Operator magnetic dipole

Analogous considerations can be used for magnetic dipole and electric qiiadnipole selection rules. The magnetic dipole operator is a vector with tln-ee components that transfonn like R, R and R. The electric... 

Suppose one first considers electric-dipole and magnetic-dipole transitions. As is now well recognized, these are the major contributors to rare-earth absorption and emission spectra. We know that the electric-dipole operator transforms as a polar vector, that is, just as the coordinates (23, 24). This means that it has odd parity under an inversion operation. On the other hand, the magnetic-dipole operator transforms as an axial vector or pseudovector and of course must have even parity (23, 24). 

As stated in an earlier paragraph, the sharp emission and absorption lines observed in the trivalent rare earths correspond to/->/transitions, that is, between free ion states of the same parity. Since the electric-dipole operator has odd parity,/->/matrix elements of it are identically zero in the free ion. On the other hand, however, because the magnetic-dipole operator has even parity, its matrix elements may connect states of the same parity. It is also easily shown that electric quadrupole, and other higher multipole transitions are possible. 

The probability of a transition being induced by interaction with electromagnetic radiation is proportional to the square of the modulus of a matrix element of the form where the state function that describes the initial state transforms as F, that describing the final state transforms as Tk, and the operator (which depends on the type of transition being considered) transforms as F. The strongest transitions are the El transitions, which occur when Q is the electric dipole moment operator, — er. These transitions are therefore often called electric dipole transitions. The components of the electric dipole operator transform like x, y, and z. Next in importance are the Ml transitions, for which Q is the magnetic dipole operator, which transforms like Rx, Ry, Rz. The weakest transitions are the E2 transitions, which occur when Q is the electric quadrupole operator which, transforms like binary products of x, v, and z. 

The structure of this contribution is as follows. After a brief summary of the theory of optical activity, with particular emphasis on the computational challenges induced by the presence of the magnetic dipole operator, we will focus on theoretical studies of solvent effects on these properties, which to a large extent has been done using various polarizable dielectric continuum models. Our purpose is not to give an exhaustive review of all theoretical studies of solvent effects on natural optical activity but rather to focus on a few representative studies in order to illustrate the importance of the solvent effects and the accuracy that can be expected from different theoretical methods. 

The origin with respect to which the electric quadrupole and magnetic dipole operators are defined is indicated by the superscript. jiPp is the /3 component of the velocity operator. The connection between the quadrupole moment referred to or - for example the centre of nuclear masses - and the EQC is... 

In group theory, the electric dipole operator transforms according to the operations of translation, and the magnetic dipole operator transforms as a rotation. It is not difficult to show that the individual transition moments are invariably orthogonal as long as the molecular point group contains improper axes of rotation. This rule is often trivialized to... 

C may be the electric dipole operator the magnetic dipole operator A, or the electric quadrupole operator (3. 

In the molecular point groups, the three trace elements of / always transform under the totaUy symmetric representation. The symmetry behaviour of the three components of (i corresponds with that of the components of the magnetic dipole operator, which transform like the rotations Ry, R. The components of 0(2) transform like those of the quadrupole moment operator, that is, like the five d orbitals. 

Here Cd is a normalization constant and Tz is the magnetic dipole operator, which often can be ignored. Thus, the basic information to be deduced from the dichroic signal at the L2,3-edges is the spin and orbital polarization,... 

We may compare the explicit non-relativistic expressions for the magnetic dipole operator... 

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