Linear vibronic constant

To this end the theory is enriched by the following set of parameters the harmonic force constants Kee (fCtt), the linear vibronic constants Xe (Xt) and eventually the quadratic ones, and the tetragonal (trigonal) distortion energy Atetr (Atrig). 

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.

Fig. 3. The tunneling splitting 8 (vertical axis) between the G and A states as a function of the linear coupling constants KG and KH for states associated with D3d wells. The two curves on the surface mark the region where the G vibronic state crosses the A vibronic state and the region where the D3d extrema are wells on the lowest APES, respectively. The region marked Overlap is where 8 has negative values and the ground state has A symmetry and the results are physically acceptable.

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 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) =... 

The quadratic vibronic constants of the e[ normal coordinates are not zero even if the linear ones are zero because in the Dsh point group E[iS> E[ first order and the second order JT effect are separated. As one component of e becomes in LS totally symmetric too, they will also mix and contribute to the totally symmetric JT coordinate. Thus, we see that considering only one normal coordinate is not enough to describe the JT effect even in this simple case. In the subsequent sections we will address this problem again, and propose how to analyse which is the contribution of the different vibrations to the total distortion of a molecule, and which of them are the most important driving force for the distortion. 

As an effect of the linear and quadratic vibronic constants the adiabatic potential surface no longer stays paraboloid-shaped. It exhibits an additional warping with several local minima out of the reference high-symmetry configuration Qq. 

In this way, the non-scalar part forms a symmetric matrix V its elements are a linear function of vibronic constants X and nuclear displacements (Sp(V) = 0). Its eigenvalues... 

This correction is responsible for the warping of the adiabatic potential surface near the nuclear configuration with electron degeneracy. The analytic form of the adiabatic potential surface becomes a non-linear (not only polynomial) function of nuclear displacements in which potential constants of two kinds occur. The force constants K , Ku, Kjik, etc. contain the pure nuclear term Fj, Fa, Fak, etc. as well as the electron-nuclear (vibronic) term X, Xa, Xak, etc. The vibronic constants Xj, Xy, etc. enter into the vibronic correction term e. They determine a warping of the adiabatic potential surface and they can couple vibration modes of different symmetry, e.g. Xy for i j. 

Vibronic interaction Force constants linear harmonic m = 1 m = 2 cubic m = 3 Vibronic constants linear quadratic n = 1 n = 2 cubic n = 3... 

The accuracy of the linear vibronic coupling model can be improved by adding diagonal quadratic terms 7." Q for the non totaUy-symmetric modes for which the diagonal linear terms vanish [63], In this case, the 7/" constants can be conveniently... 

Figure 3. Low-energy vibronic spectrum in a. 11 electronic state of a linear triatomic molecule, computed for various values of the Renner parameter e and spin-orbit constant Aso (in cm ). The spectrum shown in the center of figure (e = —0.17, A o = —37cm ) corresponds to the A TT state of NCN [28,29]. The zero on the energy scale represents the minimum of the potential energy surface. Solid lines A = 0 vibronic levels dashed lines K = levels dash-dotted lines K = 1 levels dotted lines = 3 levels. Spin-vibronic levels are denoted by the value of the corresponding quantum number P P = Af - - E note that E is in this case spin quantum number),...

Figure 5, Low-eriergy vibronic spectrum in a electronic state of a linear triatomic molecule. The parameter c determines the magnitude of splitting of adiabatic bending potential curves, is the spin-orbit coupling constant, which is assumed to be positive. The zero on the...

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... 

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