Franck-Condon Factors Research Articles

Vibronic coherent quantum beat in four-layer platinum carbonyl cluster.

Vibronic coherence has been studied for years, but direct comparisons between the rich experimental features and theory remain rare. In this work, we investigate the vibronic coherent quantum beat of a four-layer platinum carbonyl cluster [Pt3(CO)6]42- in a solution utilizing femtosecond vis-pump/vis-probe transient absorption spectroscopy. By varying the excitational wavelength, quantum beats coupled to either the electronic ground state or the excited state are selectively prepared. A 41cm-1 beat at the ground state with a phase flip at 615nm and a 28cm-1 beat at the excited state with a phase node at 735nm are observed. The beat amplitudes are asymmetric, stronger on the red side for ground state beats but weaker for excited state beats. Quantum chemistry calculations suggest that these beats result from coupling between the [Pt3(CO)6] layer motions and the electronic excitation. Theoretical model calculations for quantum beats at both electronic states are performed following the doorway-window approach. The calculations explain the oscillation frequency difference, the node positions, and the asymmetry. The beats with different frequencies result from vibronic coupling with different electronic states with the Herzberg-Teller (ground) or Franck-Condon term (excited) involved. The theoretical nodes occur at absorption and fluorescence centers, respectively, although experimental results show a slight blueshift. Quantum window operator calculations link the beat amplitude asymmetry to the Franck-Condon factor matrix imbalances, with the number of nodes dependent on the electronic dephasing rate. The theoretical insights for quantum beats are expected to be general, potentially helpful for the interpretation of observations in other systems.

Faraday rotation method improves the upper limit of the electron electric-dipole-moment sensitivity.

The electron electric-dipole-moment (eEDM) is a powerful tool for exploring new particles. The candidates for eEDM search are heavy atoms and their molecules, which are well known for the obvious relativistic effect. Lead atom is considered to be the most ideal relativistic atom [Park et al., Nat. Commun. 11(1), 815 (2020)]. PbH molecule is an important representative of the Pb compound and is considered a cold candidate molecule due to the high diagonal Franck-Condon factors. We systematically investigated the (eEDM) searches of PbH using a two-component approach. The parity- and time-reversal symmetry violation constants of ground and excited states, including internal effective electric field Eeff, electron-nucleon scalar-pseudoscalar interaction constant WP,T, and nuclear magnetic quadrupole moment, were obtained and compared to other molecules. In addition, we designed two experimental methods to measure the sensitivity of the eEDM, indicating that the Faraday rotation method could greatly improve its sensitivity.

Theoretical study on the spectroscopic and transition properties of phosphorus dimer cation

The potential energy curves of 18 Λ-S states of the P2 + cation have been constructed using the internally contracted multi-reference configuration interaction approach combined with the Davidson correction. The basis-set extrapolation to the complete basis set limit is performed. The relativistic effect is considered in calculations. Treatment of scalar relativistic effect at the third-order Dkroll-Hess approximate and spin-orbit coupling with the state-average Breit-Pauli operator is made. Based on the obtained potential energy curves, the spectroscopic parameters for all bound Λ-S and Ω states are determined. These results are in agreement with experiment where the spectroscopic data are available. The transition dipole moments are calculated. The properties of the dipole-allowed transitions among doublet or quartet Λ-S states are evaluated, from which Einstein coefficients, Franck-Condon factors and radiative lifetimes are estimated. The spin-orbit coupling effect on the transition properties of the 12Πg→X2Πu system is also studied. The spectroscopic and transition properties reported in this study can be used to guide the detection of phosphorus dimer cation in spectroscopy experiments.

Theoretical spin-orbit laser cooling for AlZn molecule.

A spin-orbit coupling electronic structure study of the AlZn molecule is conducted to investigate the molecular properties of the low-lying electronic states and their feasibility toward direct laser cooling. This study uses the complete active-space self-consistent field level of theory, followed by the multireference configuration interaction method with Davidson correction (+Q). The potential energy and dipole moment curves and the spectroscopic constants are computed for the low-lying doublet and quartet electronic states in the 2S+1Λ± and Ω(±) representations. The transition dipole moments, the Franck-Condon factors, the Einstein coefficient, the radiative lifetimes, the vibrational branching ratio, and the slowing distance are determined between the lowest spin-orbit bound electronic states. These results show that the molecule AlZn has a high potential for laser cooling through the X2Π1/2 → (2)2Π1/2 transition by utilizing four lasers at a wavelength in the ultraviolet region, reaching a sub-microkelvin temperature limit.

Theoretical study on the spectrum properties of tellurium iodide cation

The molecular potential energy function plays an important role in many fields. In this paper, the icMRCI + Q method was utilized to compute the potential energy and dipole moments for 22 Λ-S states and 51 Ω states of the TeI+ system. Two basis sets (AVQZ-PP and AWCVQZ-PP) were employed to compute the TeI+ system, with the results indicating that the AVQZ-PP basis set yielded more accurate results. Therefore, all calculations in this paper are based on this basis set. Furthermore, to ensure the accuracy of the results, a comparison was conducted on the spectral parameters of the ground state and two excited states of the molecular ion within the same main group. Given the significant impact of spin–orbit coupling, as indicated by the calculated SO matrix elements, our discussion will predominantly center on the avoidance of crossovers in the Ω states. Finally, the Franck-Condon factors, Einstein coefficients and radiative lifetime between these two states were calculated from the data of the transition between the X3Σ0+-↔11Σ0++ of the TeI+ molecule.

Structure, and the laser cooling scheme for the XF− (X=Li, Na, K, Rb) molecule

For laser-cooling considerations, we have theoretically investigated the electronic structures of the XF− (X=Li, Na, K, Rb) molecule. The potential existence of anions is initially examined through the analysis of their affinity energies and dipole moments of the corresponding neutral molecules. Subsequently, we excluded the effect of spin–orbit coupling on the molecular. Then the calculations reveal that XF− (X=Li, Na, K, Rb) exhibits a highly diagonalized Franck-Condon factor and a short spontaneous radiation lifetime, facilitating the design of laser cooling schemes for each molecular ion. The findings provide an important reference for laser cooling experiments involving alkali metal fluoride anions.

Cyanonaphthalene and cyanonaphthyl radicals: Vibrational structures via computed negative ion photoelectron spectra and thermochemistry of 1- and 2-cyanonaphthalene.

A double harmonic oscillator model is applied to compute the negative ion photoelectron spectra (NIPES) of the 1- and 2-cyanonaphthalene (CNN) radical anions. The computed Franck-Condon factors utilize optimized structures and harmonic vibrational frequencies obtained using density functional theory with the B3LYP 6-311++G (2d,2p) basis set while considering the mode-mixing Duschinsky effects. To test the accuracy of our model, the NIPES of α and β naphthyl radical anions were computed, and a strong agreement between the slow electron velocity-map ion imaging spectra and the predicted spectra was found. The adiabatic electron affinities (EAs) of the ground singlet states (S0) in 1-CNN and 2-CNN are 0.856 and 0.798eV, respectively. The origin of the lowest-lying triplet (T1) states in 1-CNN and 2-CNN is found to be 3.226 and 3.266eV, resulting in singlet-triplet energy splittings (ΔEST) of 2.370 and 2.468eV, respectively. Both the NIPES for electron detachment to the S0 and T1 states exhibit well-resolved vibrational features, allowing for the assignment of several vibrational fundamental frequencies. Following deprotonation, several isomers are formed, with the most stable deprotonated radical anions in 1-CNN and 2-CNN, corresponding to the removal of the most acidic proton, with EAs of 2.062 and 2.16eV. The rich spectroscopic and thermochemical data obtained in the current study make the CNN radical anions and their deprotonated species interesting systems for investigation in gas-phase, negative-ion experiments.

Vibrational-Mode-Selective Modulation of Electronic Excitation.

Vibrational-mode-selective modulation of electronic excitation is conducted with a synchronized femtosecond (fs) visible (vis) pulse and a picosecond (ps) infrared (IR) pulse. The mechanism of modulation of vibrational and vibronic relaxation behavior of excited state is investigated with ultrafast vis/IR, IR/IR, and vis-IR/IR transient spectroscopy, optical gating experiments and theoretical calculations. An organic molecule, 4'-(N,N-dimethylamino)-3-methoxyflavone (DMA3MHF) is chosen as the model system. Upon 1608 cm-1 excitation, the skeleton stretching vibration of DMA3MHF is energized, which can significantly change the shape of the absorption, facilitate the radiative decay and promote emission from vibrational excited states. As results, a remarkable enhancement and a slight blueshift in fluorescence are observed. The mode-selective modulation of electronic excitation is not limited in luminescence or photophysics. It is expected to be widely applicable in tuning many photochemical processes.

Theoretical study of vibronic absorption spectra and Franck-Condon factors of copper and silver hydrides

The electronic absorption spectra of CuH and AgH molecules are calculated using the analytical expression for the spectral density related to the thermal dipole time correlation function. In addition, the vibronic transition probabilities between the initial and final states are determined by calculating the transition dipole moments and the Franck-Condon factors using the LEVEL program. The spectroscopic constants recently obtained from MRCI + SOC calculations are used to calculate the absorption spectral line shapes of CuH and AgH molecules and plotted in the time and frequency domains for the vibronic transitions from the ground state Ω = 0+(I) to the excited states Ω = 0+(II), 0+(III), 1(I), 1(II), 1(III), and 1(IV) for CuH and to the excited states Ω = 0+(II), 0+(III), 0+(IV), 1(II), 1(III), 1(IV), 1(V), and 1(VI) for AgH.

On the effective vibrational temperature of the source using (2)3∏ - X3∏ system of GeC molecule

In this work, the experimental potential energy curves for (2)3Π and X3Π electronic states of GeC molecule have been constructed by using Rydberg-Klein-Rees (RKR) method. The radiative transition parameters viz., Franck-Condon (FC) factor, r-centroid, electronic transition moment, band strength, relative band strength, Einstein coefficients, radiative lifetime and oscillator strength for the (2)3Π−X3Π system of GeC molecule have been estimated for the experimentally observed vibrational levels from Rydberg-Klein-Rees (RKR) potential and the estimated values are tabulated. The estimated effective vibrational temperature found as 5628 K for the (2)3Π−X3Π system of GeC molecule. The radiative transition parameters and effective vibrational temperature are evident that the possible presence of GeC molecule in solar and sunspots atmosphere. Further, these parameters are employed in rationalizations of astrochemical and astrophysical observations.

Theoretical MRCI + Q study on electronic structure and spectroscopic properties of the HX+(X = F, Cl, Br, I) cations

In this work, we have systematacially investigated the electronic structure and spectroscopic properties of the HX+ (X = F, Cl, Br, I) cations by using the highly correlated MRCI + Q approach. This is the first comprehensive ab initio study on the electronic states of the HX+ cations. The spin–orbit coupling (SOC) effect is introduced with the state interaction approach. There are total 10 Λ-S and 18 Ω states obtained in the calculation. The results show that the potential energy curves’ (PECs) shapes and structure of the electronic states of different HX+ cations exhibit the significant distinction due to the X-dependent energy-level order of the unique dissociation channel H+(1Sg) + X(2Pu). From PECs, the spectroscopic constants of the bound electronic states are determined, which are in good agreement with available observed values. Regarding HBr+ and HI+, the good agreement has been achieved only when the SOC effect is considered. The predissociation for the first excited A2Σ+ state of HCl+, HBr+, and HI+ is analyzed based on computed spin–orbit matrix elements. Around the equilibrium position, the energy splittings of X2Π are calculated to be 288, 661, 2691, and 5137 cm−1 for HF+, HCl+, HBr+, and HI+, respectively. It has been demonstrated that SOC is substantial for HX+, leading to significant changes in PECs’ shapes as well as in electronic structure. Finally, the transition properties are predicted, including transition dipole moments, Franck–Condon factors, and radiative lifetimes. Both the Ω transitions A2Σ+ 1/2-X2Π3/2 and A2Σ+ 1/2-X2Π1/2 of the HX+ cations are determined to possess the radiative lifetimes at the microsecond (µs) level.

Simulating Vibronic Spectra by Direct Application of Doktorov Formulas on a Superconducting Quantum Simulator.

In this work, a direct quantum implementation of the Doktorov formulas for calculating the vibronic spectrum of molecules under the harmonic approximation is presented. It is applied to the three-atom molecules H2O, SO2, ClO2, HS2, and ZnOH. The method solves the classically hard problem of estimating the Franck-Condon (FC) factors by using the Duschinsky matrices as the only input via the Doktorov quantum circuit. This has the advantage of avoiding basis changes, artificial squeezing parameters, and symmetry dependencies. In other words, it is a general method for three-atom molecules that can easily be generalized to bigger molecules. The results are compared with other quantum algorithms and classical anharmonic algorithms. Furthermore, the circuit requirements are studied in order to estimate its applicability on real superconducting quantum hardware.

Anthocyanins of Delonix Regia Floral Petals: A Novel Approach on Fluorescence Enhancement, Forster Resonance Energy Transfer Mechanism and Photostability Studies for Optoelectronic Applications.

In this work, we focused on extracting the anthocyanin dye in acetone, butanol, ethanol, and water solvents from Delonix regia flowers by a simple maceration extraction process. The identification of functional group analysis, vibrational studies, energy transfer mechanisms, optoelectronic properties, photostability studies, FRET-assisted potential light emissions and photometric properties of the anthocyanin dyes are successively investigated. FTIR spectroscopy and vibrational studies have confirmed the existence of polyphenolic groups in 2-phenyl chromenylium (anthocyanin) dyes. The optoelectronic results show the least direct bandgap (2.04eV), indirect bandgap (1.55eV), Urbach energy (0.380eV), high refractive index (1.20), dielectric constant (2.794), and high optical conductivity (1.954 × 103 S/m) for the anthocyanin dye extracted found in water solvent. The photoluminescence properties such as Stoke's shift, high quantum yield, and lifetime results show that anthocyanin dyes are promising candidates for red-LEDs and optical materials. The absorption and emission spectra of the anthocyanin dyes follow the mirror image rule and the Franck-Condon factor exists between vibrational energy levels corresponding to all the electronic transitions. The excellent correspondence between the absorption and emission spectra reinforces that the anthocyanins are efficient (46%) FRET probes. Further, photometric properties such as CIE, CRI, CCT and colour purity results of anthocyanins in all studied solvents revealed that this material exhibits orange to red shades (x = 0.48 → 0.54 and y = 0.36 →0.45) and is well suitable for have great potential in the manufacturing of Organic-LEDs and other optoelectronic device applications.

Probing the activated complex of the F + NH3 reaction via a dipole-bound state

Experimental characterization of the transition state poses a significant challenge due to its fleeting nature. Negative ion photodetachment offers a unique tool for probing transition states and their vicinity. However, this approach is usually limited to Franck-Condon regions. For example, high-lying Feshbach resonances with an excited HF stretching mode (vHF = 2-4) were recently identified in the transition-state region of the F + NH3 → HF + NH2 reaction through photo-detaching FNH3− anions, but the direct photodetachment failed to observe the lower-lying vHF = 0,1 resonances and bound states due apparently to negligible Franck-Condon factors. Indeed, these weak transitions can be resonantly enhanced via a dipole-bound state (DBS) formed between an electron and the polar FNH3 species. In this study, we unveil a series of Feshbach resonances and bound states along the F + NH3 reaction path via a DBS by combining high-resolution photoelectron spectroscopy with high-level quantum dynamical computations. This study presents an approach for probing the activated complex in a reaction by negative ion photodetachment through a DBS.

An approach to calculate Franck-Condon factors involving anharmonic potentials using harmonic oscillator bases

In this contribution a discrete variable representation (DVR) approach to calculate Franck-Condon factors (FCFs) is presented. The method is illustrated using a harmonic oscillator basis. The advantage of this approach is that it is possible to calculate FCFs associated with general potentials based on the knowledge of the harmonic oscillator theory. First the coordinate and momentum representations are introduced. Based on this concept, the DVR is proposed, which allows the Franck-Condon factors to be calculated in a simple form. This method takes advantage of the momentum as generator of translations to include the displacement of the potentials. The calculation of the FCFs is exemplified for transitions involving the ground state X1Σg+ and the excited state a 1Π g of the diatomic molecule 14N2 both modelled with Morse potentials.

Comparative study of the multiple wavelength lasing of nitrogen ions: the role of vibrational level-dependent photoionization

We report on an optical amplification and energy threshold of the two most prominent emission lines, 391.4 and 427.8 nm, of the cavity-less lasing of nitrogen ions pumped by femtosecond laser pulses. It was found that the two transitions both show optical amplification under a low gas pressure condition, while the 391.4 nm emission is barely amplified under high gas pressure. Moreover, the 427.8 nm emission presents a significant lower pump laser energy threshold and a larger gain factor than the 391.4 nm emission. Numerical simulations based on a three-state coupling model suggest that the smaller ionization Franck–Condon factor from the ground state of N2 to the vibrational level ν = 1 in X2Σ g + state of N2+ favors the formation of population inversion corresponding to the 427.8 nm emission. Meanwhile, the competition between the strong field ionization and excitation induced by the pumping laser requires higher laser intensity to acquire the population inversion for the 391.4 nm radiation, leading to a corresponding larger energy threshold.

Electronic structure, spectroscopy and photophysics of rubidium mono-sulphide

We investigated the nineteen first electronic states of the RbS diatomic using post Hartree-Fock configuration interaction methods at the internally contracted multi-reference configuration interaction (MRCI+(Q)) level. We thus mapped their respective potential energy curves with and without considering spin-orbit effects. After nuclear motion treatments, we derived their equilibrium distances, rotational and vibrational constants, and dissociation and electronic excitation energies. Moreover, an evaluation of the dipole moments of RbS was conducted, which in turn were used to evaluate the radiative lifetimes of the vibrational levels of RbS (X2П and A2Σ+) states. Computations show that spin-orbit interactions are important and cannot be omitted. Afterwards, and utilizing the sum rule approximation and the Franck–Condon factors, we computed both the bound-bound contributions and the bound-free terms. Our findings suggest that RbS exhibits intriguing characteristics suitable for laser cooling experiments. In sum, this work contributes to the understanding in-depth of the spectroscopy of RbS and predicts its use as a promising molecular system for laser cooling applications.

High-Resolution Rovibronic Cross Sections of the MgH+ Molecular Cation.

The high-resolution rovibronic line lists of the MgH+ molecular cation are presented in our work. The potential energy curves are calculated by the method of multireference configuration interaction and Davidson correction (MRCI+Q) with the spin-orbit coupling (SOC) effect. Spectroscopy constants are fitted and the results are in good agreement with experiments, ensuring the accuracy of the electronic structure. On account of potential energy curves and transition dipole moments, the Franck-Condon factors and Einstein coefficients of transition are obtained. These calculations are used to obtain an accurate partition functions and line lists for molecules. Using the partition functions and line lists, the absorption cross-sections under different temperatures and pressures were simulated. Our work could provide some theoretical insights into solar and cold planet spectrum.

Relativistic coupled cluster study on the spectroscopic and radiative properties of the KFr molecule and modeling of the transport properties of potassium–francium dilute gas medium

The KFr diatomic molecule represents a promising candidate for photoassociation and indirect laser cooling. It has not yet been investigated experimentally and remains the only representative of the family of alkali-metal diatomics, which is scarcely researched theoretically. The potential energy curves of the states belonging to the three lowest dissociation limits of the KFr molecule are calculated using the Fock-space relativistic coupled cluster theory for the first time. Properties and parameters, such as electronic term energies, equilibrium internuclear distances, transition and permanent dipole moments, sequences of vibrational energies, harmonic vibrational frequencies, Franck–Condon factors, and radiative lifetimes (including bound and free transitions), are predicted. We analyzed and evaluated the probabilities of the two-step schemes (optical cycles) for the transfer process of the KFr molecules from excited vibronic states to the ground vibronic state. The transport properties (diffusion, viscosity, and thermal conductivity coefficients) of two-component dilute gas media of potassium and francium atoms as functions of the translation temperature up to 3000 K are also calculated. The data obtained would be useful for laser cooling and spectral experiments with KFr molecules.

Blue-Shifted and Broadened Fluorescence Enhancement by Visible and Mode-Selective Infrared Double Excitations.

Mode-selective vibrational excitations to modify the electronic states of fluorescein dianion in methanol solutions are carried out with a femtosecond visible pulse synchronized with a tunable high-power, narrow-band picosecond infrared (IR) pulse. In this work, simultaneous intensity enhancement, peak blueshift, and line width broadening of fluorescence are observed in the visible/IR double resonance experiments. Comprehensive investigations on the modulation mechanism with scanning the vibrational excitation frequencies, tuning the time delay between the two excitation pulses, theoretical calculations, and nonlinear and linear spectroscopic measurements suggest that the fluorescence intensity enhancement is caused by the increase of the Franck-Condon factor induced by the vibrational excitations at the electronic ground state. Various enhancement effects are observed as vibrations initially excited by the IR photons relax to populate the vibrational modes of lower frequencies. The peak blueshift and line width broadening are caused by both increasing the Franck-Condon factors among different subensembles because of IR pre-excitation and the long-lived processes induced by the initial IR excitation. The results demonstrate that the fluorescence from the visible/IR double resonance experiments is not a simple sum frequency effect, and vibrational relaxations can produce profound effects modifying luminescence.