Induced dipole matrix element

Overtone band. Similar induced dipole matrix elements for the H2-He system have been derived from Table 4.5 for the overtone bands [61]. In that case, the numerical accuracy of the resulting B coefficients is limited for large vibrational excitations unless the data are supplemented by larger variations of the vibrational coordinate r. 

According to Equations (5.14) and (5.15), we see that the probability of a particular transition depends on the electric dipole matrix element /x, given by Equation (5.12). These transitions, which are induced by interactions of the electric dipole element with the electric field of the incident radiation, are called electric dipole transitions. Therefore, electric dipole transitions are allowed when p- 0. 

The results are shown in Fig. 3 and Table I. Apparently, optimal IR femtosecond/picosecond laser pulses with durations tp = 500 fs may induce nearly perfect transitions (12), (13) in the model OH. Similar examples are documented in Refs. 13, 18, and 23. A detailed discussion of the derivation of the optimal laser parameters, depending on the vibrational level Ev and the transition dipole matrix elements nvw, is also given in Refs. 13, 18, and 23. Suffice it here to say that in many (but not all) cases the optimal frequency is close (but not identical) to the resonance frequency,... 

The unusual effects induced by the quantum interference between two transitions in the multilevel systems, discussed in the previous sections, may occur only if the dipole matrix elements of the transitions involved are nonorthogonal ... 

In the following it will be reported on a straightforward method to provide information about the dipole matrix elements and phase shifts being essential for the theoretical description of the photoemission process in a relatively simple way and with a pronounced accuracy [2]. This can be achieved by means of photoelectron spectroscopy with linearly polarized light using the ability of a continuous rotation of the electric field vector. The method is exemplarily demonstrated at the system hydrogen on Gd(0001)/W(l 10) which possesses a pronounced adsorbate induced state. 

With the definition (2.65) for the dipole matrix element Di/c, the Einstein coefficient B(] of induced absorption / finally becomes... 

Induced electric dipole matrix element for a single spectral line in an oriented system... 

Induced electric dipole transitions occur much more frequently than magnetic dipole transitions and therefore the largest part of this review is devoted to the former. However, induced electric dipole transitions have the disadvantage that the knowledge of wavefunctions is not sufficient for the calculation of electric dipole intensities and a parametrization is necessary. Judd and Ofelt developed independently the theoretical background for the calculation of the induced electric dipole matrix element. Their work is known under the common name Judd-Ofelt theory (sections 5 and 7). The papers of Judd... 

Judd (1962) and independently Ofelt (1962) worked out the theoretical background for the calculation of the induced electric dipole matrix element. The basic idea of Judd and Ofelt is that the intensity of the forbidden f- f electric dipole transitions can arise from the admixture into the 4f configuration of configurations of opposite parity (e.g., 4f d and 4f " n g ). As already mentioned in the introduction, we will unravel here in detail the theoretical model developed by Judd. Special attention will be given to the dimensions, units and selection rules. Our symbolism is close to Judd s. The difference is that we represent the crystal-field coefficient by (instead of Judd s Atp), the light polarization by p (instead of Judd s q), and the additional quantum number by r (Judd s y). 

Finally, the induced electric dipole matrix element between two states B and B of the f configuration can be written as... 

Asymptotic formulae. For a discussion of induced dipoles in highly polarizable species, it is often sufficient to consider the so-called classical multipole induction approximation in its simplest form (i.e., neglecting field gradients and hyperpolarizabilities). In such a case, one needs to know only the vibrational matrix elements of the multipole moments,... 

Fig. 4.1. The four significant induced dipole components of H2-He pairs ground state vibrational average (left panel) and vibrational transition matrix element v = 0 —> v = 1 (center panel) and of the ground state vibrational average of H2-Ar pairs (right panel). Overlap components are dotted multipole-induced components are represented by solid lines dashed lines indicate the associated classical multipole component after [279, 151, 280],...

For the rotovibrational bands, the induced dipole components Bc of Eqs. 6.13 through 6.18 become the vibrational matrix element [281], Eq. 4.20. Furthermore, each XifoAL component now occurs twice, once with molecule 1 vibrating and once with molecule 2 vibrating in the final state. For like molecular pairs, this may be taken into account by removing the factors of 1/2 in Eqs. 6.13, 6.14, 6.16, and 6.17. For dissimilar molecular pairs, the factors 1/2 are absent from the equations quoted. 

For the case of induced absorption of H2-He pairs, the vibrational matrix elements of potential and dipole function are well known [151, 294]. The spectral moments Mq- M2 have been computed for the main induction components with the corrections for the vibrational dependences... 

The spectral density is defined in terms of the matrix elements of the induced dipole moment p by the golden rule , Eq. 5.3,... 

The selection rules for the Raman spectrum turn out to depend not on the matrix elements of the electric dipole moment, but on the matrix elements of the molecular polarizability, which we now define. The application of an electric field E to a molecule gives rise to an induced electric dipole moment djnd (which is in addition to the permanent dipole moment d). If E= "> 1 + yl+ >zk, then the induced dipole moment has the components... 

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