Couple moment

In the remainder of this section, we introduce the principal modeling issues related to spatial transport using moment methods. First, we discuss the realizability of the NDF and of moment sets (which are related to the numerical errors discussed above). Second, we introduce the phenomenon of particle trajectory crossing (PTC) that occurs with the inhomogeneous KE (and is exactly captured by the NDF), and describe how it leads to a closure problem in the moment-transport equations. Next, we look at issues related to coupling between spatial and phase-space transport in the GPBE (i.e. due to correlations between velocity and internal coordinates such as particle volume). Finally, we introduce KBFVM for solving the moment-transport equations in connection with QBMM, and briefly discuss how they can be used to ensure realizability as well as to capture PTC and to treat coupled moments. 

The RCMT, definitive or quick, is considered stable when (1) a minimum of seismograms of three stations azimuthaUy weU distributed (with an angular distance of about 120° each other) is available, (2) the focal mechanism remains stable during five iterations needed to determine the centroid location, (3) the total root mean square of the misfit between seismograms and synthetics averaged for all station used is lower than 0.4, (4) the difference between initial and final coordinates is lower than 0.3°, and (5) the moment tensor should have a small non-double-couple component. This last point is quantified using thresholds arbitrarily used to define a non-double-couple moment tensor (see entry on... 

Infrared and Raman spectroscopy each probe vibrational motion, but respond to a different manifestation of it. Infrared spectroscopy is sensitive to a change in the dipole moment as a function of the vibrational motion, whereas Raman spectroscopy probes the change in polarizability as the molecule undergoes vibrations. Resonance Raman spectroscopy also couples to excited electronic states, and can yield fiirtlier infomiation regarding the identity of the vibration. Raman and IR spectroscopy are often complementary, both in the type of systems tliat can be studied, as well as the infomiation obtained. 

Figure Bl.2.1. Schematic representation of the dependence of the dipole moment on the vibrational coordinate for a heteronuclear diatomic molecule. It can couple with electromagnetic radiation of the same frequency as the vibration, but at other frequencies the interaction will average to zero.

The only tenn in this expression that we have not already seen is a, the vibration-rotation coupling constant. It accounts for the fact that as the molecule vibrates, its bond length changes which in turn changes the moment of inertia. Equation B1.2.2 can be simplified by combming the vibration-rotation constant with the rotational constant, yielding a vibrational-level-dependent rotational constant. 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

Karplus M 1959 Contact electron spin coupling of nuclear magnetic moments J. Chem. Phys. 30 11-15... 

The transition between levels coupled by the oscillating magnetic field B corresponds to the absorption of the energy required to reorient the electron magnetic moment in a magnetic field. EPR measurements are a study of the transitions between electronic Zeeman levels with A = 1 (the selection rule for EPR). 

Since atomic nuclei are not perfectly spherical their spin leads to an electric quadnipole moment if I>1 which interacts with the gradient of the electric field due to all surrounding electrons. The Hamiltonian of the nuclear quadnipole interactions can be written as tensorial coupling of the nuclear spin with itself... 

There are other important properties tliat can be measured from microwave and radiofrequency spectra of complexes. In particular, tire dipole moments and nuclear quadmpole coupling constants of complexes may contain useful infonnation on tire stmcture or potential energy surface. This is most easily seen in tire case of tire dipole moment. The dipole moment of tire complex is a vector, which may have components along all tire principal inertial axes. 

Because of difficulties in calculating the non-adiabatic conpling terms, this method did not become very popular. Nevertheless, this approach, was employed extensively in particular to simulate spectroscopic measurements, with a modification introduced by Macias and Riera [47,48]. They suggested looking for a symmetric operator that behaves violently at the vicinity of the conical intersection and use it, instead of the non-adiabatic coupling term, as the integrand to calculate the adiabatic-to-diabatic transformation. Consequently, a series of operators such as the electronic dipole moment operator, the transition dipole moment operator, the quadrupole moment operator, and so on, were employed for this purpose [49,52,53,105]. However, it has to be emphasized that immaterial to the success of this approach, it is still an ad hoc procedure. 

---