Energy matrices Born-Oppenheimer approximation

In the BO representation, the nuclear kinetic energy matrix is not diagonal because of the nuclear coordinate dependence of the wavefunction. The off-diagonal elements of the nuclear kinetic energy are non-adiabatic couplings. In order to discuss the relationship between vibronic coupling and non-adiabatic coupling, we present the Born-Oppenheimer approximation. 

Fig. 1. The molecular energy level model used to discuss radiationless transitions in polyatomic molecules. 0O, s, and S0,S are vibronic components of the ground, an excited, and a third electronic state, respectively, in the Born-Oppenheimer approximation. 0S and 0 and 0j are assumed to be allowed, while transitions between j0,j and the thermally accessible 00 are assumed to be forbidden. The f 0n are the molecular eigenstates...

It is usually assumed that the electronic coupling matrix element is a constant across the reaction coordinate. Since the electronic wavefunction is a function of both the electronic and nuclear coordinates, even in the Born-Oppenheimer approximation, it is not surprising that in some systems the assumption that the nuclear and electronic coordinates are independent (the Condon approximation) is not appropriate. The most obvious example of the failure of this approximation is for a system in which the matrix element is dominated by superexchange contributions, since the vertical energies, Adb and Eba. vary with the nuclear coordinates. There are other, probably less obvious kinds of such vibronic coupling ... 

At first sight it might be surprising that the Hessian matrix, which after all in oh initio molecular orbital theory is inherently quantum mechanical, is amenable to a purely classical treatment. This is because the Born-Oppenheimer approximation (Born and Oppenheimer 1927 Wikipedia 2010) allows for a pretty good separation of the electronic and nuclear motion, allowing the latter to be treated classically. A quantum mechanical description of the simple harmonic oscillator leads to quantized energy levels given by... 

The calculation of the second derivative, here, has no serious complications. We used the formalism described, based on the knowledge of the first derivative and the rotation matrix. As expected, this quantity is as large as the first derivative. This term multiplied by corresponds to an energy correction omitted in the Born-Oppenheimer approximation, which contributes to the so-called adiabatic correction. Other lesser corrections [42], resulting from the radial dependence of the derivative of the electronic energy in the diabatic representation can be evaluated using the Virial theorem. The second derivative is not negligible and it will consequently contribute to the nonradiative lifetime. We display in... 

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. 

In Chapters 4 and 5 we made use of the theory of radiationless transitions developed by Robinson and Frosch." In this theory the transition is considered to be due to a time-dependent intramolecular perturbation on non-stationary Born-Oppenheimer states. Henry and Kasha > and Jortner and co-workers< > have pointed out that the Born-Oppenheimer (BO) approximation is only valid if the energy difference between the BO states is large relative to the vibronic matrix element connecting these states. When there are near-degenerate or degenerate zeroth-order vibronic states belonging to different configurations the BO approximation fails. 

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