Interpolation formulae for Franck-Condon factors

Formulae are presented which describe the variation of q_v^′v^″ with a transition parameter characteristic of the band system for small values of vibrational quantum number. The formulae place the previously discussed empirical interpolation graphs for Franck-Condon factors on a firmer theoretical basis.     

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.     

Electronic transition-moment variation in the A1π-X1Σ system of the GeO molecule

The spectrum of GeO was excited in a carbon arc and the relative band intensities were measured by using photographic photometry and were then interpreted in terms of Franck-Condon factors. It was found that the electronic transition moment varies as Re(r)=constant×(1-0.476r-0.061r²), where 1.62?r, Λ?1.82. Using this relation, smoothed band strengths, band-oscillator strengths and Einstein coefficients of the bands were calculated using expressions given by Penner. The effective vibrational temperature is estimated to be 1090 K. The average life times of the v′=0 and 1 vibrational levels have also been estimated.     

Intensities of the laser-induced fluorescence of 130Te2 and electronic transition strengths for the AOu+-XOg+ and BOu+-XOg+ systems

The intensities of the laser-induced fluorescence spectra of ¹³⁰Te₂, excited by Ar⁺ and He-Cd⁺ lasers, are measured for the AO_u⁺-XO_g⁺ and BO_u⁺-XO_g⁺ systems. The RKR potentials of the A, B and X states are calculated. These potentials are used for calculating Franck-Condon factors and r-centroids with due respect to rotation. The relative r-centroid dependences of the electronic transition strength were obtained first and then normalized in lifetimes.     

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.     

Two-exciton transition in the antiferromagnetic KNiF3 and K2NiF4

For the sharp band located at 31,200 cm^?1 in the three dimensional antiferromagnetic KNiF₃, the dependence of the oscillator strength on the Ni ion concentration and the stress-induced linear dichroic spectrum are studied. The polarization dependence of the corresponding band in the two dimensional antiferromagnetic K₂NiF₄ is also measured. The weak structure located at 30,780 cm^?1 is assigned as two-exciton transition, and the band at 31,200 cm^t?1 as a two-exciton transition accompanied with a T_1u phonon.     

Quasi-localized representation of electronic band structures in cubic crystals

Tabulated results and formulae are given for the parametrization of electronic band structures in simple cubic, body-centered cubic and face-centred cubic systems. This technique is based on the decoupling transformation and does not involve least squares fitting.A band labelling scheme is introduced which allows the decoupling transformation to be used to define the energy matrix of Wannier states in copper. This approach is used to re-examine the significance of the tight binding approximation for the d-states and to relate Fermi surface parameters directly to calculated band energies.     

Franck-Condon factors and absolute transition probabilities for the if (B3Π0+-X1Σ+) transition

Improved spectroscopic constants have been used to calculate Rydberg-Klein-Rees (RKR) potentials and Franck-Condon factors for the IF(B³Π₀₊-X¹Σ⁺) transition. The Franck-Condon factors are generally in good agreement with previously calculated values, but differ by as much as 30% for transitions from higher levels of the B-state. Several experimentally measured relative transition moment functions have been evaluated and the best scaled, so that the total transition probability calculated for each B-state vibrational level, A(v'), matched measured values. The scaled function was then used to calculate individual transition probabilities, A(v',v), for the vibronic transitions.     

The isoelectronic series ScF through ThO—I notes on the band spectra of TiO,HfO and ThO

The band spectra of TiO, HfO and ThO are treated, especially for later use in connection with the isoelectronic (in valence electrons) series ScF, TiO, ZrO, HfO and ThO. Franck-Condon factors and r-centroids for 12 HfO and ThO band systems are presented. Some recommendations for future spectral studies on UO are given, based on ThO results, since ThO apparently is the only diatomic oxide of the actinides to have known electronic band spectra.     

Computational study of the accuracy of the r-centroid approximation

The r-centroid approximation, used in the analysis of intensity measurements of electronic band spectra of diatomic molecules, allows a separation of the band strength into a pure vibrational term (the Franck-Condon factor q_v′v″) and an electronic term [the electronic transition moment Re(r)]. The r-centroid approximation is known to have limits to its applicability. However, no specific error limits have previously been given. A computational study has been performed using several molecules and analytic curves to determine specific error limits which can be assigned to either a specific band strength, or in estimating the electronic transition moment using the r-centroid approximation. Other techniques suggested for the evaluation of Re(r) have also been tested.     

Adiabatic Energies of Excited Σu States of the Hydrogen Molecule

Accurate Born-Oppenheimer potential energy curves and adiabatic corrections are computed for the six lowest ¹Σ_u states of the hydrogen molecule. For the 4¹Σ_u-6¹Σ_u states adiabatic term values of the vibrational levels supported by the potentials and the corresponding rotational constants are given. For the outer potential well of the 6¹Σ_u state, Franck-Condon factors are listed for the possible transitions to the HH?¹Σ_g state.     

Morse Franck-Condon Factors and R-Centroids for Some Lyman Bands of H2, HD and D2

Using new rotational and vibrational constants, Morse Franck-Condon factors and r-centroids are computed for 108 Lyman bands of astrophysical importance H₂, HD and D₂ molecules. Experimental oscillator strengths conjoint with Morse Franck-Condon Factors for six Lyman bands of molecular hydrogen yield radiative lifetimes for v = 0–5 levels of the B 1σ_u ⁺ state that are inconsistent with the experimental lifetime measurements. The limitation of analytic Morse function in the region of small internuclear separation for the B 1σ_u ⁺ state of H₂ is discussed. The r-centroids (r_v′v″) are found to increase with frequencies of the Lyman bands of H₂, HD and D₂. For a sequence, Δr = r_{v′+1,v″+1} - r_v′v″ is constant. A comparison of r_v′v″ values of the Lyman bands shows that r_v′vv″ of the band (v′v″) increases in going from H₂ to D₂.     

The electronic transition moment of the A1∑-X1∑ band system of BaO

The relative intensities of some 50 bands of the A¹∑-X¹∑ band system of BaO have been studied, using a source the fluorescent scattering of sunlight by a cloud of barium oxide vapor generated at an altitude of 106 km in the earth's atmosphere.At this altitude, the temperature is so low that the effects of overlapping bands are minimized and it has been possible to make accurate measurements of relative band intensities. Using recently calculated Franck-Condon factors, the variation of the electronic transition moment with internuclear distance has been determined over a wider range than was previously possible.     

Luminescence of Tl<Superscript>+</Superscript> ions in a KZnF<Subscript>3</Subscript> crystal

The luminescence spectra of a KZnF₃: Tl⁺ crystal are investigated in the energy range from 4.75 to 5.9 eV at temperatures of 10–300 K upon excitation into the A absorption band (5.7–6.3 eV). At T=300 K, the luminescence spectra exhibit an intense band with a maximum at 5.45 eV, which is attributed to single Tl⁺ ions substituted for K⁺ ions. The 5.723-eV intense narrow band observed at T<20 K is assigned to the ³Γ_1u-¹Γ_1g zero-phonon transition, which is weakly allowed by the hyperfine interaction. The luminescence decay is studied as a function of temperature. The main characteristics of the luminescence spectra are adequately described in terms of the semiclassical theory based on the Franck-Condon principle and the Jahn-Teller effect for an excited sp configuration of the Tl⁺ ion with the use of the parameters obtained earlier from analyzing the absorption spectra of the system under investigation.     

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.     

Angle-resolved photoemission from molecular N2 adsorbed on Ni(100)

Angle-resolved photoemission has been used to study the band structure of the ordered c(2 × 2) molecular nitrogen overlayer chemisorbed on Ni(100). The molecular nitrogen chemisorbs to the Ni(100) with the molecular axis perpendicular to the surface. The 1π_u, 3σ_g and 2σ_u (1π, 5σ and 4σ of adsorbed N₂) orbitals of N₂ are identified at 8.1 ± 0.05, 8.3 ± 0.1 and 12.8 ± 0.2 eV below the nickel Fermi energy at Г&#x0304;gG respectively. Angle-resolved photoemission of adsorbed molecular N₂ exhibits several satellite lines with less kinetic energy than the primary molecular orbital emission. These satellite lines are a final state effect. A k_∥ (parallel momentum vector) dependence of the satellite line intensity was observed.     

The pressure dependence of the Herzberg photoabsorption continuum of oxygen

The origin of pressure dependence in the dipole forbidden Herzberg continuum of oxygen is discussed in terms of the formation of oxygen dimers and the collision of free molecules. The small temperature dependence of the pressure coefficient indicates that the collisions of free molecules have the dominant influence. Consideration of the selection rules applying to the Herzberg systems and the strengths of the associated band systems leads to the conclusion that the pressure dependence results from enforced dipole transitions in the Herzberg III (A′³Δ_u−X³Σ_g^-) system.     

Excitation of projectile Rydberg states in the collisions Be+-He and Mg+-He, 10–75 keV

The photoabsorption coefficient of molecular oxygen has been measured at 1215.70 Å and in the ranges 1205 Å–1214 Å and 1218 Å–1225 Å with an average resolution of ±0.015 Å. The light source in this experiment was the Doppler shifted radiation obtained from Stark quenching of a metastable hydrogen beam with energies between 2 keV and 60 keV. Using observation angles of 45°, 90°, and 135° with respect to the beam the above mentioned tuning ranges are obtained. Our data join smoothly to those of Ogawa [1] in the range 1214 Å–1218 Å and are in fair agreement with earlier measurements in other laboratories. Analytical expressions for the absorption coefficient for use in geophysical applications are presented for the whole wavelength range 1205 Å–1225 Å. Rotational structure of the absorption coefficient in the range 1220 Å–1223 Å arising from the 3-0 band of theα ¹ ∑ _u ⁺ -X ³ ∑ _g ^? forbidden transition in molecular oxygen is clearly resolved in the present measurements. A transition probability ofA=4×10⁴ s^?1 is obtained for this system. A careful study of a possible pressure dependence of the absorption coefficient was made. Except from the region were rotational line absorption occurs and the single point at 1215.70 Å no measurable effect was found for pressures below 100 Torr.     

Inductive-resonant theory of nonradiative transitions in lanthanide and transition metal ions (review)

The review is devoted to the theory of nonradiative transitions in tricharged ions of lanthanides and transition metals in the condensed phase, which was proposed in 1971. The theory is based on the phenomenon of nonradiative energy transfer from an electronically excited ion to surrounding molecular groups with excitation of resonant vibrational states and makes it possible to calculate the nonradiative transition rate constant (k _nr) by a formula that is similar to the Förster formula. The primary emphasis is placed on recent experimental works that directly confirm the proposed theory. It is shown that the theory satisfactorily quantitatively accounts for (i) the effect of deuteration of molecular groups surrounding ions on k _nr, (ii) the energy gap law, and (iii) the dependence of k _nr on the distance between the ion and deactivating groups. Furthermore, it is shown that (iv) the theory makes it possible to satisfactorily quantitatively calculate in the dipole-dipole approximation the constant k _nr of the electronic transition based on the knowledge of the radiative rate constant and the vibrational absorption spectra of molecular groups in the range of overlap with the luminescence spectrum of the ion; (v) the temperature dependence of k _nr; and (vi) the anomalously low k _nr in the case where the corresponding radiative transition is caused by the magnetic rather than the electric dipole. Literature data are presented that directly experimentally support the proposed theory of nonradiative transitions. In addition, works where this approach is used to calculate k _nr of transitions in laser media are described.     

Geometry of C2N2 + and HCN+ from low-energy photoelectron spectroscopy

Quantitative Franck-Condon calculations are applied to the first two band systems of the low energy photoelectron spectrum of dicyanogen (C₂N₂) and to the first band system of hydrogen cyanide. Estimates are made of the bond length changes from the ground electronic state of the molecule to various states of C₂N₂ ⁺ and to the ground state of HCN⁺. These changes are consistent with expectations based on simple molecular orbital theory.

It is shown that there is sufficient intensity in the perturbed first system of HCN, and also that of DCN, to account for an overlapping second system due to the removal of an electron from the 5σ orbital.