Multipole Moments Magnetic

There are unfortunately various definitions of magnetic multipole moments in the literature - a recent paper by Raab16 discusses the various definitions and properties, and creates some order from an apparent chaos. 

General considerations on symmetry [12,13] lead to the result, that an atomic nucleus in a stationary state with spin quantum number / has electric and magnetic multipole moments only of order 2 with 0 < I <21. For electric multipole moments I must be even, while magnetic multipole moments require I to be odd. These rules are strictly obeyed, as long as very tiny parity non-conservation effects, due to weak interaction between nucleons, axe omitted (as is usually done for the nucleus, but see Sect. 6.3, where these effects are briefly discussed for the electronic structure). Thus,... 

With all the electrons paired and a spherically symmetric charge distribution, the noble gas atoms (to a first approximation) have no electric or magnetic multipole moments whatever. That leaves the dispersion forces to account for the binding of the cluster. The dispersion force depends on the polarizability, and since these single atoms are equally polarizable in all directions, the potential energy curve for each atom is completely isotropic it initially depends only on R. 

Response theory describes the change in observable quantities such as electric and magnetic multipole moments due to external fields. The starting point of response theory therefore is the time evolution of the expectation value (A) of an Hermitian operator A representing the observable quantity (e.g., the electric or magnetic dipole moment). In order to monitor the change in an observable quantity, we need to know the state of the molecule before the external field is switched on. As mentioned above, the electronic ground state 0) is the proper choice imder most experimental conditions. With this initial condition, the time evolution can be written as the perturbation expansion... 

The symbol p is a transition moment operator, of which there are various kinds, namely, those corresponding to changes in electric or magnetic dipoles, higher electric or magnetic multipoles, or polarizability tensors. 

We firstly define the electric and magnetic multipoles, then show how the interaction energy with external fields can be expressed in terms of the induced moments and go on to see how these are related to the field-free moments via response functions. Most of our treatment will concern the electric case, since this is the simpler, and since magnetic effects can be treated by analogy. 

Here Modpg is the module or magnitude of the polarization moment, but ipo is its initial phase at the moment of excitation. As can be seen, the magnetic field itself does not alter the absolute value of the multipole moment, changing only its phase... 

For a description of the ground state magnetic quantum beats one might conveniently use the solution of Eq. (4.10) for multipole moments aPq-Assuming that the excitation takes place by a 6-pulse at time t = 0, one may write its solution for t > 0 in the form ... 

A standard question in spectroscopy is does a given molecule have a permanent dipole moment This can usually be decided immediately by inspection, but the formal group-theoretical answer is that a molecule can exhibit a component of permanent dipole for each occurrence of the totally symmetric To in the dipole character T (/u.) = xyz = Ft- Similar reasoning can be extended to other multipole moments, polarisabilities and corresponding magnetic properties. 

The electronic magnetic multipoles (25)-(27) are unperturbed, or permanent, moment operators. In the presence of a vector potential A(r, t) (we simplify the notation, omitting the index), the canonical momentum is replaced by the mechanical momentum... 

Monatomic entities M+z consisting of one nucleus (carrying Z times the electric charge e of a proton) surrounded by K = (Z — z) electrons have been one of the major subjects for quantum-mechanical treatment. If the nucleus is treated as a geometrical point, and no attention is paid to its electric multipole moments, nor to its magnetic moments, the energy levels can be characterized by even or odd parity and by a quantum number J of total angular momentum. If the coordinates (— x, — y,... 

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