Quadratic Jahn-Teller Coupling Research Articles

Jahn-Teller effect and large-amplitude motion in [formula omitted] studied by high-resolution photoelectron spectroscopy of CH4

This article presents the current status of the knowledge of the rovibronic energy-level structure of CH4+ obtained by high-resolution photoelectron spectroscopy. The results of previous investigations are summarized and extended by new results obtained by double-resonance experiments involving vacuum-ultraviolet and mid-infrared laser radiation. These experiments have led to assignments of the nuclear-spin symmetry of 303 rovibrational levels of CH4+ with up to 3575cm−1 of internal excitation. A two-dimensional model of the pseudorotational motion of CH4+ is also presented, with which the positions and vibronic symmetry of the low-lying vibronic levels of CH4+ can be predicted. The model maps the F⊗(f⊕e) Jahn-Teller problem for the e and f2 modes corresponding to the CH bending motion onto a sphere, including the effects of both linear and quadratic Jahn-Teller coupling terms. The pseudorotation eigenstates are obtained by solving the two-dimensional Schrödinger equation using a basis of spherical harmonics.

Cation States of Ethane: HEAT Calculations and Vibronic Simulations of the Photoelectron Spectrum of Ethane.

High-accuracy ab initio calculations have been carried out on ethane and its radical cation. With the HEAT-345(Q) scheme, adiabatic ionization potentials of 11.52 and 11.57 eV are determined for the X̃ (2)Eg and à (2)A1g states, respectively, with an uncertainty of ±0.015 eV. Also considered in this report are linear and quadratic vibronic coupling involving both states. With this simple vibronic model, the photoelectron spectrum of ethane was simulated in the 11-15 eV region using linear and full quadratic Jahn-Teller coupling Hamiltonians, and with up to 70 billion direct product basis functions in a high-performance computing environment. Although the linear vibronic coupling model adequately reproduces the spectral envelope, the quadratic vibronic treatment results in much better agreement with the observed spectrum.

Jahn–Teller effects and surface interactions in multiply-charged fullerene anions and the effect on scanning tunneling microscopy images

We investigate the combined effects of Jahn–Teller (JT) coupling and interactions with a surface substrate on fullerene anions C602- to C604-. JT coupling alone causes the C60 ions to instantaneously distort from the icosahedral symmetry of the neutral molecule to a lower symmetry, with the molecule moving dynamically between a set of equivalent distortions. When adsorbed on a surface, the number of equivalent minimum-energy distortions is reduced. The implications of this on observed scanning tunneling microscopy (STM) images will be discussed, and comparisons made with existing experimental data. We show that a consistent interpretation of the images from all of the charge states of C60 can only be obtained using a JT model in which the symmetry is further reduced by surface interactions. The comparison with experimental data also allows us to determine relationships between the quadratic Jahn–Teller coupling and surface interaction parameters.

Tunneling splittings in vibronic energy levels of [formula omitted] studied by high resolution photoelectron spectroscopy and ab initio calculation

The energy levels of CD3F+(X∼2E) have been measured up to 1400cm−1 above the ground vibrational state using the one-photon zero-kinetic energy photoelectron (ZEKE) spectroscopic method. The spin-vibronic energy levels have also been calculated using an ab initio diabatic model. The potential energy surfaces of CD3F+ were calculated from those of CH3F+ using a transformation of the normal coordinates. The calculations show that tunneling splittings of vibrational energy levels exist due to the three equivalent wells caused by the linear-plus-strong quadratic Jahn–Teller coupling. The splittings are smaller than those in CH3F+. The experimental spectrum was assigned based on the fundamental vibrational modes calculated at the energy minimum. The calculated spin-vibronic energy levels are in good agreement with the experimental data. The tunneling splitting pairs for the fundamental vibrations related to the CD3 rock were observed. The first adiabatic ionization energy was determined as 101534±3cm−1 or 12.5886±0.0004eV.

Cooperative Jahn-Teller effect in a 2D mesoscopic C 60 n− system with D5d symmetry adsorbed on buffer layers using Ising EFT model

Fullerene molecules adsorbed on surfaces often show macroscopic average distortions. As charged ions C 60 are known to be Jahn-Teller (JT) active, it is suggested that these distortions could be a manifestation of cooperative JT effects (CJTE) due to interactions between neighbouring fullerene ions. In order to understand the distortion properties it is necessary to take correlations between different distortions into account. However, this can’t easily be done in the mean field approximation usually used to describe the CJTE. We therefore propose an alternative procedure to describe 2D mesoscopic islands of C60 ions in which a pseudo vector spin $$\vec S$$ is evoked to represent degenerate JT-distorted states when the quadratic JT coupling is considered. This approach is analogous to methods used for 2D magnetic systems. We then use the differential operator technique in effective field theory within the Ising approach. We include the effects of weak surface interactions and dynamic motion between equivalent distortions via terms equivalent to anisotropy and a transverse field in magnetism respectively. For distortions to D 5d symmetry, we determine single site correlations as a function of temperature, the macroscopic average distortion describing a structural phase transition, and the isothermal response function. Phase diagrams are presented for relevant cases of the system parameters.

The 3E × E, 4E × E and 5E × E Jahn–Teller Hamiltonians of trigonal systems

The vibronic Hamiltonians describing linear and quadratic Jahn–Teller coupling as well as spin–orbit coupling in M E states of trigonal systems are derived for the spin multiplicities M = 3 , 4 , 5 , starting from the microscopic Breit–Pauli spin–orbit operator. The 2 M × 2 M Hamiltonian matrices in a diabatic spin-electronic basis are obtained by the expansion of the Hamiltonian in powers of the Jahn–Teller active normal mode. It is shown that the M E × E Jahn–Teller Hamiltonians can be made block-diagonal by the transformation to an alternative diabatic basis which mixes electronic orbital and spin projections. The adiabatic potential-energy surfaces and the adiabatic electronic wave functions are obtained in analytical form. In systems with an odd number of electrons (and thus even spin multiplicity) the Jahn–Teller effect is quenched by strong spin–orbit coupling and the adiabatic potential-energy surfaces are strictly two-fold degenerate (Kramer’s degeneracy). In systems with an even number of electrons (and thus odd spin multiplicity), two of the 2 M adiabatic potentials exhibit an E × E Jahn–Teller effect which is unaffected by the spin–orbit interaction. The geometric phases of the adiabatic electronic wave functions are those of the 2E × E Jahn–Teller effect with and without spin–orbit coupling, respectively.

Calculation of the Jahn-Teller effect in benzene cation: Application to spectral analysis

Ab initio calculations have been performed for the cations of benzene, C6H6, and its fluorinated analogs, C6F6 and C6H3F3. Calculated molecular parameters characterizing the Jahn-Teller potential energy surface (PES) are very consistent with those derived from the spectra of C6F6+ and C6H3F3+. However the calculated Jahn-Teller stabilization energy for the benzene cation is roughly three times greater than that previously reported experimentally. With the aid of the calculated values, a more complete analysis of the available spectral data for C6H6+ and C6D6+ is performed, with an emphasis on the data from ZEKE experiments and IR spectra of the Ar⋅C6H6+, Ne⋅C6H6+, and Ar⋅C6D6+ complexes. The comprehensive analysis reveals Jahn-Teller activity in 3 e2g modes for C6(H/D)6+ and provides values for their vibrational frequencies, linear and quadratic Jahn-Teller coupling constants, as well as quadratic coupling constants for several other degenerate modes. These new molecular parameters are generally in good agreement with the corresponding values derived from the ab initio calculations and produce a total Jahn-Teller stabilization energy in good agreement with the computed value.

An analysis of the T⊗(e+e) two-mode Jahn-Teller system with strong coupling

Details of an approach for strongly coupled orbital triplet Jahn-Teller (JT) systems using a unitary transformation and energy minimization procedure are applied to the two-mode T⊗ (e+e) JT system in the case when both linear and quadratic coupling are included. The inclusion of quadratic JT coupling leads to a partial lift of degeneracy of the triplet ground states and the excited ones. The effect of static strains in strongly coupled T⊗ (e+e) JT system is also investigated analytically. We have shown that tetragonal trigonal and orthorhombic strains modify the structure of the system.

The Effect of an E-Type Quadratic Jahn-Teller Coupling on the Eigenstates in theT×t2Vibronic States

Numerical studies of the vibronic eigenstates in the case when an orbital triplet states is coupled linearly and quadratically to a t 2 vibrational mode have been made. For a fixed value of the linear Jahn-Teller coupling strength in the intermediate coupling region, the effects of an E -type quadratic Jahn-Teller coupling strength on the vibronic eigenstates of the Jahn-Teller system have been investigated. As the quadratic Jahn-Teller coupling strength increases, the feature of an orthorhombic Jahn-Teller distortion has been clearly shown in the vibronic properties in the sixfold degenerate ground state.

Vibronic reduction factors of the excited states in the E

Extensive numerical studies of the vibronic eigenstates in the case when an orbital doublet state is coupled linearly and nonlinearly to an e vibrational mode with cubic symmetry have been made. For a fixed value of the linear Jahn-Teller coupling in the strong-coupling region, the effects of the quadratic Jahn-Teller coupling on the vibronic properties, such as Ham's reduction factors or the reduction factors for vibrational angular-momentum operators, have been studied. At a critical value of the quadratic Jahn-Teller coupling coefficient, where the behavior of near-level crossing in the excited E states has been seen, the abrupt change of Ham's reduction factor and the fluctuations in both the admixture coefficient and the reduction factor for the vibrational angular-momentum operator have been found in the excited E state. Similar effects of the lowest-order anharmonic term on the vibronic reduction factors have been found.

Ham factors for a T×(τ2+ε) Jahn-Teller system at strong coupling

When an electronic triplet state interacts through a strong linear Jahn-Teller coupling with the five normal modes transforming as the tau 2 and epsilon representations of the cubic group, the potential energy minimum can be mapped onto a sphere in phase space. The additional effects of anharmonic coupling, quadratic Jahn-Teller coupling and unequal coupling to tau 2 and epsilon modes can all be represented by adding a potential energy that is a linear combination of the fourth- and sixth-order cubic invariant harmonic functions. Their effect can be pictured as a contour map on the spherical surface. The pattern of low-lying energy levels and the associated Ham factors have been worked out in this strong-coupling approximation. Some choices of the parameters give rise to potential surfaces with different dynamic properties from those previously studied.

Ham factors and energy levels in multimode Jahn-Teller systems

A method of calculating the energy levels and Ham factors of a Jahn-Teller system interacting with many modes of vibration is described. The multimode effect in the ground states is dealt with by first doing a transformation to concentrate most of the Jahn-Teller interaction into an effective cluster Hamiltonian, and then allowing for the remainder by introducing one extra phonon mode at a time. Calculations are done using a numerical perturbation theory: energies are calculated by the Lanczos method of diagonalising large sparse matrices, and the small changes in energy resulting from the introduction of small interactions are used to deduce perturbation parameters. This method is applied to E*( Sigma i epsilon i), with the inclusion of a quadratic Jahn-Teller coupling or warping term. Experimental results on Ni3+:Al2O3 are discussed as an example.

Effects of the quadratic Jahn-Teller coupling on the eigenstates of vibronic systems

Numerical investigations are presented for the vibronic coupling of a degenerate E electronic state to a degenerate eg vibrational mode in the case when both the linear and quadratic Jahn-Teller couplings are included. To understand the effects of the quadratic Jahn-Teller coupling, the calculations of eigenvalues and Ham's reduction factors are made for various values of the linear and quadratic terms. The Renner effect on the system has been investigated and new classification of eigenstates is proposed. The same calculations have been extended for vibronic system of a degenerate T electronic state coupled to a degenerate eg vibrational mode.

Vibronic Fine Structure of the Line Shape for E →T Transition: The GRl Band in Diamond

An optical line shape for E → T 2 transition in tetrahedral symmetry has been calculated for the case where both the ground and the T 2 excited states are coupled to an e vibrational mode. The effect of quadratic Jahn-Teller coupling in the ground state on vibronic fine structure is investigated. The calculated result has been compared with the GRl absorption band in diamond. The origin of a 40 meV sideband observed experimentally has been explored within the framework of the Jahn-Teller interaction following the suggestion by Davies and Foy. It has been shown that its energy position and relative intensity can be explained, if the quadratic coupling of e - t in the T 2 excited state is taken into account.

The Jahn–Teller effect in C6 F+6

Recent results from the laser spectroscopy of C6 F+6 are collected to derive the positions of ∼15 Jahn–Teller active vibronic levels of the four e2g vibrational modes of the ground X 2E1g state. Intensity information is also collected for a number of transitions, both in absorption and emission, involving these modes. The data are analyzed by a model including linear and quadratic Jahn–Teller coupling, and coupling among all four of these active e2g modes. The positions of these levels are predicted by the eigenvalues, and the transition intensities by the eigenvectors, of matrices of order ∼7000×7000. Quantitative agreement is obtained for both frequencies and intensities, thereby determining unperturbed oscillator frequencies, distortion constants, and stabilization energies for all the possible Jahn–Teller active modes of C6 F+6. Combination of these results with a normal coordinate analysis gives the distorted geometry of the ion at the minimum of its potential function.

Analysis of Satellite Lines of the GR1 Band in Diamond by Quadratic Jahn-Teller Coupling

The quadratic Jahn-Teller effect has been studied in the intermediate linear coupling region in the E - e system. Reduction factors, E - A separation and the zero-phonon intensity for E → T transition have been calculated by diagonalizing the vibronic Hamiltonian including both the linear and the quadratic coupling. From comparison of these calculated results with experimental data on the GR1 band in diamond, values of vibronic parameters for the GR1 center have been extracted.

Jahn—Teller effects in substituted benzene cations. IV. Intermode interactions and intensity anomalies in sym-trifluorobenzene ions

A theoretical treatment of coupling between Jahn—Teller modes is given which includes both linear and quadratic Jahn—Teller coupling. Effects of two and three mode interactions on the B̃ 2A″ 2 → X̃ 2E″ transition of the sym-trifluorobenzene ions are studied, within the linear coupling approximation, for modes 6, 7, 8 and 9. Quadratic coupling effects are considered in a companion paper (part V). The results show that discrepancies previously noted between observed relative band intensities and those calculated on the basis of single Jahn—Teller modes cannot be accounted for by multimode interactions or by Fermi resonance effects operating in conjunction with Jahn—Teller effects. Jahn—Teller combination bands are shown to have several components, the most intense of which are assigned. Particular attention is paid to various possible assignments of the 8 0,0 1, 1 2 and 6 0,0 3, 1 2 vibronic transitions in the light of the multimode interaction analyses.

Quadratic Jahn-Teller coupling and off-centre displacements for Ni2+ ions in strontium oxide

EPR spectra for isolated Ni2+ ions in Sr1-xNixO display trigonal rather than octahedral symmetry, and reveal an anomalously large splitting of (20 ± 6) cm-1 in the 3A2 ground state. These unusual features are explained using a model based on quadratic Jahn-Teller coupling and a soft, localised, off-centre, mode of vibration.

Quadratic Jahn-Teller coupling in octahedral systems

In this paper Jahn-Teller vibronic coupling calculations are presented for the general case [Γ8g × eg ] in Oh * symmetry. The hamiltonian employed is complete up to terms quadratic in vibrational coordinates qθ and qε which span the ν2(eg ) mode of MF6 systems. Calculational details are discussed. It is concluded that, in agreement with previously reported spectroscopic data for ReF6 and IrF6, quadratic terms in the vibronic interaction are essential (both qualitatively and quantitatively) to the complete understanding of intra-state vibronic coupling in transition metal hexafluorides.

Operator Form for the Linear and Quadratic Jahn—Teller Coupling in Octahedrally Coordinated Eg Electronic States

The operator form for the linear and quadratic Jahn—Teller effect in octahedrally coordinated Cu+ + is derived by consideration of the simultaneous effect of three tetragonal distortions. This leads to the same result as found by Liehr and Ballhausen, but in fewer steps. The final form of the operator is independent of the choice of phase of symmetrized distortion coordinates.