Linear vibronic coupling

With p = (Qg + Ql f being the Jahn-Teller radius, FE the linear vibronic coupling constant, GE the quadratic vibronic coupling constant, KE the force constant for the Eg normal mode of vibration, and Qo, Qe the two degenerate vibrations of eg symmetry. 

Fig. 1. Conical intersection surface topologies (top), and Renner-Teller surface topologies (bottom). Top left is a generic circular cone, such as is obtained from a Jahn-Teller problem involving only the linear vibronic coupling. Top right is a sloped conical intersection obtained in a general vibronic coupling problem where all three linear vibronic coupling constants are different. Bottom left to right show type-1, -II, -III Renner-Teller surfaces. These are obtained when only second-order vibronic coupling is included.

Table 1. A comparison of tunneling splitting in a cubic T t2 system with linear vibronic coupling calculated by different methods...

Here we consider an optical transition between Aj and E electronic states of a center of a trigonal symmetry. To describe the vibrations of the center we use the collinear-configurational approximation [27] in which only the central forces are taken into account in the optical center (taking account of deviations from this approximation, see later). If one restricts oneself to the linear vibronic coupling in the e state, then in this approximation the potential energy operators in the Ai and E electronic states can be presented in the form ... 

THEORETICAL FRAMEWORK 2.1. The linear vibronic coupling approach... 

From equation (35), the 30-warping terms arise from the third-order and fifth-order anharmonic vibration and the quadratic vibronic coupling, and the 60-warping terms arise from the sixth-order anharmonic vibration and the fourth-order non-linear vibronic coupling. 

The aim of this work is to elucidate these problems. To this end, we calculate the effective spin Hamiltonian of the 5f2—5f2 superexchange interaction between the neighboring U4+ ions in the cubic crystal lattice of UO2 and we calculate T5 <%> eg, rs f2g(l) ancl r5 f2g(2) linear vibronic coupling constants. These data are then used to draw a more definite conclusion about the driving force of the phase transition and especially about the actual mechanism of the spin and orbital ordering in U02. 

The Hamiltonian Eq. (7) provides the basis for the quantum dynamical treatment to be detailed in the following sections, typically involving a parametrization for 20-30 phonon modes. Eq. (7) is formally equivalent to a class of linear vibronic coupling (LVC) Hamiltonians which have been used for the description of excited-state dynamics in molecular systems [66] as well as the Jahn-Teller effect in solid-state physics. In the following, we will elaborate on the general properties of the Hamiltonian Eq. (7) and on quantum dynamical calculations based on this Hamiltonian. 

The effective-mode transformation described here is closely related to earlier works which led to the construction of so-called interaction modes [75, 76] or cluster modes [77, 78] in Jahn-Teller systems. The approach of Refs. [54,55,72] generalizes these earlier analyses to the generic form - independent of particular symmetries - of the linear vibronic coupling Hamiltonian Eq. (8). 

Consider the APES of a two-level system with the ground state 1 and excited state 2 and an energy gap A between them, which interact (mix) under the symmetrized nuclear displacement Qr- Using perturbation theory with respect to the linear vibronic coupling term (dH/dQr)o Qr we easily obtain [1] that the primary curvature (the curvature without vibronic coupling) of the ground state K, ... 

The diagonal part of the linear vibronic coupling constants has a clear physical meaning the force along the normal mode F from the field produced by the electronic state F. 

The atomic unit of the linear vibronic coupling constant Va is Eh/(meao) =... 

Fig. 2 Representative cuts through the potential energy surfaces of Bz+ (upperpanel or a) and its mono fluoro derivative, F-Bz+ (lowerpanel or b). The upper panel shows the results for the linear vibronic coupling model, while in the lower one the quadratic coupling terms are also included. In both panels the effective coordinate connects the centre of the Franck-Condon zone to the minimum of the intersection seam between the A and C states of F-Bz" ", and between the X and B states of the parent cation (within the subspace of JT active coordinates)...

Taking into account the linear vibronic coupling in the Hamiltonian of the system (the linear E (x) e problem), we have... 

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