Vibronic coupling constants

If the solution of the zero-order Schiodinger equation [i.e., all teiins in (17) except V(r,Ro) are neglected] yields an/-fold degenerate electronic term, the degeneracy may be removed by the vibronic coupling tenns. If F) and T ) are the two degenerate wave functions, then the vibronic coupling constant... 

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.

By expanding the potential V r,Q) in a series of small Q values one finds that the first nonzero matrix element V12 is linear in the JT case (nonlinear systems) and quadratic for the RT effect (linear molecules) [1-3]. In the case of proper JT effect the minima positions in the linear approximation are at Q0 = F/K0 (Fig. la), where F = (ll(9V/90ol2) is the vibronic coupling constant, while the JT stabilization energy is... 

Fig. 3. A plot of the second-order RF Kj (Tx (S>7 ) as a function of the vibronic coupling constant. The new results (non-Condon) as well as the analytical calculations of Bates etal. [3] and the numerical results of O Brien [9] are shown.

To date, the best experimental data in the literature pertaining to vibronic coupling constants in the free Cg0 ion are the (PES) results of Gunnarsson et al. [10]. Figure 3 shows the spectrum obtained by these workers together with a graphical... 

Full tabulations of the vibronic coupling constants and tensor operators involved in the calculation of f —> f electronic transitions in centrosymmetric compounds are available and can be requested from R.A. (Documentation I - Appendices 1 and 2), and therefore will not be repeated here. 

Jahn-Teller effect for the ground T5 state of U4+ ions calculations of vibronic coupling constants for the eg, f2g(l) and r2g(2)... 

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. 

To estimate the vibronic coupling constant, we performed CF calculations for the UOg cluster distorted along the eg, f2g( 1) and t2g(2) Jahn-Teller modes (Fig. 10). A similar approach was previously applied for calculations of vibronic coupling constants in a series of UXg octahedral complexes (X = F, Cl, Br, and I) and in the U(NCS)g cubic complex in compound (NEt4)4U(NCS)8 [14]. 

For each mode, the amplitude of the distortion corresponds to the actual displacement of oxygen atoms, 0.014 A. Then the vibronic coupling constants C were calculated from the splitting energy AE of the ground T5 level. 

The largest vibronic coupling constants are found for the eg and f2g(2) modes, while for the f2g( 1) mode it is about one order of magnitude smaller. From these data we can also estimate the JT stabilization energy per U atom, 10 cm-1 for eg, 8 cm-1 for tyjl), and only 0.5 cm-1 for 2g(l)- All these energies are smaller than the thermal energy at the transition point, TN /kH 20 cm 1. 

Instead, the actual distortion of the UOg cage in U02 just corresponds to the trigonal JT minimum of the f2g(l) mode, which has the vibronic coupling constant being more than 20 times smaller than those for the eg or t2g(2) modes. In fact, the distortion shown in Fig. 11 can be represented by the sum (Q[ly] + of... 

It is convenient to introduce the dimensionless electron-vibron coupling constant, known as the Huang-Rhys factor,... 

Note The S value is the Huang-Rhys factor, or the (vibronic) coupling constant. Frequency (cm-1). 

The lowering of the vibronic coupling constants for Ni as compared with Cu is due to configuration interaction. 

Vibronic Coupling Constant and Vibronic Coupling Density... 

Vibronic Coupling Constant and Jahn-Teller Hamiltonian... 

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