Continuous vibronic symmetries in Jahn–Teller models

Abstract

Explorations of the consequences of the Jahn-Teller (JT) effect remain active in solid-state and chemical physics. In this topical review we revisit the class of JT models which exhibit continuous vibronic symmetries. A treatment of these systems is given in terms of their algebraic properties. In particular, the compact symmetric spaces corresponding to JT models carrying a vibronic Lie group action are identified, and their invariants used to reduce their adiabatic potential energy surfaces into orbit spaces of the corresponding Lie groups. Additionally, a general decomposition of the molecular motion into pseudorotational and radial components is given based on the behavior of the electronic adiabatic states under the corresponding motions. We also provide a simple proof that the electronic spectrum for the space of JT minimum-energy structures (trough) displays a universality predicted by the epikernel principle. This result is in turn used to prove the topological equivalence between bosonic (fermionic) JT troughs and real (quaternionic) projective spaces. The relevance of the class of systems studied here for the more common case of JT systems with only discrete point group symmetry, and for generic asymmetric molecular systems with conical intersections involving more than two states is likewise explored. Finally, we show that JT models with continuous symmetries present the simplest models of conical intersections among an arbitrary number of electronic state crossings, and outline how this information may be utilized to obtain additional insight into generic dynamics near conical intersections.

Vibronic Ground-State Degeneracies and the Berry Phase: A Continuous Symmetry Perspective

“…Vibronic ground-state degeneracy (VGSD) in JT models appears frequently when linear vibronic couplings dominate [10,11] (for a recent proposal of direct noninterferometric experimental verification of VGSD, see e.g., [15,16]), although there are exceptions [17][18][19][20]. More specifically, there exists a particular class of JT models for which VGSD is guaranteed to exist whenever the adiabatic approximation (Born-Oppenheimer [21] with inclusion of Berry phase effects [22,23]) is valid [10,11,24]. These are the JT systems containing continuous symmetries and all possible couplings between JT active modes and a single electronically degenerate multiplet (at the reference geometry for a description of the JT effect, from now on denoted by JT center ) [3,4,[24][25][26][27][28][29], the simplest and most famous example being the linear E ⊗ e model (we use the standard convention where the electronic irreducible representation (irrep) is given by a capital letter and the vibrational irrep is given by a lowercase) which displays an exotic SO(2) (circular) symmetry in its potential energy surface [3,11].…”

“…More specifically, there exists a particular class of JT models for which VGSD is guaranteed to exist whenever the adiabatic approximation (Born-Oppenheimer [21] with inclusion of Berry phase effects [22,23]) is valid [10,11,24]. These are the JT systems containing continuous symmetries and all possible couplings between JT active modes and a single electronically degenerate multiplet (at the reference geometry for a description of the JT effect, from now on denoted by JT center ) [3,4,[24][25][26][27][28][29], the simplest and most famous example being the linear E ⊗ e model (we use the standard convention where the electronic irreducible representation (irrep) is given by a capital letter and the vibrational irrep is given by a lowercase) which displays an exotic SO(2) (circular) symmetry in its potential energy surface [3,11]. The most complex spinless example is the SO(5)-invariant model of the icosahedral JT problem H ⊗ (g ⊕ 2h), which contains all possible JT active modes associated with the electronic H quintuplet [9,10,29,30].…”

“…The energy gap between the electronic ground-state and any other state in the considered multiplet is also generally large enough, and the adiabatic approximation holds. In this case, VGSD in JT models with MCSs arises from the following facts [10,11]: (i) Continuous symmetry implies the space of minima of the ground-state APES is a continuous trough [24,28,35,36]. (ii) If the choice is made that the electronic ground-state wavefunction is real for any nuclear geometry, then it can only change by ±1 when transported over a loop on the trough.…”

“…where F is the reduced vibronic coupling constant (from the Wigner-Eckart theorem), and {V k } are Clebsch-Gordan matrices depending on the choice of electronic diabatic basis vectors |ψ k , k ∈ {1, 2, ..., N } forming a representation of the corresponding continuous group [28,29]. A fundamental property of the JT models with MCSs is that they have a continuous set [24,28,29,36]. In each case the electronic spectrum for any molecular geometry in the trough O is given by [3,4,10,24,57]…”

“…A fundamental property of the JT models with MCSs is that they have a continuous set [24,28,29,36]. In each case the electronic spectrum for any molecular geometry in the trough O is given by [3,4,10,24,57]…”