Electron magnetic dipole moment

In addition, Dirac s theory provides a direct explanation for the fact that the electron magnetic dipole moment is about twice the value expected classically on the basis of a spherical charged particle rotating around one... 

Table 11. CCSD results for the total Verdet constant at w = 0.11391 a.u. in the case of hydrogen fluoride. Results labeled as Unrelaxed refer to the use of the unrelaxed (one-electron) magnetic dipole moment operator together with the usual magnetic-field independent basis sets. Results labeled Relaxed include additional contributions due to orbital relaxation in the presence of the magnetic field. Results labeled LAO are those obtained when using GIAOs/LAOs ...

Prom the expansion in Eq. (5.30) we can define the components of the magnetizability and nuclear magnetic shielding tensor of nucleus K as first derivatives of the perturbed electronic magnetic dipole moment with respect to a component of the magnetic induction or the nuclear magnetic dipole moment of nucleus if ... 

Inserting now the expansion of the electronic magnetic dipole moment from Eq. (5.30)... 

In the previous section we have defined the cartesian components of the magnetizability tensor as second derivatives of the energy E B) in the presence of a magnetic induction B, Eq. (5.39), or alternatively as first derivatives of the magnetic-field-dependent electronic magnetic dipole moment ma B), Eq. (5.32). Both definitions can be used to derive quantum mechanical expressions for the magnetizability. 

This is the basic unit or measure for electronic magnetic dipole moments in the same sense that h is the measuring unit for angular momentum. 

Electronic magnetic dipole moments in molecules and atoms are measured in terms of Bohr magnetons in the same way that angular momentum is measured in terms of h. 

The interaction of the electron spin s magnetic dipole moment with the magnetic dipole moments of nearby nuclear spins provides another contribution to the state energies and the number of energy levels, between which transitions may occur. This gives rise to the hyperfme structure in the EPR spectrum. The so-called hyperfme interaction (HFI) is described by the Hamiltonian... 

The magnetic field seen by the probe neutron is solely due to the magnetic dipole moment density of the unpaired electrons. In other words, the magnetisation density is simply related to the electron spin density by a multiplicative factor, and there is no ambiguity in its definition. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

There are two contributions to the magnetic dipole moment of an electron bound to an atomic nucleus, which, in semiclassical models, are attributed to orbital motion, represented by quantum number l, and spin, represented by quantum number, v. The orbital and spin components are linked, or coupled, on isolated atoms or ions to give an overall magnetic dipole moment for the atom. The total magnetic dipole moment of the atom is given by... 

Equation (S6.1) is applicable to the salts of lanthanide ions. These have a partly filled 4f shell, and the 4f orbitals are well shielded from any interaction with the surrounding atoms by filled 5.9, 5p, and 6.9 orbitals, so that, with the notable exceptions, Eu3+ and Sm3+, they behave like isolated ions. For the transition metals, especially those of the 3d series, interaction with the surroundings is considerable. Because of this, the 3d transition-metal ions often have magnetic dipole moments corresponding only to the electron spin contribution. The orbital moment is said to be quenched. In such materials Eq. (S6.1) can then be replaced by a spin-only formula ... 

This can be roughly seen by considering the classical view of the valence electron, where this electron describes a circular orbit of radius r around the nucleus. In this case, the magnitude of the magnetic dipole moment is proportional to the area of the circular orbit and u , a so that u ,(r) = Um(—r). 

The perturbation //ss(1.2), which describes the interaction of the magnetic dipole moments associated with the spins of two electrons, is given by ... 

As discussed earlier, VCD depends on both the electric and magnetic dipole moment derivatives in a molecule. The simpler descriptions of VCD focus only on local electric dipole moment derivatives that have an overall chiral disposition. More advanced descriptions of VCD allow for induced electronic currents or charge flows in molecules, which give rise to additional magnetic dipole moment intensity. Such additional contributions are likely whenever delocaliz-able electron density is present in a molecule. 

The magnetic dipole moment of the electrons about the y axis is My, electrons = (e/2mep) Ly. ... 

The ar 1 (Ml) term thus describes the interaction of the magnetic dipole moments of the electrons and nuclei with the magnetic field (of strength IHI = Ao k) of the light (which lies along the y axis) ... 

Magnetic properties are due to the orbital and spin motions of electrons in atoms. The relation between the magnetic dipole moment p and the angular momentum J of an electron of charge e and mass m can be expressed as... 

In quantum mechanics this then becomes the vector product of the nuclear magnetic dipole moment and the distance vector between an electron i and the field-creating nucleus I (60)... 

The induced magnetic dipole moment has transformation properties similar to rotations Rx, Rt, and Rz about the coordinate axes. These transformations are important in deducing the intensity of electronic transitions (selection rules) and the optical rotatory strength of electronic transitions respectively. If P and /fare the probabilities of electric and magnetic transitions respectively, then... 

Now, let us consider a system where an achiral molecule (A) and a chiral molecule (C) have a fixed mutual orientation. An electronic transition of the achiral molecule from the ground state z(0> to the excited state Aa, higher in energy by E0a, has a zero-order (non-perturbed) electric dipole moment po0 and an orthogonal magnetic dipole moment ma0. These moments are increased in the molecular pair (A -C) by first-order dynamic coupling as ... 

---