipole operators characterizing the ion, analogous to the expansion used in the formulation of the radiation problem. One obtains... 

Let us consider the expressions for the multipole operators. These are given by Stassis and Deckman (1976a)... 

Stassis and Deckman (1976b) have also given a completely relativistic formulation of the problem, and the reduced matrix elements for the (effective) multipole operators needed in this case have also been tabulated. In addition, of course, one needs to replace the radial wavefunction f(r) appearing in the expression for the by relativistic (Dirac-Fock) radial wave functions. 

This equation also applies to the operators for the multipole moments of the individual regions. There is a degree of arbitrariness in the way that the molecule is divided into regions [7] but however the regions are defined, the multipole operators for the individual regions must be related to the multipole operators for the molecule as a whole by Eq. (16). 

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