The Quantization of the E ⊗ e Jahn–Teller Hamiltonian

Abstract

The E ⊗ e Jahn-Teller Hamiltonian in the Bargmann-Fock representation gives rise to a system of two coupled first-order differential equations in the complex field, which may be rewritten in the Birkhoff standard form. General leapfrog recurrence relations are derived, from which the quantized solutions of these equations can be obtained. The results are compared to the analogous quantization scheme for the Rabi Hamiltonian.

Jahn–Teller and Pseudo-Jahn–Teller Effects: From Particular Features to General Tools in Exploring Molecular and Solid State Properties

“…This section is to introduce the JTE and PJTE (jointly, vibronic effects) in a (somewhat pedagogical) manner (using simple cases as examples), which allow us to demonstrate the main origins of these effects and their most important features that contribute to the observable phenomena in mater, emphasizing the latest achievements. In so doing, as mentioned above, we avoid bulk mathematical deduction for the more complicated cases, which can be find in the cited publications, ,,− ,− …”

“…There is a series of other JTE and PJTE problems (some of them considered below in this review), with many applications to studies of molecular and solid state phenomena, reflected in original publications, reviews, and books cited in the introduction, notably refs , , , and − for books, refs , , , and − for major reviews, and refs − for more recent developments of the vibronic coupling theory in relation to the JTE and PJTE. In addition to the noted above E⊗e problem of systems with 2-fold degenerate states, widespread are also problems T⊗ e , T⊗ t 2 , T⊗( e+t 2 ) of cubic systems with 3-fold degenerate T terms interacting with e and t 2 displacements, and higher level of degeneracy in icosahedral systems, as well as combined and multimode JTE problems <...…”

“…The theoretical background of all these effects in atomic systems emerges from the first-principles as a delicate addition to the coupling between the electronic states and nuclear displacements, which occurs beyond the adiabatic approximation (vibronic coupling). The theory of the JTE and PJTE with many of their manifestations in molecular and solid state properties are rather well studied and continues developing. − Less attention has been paid so far to the PJTE, especially to its use as a tool in the study of the structure and properties of polyatomic syst...…”

“…As in any theoretical work, mathematical proofs of the origin of the JTE and PJTE structural instabilities based on first-principles are most important, but in detail they are of primary interest to more specialized readers and deserve separate consideration. For the more significant achievements in this field, see refs , , , and − for books; refs , , , and − for major reviews; refs − for JTE conference proceedings; and refs − …”

Ansatz for the Jahn-Teller triplet instability

“…Application of this transformation greatly enriches our understanding of Jahn-Teller Hamiltonians. It was applied successfully to the single mode Rabi model, and the dual mode E × e Jahn-Teller model [1,5,19,24]. It can be applied as well to the instabilities of the quartet spinor Γ 8 .…”