Magnetic dipole production

Here (r - Rc) (r - Rq) is the dot product times a unit matrix (i.e. (r — Rg) (r — Rg)I) and (r - RG)(r — Rg) is a 3x3 matrix containing the products of the x,y,z components, analogous to the quadrupole moment, eq. (10.4). Note that both the L and P operators are gauge dependent. When field-independent basis functions are used the first-order property, the HF magnetic dipole moment, is given as the expectation value over the unperturbed wave funetion (for a singlet state) eqs. (10.18)/(10.23). 

As shown in the Appendix (in Section V), in the C2h point group, the 1Ag - 1Bll (i.e. the monoelectronic r - jr excitation) possesses only an electric dipole moment, while in the C2V structure the electric and magnetic dipole moments, both non-vanishing, are orthogonal. In both cases the product in equation 1 leads to zero rotational strength. 

A particularly useful probe of remote-substituent influences is provided by optical rotatory dispersion (ORD),106 the frequency-dependent optical activity of chiral molecules. The quantum-mechanical theory of optical activity, as developed by Rosenfeld,107 establishes that the rotatory strength R0k ol a o —> k spectroscopic transition is proportional to the scalar product of electric dipole (/lei) and magnetic dipole (m,rag) transition amplitudes,... 

A magnetic dipole moment, when placed in an external magnetic field, will have an energy of interaction E with the field which is the negative of the product of the magnetic field H and the component of the magnetic moment along the field direction fin... 

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 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. 

Ordinary Raman scattering is determined by derivatives of the electric dipole-electric dipole tensor ae, and ROA by derivatives of cross-products of this tensor with the imaginary part G,e of the electric dipole-magnetic dipole tensor (the optical activity tensor) and the tensor Ae which results from the double contraction of the third rank electric dipole-electric quadrupole tensor Ae with the third rank antisymmetric unit tensor s of Levi-Civita. The electronic property tensors have the form ... 

Drude first proposed that the rotatory power of a dissymmetric substance could be understood if its absorption of light involved the motion of a charged particle along a helical path within the molecule [8]. This type of motion would result in the simultaneous production of an electric dipole from the translatory motion and a magnetic dipole from the rotatory motion. The model requires that the electric and dipole moments have at least some components which are collinear with each other, or else stereospecific interaction with circularly polarized light would not be possible. 

The ellipticity, or the intensity of the circular dichroism (CD) spectrum, is fundamentally characterized by the rotatory strength R, which is given by the imaginary part of the inner product between the electronic and magnetic dipole transition moment vectors ... 

Here i//0 is the ground vibrational wave function and ij/ is the wavefunction corresponding to the first excited vibrational state of the th normal mode /< is the electric dipole moment operator Qj is the normal coordinate for the /th vibrational mode the subscript 0 at derivative indicates that the term is evaluated at the equilibrium geometry. The related rotational strength or VCD intensity is determined by the dot product between the electric dipole and magnetic dipole transition moment vectors, as given in (2) ... 

Any dipolar magnetic field pattern is symmetric with respect to rotations around a particular axis. Hence, it is customary to describe the magnetic dipole moment that creates such a field as a vector with a direction along that axis. The SI units of magnetic moments are thus A m. From Eq. 8.2, the torque experienced by the magnetic moment in the external field is given by the cross product of the magnetic moment and... 

The final result is that the sensitivities can be obtained from the scalar product of the original electric field and the auxiliary electric fields, generated by elementary electric or magnetic dipole current sources, located at the receiver position, r. This result was originally demonstrated by Pellerin et al., 1993, and then a derivation based on Lorentz lemma was given by McGillivray et al., 1994. 

The dynamic characteristics of adsorbed molecules can be determined in terms of temperature dependences of relaxation times [14-16] and by measurements of self-diffusion coefficients applying the pulsed-gradient spin-echo method [ 17-20]. Both methods enable one to estimate the mobility of molecules in adsorbent pores and the rotational mobility of separate molecular groups. The methods are based on the fact that the nuclear spin relaxation time of a molecule depends on the feasibility for adsorbed molecules to move in adsorbent pores. The lower the molecule s mobility, the more effective is the interaction between nuclear magnetic dipoles of adsorbed molecules and the shorter is the nuclear spin relaxation time. The results of measuring relaxation times at various temperatures may form the basis for calculations of activation characteristics of molecular motions of adsorbed molecules in an adsorption layer. These characteristics are of utmost importance for application of adsorbents as catalyst carriers. They determine the diffusion of reagent molecules towards the active sites of a catalyst and the rate of removal of reaction products. Sometimes the data on the temperature dependence of a diffusion coefficient allow one to ascertain subtle mechanisms of filling of micropores in activated carbons [17]. 

The irreducible tensor product between two (spherical) vectors is defined in Eq. (37). An important feature of this Hamiltonian is that it explicitly describes the dependence of the coupling constants J, Am, and T, on the distance vectors rPP between the molecules and on the orientations phenomenological Hamiltonian (139). Another important difference with the latter is that the ad hoc single-particle spin anisotropy term BS2y, which probably stands implicitly for the magnetic dipole-dipole interactions, has been replaced by a two-body operator that correctly represents these interactions. The distance and orientational dependence of the coupling parameters J, A, , and Tm has been obtained as follows. 

We can deduce the symmetry of a response tensor by considering the operators that enter the numerator of its quantum mechanical expression. For example, the product of three electric-dipole transition moment operators in Eq. (14) render SFG a parity-odd and time-even process. It follows that a third-order process requires nonlocal magnetic-dipole contributions in order to be parity-odd and that a local fourth-order process is parity-odd within the electric-dipole approximation. Some pseudoscalars that arise at order n are tabulated below. 

Wave Astronomy Satellite (SWAS) show the abundance to be less than 0.1% that of CO. At this level, it does not interfere with organic synthesis. With elemental evolution, an increase of the 0/C ratio is expected. If this is not readily incorporated into the refractory solid phase, production of organic species in the interstellar molecular clouds could well be reduced. CO is an abundant molecule in strongly red shifted quasars (Downes and Solomon, 2003) (z = 2.6-6.4). Thus, it would be expected that its reaction products also are present, but harder to observe. SWAS is instrumented to measure O2 abundances and has not observed any. (O2, although lacking an electric dipole moment, has magnetic dipole transitions.) Note that it has probably not been observed in dense molecular clouds. 

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