Vibronic Energy Levels Research Articles

The vibronic properties of the T×t2Jahn-Teller system in the strong-coupling region

Numerical studies of the vibronic eigenstates in the case when an orbital triplet state is coupled linearly to a t2 vibrational mode in cubic symmetry have been made. The vibronic energy levels and eigenvectors, as a function of pure linear coupling in the strong-coupling region, have been computed again. A detailed analysis of the vibrational contributions in the vibronic eigenvectors has been made. The possibility of, and conditions under which, the symmetry is lowered from cubic to trigonal have been investigated from the properties of the vibronic eigenvectors. The effect of a local strain on a trigonal Jahn-Teller distortion is discussed from the vibronic model.

OsF6energy levels and vibronic coupling in the(dt2g)2configuration

Electronic and vibronic energy levels of the (dt 2g )2 configuration of OsF6 are studied using electronic Raman scattering and absorption spectroscopy. The ground state has been identified as Eg (crystal field split by ∼28 cm−1) and the first excited state is T1g with crystal energy levels at 14, 28, and 40 cm−1. The three other regions of the spectrum that have been observed are at 4000 cm−1, 8000 cm-1, and 17 000 cm−1 covering the remaining (dt 2g) 2 configuration leveIs. The 4000 cm−1 band consists of A1g and T1g electron states which do not interact strongly through vibronic coupling. The T1g electronic state evidences a strong (D∼2·2) linear Jahn-Teller effect for ν5 and an observable Jahn-Teller effect for ν2. The band at 8000 cm−1 consists of Eg and T2g electronic states; it is so complex and strongly coupled that little information can be extracted from the absorption data concerning the Jahn-Teller interaction in this 5-fold degenerate manifold. The feature at 17 000 cm−1 is an A1g electronic sta...

Jahn—Teller effects in substituted benzene cations. V. Quadratic coupling

A calculation model is derived for taking into account quadratic, in addition to linear, coupling Jahn—Teller effects in determining vibronic energy levels and transitions. Procedures are developed for analysis of Jahn—Teller electronic spectra on this basis and the new features, with respect to the linear coupling approximation, brought about by introduction of quadratic coupling, are discussed. Vibronic analyses of the B̃ 2 A″ 2-X̃ 2E″ transitions of 1,3,5-C 6F 3H + 3, 1,3,5-C 6F 3D + 3 and 1,3,5-C 6Cl 3H + 3 are carried out, in particular for bands involving excitation of the mode 6 vibration. Experimental evidence for quadratic Jahn—Teller effects is obtained for the sym-trifluorobenzene ions and the linear coupling parameters D 6, ω 6 and quadratic coupling parameter q 6 are derived. Two possible orders of magnitude of the quadratic coupling strength are found to be compatible with the spectra of 1,3,5-C 6Cl 3H + 3. The analyses are consistent between the three ions and are not in contradiction with the general findings based on the linear approximation alone.

Orbital angular momentum in triatomic molecules

The electronic orbital angular momentum effects in the A 1II u state of C3 have been considered. A very good fit to the observed vibronic energy levels and rotational constants has been obtained using a full matrix treatment of the orbital angular momentum in conjunction with a large amplitude formalism for the bending motion. The importance of the variation of the expectation value with bond angle is emphasized; the quality of the final fit is very much improved when this is included. Franck-Condon overlap integrals have been calculated for bands of the A 1II u -[Xtilde] 1Σ g + transition, which is the carrier of the 4050 A group of bands prominent in cometary spectra. A comparison has been made between the vibronic energy levels of linear and quasi-linear molecules in orbitally degenerate electronic states.

Orbital angular momentum in triatomic molecules

A general method is described for calculating the vibronic energy levels associated with two Born-Oppenheimer states of a triatomic molecule that become degenerate in the linear limit. The method combines Renner's matrix treatment of orbital angular momentum in linear molecules with Hougen, Bunker and Johns' formalism for large amplitude bending motions, and allows for the effects of electron spins; it applies equally well to linear and bent molecules. The effects of the non-integral nature of the matrix elements of Lz in bent molecules are included, and the calculation of relative band intensities in optical and photoelectron spectra is described.

A Calculation of the Bandshape for the Transition to the State with Jahn-Teller and Spin-Orbit Couplings

The optical absorption bandshape for the transition from an s-like ground state to a vibronically coupled p-like set of excited states with spin-orbit coupling is calculated from a numerical approach. The vibronic energy levels and eigen-states of the p-like state are obtained by a numerical diagonalization of the Jalm-Teller plus spin-orbit operators. To explain the experimental band shape in cesium fluoride, it is essential to consider the effect of a T 2g vibrational mode rather than an E g vibrational mode. The strength of Jahn-Teller coupling and the spin-orbit coupling parameter for the F-center in cesium fluoride are estimated.

Vibronic model for mixed valence compounds and spin crossover systems

The vibronic coupling model for mixed valence systems recently proposed by Piepho, Krausz and Schatz [1] is described. Using the model, it is a simple process to obtain the vibronic energy levels and eigenfunctions relevant to an intervalence band. Calculations are essentially exact within the context of the model, and are not limited to the cases of very strong or very weak interactions. The eigenvalues and eigenvectors obtained may be used to determine properties associated with the mixed valence band. These include, for example, the absorption profile of the band, the degrees of delocalization and valence-trapping, and the electron transfer kinetics for the system. Very similar formalism applies to both mixed valence and spin crossover systems. Thus the applicability of the model to single-center systems with two different spin states with intersecting potential surfaces is discussed. Finally, the limitations of the mixed valence model are discussed, and methods of improving the model by more explicit consideration of the nature of the interacting centers are considered.

Studies of the Ni3+:Al2O3system by thermal conductivity measurements

A detailed investigation of the phonon-spin interaction in the system Ni3+:Al2O3 has been carried out by low-temperature thermal conductivity measurements on single crystals (0.05 to 100K). Using the theoretical model of Abou-Ghantous et al. (1974) to describe the vibronic energy levels, a complete quantitative analysis of the data has been achieved with two resonant phonon scatterings at 0.47 cm-1 and 55 cm-1, taking into account the residual strains which play a major role in the low-frequency resonance. From the measurements with applied uniaxial stress the ion-lattice coupling constant has been determined as VEb=60000+or-9000 cm-1. It has been also possible to estimate the magnitude of the E-type random strains ( approximately=10-5) and the concentration of the Ni3+ ions in the crystals (0.2 to 5 ppm weight).

Rovibronic symmetry correlation in molecular combination and dissociation. II. The O 2 + O 2 ⇌ O 4 (D 2h) system

Rovibronic, vibronic and electronic correlation diagrams are constructed for the following reactions: X 3Σ g −(O 2) · X 3Σ g −(O 2) ⇌ {X 1Ag(O 4), a 3B 2u(O 4), b 5Ag(O 4)}; X 3· g −(O 2) · a 1Σ g +(O 2 ⇌ {g 3B 1g(O 4), h 3B 3u(O 4)}; b 1Δ g(O 2) · b 1Δ g(O 2) ⇌ {A 1Ag(O 4), B 1B 1g(O 4), C 1B 3u(O 4), D 1Ag(O 4)}. The correlations are for an assumed rectangular D 2h dimer conformation with the B 2u vibration excited up to the ν = 3 state and with the rotations excited up to the j = 3 state. Approximate values of the vibronic energy levels, for both sides of the reactions, are presented. A peculiarity of the system due to the zero nuclear spin of the oxygen atom is pointed out. The possibility of level crossings due to orbit—rotation, spin—rotation and orbit—vibration interactions is discussed.

Vibronic Interaction in Molecules and Ions. I. The first order Jahn-Teller interaction in doubly degenerate electronic states of C3ν type molecules

Expressions relevant to optical and electron spectroscopy are presented for calculation of the vibronic energy levels and Franck-Condon factors due to excitation of a Jahn-Teller active e vibrational mode in an electronic E state of C3v type molecules. Both A → E and E → A electronic transitions are investigated with allowance of different vibrational energies in the initial and final states. Energy levels and Franck-Condon factors including consideration of hot band transitions are explicitly given for some selected cases. Application is made to the fourth electron band at 18 eV of the electron spectrum of SO3. The vibronic band structure is well reproduced when the Jahn-Teller coupling parameters D3 and D4 of the ν3 and ν4 active modes are 0.06 and 0.35, respectively. A theoretical estimation is also made of the Jahn-Teller stabilization energy of SO3+. Finally, the X → B Rydberg transition of CF3I is discussed and a value of 0.01 is suggested for the Jahn-Teller coupling parameter D6.

Anharmonic potentials and vibronic energy levels for closely coupled molecular electronic states

New anharmonic adiabatic potentials are derived to represent pairs of closely coupled electronic states subject to a pseudo Jahn—Teller or pseudo Renner—Teller effect. These potentials are combined with harmonic diabatic potentials to construct a model that yields exact vibronic energy levels for arbitrary coupling strength.

Vibronic relaxation in the S 1 state of rhodamine dye solutions

The relaxation of vibronic energy levels was investigated by two different techniques. (A) The transmission of pico-second pulses was measured as a function of the incident peak intensity and (B) the transmission of a second probe pulse was studied versus delay time. Population lifetimes of the excited levels between 0.5 ps and 1.0 ps were found for rhodamine 6G and rhodamine B in the solvents water, ethanol, ethylene glycol and glycerol. The lifetimes are independent of solvent viscosity.

Time-resolved spectroscopy with subpicosecond ultraviolet pulses

The relaxation of vibronic energy levels was investigated by two different techniques. (A) The transmission of pico-second pulses was measured as a function of the incident peak intensity and (B) the transmission of a second probe pulse was studied versus delay time. Population lifetimes of the excited levels between 0.5 ps and 1.0 ps were found for rhodamine 6G and rhodamine B in the solvents water, ethanol, ethylene glycol and glycerol. The lifetimes are independent of solvent viscosity.

Isotope effect on the hyperfine splittings of the deuteriobenzene anions: An example of the dynamic Jahn–Teller effect

The vibronic wavefunctions and energy levels of the benzene, benzene−1−d, benzene−2,6−d2, benzene−1,4−d2, and benzene−1,3,5−d3 anion radicals have been obtained and used to determine the effect of deuterium substitution on the degenerate energy levels and spin densities of the benzene anion. It is shown that a dynamic Jahn−Teller effect treatment is required for these systems to obtain a quantitative interpretation of the consequences of deuterium substitution. The ground vibronic states of benzene− and benzene−1,3,5−d3− are degenerate. In benzene−2,6−d2−, the symmetric state (A1 under C2v) is 22 cm−1 below the antisymmetric state (B1 under C2v). In benzene−1−d− and benzene−1,4−d2−, the antisymmetric states are lower by 9 and 39 cm−1, respectively. The benzene−2,6−d2− and benzene−1,4−d2− hyperfine coupling constants and their temperature dependence calculated using the vibronic wavefunctions and energy splittings are in very good agreement with the experimental values. For benzene−1−d, the experimental splitting is estimated to be 20 cm−1, about twice as large as the calculated value, and agreement between the experimental and calculated hyperfine constants is less satisfactory. The vibronic energy splitting are interpreted in terms of the effect of deuterium substitution on the vibrational potential energies corresponding to the symmetric and antisymmetric electronic states. Calculations of the expectation values of the unpaired electron energy over the neutral molecule and the two anion Born−Oppenheimer state vibrational functions show that changes in the average values of the out−of−plane bending coordinates upon deuteration lead to a splitting of the vibrational potential energies. To a first approximation, the differences among the anions are a function of the unpaired orbital bond orders at the deuterated positions in the two electronic states. If the substituted bond order is large and negative, deuterium substitution increases the vibrational potential energy, but if it is approximately zero, the potential energy is essentially unchanged.

Jahn-Teller effect in the cyclopentadienyl radical: an example of three mode coupling

The cyclopentadienyl radical C 5H 5· vibronic wavefunctions and energy levels are calculated and used to discuss the vibrational structure of the allowed 2A″ 2 ← 2E″ 1 electronic transition in C 5H 5 and allowed A←E transitions in similar systems in which Jahn-Teller coupling occurs through two or three vibrational modes. As has been pointed out by Alpert and Silbey, the vibrational pattern predicted for single mode coupling is markedly distorted. With larger coupling parameters than those used by Alpert and Silbey, in an E ← A transition (e.g., the benzene Rydberg states) progressions in the individual coupling vibrations cannot be distinguished. In an A ← E transition (e.g., C 5H 5·), the higher progression members lose intensity and combination levels in the coupling vibrations appear. In both cases, a complex pattern of 1-1 hot band splittings results. Comparison is made with the experimental C 5H 5· spectrum, and assignments are suggested for three of the observed A″ 2 state frequencies.

Vibronic spectra and energy levels of polyatomic molecules. A physical chemistry experiment

In the experiment described here, the student measures and assigns the vapor phase vibronic spectrum of an aromatic and an azaaromatic molecule.

Magnetic Anisotropy due to Dynamical Jahn-Teller Effect in dε1 and dε5 Ions

Effects of Jahn-Teller distortions on the E2 ground state of dε1 and dε5 ions with tetragonal symmetry are investigated. For C4 symmetry the electronic system in the E state interacts only with a nondegenerate B-vibration mode to remove its electronic degeneracy. The resulting vibronic energy levels are all doubly degenerate, the same as in the absence of Jahn-Teller effect. The obtained results are applied to interpreting the anisotropy of g factors of the low-spin ferrihaemoglobin compounds and it is concluded that a combined action of the dynamical Jahn-Teller effect and the rhombic crystal field may be important in producing the large anisotropy within the haem plane.

THE CORRELATION OF THE SYMMETRY CLASSIFICATIONS OF STATES OF NONRIGID MOLECULES

The rovibronic and vibronic energy levels of some nonrigid molecules occur in nearly degenerate sets, whose splitting is due to tunneling between symmetrically equivalent regions of vibrational phase space. When the splitting is negligible, it is convenient to classify the different sets of energy levels according to a symmetry group that is determined by any one of the symmetrically equivalent regions, whereas when the splitting is appreciable, the individual split levels can be classified further according to a larger group that is determined in a similar way by a region containing all these equivalent regions. The former group is a subgroup of the latter.A simple correlation rule is proved, by means of which the symmetry species (in the larger group) of the individual split levels can be obtained from the symmetry species (in the smaller group) of the nearly degenerate set of levels. Some examples of the application of this rule to rovibronic and vibronic species are given.

Vibronic Coupling. III. Adiabatic Potentials and Vibronic Functions

General vibronic equations are given for the coupling of two electronic states in terms of their adiabatic potentials. It is shown that in addition to the adiabatic potentials it is necessary to know U(Q), a unitary transformation that depends on the vibrational coordinates Q. It is suggested that U(Q) be obtained from the ``crude adiabatic'' model. The appropriate U(Q) for the Renner and Jahn—Teller effects are given. A more extensive discussion is given of the pseudo Jahn—Teller effect described in previous papers of this series. Adiabatic potentials are given and it is shown how vibronic energy levels may be related to these potentials.

Ligand Field Spectra of Some New Laser Chelates

The high-resolution emission spectra from microcrystals of three new europium benzoylacetonate chelates, EuB4HP, EuB4HM, and EuB4HD, are presented and discussed in terms of the ligand field symmetries. It is established that the ligand fields about the Eu3+ in these compounds must be of one of the point symmetries C1, C1h, C2, or C2v; however, the exact symmetry in each case cannot be determined from the evidence presented. The emission spectra of EuB4HP and EuB4HM are found to be identical, indicating that the ligand field magnitudes and symmetries are the same for these cases. The emission spectra of EuB4HD, on the other hand, is considerably different, indicating that the magnitude and/or the symmetry of the ligand field in this compound must be different from the other chelates. A vibronic energy level for EuB4HP, probably corresponding to an europium molecular vibrational mode, was determined to be 37 cm—1 above the 5D0 energy level, and 33 cm—1 above the 7F0 energy level. A general tendency is noted in these spectra for the lines terminating on the higher ligand energy levels to be shaded toward longer wavelengths. This may be attributed to the greater number of vibronic states which are closer and more thoroughly mixed with the higher-lying ligand field states.