Magnetic dipole transition moments equations

We note again that the magnetic dipole operator is pure imaginary, and the multiplication by i results in a real value. In the denominator of eq. (20) we have also neglected the magnetic dipole transition moments in eq. (14) compared to the usually larger electric dipole moments. This equation is sometimes written as... 

The approximation inherent in Equation [5b] means that the nuclear kinetic operator, -hyi) d ldQ ), has no influence on the electronic wavefunction, namely, the terms, -h d g/dQ)ei[S/dQi) and (-fi2/2)(a2 aQ2), which operate on the electronic part of the wavefunction, are omitted. This implies that infinitesimal changes in the nuclear configuration do not afiect the electronic wavefunction, g. This approximation is, however, not good enough in evaluations of the magnetic dipole transition moments. The disturbance to the electronic state caused by changes in the nuclear configuration has to be taken... 

The theoretical foundation for all chiroptical techniques lies in the Rosenfeld equation (2) which expresses the rotatory strength of a transition, assumed to be between states 0 and n, as the imaginary part of the scalar product of the electric dipole and magnetic dipole transition moments for the transition. 

There is only one other ab initio implementation of the theory of optical activity to calculate optical rotatory strengths, that due to Hansen and Bouman, based on the random-phase approximation (RPA) and implemented in the program package, RPAC. The RPA method is intended to include those first-order correlation effects that are important both for electronic transition intensities and for excitation energies. The electric and magnetic dipole transition moments in RPA are given by equations (14), (15), and (16) (analogous to equations 7, 8, and 9, above). 

Insertion of equation (20) into equation (14) yields an expression in which the sum over states does not appear, namely the interesting result that the electronic contribution to the magnetic dipole transition moment is proportional to the overlap of two derivatives of the ground state electronic wavefunction ... 

In practice, the electric and magnetic dipole transition moments are usually expressed as summations of atomic properties, namely the atomic polar tensor (APT), P", and atomic axial tensor (AAT), MJ, which, in the VCT approach, can be extracted from equations (15) and (16) as. 

The emphasis in much of the CPL-based work developed over the past decade is a result of the unusual spectral characteristics of the luminescent chiral Ln(lll) complexes (see Chapter 1) and the technical advantages resulting from the use of CPL. One ean expect the measurement of larger values when the transitions involved are inherently weak. This can be seen from the form of Equation 3.12 where one may expect a large value when the transition considered is electric dipole forbidden, but magnetic dipole allowed. Since the magnetic dipole transition moments are typically much smaller than the electric dipole terms, the denominator in Equation 3.12 will be dominated by the first term, p. Transitions... 

In equation 23 Im means to take the imaginary part of the corresponding expression. In a series of related compounds for a particular magnetic dipole transition, as is the (n, tt ) excitation of 152-154, 156, 157, the magnetic transition moment is approximately constant, but the electric transition moment may vary widely. Then, if 0 represents the angle between the directions of the electric and magnetic transition moments, an equation... 

The value of Equation [12] is therefore at most a few times the ratio of a magnetic to an electric dipole transition moment. The largest observed values of A/// are of the order of 5x10, and values of 10 or less are more common. 

Analogously to UV/ECD, IR and VCD intensities are proportional to the quantities dipole (D) and rotational (R) strengths, respectively, which are calculated using equations (16) and (17), at the same level of theory employed in the optimization step. However, in the case of VCD, the electric (jx) and magnetic (m) dipole transition moment vectors include the wavefunctions of the ground (0) and the first excited vibrational state (1), within the ground electronic state of the molecule. Furthermore, in the harmonic approximation, the dipole strength in the normal mode Qa is proportional to ... 

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)... 

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 ... 

In this expansion the dipole-dipole term is the most prominent if donor-acceptor distance R is not too small. The dipole-dipole term represents the interaction between the transition dipole moments Md and MA of donor and acceptor molecules, respectively. The square of these transition dipoles is proportional to the oscillator strengths fy> and fA for radiative transitions in the individual donor and acceptor molecules (equation 3.73). Higher order terms such as electric dipole-electric quadrupole, electric-dipole-magnetic dipole, become important at close approach and may be effective in crystals and highly ordered array of chromophores. 

The relation between the spherical components AJ0( ) of a general tensor A of rank 2 and the cartesian components A, ( ) are given in Appendix 4. Equations (3.36) will form the basis for derivation of selection rules for rotation-internal motion transitions of SRMs presented in the next section. They also may serve for derivation of the transformation properties of the electric and magnetic dipole moment operators referred to the laboratory system (VH G... 

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