Diabatic potential curves

The coupling to the dissociative part of the Hamiltonian (represented by the repulsive diabatic potential curve in Figure 7.1) broadens the discrete energy levels with the widths depending on the coupling strength between the two manifolds. 

Fig. 15.1. Schematic illustration of an electronic transition in the neighborhood of the crossing of two diabatic potential curves VI and V. The upper part shows the splitting of the wavepacket which was originally generated by the photon in electronic state 1. The part labeled by index 1 (white) evolves on V) and the part labeled by index 2 (black) evolves on V. ...

In Section 3.3 it will be shown that, to describe perturbations which result from neglected terms in the Hel + TN (R) part of the Hamiltonian, two different types of BO representations are useful. If a crossing (diabatic) potential curve representation is used, off-diagonal matrix elements of Hel appear between the states of this representation. If a noncrossing (adiabatic) potential curve representation is the starting point, the TN operator becomes responsible for perturbations. 

Note that the negative sign of VeN implies that it contributes to energy stabilization. Crossing curves are obtained by excluding parts of the spin-orbit term, Hso, and of the interelectronic term, Vee, from the Hel operator/ The effect of Vee, discussed in Section 3.3.2, is extremely important as it compromises the validity of the single electronic configuration picture which is often taken as synonymous with the diabatic potential curve picture. 

It is not possible to give a unique definition of a diabatic potential curve without identifying the specific term in Hel that is excluded. The impossibility of identifying such a term and the consequent nonuniqueness of the a priori definition of diabatic curves is discussed by Lewis and Hougen (1968), Smith (1969), and Mead and Truhlar (1982) (See also Section 3.3.2). Diabatic curves may be defined empirically (Section 3.3) by assuming a deperturbation model [e.g., that HffiR) is independent of R or, at most, varies linearly with R]. 

Equation (3.1.2) is the nonrelativistic Hamiltonian. This means that the spin-dependent part of the Hamiltonian (Hso spin-orbit and Hss spin-spin) has been neglected. The electronic angular momentum quantum numbers, which are well-defined for eigenfunctions of nonrelativistic adiabatic and diabatic potential curves, are A, E, and 5 (and redundantly, Q = A + E). 

This latter relation means that the diabatic potential curves associated with these functions can cross. The noncrossing rule between states of identical symmetry applies only for exact solutions of the electronic Hamiltonian. 

The vibronic interaction between the level vi of the diabatic potential curve Vf(R) and the level v2 of another diabatic curve V2 R) is reduced to... 

The crucial features of a simple model for the ii-dependent characters of the Br+ (4p4 3P)+ H molecular states are obtained from four diabatic potential curves, 2,4nj and 2,4E, which correspond respectively to the four 4p electrons on Br+(3P) arranged 7r3(2n)singlet-coupled electrons in [Pg.477]

In chapter 7 the statistical adiabatic channel model (SACM) (Quack and Troe, 1974, 1975) was described for calculating unimolecular reaction rates. This theory assumes the reaction system remains on the same diabatic potential energy curve while moving from reactant to products. Two parameters, a and (3 are used to construct model diabatic potential curves. The unimolecular rate constant, at fixed E and 7, for forming products with specific energy , (e.g., a specific vibrational energy in one of the fragments) is... 

The major effort in SACM calculations is concentrated in constructing the diabatic potential curves which correlate reactant and product states, and in determining the number of open channels at a given E. A channel is open if its potential maximum is less than E. This approach has been effectively used in modeling the rotational FED of HOOH (Brouwer et al., 1987). 

X being the relevant cartesian reaction coordinate and the effective mass for motion along it. A cross-cut along f(x) yields two "diabatic" potential curves /37d/ (Pig.6)... 

A simplified solution to the problem may be found, as shown by NIKITIN et al./113/, using the linear potential functions (154.11) which represent an approximation of the "diabatic potential curves in the region of the crossing point x = x. Choosing this point as origin of the coordinate system, we set x = 0 and V-j (x ) = 2(Xq) = 0 to obtain the equations... 

We see that the semiclassical Landau- Zener theory is valid if the energy is considerably higher than the crossing point of the diabatic potential curves (x) and Y2M > so that C If the electronic coupling is weak (small values of this con-... 

The above considerations are based on the linear "diabatic potential curves (165.11). The formulas derived, however, appear to have a more general validity, as has been shown by CHRISTOV /I16/ for... 

The probability of a nonadiabatic transition is essentially determined by the local conditions at the crossing point (x = x ) of the "diabatic" potential curves (x) and Y2M9 rather than those at the transition state (x = x ), i.e., the peak of the adiabatic potential curve V(x)(Pig.21). Therefore, it is necessary to change the energy variable, using (156.111), in the integral expression (78.Ill) in order to obtain... 

The characteristic feature of the systems relevant for this article is that their diabatic potential curves are rather parallel. This feature is incorporated in the models by Demkov and Nikitin and it differs from the curve-crossing feature adopted in the Landau-Zener model. Hence, the models by Demkov and Nikitin are favorable for the cases treated here. 

It is noted that for low velocities the sharing probability is not influenced by the slope of the diabatic potential curves. From Eq. (32) follows the same expression, but the factor 1 - cos is missing in the exponent. This factor determines the diabatic energy difference in the coupling region, since A//(l c) = Ae(l - cos 6), as may readily be deduced from Eq. (15). Thus, in the low-velocity range the treatments by Nikitin and Demkov are identical provided that in Demkov s formula the diabatic energy difference AH Rc) is inserted instead of the asymptotic value Ae 2- Thus, the modified value for Ae is justified in Table 1. 

The situation is different when the diabatic potential curves cross as assumed in the Landau-Zener model. In this case the diabatic energy difference vanishes and it loses significance as a model parameter. Then, the expression (33) should be interpreted differently. As noted in Section 3.2, the Landau-Zener model implies 6 1 so that 1 - cos d [sin 0]/2 = 2[Hi2(Rc)/ Ae]. Here, it is useful to introduce the derivative of the diabatic energy difference F = dAH/dR, which may be regarded as the force holding the system in the diabatic state. Likewise, the interaction Hu pushes ... 

Fig. 5.38 Diabatic potential curves and dipole moment cimres of the LiF molecule. (Reprinted with permission from Y. Arasaki et at., J. Chem. Phys. 138, 161103 (2013)).

An estimate of a is provided by regarding the two diabatic potential curves as linear functions of R, in the vicinity of R. The two straight lines diverge because of their different slopes, intersecting at Hence if we expand each potential as a Taylor series about... 

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