Symmetric mode coupling potential

FIGURE 4.1 Contours of the symmetric mode coupling potential [Equation (4.15)] for the parameters (Oy, a) = (0.5, 1.0). The interval of contour is 0.02 and the contours with the energy higher than 0.28 are omitted. (Taken from Reference [31] with permission.)... 

Tunnel Splitting in the Symmetric Mode Coupling Potential Equation (4.15) Calculated Quantum Mechanically Numerically for Several Vibrationally Excited States. Parameter g Is Taken to be 0.04... 

Tunnel Splitting in the Symmetric Mode Coupling Potential Equation (4.15). The Second (Third) Column Is the Result by Exact Quantum Mechanical (EQM) Calculation [by the Sudden Approximation Equation (4.12)]. Parameters (u)y, a, g) Are Equal to (0.2,0.25,0.04)... 

Now the physical picture of multidimensional tunneling obtained from the above analysis is explained by taking a symmetric mode coupling model potential [30]. The Hamiltonian is given by... 

Then the scaled Hamiltonian with the symmetric mode coupling (SMC) potential is given by... 

Finally, we touch upon the adiabatic and sudden approximations in the present model. In the same way as in the case of symmetric mode coupling model, the adiabatic approximation leads to the tunneling splitting independent of Hy. This does not exhibit any characteristic behavior discussed above and can never be reliable. If we apply the sudden approximation, we also encounter a problem since the potential curve in jc direction is not symmetric except when y is zero. Thus we cannot use Equation (4.12) directly anymore. 

The situation is more subtle for the antisymmetrically coupled mode. As shown in fig. 17, this vibration, in contrast to the symmetric mode, asymmetrizes the potential and violates the resonance. This should lead to a decrease in the splitting. Consider this problem perturbatively. If the vibration and the potential V Q) were uncoupled, each tunneling doublet Eq, Ei (we consider only the lowest one) of the uncoupled potential V Q) would give rise to a progression of vibrational levels with energies... 

Typical potential energies associated with such a Hamiltonian are shown in Figure 4 as a function of the parameter 0 = x/2J. The coordinate is the antisymmetric combination. The symmetric mode clearly adds a term to the total energy independent of coupling. 

Fig. 1. Symmetric double-well potential U-(Q) for a pseudo-JT molecule with two nondegenerate electronic terms coupled to one low-symmetry mode [equation (9)]. The curve corresponds to strong coupling case with k = 4 and a relatively large energy gap, A = 12 (both in units of hcS). The dashed line represents the twofold degenerate ground-state energy level subject to a tunneling splitting.

By using the LVC model, augmented by purely quadratic couplings only for totally symmetric modes (thus adding QVC contributions) the symmetry-selection rule, (6), can be directly applied to deduce the vibronic Hamiltonian matrices for the description of the five lowest X — D doublet states of these fluorobenzene cations. We shall not write down all five matrices here, but rather provide the basic features regarding their QVC Hamiltonian. The general form of the QVC potential energy matrix, W// oro, for the above mentioned fluorobenzene cations is depicted below ... 

The sets of coupling constants and the Hamiltonians, (8), (15) define the highdimensional potential energy surfaces of the lowest five electronic states of the various cations treated. Typically 6-8 totally symmetric modes and 8-10 non-totally symmetric modes are found to have non-negligible coupling constants in the C2v systems in the two cases with higher symmetry these numbers apparently decrease, e.g. to 3 relevant totally symmetric modes for the 1,4-difluoro isomer. Only few selected constants are included in Table 3 and we refer to the original papers for full details [62,68,69]. 

Figure 3 Pseudo-Jahn-Teller (PJT) effect between % an(l T along a nontotally symmetric mode. Dashed-line parabolas uncoupled potential energy surfaces (without PJT). Full-line parabolas coupled potential surfaces (with PJT)...

Totally symmetric modes are not subject to symmetry restrictions. Their potentials may contain odd and even terms in Q so that the harmonic-oscillator approximation imposes unwarranted symmetry restrictions. Similarly, the corresponding vibronic coupling operator may contain both odd and even terms so that the distinction between pseudo-Jahn-Teller and pseudo-Renner-Teller coupling disappears. Since the potential energy minimum of a totally symmetric mode is different in different electronic states, the pseudo-Jahn Teller/Renner-Teller limit is quite different from the limiting cases discussed in Section I V,B,C. Finally, the transition moments... 

The absence of symmetry restrictions implies that it is difficult to make a strict separation of Franck-Condon and vibronic coupling effects. Suppose we have obtained accurate expressions for the adiabatic potential surfaces E (Q) and E (Q), for instance, by quantum-chemical calculations. To evaluate the effect of vibronic coupling between them, we wish to expand the adiabatic electronic wavefunctions in -independent wavefunctions. For nontotally symmetric modes the origin of this expansion is determined by symmetry, but for totally symmetric modes, it is not. It will thus be necessary to calculate for which value of Q the nonadiabatic coupling between the adiabatic potentials vanishes and to use this value as the origin. 

For practical reasons, we restrict ourselves here to potentials and/or couplings that are more symmetric than required for a general model. Our results will thus be too simple. Their main purpose is to show how totally symmetric modes differ from nontotally symmetric ones. For actual applications, additional parameters may be required. 

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