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International Journal of Quantum Chemistry

A canonical sequence approach to the E ⊗ ε Jahn–Teller effect

Vassilios Fessatidis, Jay D. Mancini, Samuel P. Bowen, William J. Massano

Published Jan 1, 2005• 3 citations
Publisher

Abstract

Abstract We study the linear E ⊗ ε Jahn–Teller effect in which a doubly degenerate fermionic (electron) level is coupled to a pair of degenerate oscillators. Because of the relative simplicity of the model, and since a complete numerical solution has been known for a long time, theorists have had at their disposal an excellent proving ground for a variety of calculational efforts. Here we apply the recently developed canonical sequence method (CSM) of Šamaj et al. J Phys 1997, A30, 1741 to investigate the ground state of this system. Comparisons are then made with other calculation schemes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

DOI
10.1002/qua.20538

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Abstract We study the linear E ⊗ ε Jahn–Teller effect in which a doubly degenerate fermionic (electron) level is coupled to a pair of degenerate oscillators. Because of the relative simplicity of the model, and since a complete numerical solution has been known for a long time, theorists have had at their disposal an excellent proving ground for a variety of calculational efforts. Here we apply the recently developed canonical sequence method (CSM) of Šamaj et al. J Phys 1997, A30, 1741 to investigate the ground state of this system. Comparisons are then made with other calculation schemes. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005
A variational study of the linear E-e Jahn-Teller effect, in which an electronic doublet interacts with a doubly degenerate vibration, has been performed using a simple correlated trial ground-state wave function--a correlated squeezed state. Our calculated variational ground-state energies are not only in good agreement with the exact diagonalization results, but they are also better than those from earlier analytical studies. This seems to suggest that the linear E-e Jahn-Teller system favors a strongly correlated squeezed ground state.
We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system models the coupling of a two-level electronic system, or qubit, to a single-oscillator mode, while the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ system, the ground states of the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ model always exhibit entanglement. For the $E\ensuremath{\bigotimes}\ensuremath{\beta}$ case we aim to clarify results from previous work, alluding to a link between the ground-state entanglement characteristics and a bifurcation of a fixed point in the classical analog. In the $E\ensuremath{\bigotimes}\ensuremath{\epsilon}$ case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.
Within the Trotter-Suzuki approximations on ${e}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}H}$, the extended projector method [Phys. Rev. A 36, 4612 (1987)] previously introduced by us is shown to be a valuable technique in investigating the general class of few-level systems coupled both linearly and quadratically to a selected number of harmonic oscillators. As an illustration we compare eigenvalues and Ham-reduction factors with the literature for the cubic linear Jahn-Teller systems T\ensuremath{\bigotimes}\ensuremath{\epsilon} and T\ensuremath{\bigotimes}${\ensuremath{\tau}}_{2}$ as well as for the system E\ensuremath{\bigotimes}\ensuremath{\epsilon} for which a quadratic warping interaction is included. For the latter system a symmetry projector is additionally developed, dealing with the transformation properties of nonlinear combinations of representational basis functions. As another illustration, the low-energy eigenstates of a two-level system quadratically coupled to a single harmonic oscillator are investigated. Within the ground state, the system becomes unstable at high coupling strengths and through the evaluation of time- and frequency-dependent correlation functions we found substantially different ground-state dynamic properties compared with those of a linearly coupled system.
The octahedral Jahn-Teller system E ⊗ is studied using generalised Glauber states which are strong-coupling eigenstates of the system. It is shown that these Glauber states become, under certain conditions, equivalent to the weak-coupling eigenstates obtainable from standard perturbation theory. Eigenstates are thus obtained for the system which can be applied using perturbative procedures over the complete range of coupling strengths. Expressions which are functions of the coupling strength are obtained for the energies, the ground state Ham factor p and the A to E absorption intensities. The results compare well with the available numerical calculations.
In this work we have applied the optimal coupled-cluster approximation to study the ground state of the E⊗(b1+b2) Jahn–Teller effect. The effectiveness of the optimal coupled-cluster approximation has been investigated for the whole range of the asymmetry parameter and various coupling strengths. It is shown that our results up to the third level of approximation are in very good agreement with the exact numerical diagonalization results and are better than those from earlier variational treatments. Furthermore, unlike previous variational treatments, the optimal coupled-cluster approximation has the advantage that the accuracy of both the ground state energy and wavefunction estimates is being taken care of and can be systematically improved. Since the mathematical treatment in this work is simple, the optimal coupled-cluster approximation could be easily extended to the studies of other fermion-boson interacting systems, e.g. the extended Jahn–Teller system.
Jahn-Teller systems exhibit strong coupling between the electronic and nuclear degrees of freedom. We analyse how linear and quadratic coupling affect the electron-nuclear entanglement in the E⊗ϵE\otimes \epsilonE⊗ϵ system. A measure of quantum entanglement, recently developed by Shimony [Ann. New York Acad. Sci. 755, 675 (1995)], is used. States in the adiabatic regime with quadratic coupling are analysed. The electron-nuclear entanglement in vibronic states of the linear Exe model is calculated numerically. The physical significance of electron-nuclear entanglement is discussed in terms of reduction factors and magnetic vibrational circular dichroism.