Physical Review Letters

Berry’s geometrical phase and the sequence of states in the Jahn-Teller effect

Frank S. Ham

Published Feb 16, 1987• 192 citations

Abstract

A proof is given for the Jahn-Teller problem of an orbital doublet (E\ensuremath{\bigotimes}\ensuremath{\epsilon}) that the order of the lowest vibronic levels is fixed by the requirement that the vibrational part of the wave function change sign under 2\ensuremath{\pi} rotation in the vibrational coordinates. This sign change in turn is a consequence of the sign change in the electronic part of the wave function, a special case of Berry's geometrical phase. Experimental confirmation of this sign change and thus of Berry's phase is available from studies of Jahn-Teller defects in crystals that reveal the sequence of these lowest levels.

DOI: 10.1103/physrevlett.58.725

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Physical Review Letters

Berry’s geometrical phase and the sequence of states in the Jahn-Teller effect

Frank S. Ham

Published Feb 16, 1987• 192 citations

Publisher

Abstract

DOI: 10.1103/physrevlett.58.725

Related papers

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C. C. Chancey, Mary Claire O’Brien

Journal of Physics A Mathematical and GeneralAug 21, 198821 citations

The origin of molecular distortions: A proposed experimental test

R. Englman, Tamás Vértesi

Bersuker and co-workers [Vibronic Interactions in Molecules and Crystals (Springer-Verlag, Berlin, 1989); Chem. Rev. 101, 1067 (2001); The Jahn-Teller Effect (Cambridge University Press, Cambridge, England, 2005); J. Chem. Phys. 124, 044321 (2006)] showed that the source of distortions from high symmetry configurations in molecules is vibrational-electronic (vibronic). It is here suggested that if one induces transitions by two consecutive operations (vibrational and electronic) performed in a given and in the reverse order, then the vibronic scenario gives different probabilities for the two orders, while an alternative scenario gives the same probability. The entanglement measure of the vibronic description is noted, and the broader aspect of the suggestion for experimentally testing state entanglements is discussed.Bersuker and co-workers [Vibronic Interactions in Molecules and Crystals (Springer-Verlag, Berlin, 1989); Chem. Rev. 101, 1067 (2001); The Jahn-Teller Effect (Cambridge University Press, Cambridge, England, 2005); J. Chem. Phys. 124, 044321 (2006)] showed that the source of distortions from high symmetry configurations in molecules is vibrational-electronic (vibronic). It is here suggested that if one induces transitions by two consecutive operations (vibrational and electronic) performed in a given and in the reverse order, then the vibronic scenario gives different probabilities for the two orders, while an alternative scenario gives the same probability. The entanglement measure of the vibronic description is noted, and the broader aspect of the suggestion for experimentally testing state entanglements is discussed.

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Berry phase and entanglement in the icosahedral H⊗(g⊕2h) Jahn-Teller system with trigonal minima

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Physical Review BJan 25, 200515 citations

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arXiv:0804.4416Physical Review ASep 22, 200851 citations

R

space. We show that this topological quantity reverts to a geometric quantity ${e}^{i\ensuremath{\gamma}}$ if the geometric phase $\ensuremath{\gamma}=\ensuremath{\oint}\mathrm{Im}\ensuremath{\langle}{\mathrm{\ensuremath{\Phi}}}_{R}|{\mathbf{\ensuremath{\nabla}}}_{\ensuremath{\mu}}{\mathrm{\ensuremath{\Phi}}}_{R}\ensuremath{\rangle}\ifmmode\cdot\else\textperiodcentered\fi{}d{\mathbf{R}}_{\ensuremath{\mu}}$ is evaluated with the conditional electronic wave function ${\mathrm{\ensuremath{\Phi}}}_{R}(r)$ from the exact electron-nuclear factorization ${\mathrm{\ensuremath{\Phi}}}_{R}(r)\ensuremath{\chi}(R)$ instead of the adiabatic function ${\mathrm{\ensuremath{\Phi}}}_{R}^{\text{BO}}(r)$. A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that cannot be removed by a gauge transformation. The exact potential energy surface is calculated and found to contain a term depending on the Fubini-Study metric for the conditional electronic wave function.

{D}_{3d}

minima. It is shown that the Berry phase of every loop on the potential energy hypersurface can be expressed in terms of these 2D triangular cross sections. Using this method we show that the absence of Berry phase for the cycles containing five trigonal wells originates from two seams of conical intersections going through these cycles. The connection between this analysis and previous treatments is established and the apparent dilemma of interpretation is resolved.