Diabatic-adiabatic coincidence

The right panel of Figure 2.9 shows that the rate of level v = 8 reaches zero twice. This multiple occurrence of the zero-width phenomenon has been related to the possibility to produce several times a diabatic-adiabatic coincidence as the intensity increases [69]. This is so because the adiabatic (vibrational) levels goes up faster than the diabatic ones and therefore a given adiabatic level can cross several diabatic levels as the intensity increases. It is to be noted that resonance coalescence, i.e., the existence of an (EP), requires an appropriate choice of both frequency and intensity, while a ZWR would show up at some critical intensity(ies), irrespective of the choice of the wavelength, for all resonances of Feshbach type. One must keep in mind that the classification into shape and Feshbach depends strongly on the wavelength. 

Diabatic surfaces were obtained for the neutral according to the scheme outlined in Sec. 5.2.1.2. For (3 < 80°, the 2 A surface approaches very close to the 3 A (both in C iv notation) and the diabatization can no longer be described by a two-state scheme. Also near j3 = 160° the 2 A and 3 A surfaces cross and again the simple phenomenological two-state diabatization scheme fails. We employ Gaussian functions to damp the computed coupling potential surface VniR) in these regions to make the diabatic curves coincide with the adiabatic. 

Equations (14-16) yield a set of diabatic. states with exactly vanishing derivative couplings for the coordinate Q . The arbitrary matrix U(Q ) can be chosen so that the diabatic states coincide with the adiabatic ones at least for Q = Q a for instance, in photodissociation or collisional problems it is desirable that the asymptotic states are diabatic and adiabatic at the same time. 

The ordinary BO approximate equations failed to predict the proper symmetry allowed transitions in the quasi-JT model whereas the extended BO equation either by including a vector potential in the system Hamiltonian or by multiplying a phase factor onto the basis set can reproduce the so-called exact results obtained by the two-surface diabatic calculation. Thus, the calculated hansition probabilities in the quasi-JT model using the extended BO equations clearly demonshate the GP effect. The multiplication of a phase factor with the adiabatic nuclear wave function is an approximate treatment when the position of the conical intersection does not coincide with the origin of the coordinate axis, as shown by the results of [60]. Moreover, even if the total energy of the system is far below the conical intersection point, transition probabilities in the JT model clearly indicate the importance of the extended BO equation and its necessity. 

In (2), Ea corresponds to the vertical IP for the state a. This is due to our particular choice of the diabatic states, chosen to coincide with the adiabatic ones at the... 

Much work has been done to predict the locations of the zeroes of the line width. Child (1976) has demonstrated that, in the case of intermediate coupling strength, a zero occurs in the linewidth whenever the energies of the diabatic and adiabatic levels coincide (and this conclusion is also valid in the case of weak coupling). However, until the work of Kim, et al., (1994) and recently of Cornett, et al., (1999), it seems that the connection between zeroes of predissociation linewidth and q reversal was not appreciated. 

Once rotational constants are obtained from the sharpest lines, trial diabatic and adiabatic potentials are refined until the rotational constants, Bd and Bad, calculated from them satisfy Eq. (7.12.5). Figure 7.37 shows that the exact solutions of the coupled equations give a width that is not proportional to (He)2 when the coupling becomes very strong. One can understand this by varying the electronic matrix element. As He increases, the adiabatic level shifts. When it has shifted into coincidence with the diabatic level, the width is zero according toEq. (7.12.6). 

Figure 7.37. Variation of r versus (He)2. The full line corresponds to the v = 21 diabatic level of a model molecule. The arrow indicates the value of He for which the adiabatic v = 0 level coincides with the diabatic v = 21 level. The dotted line gives the F-values expected from the Golden Rule formula. [Adapted from data of the model of Child and Lefebvre (1978) and unpublished results.]...

Fig. 14.24. The diabatic potential energy curves (I for the reactants and Vp for the products) pertaining to the electron transfer reaction Fe + + Fe + - Fe + - - Fe + in aqueous solution. The curves depend on the variable q = r2 > l that describes the solvent, which is characterized by the radius rj of the cavity for the first (say, left) ion and by the radius r2 of the cavity for the second ion. For the sake of simplicity we assume rj +r2= const and equal to the sum of the ionic radii of Fe + and Fe +. For several points g the cavities were drawn as well as the vertical sections that symbolize the diameters of the left and right ions. In this situation, the plots Vp and Vp have to differ widely. The dashed lines represent the adiabatic curves (in the peripheral sections they coincide with the diabatic curves).

Contrary to the analogous expression [Eq. (16)] in the diabatic basis, the result [Eq. (27)] contains also off-diagonal electronic contributions. These are expected to play an important role when both channels, corresponding to and 1 +, are open. If only VL is open, then only the first of the three terms on the right hand side of Eq. (27) contributes to the reaction probability. Even in this case, however, both terms in the diabatic analogue of Eq. (16) may play a role, because adiabatic and diabatic surfaces need not coincide asymptoticaiiy. ... 

Finally, in Fig. 9 we present the potential energy surface for the B state. Since this state is uncoupled from A and, the diabatic and adiabatic surfaces coincide. 

Far from the point or line of crossing, the diabatic and adiabatic terms virtually coincide therefore, a simple relation may be established for the transition prob-abihties between adiabatic terms and between diabatic terms. Namely, if these transition probabilities are denoted by Pi, 2 and P 2-... 

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