Vibrational Overlap Integrals Research Articles

Computer modeling of thermal quenching of chromium photoluminescence in fluoride elpasolites

Radiationless transition rates were predicted ab initio in K 2NaScF 6:Cr 3+ and K 2NaGaF 6:Cr 3+. Embedded cluster calculations provided adiabatic potential energy functions of a 1g, e g and t 2g nearest-neighbor fluorine displacements in both ground ( 4 A 2g.) and excited ( 4 T 2g.) states. Transition rates were calculated from normalized lineshape functions G (Ω) derived by evaluating vibrational overlap integrals for an arbitrary combination of linear, quadratic and anharmonic coupling. Predicted transition rates, which are lower than experimental values, were not improved by including anharmonicity.

Analytical formula for Franck-Condon factors involving the Morse potential

Vibrational overlap integrals for the Morse potential are presented by analytic functions which are written as binomial series containing polygamma function and its derivative functions. Our calculating Franck-Condon factors are compared with RKR analysis, numerical calculation by the Morse potential and experimental results by EELS and PES for the following electronic transitions: N 2( a 1Π g, C 3Π u and b 1Π u← X 1∑ + g), CO( A 1Π ← X 1∑ +) and X 2∑ +, A 2Π and B 2∑ + ← X 1∑ + in the ionization processes from N 2 and CO to N + 2 and CO +. Very satisfactory agreement among our results, RKR analysis, Morse analysis and experiments is found.

Spin–orbit perturbations between the A(2)1Σ+ and b(1)3Π states of NaK

We have studied spin–orbit perturbations between the A(2) 1Σ+ and b(1) 3Π0 states of the NaK molecule by accurately measuring the energies of mutually perturbing levels, and by measuring ratios of A(2) 1Σ+ → X(1) 1Σ+ and b(1) 3Π0 → a(1) 3Σ+ emission intensities, for five perturbed pairs. This allows two partially independent determinations of each perturbation matrix element ‖〈1 3Π0(v′t, J′)‖Ĥso‖21Σ+(v′s,J′)〉‖. From these matrix elements, and calculated vibrational overlap integrals 〈v′t‖v′s〉, the electronic part of the perturbation matrix element, Hel≡‖〈1 3Π0(v′t, J′)‖Ĥso‖2 1Σ+(v′s, J′)〉‖/‖〈v′t‖v′s〉‖, was obtained. Our results for Hel from the two methods are consistent, and independent of vibrational and rotational quantum numbers, as expected. The determined best value for Hel is (15.64±0.39) cm−1.

Theory of hydrogen-transfer reactions based on a harmonic two-mode tunneling model: an analytical treatment

Based on a hydrogen-transfer model proposed by Siebrand et al. an exact analytical expression for the temperature-dependent rate constant is derived. In this model the potential barrier separating the positions of the transferred particle in the initial and final configuration is modulated by the reactant-product oscillation, the frequency of which is assumed to be small compared to the frequencies of the collinear intramolecular vibrations of the hydrogen atom before and after the transfer. The possible changes in the latter is neglected. Moreover, adiabatic separability of the slow and fast oscillation is used. Thus, in the harmonic approximation used here, the dynamics of the system is that of two independent harmonic oscillations with frequencies ω and Ω. There is, however, explicit mode interaction in the “two-mode tunneling model” via the vibrational overlap integral of the fast H oscillations centered at distances which are specific for particular transfer reactions. The analytical results show explicitly in which way the physical parameters enter. In the symmetric case, which is met when the exothermicity Δ E of the reaction vanishes and the conditionł&ω⪢ k B T is fulfilled, the result of Siebrand et al. is regained. The variation of the transfer rate constant at T = 0 as a function of the tunneling distance, the line of the final density of states and Δ E is discussed. Moreover, numerical calculations which can be easily done on the basis of the analytical result, are performed for different combinations of values for the two frequencies involved.

Perturbations between the X2Π and a4Σ − states of the NH + ion

On the basis of previously published optical data (Colin and Douglas, Canad. J. Phys. 46, 61–73 (1968)) for three isotopes ( 14NH +, 15NH + and 14ND +) of the NH + ion, a study of the perturbations occurring between the X 2Π and a 4Σ − states has been carried out using matrix diagonalization techniques. This makes it possible to derive for both states equilibrium rotational and vibrational constants, from which RKR potential energy functions, vibrational overlap integrals, electronic interaction parameters, and a pure electronic term value for the a 4Σ − state are deduced.

An iterative franck-condon analysis procedure for the determination of ionic geometries for H 2O and H 2S

The pattern search optimization procedure is applied to a Franck-Condon analysis method with the use of the formulation proposed by Kupka and Cribb for evaluating vibrational overlap integrals [H. Kupka and P.H. Cribb, J. Chem. Phys., 85 (1986) 1303; H. Kupka and G. Olbrich, J. Chem. Phys., 80 (1985) 3163]. The method was applied to three different ionization processes accompanied by small to large changes in molecular geometry. It was found that ionic geometries thus deduced for transitions with small structural changes agree well with observation. However, the agreement is poor for transitions with large geometric changes although the method can still account for the observed vibrational intensity distributions to a certain extent. In addition, properties of the pattern search method as well as Kupka and Cribb's formulation for Franck-Condon analyses are discussed.

ELECTRON‐DONOR‐ACCEPTOR ORBITAL CORRELATIONS‐VII. INTERMOLECULAR ORBITAL (LCMO) DESCRIPTION OF PHOTOSENSITIZED CHARGE SEPARATION IN A 4‐CENTER UNIT*

Abstract— The conditions under which electronically‐excited 1: 1 complexes are expected to produce high yields of separated radical‐ions are presented with reference to the porphyrin‐quinonc system in terms of the symmetry properties of intermolecular orbitals (IMO). These conditions are translated into design properties of a 4‐center photoinduced charge separation unit DDAO in which post‐excitation energy and electron transfer processes result from diabatic internal conversions between singly‐excited configurations differentiated by the location of holes and electrons in valence and conduction IMO of the unit. It is suggested that a major role of the donor pair is the inhibition of diabatic charge recombination by the low intermolecular component of the vibrational overlap integral over isoenergetic ground (DD) and cationic (DD +) configurations.

Vibrational overlap integrals between the neutral and ion states of NH 3 and ND 3: Application to the vibrational dependence of the NH 3+( v2)+NH 3(0) symmetric charge transfer reaction

Vibrational overlap integrals have been calculated for the transition between the neutral ground state and the ion state of NH 3 and ND 3 for the ν 2 vibrational bending mode. Calculations were performed by numerically solving the Schrödinger equation. The vibrational overlap integrals were used to simulate the dependence of the cross-section for the NH 3 +( v 2)+NH 3(0)→NH 3( v 2″)+NH 3 +( v 2′) symmetric charge-transfer reaction on the initial vibrational quantum state of the ion. The cross-sections show a gradual increase with v 2, in agreement with experiment.

Exact analytic formula for calculating Franck–Condon factors using the Kratzer potential

To calculate Franck–Condon factors (i.e., squares of the vibrational overlap integrals for two electronic states of a diatomic molecule), the potential energy curve in each state is approximated by the function proposed originally by Kratzer. The resulting factors q(v′,v″) are compared with experimental data for the following band systems: CN(A 2Π–X 2Σ+), N2(B 3Πg–A 3Σ+u; a 1Πg–X 1Σ+g; C 3Πu–X 1Σ+g), and CO(A 1Π–X 1Σ+). Very satisfactory agreement between theory and experiment is observed for low values of the vibrational quantum numbers v′ and v″ (v′,v″≤3).

A rotationally-resolved Rydberg transition of I2

Rotationally-resolved bands leading to a Rydberg state R 0 u + of molecular I2 are observed in a two-stage, three-photon transition from the ground state. The R 0 u + state interacts homogeneously with high vibrational levels, νF ≈ 200–250, of an ionic state F 0 u +, the perturbation being directed by the vibrational overlap integrals towards even-numbered vibrational levels of R. Spectral constants of R 0 u + are (in cm-1): T e = 61665·15, ωe = 209.29, ωe x e = 0·859, B e = 0·03842 and αe = 1·6 × 10-4. The electronic matrix element for the R, F interaction (excluding one deviant result) is |W e| = 107 ± 1 cm-1; thus W e/ωe ≈ 0·5, corresponding to ‘intermediate’ coupling. Energy considerations indicate that R should be assigned to the 0 u + state of either the configuration (2430 Π1/2g )6pσ u , or of (2421 4Σ u -)6sσ g . This state is the first extra-valence state of I2 to be rotationally analysed.

A New Approach to the Estimation of Accuracy of Calculations of Vibrational Wavefunctions and Radiative Parameters

Abstract The modified method of solving the radial Schrödinger equation, proposed earlier by the authors, is used for developing a new approach to the estimation of accuracy of calculating the vibrational wavefunction overlap integrals for diatomic molecules. Such an approach is shown to allow to separate quantitatively three major sources of errors arising in calculations of the overlap integrals.

Theoretical and experimental studies of a new predissociation in theA2Δ state of CCl

The A 2Δ-X 2Π transition of CCl has been studied with the High Frequency Deflection (HFD) technique. Lifetimes were measured for the ν′ = 0, 1 and 2 vibrational levels of the A 2Δ state. A drastic shortening of the lifetime going from ν′ = 0 to ν′ = 1 indicated that a predissociation takes place. In order to obtain more information about this predissociation, potential energy curves were calculated with the CASSCF method and the contracted CI method for the bound X 2Π and A 2Δ states and several repulsive states. It was found that the dominating cause of the predissociation is a 2Π state going to the ground state dissociation products. Electronic matrix elements were roughly estimated and the bound-continuum vibrational overlap integrals were calculated. The A 2Δ-2Π crossing point was adjusted until satisfactory agreement was obtained between experimental and theoretical predissociation rates for ν′ = 1.

A new expression for the vibrational overlap integrals between arbitrary one-dimensional harmonic oscillators

A new, compact formula for the vibrational overlap integrals between two harmonic potentials with both arbitrary curvatures and equilibrium positions is derived in a systematic manner with its several derivative formulas. Some mathematical properties of the overlap integrals obtained are discussed.

Collision induced intramolecular vibrational energy transfer in 1B2 aniline

The technique of SVL fluorescence spectroscopy was used to observe collision-induced intramolecular energy transfer in a large polyatomic molecule in the gas phase. Vibrational energy transfer was traced from eight vibronic levels in 1B2 aniline with argon as a collision partner. Overall rates for depletion of the initial level range from 0.1 to 0.5 of the equivalent hard sphere collision rate. The vibrational levels below 800 cm−1 are found to fall into groups: energy transfer is much more efficient between levels in the same group than between levels in different groups. This pattern of energy disposal indicates that specific dynamic characteristics of the vibrational modes are as important as the energy gap and vibrational overlap integrals. Another interesting feature of the results is the importance of endoergic processes, even when exoergic pathways for vibrational energy exchange are available. Rotational relaxation was also examined, but only with low resolution, and not in a systematic study. The rate of rotational equilibration found approximates the collision rate.

Relative rate calculations for intramolecular radiationless transitions

We present a formulation of relative rate calculations for intramolecular radiationless transitions which includes many of the recent theories in the literature as special cases. Our method, which is closely related to the previous work of Siebrand, allows us to treat simultaneously the effects of anharmonicity (κ), geometry shifts (Δ Q) and frequency changes (Δω) in all the vibrations of interest. We apply this approach to the S 1 → T 1 intersystem crossing in benzene and argue that the theory can be used to determine bounds for Δω S 1→T 1 's only if Δ Q's and κ's are accurately known. The use of “local” versus “normal” modes of vibration is discussed. We conclude, from our failure to explain the emission cutoff in S 1, that the representation of vibronic interaction energies as products of “electronic factors” and vibrational overlap integrals is valid only near the origin of the excited singlet.

The effect of molecular distortion in the rates of intersystem crossing S 1 T x S 0 processes

It is shown that the interpretation of the experimental observations of Dewey and Hadley that led to the conclusion of the unexpectedly inefficient intersystem crossing S 1 ▪ T x process in 9,10-phenanthroline, and which prompted a new proposal by Azumi for the mechanism of intersystem crossing in N-heterocyclics is incorrect. The intersystem crossing process in aromatic hydrocarbons and the N-heterocyclics is briefly discussed and compared. The results indicate that the out-of-plane molecular distortions enhance the rate of intersystem crossing, S 1 → T x , mainly through electronic mixing and not the vibrational overlap integrals. The lack of phosphorescence emission for 9,10-phenanthroline in hydrocarbon solvents is attributed to a fast non-radiative T 1 ▪ S 0 channel resulting from the pseudo-Jahn-Teller effect.

Condon moments for the configurational-coordinate model

The first three moments for transitions between offset harmonic oscillators in a single configurational coordinate are derived for the Condon approximation. The derivation uses simple recursion techniques based on the Manneback recursion formulas for the vibrational overlap integrals. The first two moments are compared with moments obtained previously by quantum-mechanical transform methods and semiclassical methods.

Radiative and predissociative lifetimes of the A 2Σ+ state of OH

Lifetime measurements have been made on the A 2Σ+ state of OH by the delayed coincidence technique in conjunction with an interrupted rf discharge through water vapor. Lifetimes are reported for rotational states up to K=29 and K=20 in the ν=0 and ν=1 vibrational levels, respectively, and on the unresolved levels of ν=2. A predissociation, previously predicted from relative intensity measurements, resulting from a curve crossing by the repulsive 4Σ− state, was observed for the rotational states above K=23 of ν=0, for those above K=14 of ν=1, and for all those of ν=2. These lifetime data are discussed qualitatively in terms of the electronic-rotational perturbation matrices and the vibrational overlap integrals.

Calculation of Davydov Splitting of the First Triplet State in Pyrene

The Davydov splitting of the first triplet state of crystalline pyrene has been calculated for three different molecular orbital π-electron wavefunctions. It is found that the contribution to the splitting from charge transfer states is an order of magnitude larger than that due to neutral exciton states. With an estimated value of 0.25± 0.05 for the square of the (0, 0) vibrational overlap integral, the calculated Davydov splitting of the zero vibrational component of the first triplet state is between 1 and 2 cm−1. The Bu state is found to be primarily b polarized. These results compare favorably with the measured splitting of 2.4± 0.6 cm−1 with the b polarized state at higher energy.

Moment analyses of intensity distributions in molecular electronic spectra

The use of moments in the analysis of vibronic intensity distributions in molecular electronic spectra is explored. Theoretical relations are derived for first moments. No attention need be paid to the form of individual vibrational overlap integrals. It is shown how these relations give information on the change of geometry accompanying the transition and on the dependence of the electronic transition moment on the appropriate normal coordinate. The theory is successfully applied to the 3500 AA emission spectrum of formaldehyde and to the 3400 AA spectrum of phenanthrene. An outline is given of the extension of the theory to higher moments.