Magnetic moment operator

The electric dipole moment operator is independent of spin and, in the absence of spin-orbit coupling, the spin part of the magnetic moment operator will have no influence on MCD. We shall therefore neglect the spin part of Eq. (8) until we come to spin-orbit-induced MCD in Section II.A.4. 

The first- and second-order Zeeman effect coefficients in the expansion of equation (62) are defined by the quantum numbers which specify the atomic energy level. They are in general a function of the direction of the magnetic field with respect to the axis of quantization of the wave functions. They are obtained by the use of the magnetic moment operator for the appropriate direction, q = x,y ox z ... 

The above is derived from quantum mechanics, via a definition of the anisotropic magnetic moment operator. What are the conditions this imposes on 3 x 3 matrix g ... 

For a perturbing electric field in the v-direction we have V = W = Dv and W — Y = 0, while for a magnetic field in the v-direction we have for the imaginary magnetic moment operator W = —V = +MV and V + W = 0. A nonzero frequency couples the symmetric and the antisymmetric part of the perturbed density matrix, whereas in the static case the two equations in (16) are not coupled. For comments on the apparent lack of symmetry for the perturbation equations for static electric and magnetic fields see [46]. 

While the chemical interpretation of the e parameters is a matter of real concern to us, there are also several other difficulties which are, however, more apparent than real. Consider the question of the calculation of magnetic properties in transition metal complexes - paramagnetic susceptibilities and e.s.r. g values. In contrast to the study of eigenvalues for optical transition energies, these require descriptions of the wavefunc-tions after the perturbation by the ligand field, interelectron repulsion and spin-orbit coupling effects. In susceptibility calculations it is customary to use Stevens orbital reduction factor k in the magnetic moment operator... 

Applying the magnetic moment operator to this eigenfunction, one obtains for the moment... 

Here, m is the magnetic moment operator, and p, is the linear momentum of electron i. (If Po and Pq are real wave functions, as is generally the case, the magnetic dipole transition moment is an imaginary quantity,... 

Eq. (159) also defines a magnetic moment operator, [x, given by... 

The magnetic moment operator for electrons is M = — f (1 with the Bohr... 

In the dipole approximation, the magnetic moment operator is replaced by a weighted sum of L and S, and the weighting factors depend also on the magnitude, k, of k. The dependence is expressed in terms of spherical Bessel functions averaged over the electron radial density I 4((/ ). It is convenient to define the quantities... 

Magnetic dipole-derived intensity, which requires no parity mixing, is calculated in absolute units from the matrix elements of the magnetic moment operator and requires no additional parametrization. For d-d spectra, this source is generally unimportant. However, for f-f spectra, magnetic dipole-derived intensity must be included. As the electric dipole model only provides a relative intensity scale, a scaling parameter for the two sources must be introduced. 

Relatively modest modifications are required to extend a static linear polarizability Hartree-Fock computer code to dynamic or frequency-depen-dent polarizabilities. And if the code already handles magnetic moment operators, then the conversion to optical rotatory dispersion (ORD) or circular dichroism (CD) is also straightforward. At the third derivative level, e.g., hyperpolarizabilities, modifications are more complicated. 

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