Calculated Franck-Condon Factors Research Articles

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.

The calculation of ro-vibrational energies, franck-condon factors and r-centroids for Yttrium/Scandium oxides in D-dimension

The spectroscopic data as ro-vibrational energy values, Franck-Condon Factors and r-centroids are obtained for Yttrium/Scandium oxides with (B 2Σ+, X 2Σ+) different electronic states. Ro-vibrational energy values are calculated by bound state solution of the Schrödinger equation for the general molecular potential by applying Pekeris-type approximation to the centrifugal term in D-dimensional. Using these solutions, the relation giving the bound state energy eigenvalues and the corresponding wave functions expression are derived. The Franck-Condon factors and r-centroids values are determined by using Fraser-Jarmain method for the B 2Σ+ → X 2Σ+ electronic transitions in both molecules. All the results obtained are given numerically in the tables. The obtained values are compared with the previously get data in the literature and it is seen that the results are consistent.

Photoelectron Spectroscopy Confirms the Gas-Phase Stability of the C3O2- Anion.

The carbon suboxide anion with an asymmetric zigzag structure, C3O2-, which was recently characterized in a solid neon matrix, has not been observed in the gas phase. Here, we have experimentally generated C3O2- in a supersonic plasma jet expansion and studied it by negative ion photoelectron spectroscopy. A spectral analysis based on the calculated Franck-Condon factors, in combination with quantum chemical calculations, has for the first time yielded a determination of the electron affinity of the neutral carbon suboxide, EA = +0.48(8) eV. Molecular orbital (MO) analysis indicates that, upon linear-to-zigzag geometric change, additional σ-π MO mixing significantly reduces the energy of the singly occupied MO of C3O2- and thus stabilizes the negative ion in the gas phase. This work provides a benchmark understanding of the gas-phase stability of the C3O2- anion.

Cavity Ring-Down Spectroscopy of Anthracene, 9-Methylanthracene, and 2-Methylanthracene in Supersonic Expansion.

The S0-S1 absorption spectra of anthracene (C14H10), 9-methylanthracene (C15H12), and 2-methylanthracene (C15H12) are measured in the ultraviolet region between 330 and 375 nm (26,666 to 30,303 cm-1) with cavity ring-down spectroscopy in supersonic free-jet expansions of argon. The associated vibronic band systems and their spectroscopic assignments are discussed and compared to previous studies performed using fluorescence excitation and dispersed fluorescence techniques. Density functional theory (DFT) calculations were carried out to study the structures and evaluate the vibrational transitions of the ground and excited states. Time-dependent DFT calculations of the first electronic excited states and Franck-Condon factor calculations were carried out to assist in the assignment of the experimentally measured vibronic bands. The vibronic spectra obtained in absorption agree well with fluorescence excitation spectra in terms of peak positions but exhibit different relative band intensities. We find a very good match between experimentally obtained vibronic line positions and the peak positions of the quantum chemically calculated Franck-Condon excitation lines.

Laser-coolable AcOH+ ion for CP -violation searches

The ${\mathrm{AcOH}}^{+}$ molecular ion is identified as a prospective system to search for $\mathcal{CP}$-violation effects. According to our study ${\mathrm{AcOH}}^{+}$ belongs to the class of laser-coolable polyatomic molecular cations implying a large coherence time in the experiments to study symmetry-violating effects of fundamental interactions. We perform both nuclear and high-level relativistic coupled cluster electronic structure calculations to express the experimentally measurable $\mathcal{T},\mathcal{P}$-violating energy shift in terms of fundamental quantities such as the nuclear magnetic quadrupole moment (MQM), electron electric dipole moment ($e\mathrm{EDM}$) and dimensionless scalar-pseudoscalar nuclear-electron interaction constant. We further express nuclear MQM in terms of the strength constants of $\mathcal{CP}$-violating nuclear forces: quantum chromodynamics vacuum angle $\overline{\ensuremath{\theta}}$ and quark chromo-EDMs. The equilibrium structure of ${\mathrm{AcOH}}^{+}$ in the ground and the four lowest excited electronic states was found to be linear. The calculated Franck-Condon factors and transition dipole moments indicate that the laser cooling using an optical cycle involving the first excited state is possible for the trapped ${\mathrm{AcOH}}^{+}$ ions with the Doppler limit estimated to be $\ensuremath{\sim}$4 nK. The lifetime of the ($0,{1}^{1},0$) excited vibrational state considered as a working one for MQM and $e\mathrm{EDM}$ search experiments is estimated to be $\ensuremath{\sim}0.4$ s.

Theoretical Analysis of the Formylmethylene Anion Photoelectron Spectrum: Importance of Wolff Rearrangement Dynamics.

Photoelectron spectroscopy of a molecular anion is very useful for investigating the transition state and intermediate regions on the reactive potential energy surfaces of a neutral system. In this work, we theoretically analyzed the previously measured photoelectron spectrum of the formylmethylene anion, HCCHO-. We simulated the photoelectron spectra for both the singlet and triplet states using the semiclassical method with quantum nuclear densities and Franck-Condon factor calculations with harmonic vibrational analysis. We also performed real-time quantum dynamics calculations to elucidate the importance of the Wolff rearrangement process, which leads to the stable product ketene from the carbene intermediate on the neutral singlet potential energy surface.

Threshold photoelectron spectroscopy of the methoxy radical.

We present here a synchrotron radiation vacuum ultraviolet photoionization study of the simplest alkoxy radical, CH3O, a key reaction intermediate in atmospheric and combustion chemistry. A microwave discharge fast flow tube connected to a molecular beam sampling system is employed as a chemical reactor to initiate reactions and generate radicals. The CH3O+ cation from direct ionization of the CH3O radical is detected successfully in the photoionization mass spectrum close to its ionization threshold. In addition, after identifying and removing the contribution of the 13C-isotopic formaldehyde H2 13CO with the same isobaric mass m/z = 31, the high-resolution threshold photoelectron spectrum of CH3O is obtained and assigned with the aid of calculated Franck-Condon factors. The adiabatic ionization energy of CH3O is determined at 10.701 eV with an accuracy of 0.005 eV.

Velocity-Map Imaging and Magnetic-Bottle Photoelectron Spectroscopy of [SeCCH]-: Electronic Properties and Spin-Orbit Splitting.

The recently synthesized acetylide compound KSeCCH containing the main group element selenium within the novel and in crystalline form unprecedented [SeCCH]- anion was successfully investigated in the gas phase by high-resolution velocity-map imaging (VMI) and magnetic-bottle (MB) photoelectron spectroscopy coupled with an electrospray ionization source. Both VMI and MB spectra exhibited identical electron affinities (EA, 2.517 ± 0.002 eV), spin-orbit coupling (SOC) splittings (1492 ± 20 cm-1), and Se-C stretching frequencies (573 ± 20 cm-1) of the corresponding neutral tetra-atomic radical [SeCCH]• with the VMI spectrum possessing six times higher spectral resolution compared with the MB spectrum. These experimental values were well reproduced by calculations at the CCSD(T) level, in which both the isolated [SeCCH]- anion and the [SeCCH]• radical adopted linear geometries. The simulated spectra based on the calculated Franck-Condon factors, the SOC splitting, and the experimental line width matched well with the measured spectra. Furthermore, comparisons of the EA and SOC splitting values with the previously reported isolobal species [SeCN]• are also made and discussed. The decrease in the EA and SOC splitting of [SeCCH]• is ascribed to the differences in the electronegativities between C and N atoms as well as the electron density on the Se atom in its singly occupied molecular orbital (SOMO).

An algebraic approach to calculate Franck–Condon factors

An algebraic approach based on unitary algebras to calculate Franck–Condon factors (FCF’s) is presented. The method is based on the unitary group approach, which consists in adding a scalar boson to the $$\nu $$-D harmonic oscillator space, taking advantage of the transformation brackets connecting the energy, coordinate and momentum representations. In this scheme the solutions are given in terms of an expansion of harmonic oscillator functions. In this way the overlaps are expressed in terms of simple scalar product of the eigenvectors. As a benchmark to illustrate our approach the case of two 1D-Morse potentials is presented. Discussions concerned with both non-Condon contributions and the harmonic limit are included. FCF’s involving the 1D-Morse and asymmetric double Morse potentials are also considered. As an application, the FCF’s involving the S–S stretching in the $$\hbox {S}_2\hbox {O}$$ molecule are described in terms of two Morse potentials.

Valence Photoionization of Thymine: Ionization Energies, Vibrational Structure, and Fragmentation Pathways from the Slow to the Ultrafast.

The photoionization of thymine has been studied by using vacuum ultraviolet radiation and imaging photoelectron photoion coincidence spectroscopy after aerosol flash vaporization and bulk evaporation. The two evaporation techniques have been evaluated by comparison of the photoelectron spectra and breakdown diagrams. The adiabatic ionization energies for the first four electronic states were determined to be 8.922±0.008, 9.851±0.008, 10.30±0.02, and 10.82±0.01 eV. Vibrational features have been assigned for the first three electronic states with the help of Franck-Condon factor calculations based on density functional theory and wave function theory vibrational analysis within the harmonic approximation. The breakdown diagram of thymine, as supported by composite method ab initio calculations, suggests that the main fragment ions are formed in sequential HNCO-, CO-, and H-loss dissociation steps from the thymine parent ion, with the first step corresponding to a retro-Diels-Alder reaction. The dissociation rate constants were extracted from the photoion time-of-flight distributions and used together with the breakdown curves to construct a statistical model to determine 0 K appearance energies of 11.15±0.16 and 11.95±0.09 eV for the m/z 83 and 55 fragment ions, respectively. These results have allowed us to revise previously proposed fragmentation mechanisms and to propose a model for the final, nonstatistical H-loss step in the breakdown diagram, yielding the m/z 54 fragment ion at an appearance energy of 13.24 eV.

Laser cooling of the SiO+ molecular ion: A theoretical contribution

The feasibility of laser-cooling SiO+ molecular ion is studied by using ab initio methods. The electronic structures of 14 Λ-S states and 30 Ω states of SiO+ are computed by means of high-level configuration interaction method (including Davidson correction) with scalar relativistic correction. The calculated spectroscopic constants of bound states of SiO+ are in excellent agreement with available experimental results. The perturbation on vibrational levels of B2Σ+ is studied with the help of a spin-orbit coupling matrix elements which includes B2Σ+-14Σ−, B2Σ+-14Π, and B2Σ+-12Σ− interacting states pairs. The calculated Franck-Condon factors and branching ratio of the B2Σ+-X2Σ+ transition are highly diagonal. The radiative lifetime of B2Σ+ state is evaluated to be 72 ns, which is sufficiently short for laser cooling. On the basis of the B2Σ+-X2Σ+ transition, we propose a possible laser cooling scheme by using two lasers with wavelengths around 400 nm.

Theoretical study of the low-lying electronic states of magnesium sulfide cation including spin–orbit interaction

Highly correlated ab initio calculations have been performed for an accurate determination of electronic structures and spectroscopic features for the low-lying electronic states of the MgS+ cation. The potential energy curves for the four Λ-S states correlating to the lowest dissociation asymptote are studied for the first time. Four Λ-S states split into nine Ω states through the spin–orbit coupling effect. Accurate spectroscopic constants are deduced for all bound states. The spin–orbit couplings and the transition dipole moments, as well as the PECs, are utilized to calculate Franck–Condon factors and radiative lifetimes of the vibrational levels. To verify our computational accuracy, analogous calculations for the ground state of MgS are also carried out, and our derived results are in reasonable agreement with available experimental data. In addition, photoelectron spectrum of MgS has been simulated. The predictive results are anticipated to serve as guidelines for further researches such as assisting laboratorial detections and analyzing observed spectrum.

Spectroscopic Characterization, Computational Investigation, and Comparisons of ECX- (E = As, P, and N; X = S and O) Anions.

Three newly synthesized [Na+(221-Kryptofix)] salts containing AsCO-, PCO-, and PCS- anions were successfully electrosprayed into a vacuum, and these three ECX- anions were investigated by negative ion photoelectron spectroscopy (NIPES) along with high-resolution photoelectron imaging spectroscopy. For each ECX- anion, a well-resolved NIPE spectrum was obtained, in which every major peak is split into a doublet. The splittings are attributed to spin-orbit coupling (SOC) in the ECX• radicals. Vibrational progressions in the NIPE spectra of ECX- were assigned to the symmetric and the antisymmetric stretching modes in ECX• radicals. The electron affinities (EAs) and SO splittings of ECX• are determined from the NIPE spectra to be AsCO•: EA = 2.414 ± 0.002 eV, SO splitting = 988 cm-1; PCO•: EA = 2.670 ± 0.005 eV, SO splitting = 175 cm-1; PCS•: EA = 2.850 ± 0.005 eV, SO splitting = 300 cm-1. Calculations using the B3LYP, CASPT2, and CCSD(T) methods all predict linear geometries for both the anions and the neutral radicals. The calculated EAs and SO splittings for ECX• are in excellent agreement with the experimentally measured values. The simulated NIPE spectra, which are based on the calculated Franck-Condon factors, and the SO splittings nicely reproduce all of the observed spectral peaks, thus allowing unambiguous spectral assignments. The finding that PCS• has the greatest EA of the three triatomic molecules considered here is counterintuitive based upon simple electronegativity considerations, but this finding is understandable in terms of the movement of electron density from phosphorus in the HOMO of PCO- to sulfur in the HOMO of PCS-. Comparisons of the EAs of PCO• and PCS• with the previously measured EA values for NCO• and NCS• are made and discussed.

LEVEL: A computer program for solving the radial Schrödinger equation for bound and quasibound levels

This paper describes program LEVEL, which can solve the radial or one-dimensional Schrödinger equation and automatically locate either all of, or a selected number of, the bound and/or quasibound levels of any smooth single- or double-minimum potential, and calculate inertial rotation and centrifugal distortion constants and various expectation values for those levels. It can also calculate Franck–Condon factors and other off-diagonal matrix elements, either between levels of a single potential or between levels of two different potentials. The potential energy function may be defined by any one of a number of analytic functions, or by a set of input potential function values which the code will interpolate over and extrapolate beyond to span the desired range.

Angular and internal state distributions of H2 (+) generated by (2 + 1) resonance enhanced multiphoton ionization of H2 using time-of-flight mass spectrometry.

We report direct measurement of the anisotropy parameter β for the angular distribution of the photoelectron and photoion in (2 + 1) resonance enhanced multiphoton ionization process of H2 X (1)Σg (+) (v = 0, J = 0) molecules through the intermediate H2 E,F (1)Σg (+) (v' = 0, J' = 0) level (λ = 201.684 nm) using a time-of-flight mass spectrometer. The time-of-flight spectra were recorded as the direction of polarization of the ionizing laser was varied with respect to the flight axis of the H2 molecular beam and were fitted to an angular distribution in an appropriately rotated coordinate system with the z-axis oriented along the time-of-flight axis. The anisotropy parameter β was found to be 1.72 ± 0.13 by fitting the time-of-flight spectra and agreed with previous measurements. Using secondary ionization with a delayed laser pulse of different wavelength, we also determined the vibrational energy distribution of the ions, showing that 98% ± 4% of the ions are generated in their ground vibrational state, in agreement with the calculated Franck-Condon factors between the H2 E,F (1)Σg (+) (v' = 0) and H2 (+) X (1)Σg (+) (v″) vibrational levels.

A Study of H2O2 with Threshold Photoelectron Spectroscopy (TPES) and Electronic Structure Calculations: Redetermination of the First Adiabatic Ionization Energy (AIE).

In this work, hydrogen peroxide has been studied with threshold photoelectron (TPE) spectroscopy and photoelectron (PE) spectroscopy. The TPE spectrum has been recorded in the 10.0-21.0 eV ionization energy region, and the PE spectrum has been recorded at 21.22 eV photon energy. Five bands have been observed which have been assigned on the basis of UCCSD(T)-F12/VQZ-F12 and IP-EOM CCSD calculations. Vibrational structure has only been resolved in the TPE spectrum of the first band, associated with the X̃(2)Bg H2O2(+) ← X̃(1)A H2O2 ionization, on its low energy side. This structure is assigned with the help of harmonic Franck-Condon factor calculations that use the UCCSD(T)-F12a/VQZ-F12 computed adiabatic ionization energy (AIE), and UCCSD(T)-F12a/VQZ-F12 computed equilibrium geometric parameters and harmonic vibrational frequencies for the H2O2 X̃(1)A state and the H2O2(+) X̃(2)Bg state. These calculations show that the main vibrational structure on the leading edge of the first TPE band is in the O-O stretching mode (ω3) and the HOOH deformation mode (ω4), and comparison of the simulated spectrum to the experimental spectrum gives the first AIE of H2O2 as (10.685 ± 0.005) eV and ω4 = (850 ± 30) and ω3 = (1340 ± 30) cm(-1) in the X̃(2)Bg state of H2O2(+). Contributions from ionization of vibrationally excited levels in the torsion mode have been identified in the TPE spectrum of the first band and the need for a vibrationally resolved TPE spectrum from vibrationally cooled molecules, as well as higher level Franck-Condon factors than performed in this work, is emphasized.

Comparison of methods for calculating Franck–Condon factors beyond the harmonic approximation: how important are Duschinsky rotations?

Three different approaches for calculating Franck–Condon factors beyond the harmonic approximation are compared and discussed in detail. Duschinsky effects are accounted for either by a rotation of the initial or final wavefunctions – which are obtained from state-specific configuration-selective vibrational configuration interaction calculations – or by a rotation of the underlying multi-dimensional potential energy surfaces being determined from explicitly correlated coupled-cluster approaches. An analysis of the Duschinsky effects in dependence on the rotational angles and the anisotropy of the wavefunction is provided. Benchmark calculations for the photoelectron spectra of ClO2, HS−2 and ZnOH− are presented. An application of the favoured approach for calculating Franck–Condon factors to the oxidation of Zn(H2O)+ and Zn2(H2O)+ demonstrates its applicability to systems with more than three atoms.

Franck-Condon Factors and r-Centroids for Several Systems of CoF Molecule

Franck–Condon factors factors and r-centroids have been calculated for the G3Φ4-X3Φ4 , K3Φ4-X3Φ4, L3Φ4-X3Φ4, [20.6]3Γ5-X3Φ4 and [34.7]3Γ5-X3Φ4 bands of Cobalt monofluoride 59Co19F from the best available molecular constants using Morse and Rydberg–Klein–Rees (RKR) methods. The reliability of the obtained values was verified by comparing the results of five different programs. Based on the calculated Franck–Condon factors values, the appearance of the bands in these systems are discussed and the relationship between the r-centroids and the band-origin wavenumbers and wavelengths are also determined along with accuracy factors.

Franck-Condon Factors and r-Centroids for Several Systems of CoF Molecule

Franck–Condon factors factors and r-centroids have been calculated for the G3Φ4-X3Φ4 , K3Φ4-X3Φ4, L3Φ4-X3Φ4, [20.6]3Γ5-X3Φ4 and [34.7]3Γ5-X3Φ4 bands of Cobalt monofluoride 59Co19F from the best available molecular constants using Morse and Rydberg–Klein–Rees (RKR) methods. The reliability of the obtained values was verified by comparing the results of five different programs. Based on the calculated Franck–Condon factors values, the appearance of the bands in these systems are discussed and the relationship between the r-centroids and the band-origin wavenumbers and wavelengths are also determined along with accuracy factors.

Ab initio study on the low-lying excited states of gas-phase PH+ cation including spin–orbit coupling

Ab initio calculations have been performed on the low-lying excited and ground states of PH+. The potential energy curves (PECs) of the Λ–S states were calculated with multi-reference configuration interaction (MRCI) method along with the basis sets at 5-ξ level. In order to improve the PECs, the Davidson(+Q) correction and the Scalar relativistic effect are included. The corresponding spectroscopic constants were determined and good agreements with the available measurement were found. The interactions of the A2Δ–4Π and 12Σ+–4Π by the spin–orbit coupling (SOC) effect were well described by the spin–orbit matrix elements. The SOC effect makes the original 8 Λ–S states split into 15Ω states. The Ω=1/2 state generated from the X2Π state is confirmed to the ground Ω state. And the SOC splitting for the X2Π is calculated to be 294cm−1. The SOC effect has large effect on the PECs of the A2Δ and 12Σ+ states, leading to much more shallow potential wells as well as potential barriers. The analysis of the wavefunction for the Ω states shows that the strong spin–orbit interaction exists near the crossing points of the PECs for the Λ–S states. The transition dipole moments (TDMs) of transitions A2Δ–X2Π and 12Σ−–X2Π are evaluated with the MRCI wavefunction. Based on the TDMs along with the calculated Franck–Condon factors, the radiative lifetimes for the selected vibrational levels of A2Δ and 12Σ− states are predicted at the microseconds (μs). Good agreement with the measurement shows that the lowest vibrational level for A2Δ state is almost uninfluenced by the perturbation via the SOC effect.