Crude Born-Oppenheimer approximation matrix elements

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

There has been considerable controversy in the literature as to whether the nonradiative decay rates kn T are to be evaluated using the adiabatic Bom-Oppen-heimer (ABO) or the crude Born-Oppenheimer (CBO) approximation mo,31,39-43) In Sect. 5 it was noted that the complimentary principle of quantum mechanics requires that the rates exactly calculated within these two schemes be the same provided that 4>s in both schemes is, as expected, a reasonable approximation to the true physical state. As noted also, in both the ABO and the CBO approximations. we have the same mechanistic schemes of (f>s coupled to the effective quasicontinuum 0,. Thus, both cases represent differing, yet reasonable representations of s and 10,). In the present discussion it is necessary to consider the remainder of the vibronic states of the moleculae, 0C in addition to f coupling matrix elements are no longer the effective values, but are actual matrix elements of the perturbing Hamiltonian. 

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