Condon Integrals Research Articles

Photoelectron Spectrum of Ammonia, a Test Case for the Calculation of Franck−Condon Factors in Molecules Undergoing Large Geometrical Displacements upon Photoionization

The vibrational structure of the photoelectron spectrum of ammonia, the simplest molecule undergoing a large displacement of its equilibrium geometry upon photoionization, is analyzed by evaluating the Franck-Condon integrals at the anharmonic level of approximation. It is shown that if the rectilinear Cartesian representation of normal modes is adopted Duschinsky's transformation yields a too large displacement of the bond distance coordinate, with the appearance of several progressions which are not observed in the experimental spectrum. This apparent failure is completely corrected by the inclusion of anharmonic couplings between the principal active mode, the out of plane bending of the planar cation, and the totally symmetric stretching mode, leading to a satisfying reproduction of the observed spectrum and to a more convincing assignment of the weaker progression observed in the high-resolution spectrum.

Near-UV Absorption Spectrum of the Phenoxyl Radical and Kinetics of Its Reaction with CH3

Cavity ring-down spectroscopy combined with pulsed laser photolysis has been used to study the near-ultraviolet absorption spectrum (375−410 nm, 22B1 ← X2B1 transition) of the phenoxyl radical (C6H5O•) in 10−20 Torr of nitrogen diluent at 298 K. By using a numerical fitting routine on the basis of a modeling of chemical reaction system, the absorption cross section of the phenoxyl radical was obtained, σ = (7.7 ± 2.3) × 10-18 cm2 molecule-1 (base e) at 394.4 nm. A spectrum simulation was carried out using available Franck−Condon integrals with a 400 cm-1 Lorentzian line width, which suggests a short-lived excited state. Time-dependent density functional theory (TD-UB3LYP/aug-cc-pVTZ) calculations supported the interpretation of the absorption band for the phenoxyl radical. The rate constant of the phenoxyl radicals with methyl radicals was derived, k(C6H5O + CH3) = (6.2 ± 2.6) × 10-11 cm3 molecule-1 s-1, at 298 K in 20 Torr of nitrogen diluent.

Calculation of franck-condon integrals

As is known, the vibrational structure of molecular electron spectra in the adiabatic approximation is determined by overlap integrals for the vibrational wave functions of the combining electron states (Franck--Condon integrals) and so it becomes necessary to calculate these integrals. This problem is simplified by adopting a harmonic approximation. As a rule, this is possible for multiatomic molecules whose spectra usually include lines corresponding to the excitation of a small number of phonons of each oscillator; this corresponds to transitions between levels lying close to minima of the potential surfaces (small vibrational quantum numbers) In this case the vibrational wave function is written as the product of the wave functions of the harmonic oscillators corresponding to normal vibrations of the molecules of the combining electron states.