Crude Born-Oppenheimer approximation calculation

Force constants, crude Born-Oppenheimer approximation, hydrogen molecule, minimum basis set calculation, 545-550 Forward peak scattering, electron nuclear... 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

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. 

Our problem now is to see how this shape arises from the electronic structure of the atoms involved and to find out what we can about wave functions, energies, charge distributions and so on, all of which will be needed for an understanding of the interactions of these molecules in the liquid and solid states. Even using the Born-Oppenheimer approximation, which allows us first to solve the electronic problem with the nuclei fixed and then to use this result to determine the effective potential in which the nuclei move, exact solution of the Schrodinger equation is out of the question. It is possible, however, with relatively little labour, to see how the particular structure of the water molecule comes about and then, by refining this crude model, to calculate relevant quantities quite accurately. 

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