Comparison between Diabatic and Adiabatic Parameters

Interacting States Molecule He (cm 1) Maximum value of We(R) (hr1) Width (A) FWHM of We(R) 

If the diabatic coupling matrix element, He, is -independent, this d/dR matrix element between two adiabatic states must have a Lorentzian H-depen-dence with a full width at half maximum (FWHM) of 46. Evidently, the adiabatic electronic matrix element We(R) is not - independent but is strongly peaked near Rc- Its maximum value occurs at R = Rc and is equal to 1/46 = a/4He. Thus, if the diabatic matrix element He is large, the maximum value of the electronic matrix element between adiabatic curves is small. This is the situation where it is convenient to work with deperturbed adiabatic curves. On the contrary, if He is small, it becomes more convenient to start from diabatic curves. Table 3.5 compares the values of diabatic and adiabatic parameters. The deviation from the relation, We(i )max x FWHM = 1, is due to a slight dependence of He on R and a nonlinear variation of the energy difference between diabatic potentials. When We(R) is a relatively broad curve without a prominent maximum, the adiabatic approach is more convenient. When We (R) is sharply peaked, the diabatic picture is preferable. The first two cases in Table 3.5 would be more convenient to treat from an adiabatic point of view. The description of the last two cases would be simplest in terms of diabatic curves. The third case is intermediate between the two extreme cases and will be examined later (see Table 3.6). 

To obtain the second adiabatic electronic matrix element of Eq. (3.3.11), the ket, (d/dR) j, is expanded using the complete set, of adiabatic 

The second summation reduces to a single term because the adiabatic functions axe orthonormal (Hobey and McLachlan, 1960). In the simple case where only two electronic states interact (Eq. (3.3.13)), one can assume that the matrix elements of d/dR connecting either of these two states with other states axe negligible and, from Eq. (3.3.14), 

This calculated matrix element of d2/ dR2, acting on the electronic wavefunc-tions for the E, F and G, K states of H2 (Fig. 3.8), is displayed in Fig. 3.9 and is seen not to deviate appreciably from the derivative of a Lorentzian curve. Its contribution to the Hi,Vm-,2,Vn vibronic matrix element [Eq. (3.3.11)] is generally smaller than the contribution due to the d/dR operator acting on the electronic functions, but it is in no case negligible. 

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Table 3.5 Comparison between Diabatic and Adiabatic Parameters...

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