Dipole moments matrix element

Remembering that each of these is further split by the hyperfine interaction, there are obviously several possible transitions among these four energy levels. To find out which are important, we must evaluate the transition dipole moment matrix elements, (i Sx j), since the absorption intensity is proportional to the square of these matrix elements. The operator Sx can be written ... 

Electric field in 10 an, energies in eV, dipole moment matrix elements in Debye, charges Aq in e for the definitions of various quantities, see Table 1 Adapted from [41]... 

According to the results of the last section, if the integrals in (3.48) all vanish, then the probability for a transition between states m and n is zero. Actually, (3.47) is the result of several approximations, and even if the electric dipole-moment matrix elements vanish, there still might be some probability for the transition to occur. 

The pattern of intensities in Fig. 4.9 deserves mention. The intensities of absorption lines are proportional to the population of the lower level, and to the square of the dipole-moment matrix element (4.97). It turns out that for vibration-rotation transitions in the same band, the integral (4.97)... 

Table 10.1. Symmetry of the dipole moment matrix elements in trans-dichlorobis(ethylenediamine)cobalt(III) in D4h symmetry.

The double helix has unbalanced charge on many atoms. The displacements associated with vibrational modes generate oscillating dipole moments. These moments are constructed from our eigenvector displacements and models of atomic net charge taken from the literature (2, 3). These dipole moment matrix elements have been calculated (4). 

Each spectral line can arise from a species with a particular concentration and transition dipole moment matrix element and a particular linewidth determined by the extent of homogeneous and inhomogeneous broadening. The magnitude of absorption as a function of frequency is given by Beer s law ... 

Figure 16 Diagrams which arise in the calculation of polarizabilities. The heavy dot represents a dipole moment matrix element...

The importance of the hyper Raman effect as a spectroscopic tool results from its symmetry selection rules. These are determined by products of three dipole moment matrix elements relating the four levels indicated in Fig. 3.6-1. It turns out that all infrared active modes of the scattering system are also hyper-Raman active. In addition, the hyper Raman effect allows the observation of silent modes, which are accessible neither by infrared nor by linear Raman spectroscopy. Hyper Raman spectra have been observed for the gaseous, liquid and solid state. A full description of theory and practice of hyper-Raman spectroscopy is given by Long (1977, 1982). 

Boltzmann averaged density of states. By introducing appropriate dipole moment matrix elements, the infrared spectrum can be obtained. Reimers and Watts [28] obtained very good agreement with experimntal infrared spectra for the intramolecular stretch and bend vibrations, although considerable development is needed to reproduce the intensities of overtone and combination bands. 

The general microscopic expression for the nth-order susceptibility contains n + 1 dipole moment matrix elements, involving n intermediate states. For the linear susceptibility there is only one intermediate state, and if the latter is a hybrid one, the corresponding dipole matrix elements are determined mainly by the Frenkel component of the hybrid state. Thus, the linear susceptibility of the hybrid structure contains the factor (dp/ap)2, as is seen from eqn (13.77). For the second-order nonlinear susceptibility x one must have two intermediate states or three virtual transitions. One of them may be a hybrid one, and as long... 

Bmn in Equation 1.17 can be replaced in terms of the sum of the squares of the dipole moment matrix elements (Equation 1.3) and the equation modified by the Lorentz function, yielding the general form of the equation for a the absorption coefficient at any point on the spectral profile ... 

The natural linewidth of a molecular spectral line in the MMW region is related inversely to the spontaneous emission coefficient of the upper state (Equation 1.5) and consequently imparts a Lorenteian shape to the line profile (Equation 1.32). As the upper state can radiate to more than one lower state, the actual natural linewidth is related to the sum of the squared dipole moment matrix elements of the states involved. In any case its contribution to the overall linewidth is negligible in comparison with the other broadening contributions at 100 GHz it would be 10 Hz. 

Here R" symbolizes the positions of molecular centers of mass and A the orientations of all molecules in the crystal. The absorption intensity is determined by the transition dipole moment matrix element ... 

As in the case of the dipole moment matrix element (4.5), we obtain on substitution of (4.33) into (4.32) a product of the well-known harmonic oscillator transition moment matrix element and the sum of polarizability derivatives ... 

In the above, /(r) = (c/87r) o( ) the intensity of the laser beam at the point r Is = cl4n) h yld) is the saturation intensity d=d- e is the projection of the dipole moment matrix element of the polarization vector e of the laser beam A is the detuning of the laser field frequency cu with respect to the atomic transition frequency tuo, that is, A = lo — loo, and the quantity 27 defines the rate of spontaneous decay of the atom from the upper level e) to the lower level g), that is, the Einstein coefficient A. Figure 5.6 shows the dependence of the radiation pressure force and the gradient force on the projection v =v of the atomic velocity on the propagation direction of a Gaussian laser beam for the case of strong saturation of the D-line of Na. 

The Einstein coefficient Ba in equation (98) can thus be related to the square of the transition dipole moment matrix element by equation (99), in the electric dipole approximation ... 

The intensity of a vibronic band is proportioned to the square of the electronic dipole moment matrix element between the initial and final states. These states are well described by Bom-Oppenheimer products of electronic and nuclear functions. For a symmetry allowed transition the nuclei coordinate dependence of the electronic function can be ignored so that the intensity of a vibronic band can be factored into a purely electronic factor that is independent of nuclear coordinate times the square of the overlap of the vibrational functions (Franck-Condon factor). 

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