Lorentzian function, energy-dependent

Long-range order theory, 35 4-5 Looper s walk capping, 32 438-445 Lorentzian energy averaging, 34 217 Lorentzian function, energy-dependent, 34 243 Losod, 33 215, 224, 258 Low-coordinated transition-metal ions, 34 133 Low energy... 

Fig. 7.6. (a) Energy dependence of a Lorentzian line-shape function with width KT centered at the resonance energy (Ei + 6E). (b) Partial photodissociation cross sections a(E,0) as given by (7.23). All of them have the same width hT the values at the maximum scale like the partial decay rates Tp. 

In indirect methods, the resonance parameters are determined from the energy dependence of the absorption spectrum. An important extra step — the non-linear fit of (t E) to a Lorentzian line shape — is required, in addition to the extensive dynamical calculations. The procedure is flawless for isolated resonances, especially if the harmonic inversion algorithms are employed, but the uncertainty of the fit grows as the resonances broaden, start to overlap and melt into the unresolved spectral background. The unimolecular dissociations of most molecules with a deep potential well feature overlapping resonances [133]. It is desirable, therefore, to have robust computational approaches which yield resonance parameters and wave functions without an intermediate fitting procedure, irrespective of whether the resonances are narrow or broad, overlapped or isolated. 

For the purpose of comparison with the measured absorption coefficient, the theoretical spectra are convoluted with a Lorentzian broadening function F(E). This function is the sum of two terms. The first takes account of the core hole width and the second term is the width of the excited band energy, which is a function dependent on the mean free path of the excited electrons, and takes account of the photoelectron inelastic scattering which is energy dependent and varies for each material as shown in Fig. 1. Note that in this theory any broadening effect due to the experimental resolution and many-body effects, such as the influence of the core hole on the band states, are not included. 

Fig. 23. Temperature dependence of the half-width at half-maximum (HWHM) F of the Lorentzian function fitted to the E-process at Q=1.73 A K Numerical values in the figure are apparent activation energies from the straight lines. ( ) PB, ( ) PIB, ( ) PCP, (x) PE. (Reprinted with permission from [130]. Copyright 1999 American Chemical Society,Washington)...

The width of an absorption band for an individual vibronic transition depends on how long the excited molecule remains in the state created by the excitation. According to Eq. (2.70), the spectrum for excitation to a state that decays exponentially with time should be a Lorentzian function of frequency. The shorter the lifetime of the excited state, the broader the Lorentzian (Fig. 2.12). A variety of processes can cause an excited molecule to evolve with time, and thus can broaden the absorption line. The molecule might, for example, decay to another vibrational state by redistributing energy among its internal vibrational modes or by releasing... 

The steady-state rate of population of state 2 thus has a Lorentzian dependence on the energy gap 12. As we discussed in Qiap. 2, the Lorentzian function can be equated to the homogeneous distribution of 12 when the mean value of 12 is zero and state 2 has a lifetime of 2/2. Note that, according to Eq. (10.29b), T2II — when pure dephasing is negUgible. If we identily the time ccaistant T2 in Eq. (10.35) with 2T in Eq. (2.71), and identify the energy difference 12 with ( — ), then the factor in the second set of parentheses in Eq. (10.35) must be lulh times the distribution function Re[G( )] in Eq. (2.71). 

Note The real collision-induced line profile depends on the interaction potential between A and B. In most cases it is no longer Lorentzian, but has an asymmetric profile because the transition probability depends on the intemu-clear distance and because the energy difference A (/ ) = Ei R) — Ek(R) is generally not a uniformly rising or falling function but may have extrema. 

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