Uniform approximate Franck-Condon matrix elements for bound-continuum vibrational transitions

Franck-Condon matrix elements are calculated approximately for vibrational transitions of a diatomic molecule from a bound electronic potential curve to a purely repulsive curve. The bound states are approached by exactly normalized Miller-Good wavefunctions uniform in both turning points. For the continuum wavefunction a single turning point uniform Airy approximation is taken. The resulting Franck-Condon matrix element is approximately done in closed form with the help of a new canonical integral for a product of harmonic oscillator wavefunctions and Airy functions. The degree of agreement with a closed form exact result is qualitatively discussed for transitions from the ground state of a Morse curve to the continuum of a particular repulse exponential curve.Dedicated to Professor Hermann Hartmann on the occasion of his 65th birthday.     

Ab initio simulation of the photoelectron spectra of HCO− and DCO−

Large scale, vibrational CI calculations using a global ab initio potential energy surface are used to calculate multidimensional Franck-Condon overlaps from the ground vibrational state of HCO⁻ and DCO⁻ to all final bound and several quasibound vibrational states of HCO and DCO. The resulting Franck-Condon spectra are compared with recent experimental photoelectron spectra of HCO⁻ and DCO⁻.     

The photodissociation of H2S: A study of indirect photodissociation

We have performed calculations for the photodissociation of H₂S using surfaces constructed to test a model proposed by van Veen et al., in which the dissociation occurs via predissociated levels of the bound ¹B₁ excited state. Total Franck-Condon factors for the photodissociation and partial Franck-Condon factors for the product vibrational distributions are presented.     

Semiclassical bound-continuum Franck-Condon factors uniformly valid at 4 coinciding critical points: 2 Crossings and 2 turning points

A uniform semiclassical approximation to bound-continuum Franck-Condon factors is derived by applying differential topological mapping techniques on a suitable three dimensional integralrepresentation. This approach even holds uniformly if two potentialcurve intersections (real or complex conjugate) and two turning points come close or coincide. The resulting Franck-Condon matrix element is expressed in terms of two generic swallowtail (A₄) integrals whose unfolding parameters are obtained from a single algebraic equation amenable to fast standard routines. Transitional approximations to this result are shown to cover all previously known approaches and to lead to a generalization of a formula of K. Sando and F. H. Mies [1]. A simple and fast trapezoidal method to evaluate the generic swallowtail integrals and other generic integrals of odd determinacy is presented, which even permits the derivation of numerical error bounds.     

Vibrational levels and Franck–Condon factors of diatomic molecules via Morse potentials in a box

This work deals with two shortcomings in the use of Morse potentials to describe energy spectra and transitions of diatomic molecules: (1) Morse's well-known “exact” solution for purely vibrational states includes the unphysical region – ∞ < r < 0 of the internuclear separation, and (2) Franck-Condon factors are evaluated in harmonic and anharmonic approximations to the Morse potentials. The method of confining the molecule in a spherical box is developed to obtain (1) purely vibrational energy spectra and eigenvectors of Morse potentials in the physical region 0 ? r < ∞, and (2) the corresponding Franck-Condon factors without any additional approximations. The method is applied to Li₂, N₂, CN, and CO molecules. © 1995 John Wiley & Sons, Inc.     

Franck-Condon matrix elements for bound-continuum vibrational transitions calculated by numerical integration and basis set expansion techniques

Franck-Condon factor distributions for bound-to-continuum transitions of one-dimensional vibrational states are calculated by a) using numerical integration, b) employing a finite number of square integrable harmonic oscillator functions. The methods are generally applicable to any kind of bound or repulsive potential involved. Results are presented and compared to model potential calculations previously reported by Krüger [1].     

Interpretation of resonance cars and shpolskii spectra with calculated molecular geometries,vibrational frequencies and relative intensities: Chrysene in it's lowest excited singlet and triplet state

Coherent anti-Stokes Raman scattering (CARS) spectra of excited molecules as well as Shpolskii spectra provide information about geometry changes between ground and excited states. Vibrational frequencies and relative intensities from recently obtained CARS spectra of the chrysene S₁ and T₁ state and earlier observed Shpolskii spectra are interpreted in terms of molecular geometry and force-field changes by means of quantum-chemical consistent force field (QCFF) and Franck-Condon factor calculations. The comparison of observed and calculated relative intensities indicates a coupling between the S₁ and S₂ state enhancing some of the vibrational radiative singlet transitions both in absorption and fluorescence spectra whereas within the phosphorescence spectra proportionality to calculated Franck-Condon factors is obeyed. The T₁ state is the more loosely bound state and its geometry change is different from that of the S₁ state. The resonance CARS transitions in the S₁ state are assigned to totally symmetric vibrations getting their intensity by a coupling scheme analogous to the A term of the resonance Raman effect: the relative intensity of a transition is shown to be proportional to the Franck-Condon factor to the higher excited state and to the squared vibrational frequency. Using this relation this state can be identified by means of its finger-print-like intensity pattern.     

Vibrational-rotational levels and Franck-Condon factors of diatomic molecules via Morse potentials in a box

Pekeris' solution for vibrational-rotational levels of the Morse potential includes the unphysical domain − ∞ < r < 0 of the internuclear separation, just like Morse's solution for vibrational levels. In this article, the method of confinement in a box is implemented to obtain the vibrational-rotational J = 1 levels of Morse potentials in the physical domain 0 < r < ∞, as well as the corresponding Franck-Condon factors. Applications to Li₂ and AIO are presented as illustrations. © 1996 John Wiley & Sons, Inc.     

Liquid single crystal elastomers (lsce) - mechanical optical and electric properties

Liquid crystal elastomers can be macroscopically ordered with respect to the director by applying a mechanical field similar to electric/magnetic field effects of low molar mass liquid crystals. Introducing network anisotropy a priori by the synthesis, uniformly aligned nematic or smectic elastomers are available without external mechanical field. These LSCE's combine optical properties of single crystals with entropy elasticity of elastomers. Due to uniform director orientation without defects, the LSCE's exhibit excellent transparency which makes them applicable for optical elements. Non-linear optical properties can be easily optimized by attaching suitable I.c.-moieties to the polymer network. On the other hand, due to the rubber elasticity of the LSCE's, electromechanical effects can be observed, e. g. piezoelectricity of chiral smectic C-LSCE's.     

Extensive theoretical studies on the low-lying electronic states of indium monochloride cation, InCl+

The global potential energy curves for the 14 low-lying doublet and quartet Lambda-S states of InCl+ are calculated at the scalar relativistic MR-CISD+Q (multireference configuration interaction with single and double excitations, and Davidson's correction) level of theory. Spin-orbit coupling is accounted for via the state interaction approach with the full Breit-Pauli Hamiltonian, which leads to 30 Omega states. The computed spectroscopic constants of nine bound Lambda-S states and 17 bound Omega states are in good agreement with the available experimental data. The transition dipole moments and Franck-Condon factors of selected transitions are also calculated, from which the corresponding radiative lifetimes are derived.     

Interaction of NH({X}^{3}{\Sigma}^{-}) with Rb and Cs atoms: similarities and differences from an highly correlated ab initio study

The interactions of Rubidium and Cesium atoms with the NH radical at its equilibrium geometry are investigated over several interpartner distances and for a variety of angular orientations of the atoms with respect to the intermolecular bond. Several electronic states are computed at the CASSCF level and the lowest four are analysed in greater detail at the MRCI level in order to finally obtain orientational potential energy surfaces. The evaluation of the involved Franck-Condon factors reveals the possibility of photoassociation mechanisms into bound states of the two complex triatomics and further suggests stable complex configurations in the lowest quartet states from DVR calculations of the J = 0 relevant bound states. The work above exemplifies the recent interests of the theory group in Rome on ultracold dynamics and interaction processes which are also briefly reviewed in the introduction.     

Generating functions and recurrence relations for harmonic oscillator matrix elements

A simple and straightforward way of obtaining generating functions and recurrence relations for harmonic oscillator integrals is presented. The method, which is based on the properties of unnormalized coherent states and canonical transformations, allows a unified treatment of several different physical problems. Matrix elements of Gaussian and exponential functions, Franck-Condon overlaps, and transition probabilities for time-dependent quadratic Hamiltonians are discussed as illustrative examples.     

Matrix elements for internal conversion

The usual separation of internal conversion matrix elements into a momentum integral of the promoting modes and an overlap integral of the accepting mode is shown to introduce large errors. The proper procedure leads to the replacement of Franck-Condon factors by integrals involving the square root of an energy operator.     

Ground and valence-excited states of SF: a multireference configuration interaction with single and double excitations+Q study

Ab initio calculations on the ground and valence-excited states of the sulfur monofluoride radical have been performed using entirely uncontracted all-electron augmented correlation consistent polarized valence quintuple zeta basis sets and the internally contracted multireference configuration interaction with single and double excitations method and Davidson correction (+Q). Potential-energy curves of all valence electronic states and the spectroscopic constants of several bound states are fitted. It is the first time that the entire 27-omega states generated from the 12 valence lambda-S states which come from the S(3P(g)) and F(2P(u)) atomic states of SF radical have been studied theoretically. The effects of spin-orbit coupling and the avoided crossing rule between omega states of the same symmetry are analyzed. The calculated results reproduce well the available experimental values and predict the properties of several bound excited states that have never been observed in experiment. The transition properties of the dipole-allowed transitions from bound excited states to the ground state are predicted for the first time, including the transition dipole moments, the Franck-Condon factors, and the radiative lifetimes.     

Quasi-one-dimensional systems and localized defects: Some results of the Green's matrix formalism

The application of the self-consistent field (SCF ) local-impurity formalism to quasi-one-dimensional systems is discussed. We describe a general procedure for an accurate numerical determination of the Green's function matrix elements of the unperturbed system. An application to a local impurity in a model chain with two orbitals per unit cell is reported. The changes in the charge-bond-order matrix and in local and total density of states due to the impurity are discussed with special emphasis on the changes at the critical points (van Hove singularities) at the band edges. The Green's matrix approach is used to reexamine long-range Friedel oscillations caused by an impurity in a strictly one-dimensional metal. The extent of the long-range tail of the perturbed charge density is in an inverse relation to the localization length of the impurity state: the stronger the perturbation the more localized is the bound state and the more extended are the oscillations in the charge distribution. The results for the model chain with two orbitals per unit cell indicate that the impurity-induced change in charge distribution may be locally screened by redistribution of the population of the on-site orbitals, therefore damping possible oscillations and leading to a faster decay than in strictly one-dimensional systems.     

First principles simulation of the UV absorption spectrum of ethylene using the vertical Franck-Condon approach

A new method which we refer to as vertical Franck-Condon is proposed to calculate electronic absorption spectra of polyatomic molecules. In accord with the short-time picture of spectroscopy, the excited-state potential energy surface is expanded at the ground-state equilibrium geometry and the focus of the approach is more on the overall shape of the spectrum and the positions of the band maxima, rather than the precise position of the 0-0 lines. The Born-Oppenheimer approximation and the separability of the excited-state potential energy surface along the excited-state normal mode coordinates are assumed. However, the potential surface is not necessarily approximated as harmonic oscillator potentials along the individual normal modes. Instead, depending upon the nature of the potential surface along a particular normal mode, it is treated either in the harmonic approximation or the full one-dimensional potential is considered along this mode. The vertical Franck-Condon approach is applicable therefore even in cases where the excited state potential energy surface is highly anharmonic and the conventional harmonic Franck-Condon approach is inadequate. As an application of the method, the ultraviolet spectrum of ethylene between 6.2 eV (50,000 cm(-1)) and 8.7 eV (70,000 cm(-1)) is simulated, using the Similarity Transformed Equation of Motion Coupled-Cluster method to describe the required features of the potential energy surfaces. The spectrum is shown to be a result of sharp doublet structures stemming from the pi --> 3s (Rydberg) state superimposed on top of a broad band resulting from the pi --> pi* (valence) state. For the Rydberg state, the symmetric C=C stretch and the torsion mode contribute to the spectrum, while the broad valence band results from excitation into the C=C stretch, CH2 scissors, and the torsion mode. For both states, the potential along the torsion mode is highly anharmonic and the full treatment of the potential along this mode in the vertical Franck-Condon method is required.     

Time-resolved laser-induced fluorescence of the iodine B3π(0u+) state: Study of the second maximum of predissociation

Fluorescence decay lifetimes for specific rovibrational levels of the I₂ B³π(0_u⁺) state, have been obtained by time-resolved laser-induced fluorescence. Gyroscopic and hyperfine predissociation parameters C_v² and a_v², for vibrational levels of the B³π(0_u⁺) state around the second maximum of predissociation, v' = 24, 25, have been determined. Results are discussed in terms of the dependence of the predissociation paramaters on the Franck-Condon densities and the corresponding electronic matrix elements.     

Analysis of low Z elements in various environmental samples with total reflection X-ray fluorescence (TXRF) spectrometry

Recently there is a growing interest in low Z elements such as carbon, oxygen up to sulphur and phosphorus in biological specimen. Total reflection X-ray fluorescence (TXRF) spectrometry is a suitable technique demanding only very small amounts of sample. On the other side, the detection of low Z elements is a critical point of this analytical technique. Besides other effects, self absorption may occur in the samples, because of the low energy of the fluorescence radiation. The calibration curves might be not linear any longer. To investigate this issue water samples and samples from human cerebrospinal fluid were used to examine absorption effects. The linearity of calibration curves in dependence of sample mass was investigated to verify the validity of the thin film approximation. The special requirements to the experimental setup for low Z energy dispersive fluorescence analysis were met by using the Atominstitute's TXRF vacuum chamber. This spectrometer is equipped with a Cr-anode X-ray tube, a multilayer monochromator and a SiLi detector with 30 mm² active area and with an ultrathin entrance window. Other object on this study are biofilms, living on all subaqueous surfaces, consisting of bacteria, algae and fungi embedded in their extracellular polymeric substances (EPS). Many trace elements from the water are bound in the biofilm. Thus, the biofilm is a useful indicator for polluting elements. For biomonitoring purposes not only the polluting elements but also the formation and growth rate of the biofilm are important. Biofilms were directly grown on TXRF reflectors. Their major elements and C-masses correlated to the cultivation time were investigated. These measured masses were related to the area seen by the detector, which was experimentally determined. Homogeneity of the biofilms was checked by measuring various sample positions on the reflectors.