The line strength

The starting point of theoretical calculations of the oscillator strengths and transition probabilities of spectral lines are the wavefunctions of the energy levels involved. 

In these calculations it is often found to be convenient to compute first a quantity called the line strength  

This is the quantum-mechanical analogue of the square of the classical electric dipole moment, 2.q (equation (4.13)). 

The line strength is symmetrical with respect to the initial and final states and is often quoted in atomic units, a e = 6-459 x lo cm esu = 7-188 x 10 m C. Once the line strength is known it is a simple matter to combine equations (4.29), (4.31), and (4.35) to obtain the oscillator strengths for absorption or emission  

Since explicit wavefunctions are available only for hydrogenic systems, it is in these cases alone that exact values for the f-values and transition probabilities can be calculated. To illustrate the use of the expressions derived in preceding sections, we consider the first line in the Lyman series, 2 P+1 S, which in atomic hydrogen has 

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In the ideal case for REMPI, the efficiency of ion production is proportional to the line strength factors for 2-photon excitation [M], since the ionization step can be taken to have a wavelength- and state-mdependent efficiency. In actual practice, fragment ions can be produced upon absorption of a fouitli photon, or the ionization efficiency can be reduced tinough predissociation of the electronically excited state. It is advisable to employ experimentally measured ionization efficiency line strengdi factors to calibrate the detection sensitivity. With sufficient knowledge of the excited molecular electronic states, it is possible to understand the state dependence of these intensity factors [65]. 

The peak absorption (scattering) cross sections are thus useful comparative measures of detectivity because the latter is a product of the line strength and the practical line resolution. 

This case is shown schematically in Fig. 5c. In Eq. (50), qj. are generalized y-photon asymmetry parameters, defined, by analogy to the single-photon q parameter of Fano s formalism [68], in terms of the ratio of the resonance-mediated and direct transition matrix elements [31], j. is a reduced energy variable, and <7/ y, is proportional to the line strength of the spectroscopic transition. The structure predicted by Eq. (50) was observed in studies of HI and DI ionization in the vicinity of the 5<78 resonance [30, 33], In the case of a... 

As in the previous case of infrared transitions, one wants to calculate the line strengths S(v,J —> v, J ) defined in Eq. (2.127). For Raman transitions there are two contributions, as discussed in Chapter 1. The so-called trace scattering is induced by the monopole operator... 

It is convenient to write the line strength for trace scattering in general as... 

The intensity of an electric dipole transition in absorption or emission depends, on one hand, on factors particular to the experiment measuring the intensity, e.g., the number density of molecules in the initial state of the transition and, for absorption experiments, the absorption path length and the intensity of the incident light. On the other hand, the intensity involves a factor independent of the experimental parameters. This factor, the line strength 5(f <— i), determines the probability that a molecule in the initial state i of the transition f <— i will end up in the final state f within unit time. 

If we assume that the initial state i and the final state f are both non-degenerate, then the line strength of the electric dipole transition between them [3] is given by... 

The energy density function p v) is defined so that dE—p v)dv is the amount of available radiation energy per unit volume originating in radiation with frequency in the infinitesimal interval [v,v + dv]. Thus, p v) is expressed in the SI units J/(m Hz) = J s/m, so that Bg and Bg have the SI units m /(J s ). Ag is expressed in s The Einstein coefficients defined in this manner are related to the line strength by... 

In the present section, we obtain an expression for the line strength in equation (4) in a form suitable for numerical calculation. This derivation closely follows the theory developed in Refs. [18,19] and in Chapter 14 of Ref. [3], and so we give only an outline here. 

The parameterized, analytical representations of fi, ., fiy, fifi determined in the fitting are in a form suitable for the calculation of the vibronic transition moments V fi V") (a—O, +1), that enter into the expression for the line strength in equation (21). These matrix elements are computed in a manner analogous to that employed for the matrix elements of the potential energy function in Ref. [1]. 

As detailed in Section 2, we have derived and programmed the expression for line strengths of individual rotation-vibration transitions of XY3 molecules the line strengths depend on the vibronic transition moments entering into equation (70). With the theory of Section 2, we can simulate rotation-vibration absorption spectra of XY3 molecules. In computing the transition wavenumbers, line strengths, and intensities we use rovibronic wavefunctions generated as described in Ref. [1]. 

The symbol for the energy level is denoted as Ij. The line strength takes the form ... 

In this coupling scheme the symbol for a level is denoted as K j. The line strength... 

As in the case of LS coupling, when there are N equivalent electrons in the outer shell, both the line strength and the oscillator strength should be multiplied by N as well as by the corresponding CFP [ 10,12]. As in the LS scheme the two forms of the electric dipole length transition operator have been employed here in the calculation of the radial transition integral, I nl, n l ). 

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