Franck-Condon Factors and r-Centroids of the B-X Bands of Be18O Molecule

Abstract

The Franck-Condon factors, q_v′,v″ and r-centroids, r_v′,v″, for the bands of the B¹Σ⁺ ? X¹Σ⁺ transition of Be¹⁸O molecule (420–520 nm) are determined. A comparison of calculated q_v′,v″ values with the experimental estimated intensities of the bands shows reasonable agreement.     

Franck-Condon Factors and r-Centroids for the (B [sbnd] X) and (C [sbnd] X) Bands of the 107Ag 109Ag Molecule

Arrays of Franck-Condon factors q(v′, v″) and r-centroids r(v′, v″) were computed using Morse potentials for C¹Π [sbnd] X¹Σ⁺ _g and B O⁺ _u [sbnd] X¹Σ⁺ _g bands of the ¹⁰⁷Ag ¹⁰⁹Ag molecule.     

Comparison of photoelectron intensities and Franck-Condon factors in the photoionization of H2, HD and D2

The relative intensities of vibrational bands corresponding to the photoionization reactionX¹Σ_g⁺(υ″ = 0) + hv → X²Σ_g⁺(υ′ = 0, 1, 2 …) + e^? have been measured for H₂, HD and D₂, using He I radiation and a cylindrical mirror analyzer. These relative intensities differ significantly from squared overlap integrals (Franck-Condon factors) based on accurate potential curves for X¹Σ_g⁺ and X²Σ_g⁺, but are in good agreement with calculations performed by Itikawa which include the variation of transition moment with internuclear distance and the kinetic energy of the departing electron.     

Spontaneous radiative dissociation in molecular hydrogen

The spontaneous radiative dissociations of the discrete vibrational levels of the B¹Σ⁺_u electronic states of H₂, HD and D₂ of the C¹Π_u electronic state of H₂ into the vibrational continuum of the ground X¹Σ⁺_g state are calculated as a function of the emission wavelength. The fluorescent spectra of HD in the Lyman system and of H₂ in the Werner system resulting from an excitation source uniform in wavelength are predicted. The vibrational radiative lifetimes are tabulated as are the fractions of radiative decays that lead to dissociation. The effects of centrifugal distortion are discussed briefly. An appendix describes a sum rule used to check the numerical accuracy of the calculations.     

Transition moment variation in the A2Σ+ → X2Πi transition of HCl+, DCl+ and HBr+

Relative emission intensities of sixteen bands of HCl⁺ (A²Σ⁺ - X²Π_i), four bands of DCl⁺ (A²Σ⁺ - X²Π_i), and 5 bands of HBr⁺ (A²Σ - X²Π_i) have been made using both ion-beam excitation and microwave discharge sources. Intensities were determined by comparison with computer-generated spectra. Treatment of the data within the r-centroid approximation shows that in HCl⁺ the electronic transition moment decreases strongly at large $\text{r}{}_{\mathrm{v\prime v″}}$ [$\text{Re}\text{α}\text{exp}\text{(?3.6}\text{r}{}_{\mathrm{v\prime v″}}\text{)}\text{for 1.44}\text{A}\text{?}\text{< r}{}_{\mathrm{v\prime v″}}\text{< 1.82}\text{A}\text{?}$] but levels off at shorter $\text{r}{}_{\mathrm{v\prime v″}}$. DCl⁺ data agree quantitatively with HCl⁺. The variation in the HBr⁺ moment is similar, with $\text{R}\text{e}\text{α}\text{exp}\text{[?4.5}\text{r}{}_{\mathrm{v\prime v″}}\text{]}\text{for 1.58}\text{A}\text{?}\text{<}\text{r}{}_{\mathrm{v\prime v″}}\text{< 1.78}\text{A}\text{?}$.     

Combined fitting of optical and millimeter wave data: The linked A2Π-X2Σ+ and B2Σ+-X2Σ+ systems of Ca79Br and Ca81Br

Adiabatic corrections to the Born-Oppenheimer potential energy curve for the $\text{B″}{}^1\text{Σ}{}_{\mathrm{u}}{}^{\mathrm{+}}$ state of the hydrogen molecule has been computed using a 60-term variational wavefunction in the form of an expansion in elliptic coordinates and depending explicitly on the interelectronic distance. For three internuclear separations new Born-Oppenheimer energies are also given which represent corrections of the previously published values. For H₂ the adiabatic correction to the dissociation energy amounts to 57.5 cm^?1. The computed dissociation energies for H₂, HD, and D₂ are by 23–25 cm^?1 smaller than the experimental values, the discrepancy being probably due to the convergence error of the Born-Oppenheimer potential. A few lowest vibrational levels and vibrational quanta have been calculated for H₂, HD and D₂, and compared with the experimental results.     

Quantum mechanical calculation of rotational constants and Franck-Condon factors of diatomic molecules

In this paper, the question of reliability of the Morse potential as a potential curve for a diatomic molecule is investigated on the basis of calculating the rotational constant. It is shown that the Morse potential describes well potential curves of X¹Σ _q ⁺ and B¹Π_u electronic states of a Na₂ molecule. Calculations of Franck-Condon factors for X¹Σ _q ⁺ ? B¹Π_u band of a Na₂ molecule using wave functions of the Morse potential confirm the known correlation between the values of Franck-Condon factors and rotational constants of combined electronic states.     

A new emission band spectrum of sulphur

A new emission band system of S₂ has been obtained in the region λ 3050–λ 2670 when sulphur is excited in a 30 mc/S. high frequency discharge from a 1/2 kW oscillator. The bands appear single sharp headed and are degraded towards red. Analysis of these new bands as belonging to a single system has led to the following vibrational formula.ν=36624·7+428·5(v′+1/2)?3·45(v′+1/2)² ?699·2(v″+1/2)+3·2(v″+1/2)². This system disignated as (b?x) is tentatively assigned to the electronic transition¹∑ _u ⁺ ?¹∑ _g ⁺ . The¹∑ _g ⁺ (x) state is found as the common lower state of three of the far ultraviolet systems of S₂ recently reported byTanaka andOgawa.     

C2H221分子转动角动量定向分布(Orientation)及其碰撞弛豫和转移

利用圆偏振激光受激Raman抽运,以 C₂H₂分子为样品选择性地制备了它的电子基态单一转动态(X¹Σ⁺_g,ν″₂＝1,J″的角动量定向布居(orientation)．并从圆偏振紫外激光诱导的A¹A_u(ν′₃＝1)←X¹Σ⁺_g(ν″₂＝1)的荧光（谱）,直接测定了 C₂H₂(X¹Σ^＋_g,ν″₂＝1,J″＝4,7,8,…,13）的角动量定向布居值．从时间分辨的荧光信号谱测定了角动量定向布居的碰撞弛豫速率常数,同时还研究了由各初始激励的转动态向其他邻近转动态碰撞诱导的角动量定向布居转移． 关键词：     

Vibronic intensity distribution in the B0u-X0g and B1u-X1g transition of Se2

The B0_u⁺-X0_g⁺ and B1_u-X1_g transitions of ⁸⁰Se₂ and ⁷⁸Se₂ are reinvestigated. Several new bands are reported, especially for the B1_u-X1_g 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.     

Relation between classical and quantum formulations of the Franck-Condon principle: The generalized r-centroid approximation

The classical transition point $\text{r}{}_{\mathrm{v\prime v″}}{}^1$ for the (v′, v″) band is introduced as the point where the kinetic energy does not change during the transition (classical statement of the Franck-Condon principle). It is shown that when the r-centroid approximation is valid, it implies that the r centroid $\text{r}{}_{\mathrm{v\prime v″}}$ equals $\text{r}{}_{\mathrm{v\prime v″}}{}^1$, allowing a new connection to be made between the classical and quantum statements of the Franck-Condon principle. The r-centroid approximation fails for some classically allowed bands having large Franck-Condon factors. This occurs when the (v′, v″) band has more than one classical transition point. A generalization of the r-centroid approximation permits such bands to be used in determining the variation of the electronic transition moment with internuclear distance.     

Lyman transitions of high rovibrationally excited H2, HD and D2 molecules

Energies and probabilities of Lyman transitions of high rovibrationally excited H₂, HD and D₂ molecules have been measured and compared with calculations. The experimental results are obtained from laser-induced fluorescence spectra that have been recorded in the spectral range from 60 500 to 83 500 cm⁻¹, covering 2/3 of the hydrogen Lyman band system. The necessary vacuum-UV radiation is produced by stimulated anti-Stokes Raman scattering, providing a widely tunable radiation source with narrow spectral bandwidth to resolve single Lyman transitions. The highest internal energies of detected hydrogen isotopologues are close to the dissociation limit. This extends the available data base of Lyman transitions from and to higher rotational states (J > 10) of HD and D₂.     

Spectre de photoélectrons et calcul des facteurs de franck-condon pour H2O,D2O,HDO

The first band of the photoelectron spectrum of HDO has been recorded. In agreement with the selection rules of the group theory, the fundamental terms of the three symmetric vibrations of HDO (C_s symmetry) have been observed. Taking the geometry of the ion as parameters, the Franck-Condon factors for the ionization of H₂O, D₂O and HDO have been calculated. The geometry of the H₂O⁺, D₂O⁺, HDO⁺ ions (ground state) have been determined accurately by comparison of the calculated results with the corresponding photoelectron spectra. This geometry is approximately the same for the three ions: r_OH  1,00 Å and < HOH  110°.     

Stoßdissoziation von H 2 + - und D 2 + -Ionen

The angular and energy distribution of protons produced by collision-induced dissociations of H ₂ ⁺ ions with energies of 10 and 20 keV were measured in a parabola spectrograph. From these measurements the velocity distribution of the protons in the center of mass system of the H ₂ ⁺ ion can be calculated. This gives information about the type, the abundance, and the anisotropy of the processes involved. The most frequent transitions leading to dissociations are the excitation of the 2pσ_u state, the ionisation of the H ₂ ⁺ ion, the transition into the vibrational continuum, and the electron capture into the 1³ σ _u ⁺ state of the hydrogen. It is shown that the cross section for an electronic transition depends on the velocity of the ion, the distance of the nuclei in the ion, the angle between the internuclear axis and the direction of the primary ion beam, and the excitation energy of the target. The fraction of protons produced by vibrational excitation increases with increasing atomic number of the target. Concerning electronic transitions D ₂ ⁺ ions equal H ₂ ⁺ ions of the same velocity.     

Studies of labile molecules with a tunable dye laser: Laser excitation spectrum of BrF (B3Π(0+)-X1Σ+)

The B³Π(0⁺) ← X¹Σ⁺ absorption spectrum of BrF (4850–5200 Å) has been observed by the technique of laser emission spectroscopy. Fluorescence was excited by a pulsed, scannable dye laser with a 0.1 Å bandwidth. Rotational analysis has been carried out for six bands of the v″ = 0 progression (8 ≥ v′ ≥ 3) of ⁷⁹BrF and ⁸¹BrF. Rotational constants for the B³Π(0⁺) state are reported for the first time. RKR potential energy curves for both states, and an array of Franck-Condon factors and r-centroids for the transition, have been calculated. Bands with v′ > 8 were not observed in fluorescence owing to the onset of predissociation near J′ = 28 of the v′ = 8 level. An upper limit for the ground state dissociation energy is D₀″ (BrF, X¹Σ⁺) ≤ 20 880 cm^?1.     

A spectroscopic study of the D(0u+) ion-pair state of Br2 by the optical-optical double-resonance technique

An optical-optical double-resonance technique has been applied to study the D(0_u⁺) ion-pair state of Br₂ in a one-photon resonant three-photon absorption. The OODR transition proceeds through the high vibrational level of the B³Π(0_u⁺) state, which compromises a large Franck-Condon shift required for the excitation of Br₂ from the X¹Σ_g⁺ state to the D(0_u⁺) state. Dunham parameters of the D(0_u⁺) state, based on a global least-squares fit of 407 transitions (v′ = 0–16, J′ = 17–115), are Y₀₀ = 49928.443(41), Y₁₀ = 134.467(19), Y₂₀ = ?8.71(27) × 10^?2, Y₃₀ = ?3.36(10) × 10^?3, Y₀₁ = 4.2382(15) × 10^?2, Y₁₁ = ?1.061(36) × 10^?4, Y₂₁ = ?2.00(27) × 10^?6, and Y₀₂ = ?1.93(11) × 10^?8 for ⁷⁹Br₂ (all in cm^?1, and 3σ in parentheses). The single rovibronic fluorescence spectrum of the D(0_u⁺) state shows a transition terminating on the X¹Σ_g⁺ ground state, and establishes the absolute v′ numbering on the basis of the Franck-Condon factor calculations. The v′ = 2 and 3 levels of the D(0_u⁺) state are strongly perturbed due to the heterogeneous interaction with the 1_u state correlating with the same ionic products of the D(0_u⁺) state at the dissociation limit, $\text{Br}{}^{\mathrm{?}}\text{(}{}^1\text{S) +}\text{Br}{}^{\mathrm{+}}\text{(}{}^3\text{P}{}_2\text{)}$.     

Experimental determination of the potential energy curves of the I(3/2u) and I(3/2g) states of Kr+ 2

The I(3/2u) and I(3/2g) states of Kr⁺ ₂ have been investigated by pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy following (2 + 1′) resonance-enhanced multiphoton excitation via the 0⁺ _g Rydberg state located below the Kr?([4p]⁵5p[1/2]₀) + Kr(¹S₀) dissociation limit of Kr₂. From the positions of a large number of vibrational bands in the spectra of the ⁸⁴Kr₂ and ⁸⁴Kr-⁸⁶Kr isotopomers, the adiabatic ionization potentials (IP(I(3/2u)) = 112672.4 ± 0.8cm^?1, IP(I(3/2g)) = 111 395.0 ± 1.4cm^?1), the dissociation energies (D ⁺ ₀(I(3/2u)) = 368.8 ± 2.0cm^?1, D ⁺ ₀(I(3/2g)) = 1646.2 ± 2.3cm^?1) and vibrational constants for both ionic states have been determined. Potential energy curves have been extracted which perfectly reproduce all experimental observations and are accurate over a wide range of energies and internuclear distances. The equilibrium internuclear distances (R ⁺ _e(I(3/2u)) = 4.11 ± 0.04 Å, R ⁺ _e(I(3/2g)) = 3.35 ± 0.10 Å) have been derived by comparing the intensity distribution in the PFI-ZEKE photoelectron spectra to calculated Franck-Condon factors. The dissociation energy of the I(3/2g) state and the equilibrium internuclear distance of the I(3/2u) state differ markedly from previously reported values.     

Photoionization and photoelectron angular distribution for H2

Photoionization of H₂(¹Σ_g⁺) in a vibrational υ″ and rotational N″ state into H₂⁺(²Σ_g⁺) in a vibrational υ′ and rotational N′ state is studied theoretically. The differential cross section, after summing over the final states, is expressed in the well-known simple form of $\text{(}\text{σ}{}_{\mathrm{T}}\text{4π}\text{)[1 + βP}{}_2\text{(}\text{cos}\text{θ)]}$. Parallel expressions are obtained for H₂⁺ in a specific υ′ state (in terms of σ(υ′) and β(υ′)) and for H₂⁺ in a rotational fine level υ′N′ (in terms of σ(υ′N′) and β(υ′N′)). Asymmetry parameters β, β(υ′) and β υ′N′), which are expressed in terms of Racah and Clebsch-Gordan coefficients and electronic transition moments, can be reduced approximately to 2 lineary polarized light and to -1 for unpolarized light. Using single-center electronic wave functions and including partial eaves l = 1, 3, and 5, σ(υ′) and β(υ′) are computed as a function of υ′ at 584 Å. The computed σ(υ′) divided by the Frank-Condon overlap, in agreement with experimental results, increases monotonically with υ′; σ_T and β are computed in the incident photon energy range of 600–4000 Å and the results compare favorably with previous calculations.     

Energy shifts and forbidden transitions in H+ 2 due to electronic g/u symmetry breaking

A full account is given of calculations and measurements of transition frequencies and intensities of the forbidden pure rotation transition (v = 19, N = 1)-(v = 19, N = 0) in the ground electronic state (1sσ_g) of H⁺ ₂. The transition has measurable intensity because of ortho-paru mixing that arises from electronic g/u symmetry breaking caused by the Fermi contact hyper-fine interaction. Measurements of the transition were made in both single and double resonance using a fast ion beam/microwave spectrometer. The transition frequency was determined to be at 14961.7 ± 1.1 MHz (95% confidence, 5 measurements), in excellent agreement with the theoretical prediction of 14960 ± 3 MHz. The intensity of the transition relative to the allowed 1sσ_g (v = 19, N = 1)-2pσ_u,(v = 0, N = 2) transition was estimated from the available measurements to be 8000, in reasonable agreement with the theoretically predicted value of ?3000.     

The lifetime of the 2pπu → 1sσg transition of the F2+ center in KCl

The lifetime of the 2pπ_u → 1sσ_g transition for the F₂⁺ center in KCl has been measured (15.8 ± 0.7 ns at 8 K) and found to be constant over the temperature range (8–90 K) investigated. No contradiction is found with the Aegerter and Lüty quantum yield data. A forecast for the lifetime of the 2pσ_u → 1sσ_g transition is also made, within the framework of the H₂⁺ model for the F₂⁺ center in alkali halides.