Y M Liu

A scale transformation in Jahn - Teller systems

The Jahn - Teller (JT) system has been studied previously by many authors. It is well known that the potential energy surface for this system contains four equivalent wells in strong coupling. The wells are not isotropic. In the strong coupling limit, the vibrational t-mode splits into an -mode of frequency and an e-mode of frequency . However, it is difficult to incorporate this anisotropic effect into analytical models. Previously, the current authors have used a unitary shift transformation and energy minimization procedure to model many moderately to strongly coupled JT systems. However, the part of the Hamiltonian which produces the anisotropy was not treated fully. We now present a modification of this procedure for the system in which a scale transformation is applied in addition to the shift transformation. This is shown to introduce anisotropy automatically into the problem. We show that the correct frequencies are obtained in the infinite coupling limit. Symmetry-adapted combinations of the states associated with the wells are taken to obtain expressions for the ground state and inversion level. The inversion splitting between them is compared with existing results. We then discuss how the scale transformation method can be applied to other JT systems (for which the limiting frequencies are unknown), such as those in the symmetry which applies to the molecule.

Anisotropy and the inversion splitting in the Jahn - Teller system

We present an analysis of the Jahn - Teller (JT) system in which both possible h-type quadratic terms are considered. It is well known that this results in pentagonal or trigonal minima on the potential energy surface, depending on the magnitudes of the coupling constants. Although the positions of the minima with quadratic couplings are known, the anisotropic effects which occur due to the lifting of the degeneracy of the h-vibrations by the quadratic coupling have not been studied before. Such effects have previously been found to be important in cubic systems. We investigate the nature of the minima by evaluating the curvature of the potential energy surface, and hence we determine the frequencies of the local vibrational modes in the strong-coupling limit as a function of the quadratic coupling strengths for the first time. We find that, in the linear coupling limit, the frequency of one of the e-modes at the minima tends to zero. This is as expected because, in this limit, the minimum-energy surface is a trough joining the and points. A scale transformation method is then used, which allows the anisotropy effect to be incorporated into the states associated with the wells. States having the required icosahedral symmetry of the system as a whole are then written down in terms of the anisotropic states. Specific results will be given for the dependence of the inversion splitting on the anisotropy. The new states are of significance because they are necessary for the calculation of further properties of these systems, such as reduction factors. The system is also a possible model for the ground state of the anion.

An analytical model for the Jahn - Teller system

The Jahn - Teller (JT) problem is investigated analytically using a unitary transformation method. Minimization of the adiabatic energy surface for this problem results in wells of either or symmetry, depending on the coupling strengths. The dynamic JT problem is then solved in the tunnelling regime using projection operators to find symmetrized combinations of the states associated with the wells. By analogy to other JT systems, the ground state would be expected to have the same degeneracy as the original orbital state, and thus to be an H-type quintet. However, it is found that there are a range of couplings strengths for the g and h modes for which the tunnelling ground state for the wells can be an A-type singlet. A similar result was recently found for the pure JT system. It is also found that for wells, the limiting value of the tunnelling splitting between the H and A states for a pure system tends to in weak coupling, whilst for a pure system it tends to . For systems coupled to both modes, the value of the tunnelling splitting strongly depends upon which of the two modes is dominant. Both the level ordering in strong coupling and the anomalous behaviour in weak coupling can be shown to be fundamental symmetry properties of these JT systems, and not consequences of the details of our model. The JT systems studied here are possible models for the ground state of the cation and for an excited state of the anion .

First-order Reduction Factors for the T1u ࣹ hg Jahn-Teller System*

It is now well-known that the electron-phonon interaction, and hence the Jahn-Teller (JT) effect, is important in understanding the properties of C60 and related molecules. The T,„®he JT system is of particular importance as this will be the JT effect displayed by C«, molecules doped with an electron. In this paper, first-order JT reduction factors are derived for the T,u®hs Jahn-Teller system using symmetry-adapated vibronic ground and inversion states pertaining to either pentagonal (Z>M) or trigonal (Dyj) wells in the potential energy surface. Analytical expressions are obtained for all possible first-order reduction factors for T1g and Hs operators both between and within the pentagonal and trigonal wells. The results are formulated in terms of reduced matrix elements, so that the Clebsch-Gordan coefficients for I,, symmetry can be used throughout the calculations.

Investigation of an Orthorhombic Cr3+-like Centre in GaP*

Details are presented describing further experiments and theoretical studies which have been carried out on a Cr3+-like centre in GaP. This centre is responsible for one group of lines selected from the complex thermally-detected electron paramagnetic resonance (TDEPR) spectra seen at X band, from samples of GaP doped with chromium. It was identified previously as a Cr1+ ion and its isofrequency diagrams were in good agreement with those predicted from a T<8>e dynamic Jahn-Teller model for low values of the magnetic field (£0.3 in the range of frequencies investigated). However, problems arose at high-

Further studies of a -like centre in GaP

A centre in GaP having a -like behaviour has been studied both experimentally and theoretically. It was identified originally by thermally detected (TD) EPR experiments carried out at liquid helium temperatures and correlated with the optical absorption 1.03 eV zero phonon line from the isolated substitutional ion. Subsequent TD-EPR experiments on different types of samples and sub-band-gap illumination effects show that, although the centre is -like, it can not be the isolated ion. A theoretical model is proposed in which this -like ion is described by a strain-stabilized Jahn-Teller (JT) model associated with wells that have orthorhombic symmetry in the potential energy surface. On fitting the isofrequency curves to all the relevant TD-EPR data, values for the second-order JT reduction factors are obtained which indicate that this -like centre is strongly coupled to its surroundings. The electronic structure of this centre has still to be established.

INVERSION SPLITTINGS AND REDUCTION FACTORS FOR THE H (H G) JAHN-TELLER SYSTEM

case of D}d minima by introducing tunneling between different minima. Projection-operator techniques are used to find the required symmetrized tunneling states, allowing the corresponding inversion splitting and reduction factors to be calculated. The highest occupied molecular orbital (HOMO) in the fullerene C«, is of H symmetry, so that the work presented here represents a possible model for the ground state of the cation C¿,.

The effects of anisotropy in Jahn-Teller systems: application to the icosahedral H ⊗ (h ○+ g) system

Jahn–Teller (JT) coupling between electronic motion and lattice or molecular vibrations results in an adiabatic potential energy surface that contains either wells or troughs of minimum-energy points. When wells are lowest in energy, the system will vibrate about the minimum-energy points. This vibration must be taken into account when describing the quantum mechanical states of the system. In general, the wells will be intrinsically anisotropic. This anisotropy alters the vibrational frequencies and hence the positions of the energy levels, and can be particularly significant when the barriers between wells are shallow. In this paper, we will show how anisotropic states and their energies can be calculated using two unitary transformations. The first locates minima on the adiabatic potential energy surface, and the second accounts for anisotropy in the shape of the minima. The method is developed in a way general enough to allow it to be applied to any linear JT problem. The theory is then applied to the icosahedral H ⊗ (h ⊕ g) JT system. The results obtained will help the understanding of, for example, the effects of vibronic coupling in positively charged fullerene ions.

Jahn-Teller effects in coexisting tetragonal and trigonal systems

The strongly coupled Jahn-Teller (JT) system is studied in which an ion in an orbital T1 triplet state is coupled to both e and t2 modes of vibrations of its neighbours. Such a system is usually considered to be either a T(X)(e+t2)JT system, in which orthorhombic minima in the five-dimensional Q-space are lowest in energy, or a T(X)d system, which has a trough of lowest energy. However, it is possible also for the tetragonal and trigonal minima, usually associated with the T(X)e and T(X)tJT effects respectively, to coexist with very similar energies to each other (and be of overall lowest energy) when the bilinear term of the vibronic interaction is present. This situation is described in this paper. A set of vibronic ground states is obtained by mixing the symmetry-adapted vibronic T1 ground states of the T1(X)e and T1(X)t2JT systems. This is different to the set of states associated with the orthorhombic minima. Analytical expressions for the first- and second-order JT reduction factors are also derived for the coexisting system. As a consequence of this analysis, an improved version of the theory of second-order reduction factors is obtained. The reduction factors are compared to those of existing numerical calculations for the T1(X)dJT system and it is shown that very good agreement is obtained between the two in the strong-coupling limit.

A general formalism for the calculation of second-order Jahn-Teller reduction factors

A general theory for the calculation of second-order Jahn-Teller reduction factors for strongly coupled vibronic systems has already been developed. It was based on symmetry arguments and gave results applicable to orbital triplet systems of all symmetries. This paper describes further developments and improvements that have been made in the general theory. As before, symmetry arguments dominate the analysis, which has two distinctive features. Firstly, by using a more fundamental definition of the orbital operators required than that used previously, it is shown how the problems encountered previously in attempting to apply the previous formalism to orbital doublet E(X)e systems are avoided. Secondly, the derivation of general formulae from which symmetry-adapted phonon states may be derived is presented. P is shown that their use in preference to the symmetry-adapted vibronic states used before simplifies the calculation of the oscillator overlaps required. Also, excited-stare energies may be obtained directly as they can be expressed as the sums of various reduced matrix elements among the excited phonon states. As an example, the general method is presented in detail for the strongly coupled E(X)e Jahn-Teller system.