Coupling quadratic

If the quadratic coupling cannot be neglected, the potential surface acquires three minima at wave functions corresponding to the two branches are... 

Quadratic coupling, geometric phase theory, Jahn-Teller effect, 22-23 Quantal adiabatic phase geometric phase theory, 2... 

External stress, locally applied, can have nonlocal static effects in ferroelastics (see Fig. 4 of Ref. [7]). Dynamical evolution of strains under local external stress can show striking time-dependent patterns such as elastic photocopying of the applied deformations, in an expanding texture (see Fig.5 of Ref. [8]). Since charges and spins can couple linearly to strain, they are like internal (unit-cell) local stresses, and one might expect extended strain response in all (compatibility-linked) strain-tensor components. Quadratic coupling is like a local transition temperature. The model we consider is a (scalar) free energy density term... 

Fig. 2a-c. Two-dimensional potential energy surfaces. The Qx coordinate corresponds to the low frequency mode, the Qy coordinate corresponds to the high frequency mode. The forms of the coupling are a kxy = 0 (no coupling), b k>y = xy2 (quadratic coupling), and c kxy = xy (linear/Duschinsky coupling... 

Fig. 4. Autocorrelation functions plotted versus time for the spectra described in Fig. 3. The autocorrelation function shown by the middle curve corresponds to the reference spectrum (Fig. 3a), k.j, = Ocm-1, the autocorrelation function shown by the lowest curve corresponds to a positive displacement in the quadratically coupled potential surface, and the autocorrelation function shown by the top curve corresponds to a negative displacement in the quadratically coupled potential surface...

Fig. 6a, b. Calculated emission spectra showing the effects of changing the force constant of the initial wave packet placed on the quadratically coupled surface with dx = 1.44, and Ay = 1.53. The excited state force constants are a 540 cm 1 and b 135 cm 1... 

Fig. 8a, b. Comparisons of the calculated (broken lines) and experimental (solid lines) emission spectra of frozen Me2Au(hfacac) at 15 K. a The best fit" using the uncoupled harmonic potential surface, b The best fit using the quadratically coupled potential surface... 

Fig. 2. Computed energy levels versus the quadratic coupling constant g, 0 < g < 1 in the case of orbital p-like functions.

The energies of the vibronic states have been calculated analytically by the shift transformation method developed initially by Bates et al. [24]. Consider now the quadratic coupling term as a perturbation on the linear system. The resultant energies have similar forms to those of the linear system namely ... 

The two 2x2 matrices have zero trace following the centre of the gravity theorem [13]. The second term in equation (5), H2, corresponds to quadratic coupling and, according to symmetry requirements, depends on two independent constants,... 

According to equation (9) the departures from the linearity are thus different for alg ( 3z2 r2) and blg ( x2 — y2) orbitals. Usually the isotropic term involving Vs in equation (7) is not considered in the description of the quadratic coupling [1,3,8] but included implicitly within the quadratic term of equation (1). Although both terms are formally equivalent, the removal of Vs(Ql + Qj)I term in equation (7) leads however to sb — s0 — VQg = s0 — sa — VQe, which is in general wrong. 

This result implies that the minimum associated with the 3z2 — r2 electronic state is the lowest one if the anharmonic term dominates and A3 > 0 or if the quadratic coupling is the leading term and Vq is positive. The curvature at the minimum of E(3z2 r2 Qe) and E(x2 — y2 Qe) curves also depends on both the anharmonic and quadratic coupling terms... 

Therefore if quadratic coupling can be neglected and A3 > 0 the curvature for the 3Z2 — r2 electronic state is softened, while that for the x2 — y2 state increases. Of course the same effect is produced if A3 is negligible and Vq > 0. The present description can be applied to a good extent to the case of substitutional d9 ions in cubic lattices with octahedral coordination. Under a Qe distortion there are two electronic states in D4h in Blg the hole is in blg ( x2 — v2 ), while in Alg it is placed in the al ( 3z2 — r2) orbital. Bearing in mind this fact the E Blg Qe) curve for d9 ions looks similar to E(3z2 — r2 Qe) for d7 ions. 

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