Rotational Franck-Condon factors

Here, I(co) is the Fourier transform of the above C(t) and AEq f is the adiabatic electronic energy difference (i.e., the energy difference between the v = 0 level in the final electronic state and the v = 0 level in the initial electronic state) for the electronic transition of interest. The above C(t) clearly contains Franck-Condon factors as well as time dependence exp(icOfvjvt + iAEi ft/h) that produces 5-function spikes at each electronic-vibrational transition frequency and rotational time dependence contained in the time correlation function quantity <5ir Eg ii,f(Re) Eg ii,f(Re,t)... 

The emission spectrum observed by high resolution spectroscopy for the A - X vibrational bands [4] has been very well reproduced theoretically for several low-lying vibrational quantum numbers and the spectrum for the A - A n vibrational bands has been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrationm and rovibrational transitions appear to be the largest for the X ground state followed by those... 

In the quantum mechanical description (in continuation of Box 2.2), the wavefunction can be described by the product of an electronic wavefunction VP and a vibrational wavefunction / (the rotational contribution can be neglected), so that the probability of transition between an initial state defined by ViXa and a final state defined by TQ/b is proportional to electron coordinates, this expression can be rewritten as the product of two terms < f i M vP2> 2 Franck-Condon factor. Qualitatively, the transition occurs from the lowest vibrational state of the ground state to the vibrational state of the excited state that it most resembles in terms of vibrational wavefunction. 

A number of techniques have been used previously for the study of state-selected ion-molecule reactions. In particular, the use of resonance-enhanced multiphoton ionization (REMPI) [21] and threshold photoelectron photoion coincidence (TPEPICO) [22] has allowed the detailed study of effects of vibrational state selection of ions on reaction cross sections. Neither of these methods, however, are intrinsically capable of complete selection of the rotational states of the molecular ions. The TPEPICO technique or related methods do not have sufficient electron energy resolution to achieve this, while REMPI methods are dependent on the selection rules for angular momentum transfer when a well-selected intermediate rotational state is ionized in the most favorable cases only a partial selection of a few ionic rotational states is achieved [23], There can also be problems in REMPI state-selective experiments with vibrational contamination, because the vibrational selectivity is dependent on a combination of energetic restrictions and Franck-Condon factors. 

If really good wavefunctions can be employed, then the results are convincing. Wolniewicz,175 with very accurate wavefunctions for H2, has calculated transition probabilities for the B-X,C-X and E,F-B systems. He has even considered individual vibrational and rotational lines and has shown that owing to significant variation of the electronic moments with intemuclear distance, the use of Franck-Condon factors is not permissible. 

The wavelength of the fluorescence will show a progression into the red of the transitions to the different vibrational levels of the ground state, the intensities of these vibrational transitions will be governed by the Franck—Condon factors and transition probabilities. The number of rotational lines associated with each vibrational transition depends on the nature of the two electronic states concerned. If a ground state vibrational level above v" = 0 was excited initially, then there will also be fluorescence at shorter wavelengths to the exciting line. 

It is found that the lifetimes change as v and J vary in the upper state. For example, Capelle and Broida found that the lifetime decreased from 1420 to 690 ns as v decreased from 40 to 21, a drop that can be accounted for by more favorable overlap of ground and excited state wavefunctions (Franck-Condon factors) for lower v levels. The effect of a change in rotational state is less Castano, Martinez, and Martinez found for the v = 25 level that the lifetime decreased from 745 ns to 625 ns as J increased from 0 to 106. Such a shortening of the lifetime is consistent with enhanced predissociation at higher rotational levels due to bond lengthening by the increased centrifugal force. 

Although Dispersed Fluorescence (DF) spectroscopy is probably better classified as a form of double resonance spectroscopy, DF is discussed here because it is a form of emission spectroscopy where all of the emission originates from a single, laser-populated, upper electronic-vibrational-rotational level, (e, v, J ). A DF spectrum typically contains two [R J" = J — 1), P(J" = J + 1)] or three [i ( J — 1), Q(J ), P J +1)] rotational transitions per electronic-vibrational e",v" level. Often there is a progression of vibrational bands, [ v, v" = n), (v, v" = n + 1),. .. (v, v" = n + to)] where v" = n is the lowest vibrational level (band farthest to the blue) and v" = n + m is the highest vibrational level observable (limited either by the detector response or Franck-Condon factors) in the DF spectrum (see Fig. 1.8 and Fig. 1.15). 

The collision-assisted predissociation in iodine B O + state merits a detailed discussion. It is well known that B state is weakly coupled to the dissociative A 1m state by rotational and hyperfine-structure terms in the molecular Hamiltonian. The natural predissociation rate strongly depends on the vibrational quantum number (pronounced maxima for o=5 and u = 25, a minimum for u= 15), this dependence being due to a variation of the Franck-Condon factor. " The predissociation rate is enhanced by collisions. In absence of a detailed theoretical treatment of the colhsion-assisted 12 predissociation, one can suppose that the asymmetric perturbation (breakdown of the orbital symmetry) in the collisional complex affects electronic and rotational wavefimctions but does not change the nuclear geometry. 

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