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The computed vertical energy, Ev,a/f, from the equilibrium geometry of the initial electronic state is frequently considered as representative of the experimental excitation/emission energy, Eabs/fl = hc/λmax. Application of the quantum mechanical version of the Franck–Condon principle does not involve precise specification of nuclear positions before, after, or during an electronic transition. Moreover, the duration of an electronic transition is not experimentally accessible in spectra with resolved vibrational structure. It is shown that computed vibronic spectra based on TDDFT methods and application of quantum mechanical FC analysis predict Eabs = hc/λmax with a 10-fold improvement in accuracy compared to Ev,a for nine cyanine dyes. It is argued that part of the reason for accuracy when this FC analysis is compared to experiment as opposed to Ev,a/f is the unspecified verticality of transitions in the context of the quantum version of the FC principle. Classical FC transitions that preserve nuclear kinetic energy before and after an electronic transition were previously found to occur at a weighted average of final and initial electronic state molecular geometries known as the r-centroid. Inspired by this approach a qualitative method using computed vertical and adiabatic energies and the harmonic approximation is developed and applied yielding a 5-fold improvement in accuracy compared to Ev,a. This improvement results from the dominance of low frequency vibronic transitions in the cyanine dye major band. The model gives insight into the nature of the redshift when qPCR dye EvaGreen is complexed to λDNA and is applicable to the low frequency band of similar non cyanine dyes such as curcumin. It is found that the computed vibronic cyanine dye spectra from time-dependent FC analysis at 0 K and 298 K show decreased intensity at higher temperature suggestive of increased intensity with restricted motion shown when cyanine dyes are used in biomedical imaging. A 2-layer ONIOM model of the DNA minor groove indicates restricted motion of the TC-1 dye excited state in this setting indicative of enhanced fluorescence.

The Franck–Condon Principle (FCP), the key concept for describing vibronic spectra, results from the large disparity between the time scales of electrons and nuclei in molecules. Yet in an undergra...

Vibrationally resolved electronic spectra of polyatomic molecules provide valuable information about the quantum properties of both electrons and nuclei. This chapter reviews the recent progress in ab initio semiclassical calculations of such spectra, based on the thawed Gaussian approximation and its extensions. After reviewing molecular quantum dynamics induced by the interaction with electromagnetic field and the most common semiclassical approximations to quantum dynamics, we explain details of the thawed Gaussian approximation and its variants. Next, we discuss the time-dependent approach to steady-state and time-resolved electronic spectroscopy, and review several standard models that facilitate interpreting vibrationally resolved electronic spectra. Finally, we present the on-the-fly ab initio implementation of the thawed Gaussian approximation and provide several examples of both linear and pump–probe spectra computed with this methodology, which, at a low additional cost and without sacrificing the ease of interpretation, outperforms the standard global harmonic approaches.

Franck-Condon factor arrays have been computed numerically and are displayed to highest known vibrational quantum numbers for the following C2 band systems: Freymark:(e1Σg|+-b1Πu)Mulliken:(d1Σu+-x1Σg+)Fox-Herzberg:(B3Πg-X'3Πu)Deslandres-d'Azambuja:(c1Πg-b1Πu)Swan:(A3Πg-X'3Πu)Phillips:(b1Πu-x1Σg+)Ballik-Ramsay:(A'3Σg--X'3Πu).

This work critically examines attempts to model the fine structure apparent in electronic spectra of cyanine dyes and their analogues. Numerous computational studies reported over the past decade attribute the origin of sub‐bands and their relative intensities to vibronic transitions in which the relevant electronic transition is coupled, irrespective of symmetry, with a collection of vibrations. It is contended that this type of approach is not supported by experimental evidence. An argument is reiterated for a more appropriate model that adheres closely to fundamental principles and fits the data. It stipulates that essentially just one symmetric vibration, carbon–carbon bond stretching of the cyanine polymethine chain, dominates the coupling and is responsible for the observed fine structure. Furthermore, it is pointed out that the intensities of the sub‐bands are readily explained by means of the Franck–Condon principle.

Calculations of the adiabatic potential energy curves and the transition dipole moments between the ground (A1Σ+) and the first excited (A1Σ+) states have been determined for the LiCs and NaCs molecules. The calculations are performed using an ab initio approach based on non-empirical pseudopotentials for Cs+, Li+ and Na+ cores, parameterized l-dependent polarization potentials and full configuration interaction calculations. The potential energy curves and the transition dipole moment are used to estimate the radiative lifetimes of the vibrational levels of the A+Σ+ state using the Franck–Condon (FC) approximation and the approximate sum rule method. The radiative lifetimes associated with the A+Σ+ state are presented here for the first time. These data can help experimentalists to optimize photoassociative formation of ultracold molecules and their longevity in a trap or in an optical lattice.

Band shape is an essential ingredient in the simulation of electronic absorption spectra. The excitation of multiple series of vibrational levels during an electronic excitation is a main contributor to band shapes. Here we present two simple models based on the Franck-Condon displaced-harmonic-oscillator model. The models are both derived from the time-dependent formulation of electronic spectroscopy. They assume that the transition dipoles do not depend on geometry and that the potential energy surfaces are locally quadratic; one model is second order in time and is called LQ2, and the other is third order in time and is called LQ3. These models are suitable for simulating the unresolved vibronic band shapes of electronic spectra that involve many vibrational modes. The models are straightforward and can be easily applied to simulate absorption spectra that are composed of many electronic transitions. As compared to carrying out molecular dynamics simulations, they require relatively few electronic structure calculations, and the additional cost for constructing the spectra is negligible. Therefore, the models are suitable for simulating the spectra of complex systems such as transition-metal complexes.

Knowledge of the Franck-Condon factors (FCFs) and related quantities is essential to understand and to estimate many important aspects of the astrophysical molecules, such as kinetics of the energy transfer, radiative lifetimes, band intensity, and vibrational temperatures. In this view, we propose a new analytical formula of the Franck-Condon integral for two-dimensional harmonic oscillators taking into account the Duschinsky effect. This method is based on the use of the binomial expansion theorem and the Hermite polynomials. With the formula obtained, the FCF of any transition can be computed independently. In this study, the method for FCF calculations was applied to the NO2 molecule. © 2017 Published by NRC Research Press.

Vibronic interactions in the two energetically lowest electronic states (X2Πg–A2Πu) of the diacetylene radical cation (C4H2•+) are theoretically examined here. The spectroscopy of these two electronic states of C4H2•+ has been a subject of considerable interest and measured in the laboratory by various groups. Inspired by numerous experimental data, we attempt here a detailed investigation of vibronic interactions within and between the doubly degenerate Π electronic states and their impact on the vibronic structure of each state. A vibronic coupling model is constructed in a diabatic electronic basis and with the aid of ab initio quantum chemistry calculations. The vibronic structures of the electronic states are calculated by time-independent and time-dependent quantum mechanical methods. The progression of vibrational modes in the vibronic band is identified, assigned, and compared with the literature data. The nonradiative internal conversion dynamics is also examined and discussed.

The basics of the quantum mechanical theory of the light absorption process, the simplifications of the theory in form of models and their application to dyes are reviewed. The factors governing the electronic transition energy, the intensity of the electronic transition and the vibrational fine structure of the absorption bands are examined.

We examine the influence of bending of pi-conjugated chromophores on photoluminescence (PL) by spectrally resolving the depolarization of fluorescence on the single-molecule level. The effect of excited-state mixing mediated by molecular vibrations is manifested in the departure from the usual achromatic linear dichroism of fluorescence, with the polarization anisotropy decreasing in the vibronic progression. Bent chromophores reveal an overall increase in vibronic PL intensity with polarization orthogonal to the molecular long axis. This manifestation of the Renner-Herzberg-Teller (RHT) effect illustrates the breakdown of the Franck-Condon principle in macromolecules used in organic electronics, providing information on the orientation of transition-dipole moments and the origin of spectral broadening. While some of the spectral signatures of the RHT effect appear similar to those of H aggregation in molecular dimers, discrimination between the two phenomena is straightforward since H aggregation does not induce anomalous linear dichroism.

We review the recent two topics of optical excitation and relaxation dynamics, newly unveiled by the time- and momentum-resolved photo-electron emission from the conduction band of GaAs. One is the real-time collective relaxation dynamics, resulting in the Fermi degeneracy formation in the Γ valley. We show that it takes almost infinite time to realize the exact Fermi degeneracy, due to a restricted selection rule for the intravalley transition of the photo-excited electrons. The other is the spontaneous and instantaneous intervalley transition from the Γ valley to the L one. By considering the electron-phonon coupling before the photo-excitation, such spontaneous intervalley transition is realized within the framework of the Franck–Condon principle of the photo-excitation.

Nuclear vibrations play a prominent role in the spectroscopy and dynamics of electronic systems. As recent experimental and theoretical studies suggest, this may be even more so when vibrational frequencies are resonant with transitions between the electronic states. Herein, a vibronic multilevel Redfield model is reported for excitonically coupled electronic two‐level systems with a few explicitly included vibrational modes and interacting with a phonon bath. With numerical simulations the effects of the quantized vibrations on the dynamics of energy transfer and coherence in a model dimer are illustrated. The resonance between the vibrational frequency and energy gap between the sites leads to a large delocalization of vibronic states, which then results in faster energy transfer and longer‐lived mixed coherences.

Previous work (Romanskii, 2018) was devoted to research in the framework of the modified Marcus model (calculation level UCCSD(T) 6-311++G**//B3LYP/6-31+G**) on the reaction kinetics of methane with a methyl radical in the temperature range of 10–800 K. In the present work, these studies are extended to the region of high temperatures, up to 2000 K. Analysis of data on the kinetics and KIE in the temperature range of 250–2000 K is carried out in comparison with the data of studies performed using the quantum scattering method (Remmert et al., 2009). The abnormally high values of kH/kD observed in our work are considered a consequence of a violation of the generalized Frank Condon principle (GFCP) for the deuterium atom transfer reaction. The influence of steric factor on the value of kH/kD is quantitatively analysed; the similarity of the obtained picture to similar observations for the reaction of enzymes is noted.

Spectroscopic studies of metal complexes involve the interpretation of bands (usually in the visible or ultraviolet region) that result from electronic transitions. In a field generated by ligands, which are electron pair donors, the d orbitals become split in energy, as do the spectroscopic states arising from them. Transitions are generally possible between states having the same multiplicity. In this chapter, the interpretation of such spectra is discussed. Also, it is sometimes possible for bands to occur as a result of a shift in electron density (charge transfer). Such a shift may be from the metal to the ligands (M → L) or from ligands to the metal (L → M) and these transitions are discussed. Some ligands such as CN − or CO have empty π⁎ orbitals that can accept electron density from the metal, and this give rise to a phenomenon known as back donation.

Using density functional theory and its time-dependent extension for excited states, the S0 →S1 high-resolution vibronic absorption and electronic circular dichroism spectra of (R)-(+)-1-phenylethanol are computed and compared to experimental spectra measured in jet-cooled conditions in the region within 1000 cm-1 of the 0-0 transition. The agreement between theory and computation is satisfactory and allows a confident assignment of several experimental bands in terms of fundamentals of different modes. Cases are documented for which the analysis of optical anisotropy factors, owing to their signed nature, remarkably enhances the possibility of a robust assignment of the experimental absorption bands. Computational analysis shows that the experimental spectra are dominated by Herzberg-Teller contributions and that the electronic circular dichroism spectrum and the anisotropy factors are also strongly modulated by the effect of Duschinsky mixings.

The very broad emission bands of organic semiconductor materials are, in theory, suitable for achieving versatile solid-state lasers; however, most of organic materials only lase at short wavelength corresponding to the 0-1 transition governed by the Franck-Condon (FC) principle. A strategy is developed to overcome the limit of FC principle for tailoring the output of microlasers over a wide range based on the controlled vibronic emission of organic materials at microcrystal state. For the first time, the output wavelength of organic lasers is tailored across all vibronic (0-1, 0-2, 0-3, and even 0-4) bands spanning the entire emission spectrum.

The density functional theory (DFT) based calculations were carried out to study anion photoelectron spectroscopy of ZnO clusters. Detailed calculations were performed to study the molecular orbital configurations, molecular charge distributions, Mullikan’s atomic charge distributions, isosurface plots, thermodynamic properties, vibrational spectra and Franck Condon spectra of geometrically optimized ground states of neutral and anion structures of ZnO and (ZnO)2 clusters. The results indicated that the symmetric structure of ZnO cluster shows only one IR active mode whereas non-polar structure of (ZnO)2 shows six modes of vibration including three IR active modes. The Franck–Condon spectrum of these structures revealed that an increase in number of atoms in cluster causes an increase in number of Franck–Condon factors with a smaller overlap integral and an increase in depth of the potential well. Zero to zero ground state vibronic transition was observed to be most probable among all vibronic transitions with a Franck–Condon factor of 0.987 for ZnO and 0.982 for (ZnO)2 clusters.

The giant natural optical activity of vibronic replicas of the f–f5I8 → 5F2 transition in HoAl3(BO3)4 is observed. It exceeds the optical activity related to purely electronic transitions by two orders of magnitude and corresponds to the almost complete circular polarization of the transitions. This effect is explained by the local distortions of the crystal in the excited electronic state, which lead to the mixing of electron and vibrational wavefunctions and, as a consequence, to the delocalization of the vibronic wavefunction and the enhancement of spatial dispersion, which is responsible for the optical activity.

Vibronic coupling of the 4f–5d electronic transitions is investigated using Franck–Condon theory of vibronic interactions. To perform quantitative analysis of partially resolved experimental spectra given arise from a large number of vibrational modes and multiple electronic states, a semi-empirical model is established based on frequency degeneracy and line broadening. Instead of taking account all participating modes without experimental information on vibrational frequencies and vibronic coupling strength, this model includes a restricted number of degenerate modes with characteristic frequencies and line widths to simulate the experimental spectra. It is demonstrated that this model can be satisfactorily applied to quantitate the luminescence spectra of Ce3+ in Y3Al5O12 (YAG) and evaluate electronic energy levels, frequencies of leading vibrational modes and their vibronic coupling constant.

Single vibronic level fluorescence spectra of the electronic origin and of seven vibronic bands between 0,0 and 0,0 + 1265 cm −1 have been measured and analyzed by means of a combined Franck–Condon/rotational constants fit. The rotational constants in ground and lowest electronically excited singlet state of four different isotopologues have been taken from previous rotationally resolved measurements of Schmitt et al. (2006). The intensities of 182 vibronic emission bands and of 8 rotational constants have been used for a fit of the complete heavy atom geometry changes upon electronic excitation. Vibronic modes, about 1000 cm −1 above the electronic origin, show strong deviations from Franck–Condon behavior in emission. Herzberg–Teller coupling contributes to this effect. 1300 cm −1 above the origin, we observe the onset of intramolecular vibrational redistribution in the emission spectra.

Photochemical reactions of organic molecules are now becoming important tools to improve the quality of our lives through the search of ‘green’ solutions of modern life. Photochemical reactions occur in light-sensitive organic molecules, in which organic molecules are excited or activated by absorption of light or photon. In excited molecules, the vertical electronic transitions, known as Franck Condon transitions, occur in the excited molecules, where the atomic nuclei remain in stationary states (known as Franck Condon Principle). Usually spin allowed π → π* and spin forbidden n → π* electronic transitions occur in most cases of photo-excited organic molecules. A fraction of the excited molecules undergo photochemical reactions and the rest are deactivated to ground states through the intramolecular and intermolecular photophysical relaxation processes, depicted in Jablonski diagram. In intramolecular relaxation processes, the activated molecules are deactivated by emitting photons through spin allowed fluorescence (S1 → S0) and spin forbidden phosphorescence (T1 → S0) processes. In intermolecular relaxation processes, the activated molecules transfer their excess energy to the ground state molecules in quenching processes through formation of the exciplexes/excimers. The quenching rate can be determined by the Stern–Volmer equation. Intermolecular electronic energy transfer from activated molecules to the ground state molecules occurs through long-range singlet–singlet and short-range triplet–triplet energy transfer processes. The former process is known as FRET process and the latter is known as Dexter energy transfer process. The mechanism of these photo-induced electron transfer (PET) processes and their applications in tumor therapy and other fields are highlighted.

Calculations of the vibronic structure in electronic absorption and fluorescence spectra of a large polyatomic molecule based on time- and frequency-domain approaches beyond the Condon approximation are developed. In time-domain approach, the molecular spectra are expressed in terms of Fourier transforms of appropriate time correlation functions, which are evaluated in closed form for displaced–distorted–rotated harmonic potential surfaces at finite temperature. In frequency-domain approach, use is made of the Franck–Condon factors which were evaluated in closed form for the same potential surfaces in our previous work (J. Molec. Spectrosc. 194 (1999) 179). The vibronic structure in dipole-forbidden (but vibronic-allowed) 11Ag ↔ 21Ag and dipole-allowed 11Ag ↔ 11Bu electronic transitions of all-trans-1,3,5,7-octatetraene have been calculated and compared with experimental data as illustrations of the approaches. Our vibronic structure calculations support the hypothesis that a sudden opening of a non-radiative decay channel occurs at 21Ag vibronic levels with energies larger than about 2100 cm−1.

Efficient quantum dynamical and electronic structure approaches are presented to calculate resonance Raman spectroscopy (RRS) with inclusion of Herzberg-Teller (HT) contribution and mode-mixing (Duschinsky) effect. In the dynamical method, an analytical expression for RRS in the time domain is proposed to avoid summation over the large number of intermediate vibrational states. In the electronic structure calculations, the analytic energy-derivative approaches for the excited states within the time-dependent density functional theory (TDDFT), developed by us, are adopted to overcome the computational bottleneck of excited-state gradient and Hessian calculations. In addition, an analytic calculation to the geometrical derivatives of the transition dipole moment, entering the HT term, is also adopted. The proposed approaches are implemented to calculate RR spectra of a few of conjugated systems, phenoxyl radical, 2-thiopyridone in water solution, and free-base porphyrin. The calculated RR spectra show the evident HT effect in those π-conjugated systems, and their relative intensities are consistent with experimental measurements.

The water molecule, rotationally state selected in the third and fourth OH stretching overtone (‖04〉−, ‖05〉−) and stretch–bend combination (‖04−2〉) levels, has been photodissociated via the A state at λ≂282 nm. The photofragment rotational state distributions, determined by OH(A–X) laser induced fluorescence (LIF), differ from those reported previously by Andresen and co‐workers, in which water was initially prepared in the ‖01〉− level and photodissociated at 193 nm, and from those by Crim and co‐workers, in which H2O was photodissociated via the ‖04〉− level at wavelengths shorter than 282 nm. These differences become more pronounced with increasing angular momentum in the parent water molecule and with an increasing number of quanta in the intermediate OH stretching overtone state. The Franck–Condon theory of Balint‐Kurti, previously employed successfully to account for the product state distributions arising from the 193 nm photodissociation of H2O‖01〉−, qualitatively reproduces the trends observed in the present study if it is assumed that dissociation occurs preferentially from extended RH‐OH configurations of the ‖04〉− and ‖05〉− overtones. The product OH state distributions are thus shown to be sensitive indicators of the bending and rotational motions of H2O(X) in the wide amplitude stretching region of the ground state surface.

The concept of a potential-energy surface (PES) is central to our understanding of spectroscopy, photochemistry, and chemical kinetics. However, the terminology used in connection with the basic approximations is variously, and somewhat confusingly, represented with such phrases as "adiabatic", "Born-Oppenheimer", or "Born-Oppenheimer adiabatic" approximation. Concerning the closely relevant and important Franck-Condon principle (FCP), the IUPAC definition differentiates between a classical and quantum mechanical formulation. Consequently, in many publications we find terms such as "Franck-Condon (excited) state", or a vertical transition to the "Franck-Condon point" with the "Franck-Condon geometry" that relaxes to the excited-state equilibrium geometry. The Born-Oppenheimer approximation and the "classical" model of the Franck-Condon principle are typical examples of misused terms and lax interpretations of the original theories. In this essay, we revisit the original publications of pioneers of the PES concept and the FCP to help stimulate a lively discussion and clearer thinking around these important concepts.

This chapter starts with the time-dependent Schrodinger equation and based on the semiclassical theory provides a detailed account of interaction of radiation and matter culminating into electronic transitions of valence electrons, core electrons and Rydberg transitions, and interaction between the electronic and vibrational motions. After having derived the Fermi's Golden rule, selection rules for transitions in diatomic and polyatomic molecules have been worked out. Electronic spectra of polyatomic molecules have been treated in detail and relationship established between the oscillator strength and intensity of absorption bands. Quantum mechanical treatment of the Franck–Condon principle for vibronic coupling has been given and used to understand intensity distribution in vibrational band intensities in electronic transitions. In order to predict the energy, properties, and reactivity of molecules in their excited states and electronic transitions, CIS (configuration interaction-singles) method based on Hartree–Fock theory and time-dependent density functional theory have been described with illustrative examples. A comparative account of methods used for the study of Rydberg transitions and core electronic transitions has also been given.

The potential energy curves of the low-lying electronic states correlating up to the limit K(4p) + Rb(5s) of KRb molecule have been calculated using the multi-reference perturbation theory method at the CASSCF/ XMCQDPT2 level of theory without and with spin–orbit coupling. The calculated parameters of the ground X 1 R + state are in the best agreement among all previously performed ab initio calculations for the KRb molecule. The calculated vibrational intervals of the ground electronic term of the 39 K 85 Rb molecule describe the experiment with the accuracy within ±1 cm ?1 . The calculated intensities of the 2 1 R + (v 0 = 3, J 0 = 26) ? X 1 R + (v 00 = 0...24, J 00 = 25, 27) transitions satisfactory reflect the experimentally observed intensities distribution.

The sensitivity of vibronic calculations to electronic structure methods and basis sets is explored and compared to accurate relative intensities of the vibrational bands of phenylacetylene in the S(1)(A(1)B(2)) ← S(0)(X(1)A(1)) transition. To provide a better measure of vibrational band intensities, the spectrum was recorded by cavity ringdown absorption spectroscopy up to energies of 2000 cm(-1) above the band origin in a slit jet sample. The sample rotational temperature was estimated to be about 30 K, but the vibrational temperature was higher, permitting the assignment of many vibrational hot bands. The vibronic structure of the electronic transition was simulated using a combination of time-dependent density functional theory (TD-DFT) electronic structure codes, Franck-Condon integral calculations, and a second-order vibronic model developed previously [Johnson, P. M.; Xu, H. F.; Sears, T. J. J. Chem. Phys. 2006, 125, 164331]. The density functional theory (DFT) functionals B3LYP, CAM-B3LYP, and LC-BLYP were explored. The long-range-corrected functionals, CAM-B3LYP and LC-BLYP, produced better values for the equilibrium geometry transition moment, but overemphasized the vibronic coupling for some normal modes, while B3LYP provided better-balanced vibronic coupling but a poor equilibrium transition moment. Enlarging the basis set made very little difference. The cavity ringdown measurements show that earlier intensities derived from resonance-enhanced multiphoton ionization (REMPI) spectra have relative intensity errors.

On the basis of the harmonic oscillator approximation, the two-dimensional four-mode Franck–Condon vibrational overlap integral under mode-mixing effects has been reduced to a product expression of one-dimensional Gaussian integrals with the aid of expanding Hermite polynomials. A more general analytical expression for the calculation of the two-dimensional four-mode Franck–Condon overlap integrals was derived by directly calculating the Gaussian integrals. This analytical expression was applied to investigate the photoelectron spectra of considering mode-mixing and hot band effects. Geometry optimizations and harmonic vibrational frequency calculations were performed on the 2A1 state of NO2 and the 1A1 state of . Franck–Condon analyses and spectral simulations were carried out on the NO2( 2A1)– ( 1A1) photodetachment processes. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectra. In addition, the equilibri...

A commercial mass spectrometer coupled with a spectrograph and charge-coupled device (CCD) was used to obtain the collision-induced emission (CIE) spectra (190–1020 nm) from 4 to 8 keV collisions between projectile He + ions and CO 2 target gas (He + /CO 2 ). The observed emissions were from the A 2 Π u + and B 2 Σ u + states of CO 2 + , excited states of He, along with excited fragments. The relative vibrational population of the A 2 Π u + state were estimated and compared with our previous CO 2 + /He studies. The results indicate that charge transfer ionization of CO 2 in He + /CO 2 collisions is similar to that of photoexcitation, while in the case of CO 2 + /He, the population is more evenly distributed due to intramolecular vibrational redistribution (IVR) after the collision. Adiabatic potential energy surfaces calculated at the CISD/6-311+g(2df) level of theory for CO 2 + /He collisions, show the curve crossing of the CO 2 + A 2 Π u + and CO 2 + B 2 Σ u + states, in agreement with our observation.

A theoretical model is developed for simulating optical spectra of uranyl in complexes. The spectral profiles of electronic and vibronic transitions of UO 2 2 + bonded to equatorial ligands are evaluated based on the Huang–Rhys theory of vibronic coupling in solids. This model assumes that only the symmetric O U O stretching mode exhibits multi-phonon progression, whereas all other local modes are not Franck–Condon allowed and treated as false origins of the progressing mode. A simulation of vibronic transitions in the UO 2 (NO 3 ) 2 (TBP) 2 complex is compared with absorption spectrum to determine the excited state energy levels, frequencies of local vibration modes and vibronic coupling strength.

Calculations of Franck–Condon factors are crucial for interpreting vibronic spectra of molecules and studying nonradiative processes. On the base of the closed form expression of the Franck–Condon integrals between arbitrary multidimensional harmonic oscillators under the Duschinsky mixing effects, a more general algebraic expression for the calculation of the three-dimensional four-mode Franck–Condon factors was derived straightforwardly and applied to study the photoelectron spectrum of D 2 CO + ( B ∼ 2 A 1 ) . Geometry optimization and harmonic vibrational frequency calculations were performed on the X ∼ 1 A 1 state of D2CO at B3LYP, QCISD, CCSD and CASSCF levels, and the B ∼ 2 A 1 state of D2CO+ at CIS, TD-B3LYP, and CASSCF levels. Franck–Condon analyses and spectral simulations were carried out on the D 2 CO + ( B ∼ 2 A 1 ) - D 2 CO ( X ∼ 1 A 1 ) photoionization process. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectrum. In addition, the equilibrium geometric parameters of the B ∼ 2 A 1 state of D2CO+ were obtained in the spectral simulations.

Explicitly electron correlating coupled cluster calculations, CCSD(T)-F12a, were performed to determine three-dimensional potential energy hypersurfaces of disulphanide and disulphanyl in an automated approach. Surfaces for different electronic states were used in a Watson rovibrational Hamiltonian ansatz to obtain the correlated anharmonic vibrational wavefunctions. Subsequently the anharmonic Franck–Condon overlap integrals were evaluated. The computed Franck–Condon profiles were compared to experimental photodetachment-photoelectron spectra and confirm essentially the assignments made previously. The profiles indicate, however, additional weaker, and as of yet unresolved, additional features.

Based on the contour integral expression of Hermite polynomials, a closed form for the Franck–Condon integrals between multidimensional harmonic oscillators has been presented under the Duschinsky mixing effects. Furthermore, an exact analytical expression for calculation of the three-dimensional six-mode Franck–Condon integral has been derived. In addition, a general explicit formula to calculate the three-dimensional Franck–Condon factors has also been given and applied to study the intensity distribution in the photoelectron spectrum of the H2CO+( 2 B 1)–H2CO( 1 A 1) transition of formaldehyde. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in excellent agreement with the observed spectrum.

Geometry optimization and harmonic vibrational frequency calculations were performed on the X ˜ 2 A ″ states of CH3OO/CD3OO and X ˜ 1 A ′ states of CH3OO−/CD3OO− at the B3LYP, QCISD and CCSD levels. The electron affinity energies of CH3OO/CD3OO were calculated at various theory levels. Franck–Condon analyses and spectral simulations were carried out on the X ˜ 2 A ″ - X ˜ 1 A ′ photodetachment processes. The spectral simulations of vibrational structures based on the computed Franck–Condon factors are in agreement with the observed spectra.

The corrections to conventional Franck-Condon factors required by the assumption that an electronic transition takes place in a non-zero time t(0) are investigated using a theory based on the sudden approximation. An explicit expression using a harmonic oscillator model is given for the vibrational progression k'

This paper carries out ab initio calculations to study the 80 Se2(X3?g-) state and 80 Se2+(X2 IIg), 80 Se2+(a4 IIg) states by using completed active space self-consistent field and multi-reference second order perturbation theory. The electronic curves of these states including spin?orbit coupling are calculated, and then the spectroscopic parameters are obtained. The photoelectron spectra of 80 Se2 molecule in gas phase are assigned according to Franck?Condon analysis based on calculated potential energy curves. The ionization energies of 80 Se2 molecule are determined by the present calculation.

The B0u+–X0g+ and B1u–X1g transitions of 80Se2 and 78Se2 are reinvestigated. Several new bands are reported, especially for the B1u–X1g system, their vibrational numbering being confirmed by isotope shift studies. The Franck–Condon factors and r-centroid values are evaluated. By correlating the theoretically computed intensities with observed values, the internuclear distance dependence of the electronic transition moments of these two systems is determined for the first time.

We have theoretically studied the absorption vibronic spectra with the resonance two-photon (R2P) and non-resonance two-photon (NR2P) mass analyzed threshold ionization (MATI) spectroscopy. The theory uses the inverse Born–Oppenheimer approximation (IBOA) to establish a proper basis set. To analyze the MATI vibronic spectra, we have calculated the Franck–Condon factors involved in the vibronic transitions. Several experimental spectra are analyzed using this theory with emphasis on the importance of the resonance intermediate states. The long vibrational progression in a MATI spectrum can be partly attributed to the result of including the anharmonic correction in the calculated Franck–Condon factors. The experimentally observed isotope effect is also analyzed.

We present a method of transferring a cold atom between spatially separated microtraps by means of a Raman transition between the ground motional states of the two traps. The intermediate states for the Raman transition are the vibrational levels of a third microtrap, and we determine the experimental conditions for which the overlap of the wave functions leads to an efficient transfer. There is a close analogy with the Franck-Condon principle in the spectroscopy of molecules. Spin-dependent manipulation of neutral atoms in microtraps has important applications in quantum information processing. We also show that starting with several atoms, precisely one atom can be transferred to the final potential well hence giving deterministic preparation of single atoms., 8 pages, 6 figures. Accepted for publication Journal of Physics B (Atomic, molecular and optical physics)

A theoretical treatment is presented of the relationship between absorption and emission maxima for (0–0) electronic transitions in transition metal complexes such as the ruthenium(II) polypyridyl chromophores used as dye sensitizers in photovoltaic cells, and the corresponding shift in standard redox (or half-wave) potential of these ionic complexes. It is shown that the potential shift is proportional to the average of absorption and emission wavenumbers for oxidized or reduced forms, a result that follows from a detailed analysis of medium relaxation effects, based on the Onsager cavity model, and attendant solvatochromic shifts. It is also shown how additional thermodynamic properties (reaction entropies and enthalpies) pertaining to the half reaction if the reactants are present in solution can be obtained from observation of the variation of spectra with temperature. For the case of reactants present in an insoluble monolayer, a general expression is derived for the Gibbs energy change that results from the promotion of all absorbing species to an excited state, and the changes in Gibbs energy resulting from the processes that return the system to ground-state equilibrium.

30K-photocapacitance and excitation photocapacitance methods were applied to reveal the dislocation-induced deep levels in coalescent epitaxial lateral overgrowth layers of InP. Point-contact Schottky barrier junctions with small junction areas were formed on dislocated and dislocation-free regions. In the dislocation-free layers, the dominant deep level was located at Ec-1.30 eV, whereas in the dislocated area, dominant deep levels were detected at Ec-0.86 eV and 1.05 eV. A neutralized state was also detected at 0.66 eV+Ev. Excitation photocapacitance results have shown that the defect configuration coordinate diagram of the dislocation-induced deep levels was considered with large Frank-Condon shifts of 0.28 eV. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The absorption and luminescence spectra of a LiBaF3:Pb2+ crystal are studied in the temperature range 10–300 K. At 300 K a structureless absorption band (A band) is found in the energy range 1.4–6.6 eV. The luminescence spectra at 300 and 10 K are composed of two A' and A1 bands with maxima at ~5.0 and ~4.0 eV, respectively. The A' band originates from the 'regular' Pb2+ centres. The A' band luminescence decay is single-exponential in the whole temperature range 10–300 K. The properties of the observed absorption and A' luminescence bands are described well within the semiclassical theory based on the Franck–Condon principle and the Jahn–Teller effect in the excited sp configuration. The possible nature of the A1 band luminescence is discussed.

A vibronic theory of unpolarized infrared absorption and vibrational circular dichroism (VCD) is presented that is valid in the limit of vibrational resonance with low-lying electronic states (LLESs). The theory is developed within the complete adiabatic vibronic coupling formalism that describes both the correlation of electron density to nuclear positions and the correlation of electron current density to nuclear velocities. It is found that additional terms contribute to the electric-dipole and magnetic-dipole transition moments that become zero in the limit where all excited electronic states in the molecule are much higher in energy relative to fundamental vibrational transitions. Two correction terms appear that are extensions of the lowest-order non-Born−Oppenheimer expansion term when vibronic detail is included in the energy denominators. One term is a resonance term with respect to the energy difference between the electronic and vibrational transitions, and the other is a nonresonance term. Und...

It is well-known that vibronic interactions can be modelled in terms of an effective Hamiltonian incorporating first and second-order reduction factors (RFs), particularly when analysing the spectroscopic properties. Measurements and calculations of the RFs as a function of the strength of vibronic coupling are therefore of much interest. In this paper, we develop a new general method for determining second-order RFs (soRFs) from the strength of the Jahn–Teller (JT) coupling for systems in which electron orbital degeneracy or pseudo-degeneracy exists. These include in particular the fullerene molecule C60, pseudo-Jahn–Teller molecules and impurity centres in crystals. In order to calculate the important soRFs for intermediate to strong coupling, it is necessary to determine non-Condon corrections to the strong coupling values obtained using the Franck–Condon (FC) approximation. This gives an additional contribution to the nuclear polarizability of the system, thus enabling the electrons to follow the nuclear vibrations. These non-Condon corrections are derived using perturbation theory and are found to be inversely proportional to the square of the JT energy. The validity of the approximation is first tested in the cubic JT system due to its relative simplicity. It is found that the results are closer to those obtained earlier by numerical methods than the analytical FC values alone. Results are then presented that are applicable to C60− anions.

Theoretical modeling of multi-phonon vibronic spectra is presented based on the Huang–Rhys theory of ion–phonon interaction in a single-mode lattice. The single-mode spectral function is modified for applications to vibronic spectra involving multiple vibrational modes. It is shown that, for lanthanide and actinide ions in crystals, f–d electronic transitions exhibit two types of vibronic couplings. One type of vibronic coupling to local and pseudo-local modes obeys Frank–Condon selection rules and has a sharp line width, whereas in the second type of vibronic coupling electronic transitions couple to lattice modes in which selectivity is absent, thus forming a broad vibronic band. The model calculation reproduces very well the experimental spectra of 5f–6d transitions of Pa 4+ in Cs 2 ZrCl 6 and 4f–5d transitions of Ce 3+ in Cs 2 NaYCl 6 .

We investigate the vibrational structure of electronic spectra for the transition from the nondegenerate A state to E states in E g ⊗ e g Jahn–Teller systems with the trigonal field included. By extending the analysis by Longuet-Higgins et al. [Proc. R. Soc. London, Ser. A 244 (1958) 1] to this classically chaotic system, we find that the triple-humped structure is manifest with increasing anharmonicity. Such a structure cannot be derived using the semiclassical Franck–Condon (FC) approximation, and is caused by the chaos-induced breaking of the FC principle.