Vibronic Coupling Effects Research Articles

The effect of vibronic coupling on N.M.R. chemical shifts

A method is developed whereby the effect of vibronic coupling on the N.M.R. chemical shift in a transition metal ion paramagnetic system may be explored. The specific case of a d 1 transition metal ion vibrating in a strong crystal field of octahedral symmetry is examined. The results show that in certain regions within the paramagnetic system vibronic coupling may affect significantly the N.M.R. chemical shifts.

Dynamical calculation of satellite intensities

A formalism to calculate molecular electronic spectra is developed which takes account of electronic configuration interaction as well as of vibronic coupling between the electronic states. The formalism allows the dynamical calculation of electronic spectra, i.e., the inclusion of the effects of the nuclear kinetic energy operator on the electronic motion. When neglecting the dynamical effects, the usual statical adiabatic and Franck-Condon approximations are obtained as special cases. The formalism is applied to cyanogen the photoelectron spectrum of which exhibits an unexplained peculiar satellite structure. Ab initio calculations are performed which show that vibronic interaction between close lying 2Σg and 2Πu states takes place through the bending vibration. The vibronic coupling effects are enhanced by the simultaneous excitation of the totally symmetric C–N stretching vibration. Guided by these results a dynamical calculation is performed which reproduces nicely the experimental spectrum. The adiabatic and Franck-Condon approximations turn out to be inapplicable. As a consequence of the strong vibronic interaction with the 2Σg state, the 2πu state has a strongly bent equilibrium geometry.

Two-photon ionization spectrum of the 1Lb ← S0 transition in toluene

The multiphoton ionization (MPI) spectrum of toluene arising from the 1B2 (1Lb) valence state has been investigated. The state participates as a two-photon resonance. A total of nine excited state fundamentals have been characterized, including three non-totally symmetric vibrations. The toluene MPI spectrum shows a strong resemblance to the two-photon fluorescence excitation spectrum with the strongest transitions taking place to the origin and excited state modes ν1(a1), ν12(a1) and ν14(b)2). The intensities of the observed fundamentals are rationalized in terms of Franck-Condon and vibronic coupling effects. A major conclusion is, that the primary mechanism for the activity of ν12 is vibronic coupling.

Spectroscopic effects of vibronic coupling between nearby 3nπ* and 3ππ* states of dimethylbenzaldehydes in a durene matrix

A vibrational analysis is presented of high resolution ππ* phosphorescence spectra of the 2,4-, 2,5- and 3,4-dimethylbenzaldehyde-1 h 1 and −1 d 1, guests in durene host. As previously reported for the fully hydrogenated molecules, the energy gaps between the zero-point levels of the 3ngp*, and 3ππ* states of the guests deuterated on the aldehydic group are determined from the temperature dependence of their phosphorescence spectra and lifetimes. These vibronic energy gaps tend to be larger in the deuterated species.The existence of the vibronic coupling between 3nπ* and 3ππ* states of dimethylbenzaldehydes is suggested by the important activity, in the ππ* phosphorescence spectra, of nontotally symmetric modes involving the aldehydic group and the aromatic ring. Among these vibrations, the most active are the aldehydic CH and CD wagging modes which also show overtone activity. The intensities of the induced bands show a strong deuterium effect which in the 2,4 and 2,5 compounds is anomalous in that the deuterated species shows the higher induced intensity. The observation of phosphorescence spectra from each molecule in two different sites allows an assessment of the crystal field effect which is found to be moderately strong. These observations can be understood qualitatively on the basis of weak pseudo Jahn-Teller-like coupling between the 3ππ* and 3nπ* states which is subject to interference from Herzberg-Teller coupling involving at least one additional state. No convincing evidence is found for nonplanar distortions of the guest molecules.

Vibrational State Dependence of the Photoelectron Angular Asymmetry Parameter caused by Vibronic Coupling

It is shown that the dependence of the angular asymmetry parameter β on the vibrational state, which has been found in the photoelectron spectra of some polyatomic molecules, can be explained by vibronic interactions. It is proposed that measurements of β as a function of the vibrational state may be used as a tool to analyze vibronic coupling effects in photoelectron spectra.

Resonance Raman spectra of copper tetraphenylporphyrin: Effects of strong vibronic coupling on excitation profiles and the absorption spectrum

Raman excitation profiles of copper tetraphenylporphyrin have been obtained and interpreted in terms of strong vibronic coupling. The Duschinsky mixing of normal modes in the excited Q electronic state has a dominant effect on the excitation profiles and the absorption spectrum. A general theoretical technique for calculating the vibronic states of molecules is described and applied to metalloporphyrins. The analytical expressions obtained from a new perturbation treatment of the vibronic problem permit a detailed physical understanding of strong coupling influences and interference effects observed in copper tetraphenylporphyrin excitation spectra. A modified interpretation of the Q band absorption is indicated.

Vibronic coupling effects in the photoelectron spectrum of ethylene

The vibrational structure of the first band in the photoelectron spectrum of ethylene is calculated taking into account the vibronic coupling between the ground state and first excited state of the ion. The vibronic Hamiltonian describes linear coupling to the totally symmetric vibrational modes ν1–ν3 as well as to the non-totally symmetric torsional mode ν4. The energies and coupling constants entering the calculation are computed by ab initio Hartree–Fock and many-body methods. Qualitative agreement between the theoretical and the experimental spectrum is found. By slightly readjusting some of the parameters, the experimental spectrum can be reproduced accurately. It turns out that nonadiabatic and intensity borrowing effects are small. The vibronic coupling results mainly in a pronounced anharmonicity of the adiabatic potential energy surface. In particular, a nonplanar equilibrium geometry is found for the ionic ground state, the equilibrium torsional angle being ∼25°. Although the corrections to the Franck–Condon principle are small, the calculation of the vibrational structure is greatly complicated by the nonseparability of the totally symmetric and the non-totally symmetric vibrations. A decoupling procedure is presented which approximately makes possible the separate treatment of the modes. The results obtained with this procedure are in good agreement with the full vibronic treatment for ethylene.

Jahn—Teller effect induced by non-degenerate vibrational modes in cumulenes

The theory of the Jahn—Teller effect induced by non-degenerate vibrational modes is briefly discussed and applied to calculate photoelectron spectra of allene and pentatetraene. Good agreement with the observed spectra is obtained. In both cases it is found that the Jahn—Teller effect is caused by the torsional mode and the antisymmetrical CC stretching mode. The Jahn—Teller effect helps to understand the existence of strong vibronic coupling effects in the related molecules ethylene and butatriene, where no Jahn—Teller effect can be present.

Resonance Raman excitation profiles and depolarization dispersion curves, and their use in the analysis of vibronically coupled excited states

The relationship between vibronic coupling and resonance Raman scattering is studied on a model system consisting of two excited electronic states coupled by a non-totally-symmetric mode of vibration. Scattering cross sections and depolarization ratios are calculated as a function of the wavelengths of the incident and scattered light, giving rise to Raman excitation profiles and Raman spectra, respectively. The model system allows exact solution for arbitrary values of the electronic energy gap between the coupled states, of their vibrational frequencies, and of linear and nonlinear adiabatic coupling parameters. The results are reported in the form of three-dimensional graphs for combinations of parameter values typical for weak, strong, and intermediate vibronic coupling. They are compared with previous results, based on simpler models and/or more approximate solution methods. In addition, they are related to the results obtained for the molecular dimer [J. Chem. Phys. 63, 5475 (1975)], which is the prototype of an exactly solvable vibronic coupling model. It is shown that resonance Raman spectroscopy generally allows one to probe more deeply into vibronically coupled states than conventional absorption–emission spectroscopy and thus provides a more powerful tool for their spectroscopic analysis. This is particularly significant in the region intermediate between weak and strong vibronic coupling where interference between Franck–Condon and vibronic coupling effects or between adiabatic and nonadiabatic coupling (in the adiabatic Born–Oppenheimer representation) may produce highly irregular spectra. The basic advantage of resonance Raman scattering over absorption spectroscopy is that one can obtain a higher effective resolution of excited state spectra, namely, by studying each vibrational mode separately, by comparing the excitation profiles of fundamentals and overtones, and by investigating the dispersion of the depolarization ratio.

Jahn-Teller effect in EPR spectra: Multistate theory forE2orbital states in cubic symmetry

The Jahn-Teller effect for an orbitally doubly degenerate state is considered for the case of cubic symmetry. Reduction factors are defined which describe the matrix elements of the orbital operators for $E$ doublets between the various vibronic states. For the lowest six vibronic states $E$, ${A}_{1}$, ${A}_{2}$, and $E$, the reduction factors are calculated for the cases of no or weak linear vibronic coupling and strong linear vibronic coupling. In the latter case the effects of anharmonic potential energy and nonlinear vibronic coupling are also included in the reduction factor calculations. Using the reduction factors to determine the matrix elements of the orbital operators within the vibronic states, an effective Hamiltonian is formulated which describes the mixing of the vibronic levels. This defines the multistate theory in which several vibronic levels are included in the manifold of states used to describe electronic interactions. Electron-paramagnetic-resonance spectra from $^{2}E$ orbital states are predicted for a six-state vibronic manifold by calculating the strain dependence of the $g$ factor. Particular attention is given to those spectral fectures appearing in the ground state which are the result of the second excited singlet. The relationship between $p$-type and $q$-type reduction factors is discussed.

Vibronic Effects in Mossbauer Spectra: The 57Fe Quadrupole Splitting in FeCO3

The methods developed by Orbach and coworkers for calculating spin-lattice relaxation rates have been generalized to enable calculation of the effects of vibronic admixture between low-lying electronic states on observables, such as the Mossbauer quadrupole splitting, associated with magnetic ions in crystals. Emphasis is placed on the need to take proper account of the symmetry properties of the phonons until a late stage in the calculation. Application is made to the temperature dependence of the 57Fe quadrupole splitting in FeC03 , and it is concluded that, in general, vibronic coupling effects must be considered before static splitting parameters are extracted from experimental data relating to the temperature-dependent populations of electronic levels. The superposition model of the crystal field is employed to estimate the vibronic coupling parameters.

Vibronic coupling effect on the paramagnetism of deoxygenated hemoglobin

A square pyramidal iron(II) complex coupled with a β 1 vibrational mode is considered as a model for deoxyhemoglobin. The calculations performed on this model show that the introduction of the vibronic coupling may allow a strong reduction of the rhombic splitting and account for the different magnetic moment in phosphate-free and phosphate-bound deoxyhemoglobin.

Hypersensitivity in the ƒ-ƒ transitions of hexabromo- and hexachlorneodymium(III)

Oscillator strengths of the ƒ-ƒ transitions of hexabromoneodymium(III) are reported. These and the oscillator strengths of hexachloroneodymium(III) are analyzed to give the parameters of the Judd-Ofelt theory. The results are discussed with relation to the effects of symmetry, vibronic coupling, and covalency on the intensities of ƒ-ƒ transitions.

Study of the Franck-Condon and Herzberg-Teller Approximations

Using the Born-Oppenheimer parameter lambda = (m/M)(1/4) as a perturbation parameter, we find that for allowed transitions, the zeroth order approximation of the spectral intensity gives rise to the Condon approximation, the first order vibronic coupling and anharmonic effect appear in the first order approximation of the spectral intensity, which gives us the non-Condon scheme, and only the intensity of the transitions of totally-symmetric modes with nonvanishing normal coordinate displacements is affected by the inclusion of the first order vibronic coupling and anharmonic effect. For symmetry-forbidden but vibronic-allowed transitions, the first nonvanishing term of the spectral intensity is second order with respect to lambda and the B-O couplings do not appear in the calculation of the spectral intensity until the fourth order approximation with respect to lambda; in this case, other high order vibronic couplings and anharmonic effect are competing with the B-O couplings.

The electronic emission spectrum of ionized nitrous oxide, N 2 O + : Ã 2 Σ + —X 2 Π

The emission spectrum of the transition A 22 +- X 2n between the two lowest states of the ion of N2O have been photographed under high resolution and analysed in both of the two isotopes N2ieO+ and N2lsO+. The principal vibrational structure yields the stretching fundamentals in both states (table 11), from which complete sets of quadratic valence stretching force constants are obtained, albeit uncorrected for anharmonicities (table 14). The excited state A2E is more tightly bound than the neutral NaO in its ground state. Sequences in the bending vibration show the effects of moderate Renner-Teller vibronic coupling in the orbitally degenerate 2II-state (e = — 0.19); and together with a few weaker bands of different polarization involving Av2 = ± 1 observed presumably through Herzberg—Teller vibronic coupling, these transitions make possible a complete analysis of the Renner coupling in v = 1 (table 10) and hence yield the bending frequencies and force constants. This analysis provides one of the best tests so far of the Renner-Pople-Hougen theory of these couplings. Rotational analysis gives zero-point moments of inertia I0 from which r0-structures are calculated for both states (table 13). Finally, the force constants and bond distances of the ion are compared with those of the neutral molecule, and the states of the ion correlated with those of possible dissociation products (table 16).

Hyperfine quadrupole splitting and mössbauer spectra for an orbital doublet

AbstractA theoretical investigation is made of the effect of vibronic coupling on the line shapes of Mössbauer spectra in an orbital degenerated E‐term system including relaxation transitions among the vibronic states and magnetic hyperfine interaction. The vibronic interaction is taken into account in the limiting case of strong linear coupling and arbitrary second‐order contribution. The quadrupole splitting of the energy levels for a nuclear spin I = 3/2 and the transition probabilities for I = 1/2 → I = 3/2 transitions were obtained. The effect of relaxation transitions on the line shape is studied on the basis of a generalized Kubo‐Andersen model. For a set of actual relaxation and vibronic parameter values the Mössbauer spectra are calculated numerically with a computer. The possibilities of comparison with experiment are discussed.

Non-Born—Oppenheimer vibronic coupling and deuterium isotope effect on T1 → S0 radiative transitions of aromatic hydrocarbons

Abstract Radiative lifetimes of the lowest triplet states are determined for several aromatic hydrocarbons from the measured fluorescence quantum yield, phosphorescence quantum yield, and lifetime of phosphorescence. It is found that the radiative lifetime is considerably longer for perdeuterated naphthalene and phenanthrene than for the corresponding perprotonated species. A theoretical explanation is presented which involves a new non-Born—Oppenheimer coupling mechanism.

Vibronic and spin-orbit coupling effects in the magnetic circular dichroism of nickel(II) bromate hexahydrate

The single-crystal magnetic circular dichroism spectrum of [Ni(H 2O) 6](BrO 3) 2 is found to be strongly temperature dependent with an extensive vibronic structure in the ‘red’ band. The main features of the spectrum consist of C terms exhibiting first- and second-order spin-orbit splitting and progressions in the nickel-oxygen stretching mode built upon a double false origin.

Effects of Vibronic Coupling on the Rate of Spreading of an Exciton Wave Packet: Higher-Order Effects

The results presented here are an extension to higher order of the computation of the rate of speading of an exciton wave packet in an infinite one-dimensional crystal. Attention was paid to both the initial conditions and the effects of phonon–exciton coupling. Strong nearest-neighbor coupling was assumed, and agreement with the Franck–Condon principle was explicitly required. Previous calculations by one of the authors carried through second-order perturbation (all odd-number orders being zero) showed a time-dependent decrease of the rate of spreading. For large t (time), this decrease goes at t, lnt, or a constant, depending on whether ℏω < 2W, ℏω = 2W, or ℏω > 2W. Here W is the intermonomer coupling energy and ℏω is the vibrational quantum of energy. In this paper we pursue this calculation to fourth order, showing that the rate of spreading is now increased by a term which goes at t2 or t2lnt depending on whether ℏω ≤ 2W or ℏω < 2W, respectively. We have calculated the limits of z = (2W / ℏ)t for which the results are valid and also computed the successive terms of the rate of spreading for typical values of the parameters σ = ℏω / 2W and λ = KQ02 / 2W. Finally, we discuss the behavior of the rate of spreading, υ, as a function of λ, z, and σ and also suggest some possible experimental verification of the results of this work.

Electron Spin Resonance of the Mono-tert-butyltropenyl and Tri-tert-butyltropenyl Radicals: A Study of Vibronic Near Degeneracy

The mono-tert-butyltropenyl and tri-tert-butyltropenyl radicals have been prepared by thermal and photochemical cleavage of di-tert-butylbitropenyl and hexa-tert-butylbitropenyl. The syntheses of these dimers are reported. Electron spin resonance spectra of the tert-butyltropenyls (tert-butyl-2,4,6-cycloheptatrien-1-yls) have been investigated. Proton hyperfine splittings of both radicals are markedly temperature dependent. In addition, for tri-tert-butyltropenyl, the g value and the linewidth are temperature dependent and the proton splittings vary slightly with solvent. The experimental temperature variations are ascribed to vibronic near degeneracy in the tert-butyltropenyls, and a model is proposed in order to calculate these effects. Approximate spin densities for the two nearly degenerate states are given by ρkkπ ≈ (1 / 7)[1 ∓ P cos(8πk / 7)], where the minus sign corresponds to the ground state, ρkkπ is the diagonal element of the pi-spin-density matrix for carbon atom k, and P is a parameter which combines the effects of electron correlation and vibronic coupling. Splitting calculations using these values of ρkkπ and McConnell's relationship lead to good agreement with experiment. For tri-tert-butyltropenyl it is found that P = 1.0 ± 0.1, and the energy splitting between nearly degenerate states (ΔE) is approximately 200 ≤ ΔE ≤ 300 cm− 1. In the case of mono-tert-butyltropenyl a range of 100 ≤ ΔE ≤ 180 cm− 1 corresponds to P = 1. Consideration of the Colpa–Bolton and Giacometti–Nordio–Pavan extensions of McConnell's relationship leads to negligible modifications in the hyperfine splittings. On the basis of Hückel MO calculations in which the alkyl-group perturbation is treated as an inductive effect, it is concluded that the tert-butyl group does not appreciably affect the pi-electron wavefunction but does modify the Coulomb integral on the adjacent carbon by approximately − 0.02 β. It is found that the temperature dependence of the unresolved tert-butyl proton splittings gives rise to a major contribution to the temperature variation of the linewidth in tri-tert-butyltropenyl. Analysis of the temperature dependence of the g value of this radical on the basis of a simple two-level scheme leads to approximate values for (gE–gG) and gG, where gG and gE designate the g values in the ground and excited vibronic levels.