Born-Oppenheimer beyond

In this section the Born-Oppenheimer approximation will be presented in what is necessarily a very simplified form. It has already been introduced without justification in Section 6.5. It is certainly the most important - and most satisfactory - approximation in quantum mechanics, although its rigorous derivation is far beyond the level of this book. Consider, therefore, the Mowing argument... 

Molecular Dynamics Beyond the Born-Oppenheimer Approximation. 

H. Koppel, W. Domcke, and L. S. Cederbaum, Multimode molecular dynamics beyond the Born-Oppenheimer approximation, Adv. Chem. Phys. 57, 59-246(1984). 

A detailed discussion of the theoretical evaluation of the adiabatic correction for a molecular system is beyond the scope of this book. The full development involves, among other matters, the investigation of the action of the kinetic energy operators for the nuclei (which involve inverse nuclear masses) on the electronic wave function. Such terms are completely ignored in the Born-Oppenheimer approximation. In order to go beyond the Born-Oppenheimer approximation as a first step one can expand the molecular wave function in terms of a set of Born-Oppenheimer states (designated as lec (S, r ))... 

At this stage we are at the very beginning of development, implementation, and application of methods for quantum-mechanical calculations of molecular systems without assuming the Born-Oppenheimer approximation. So far we have done several calculations of ground and excited states of small diatomic molecules, extending them beyond two-electron systems and some preliminary calculations on triatomic systems. In the non-BO works, we have used three different correlated Gaussian basis sets. The simplest one without r,y premultipliers (4)j = exp[—r (A t (8> Is) "]) was used in atomic calculations the basis with premultipliers in the form of powers of rj exp[—r (Aj (8> /sjr])... 

Much of the detailed attention paid to calculations beyond the Born-Oppenheimer limit has applied to one- and two-electron molecules the literature up to 1980 was reviewed by Bishop and Cheung [42], and for the hydrogen molecular ion more recent summaries have been given by Carrington and Kennedy [43], Carrington, McNab and... 

Positrons have the charge of protons and the mass of electrons. Quantum chemists who take an interest in mixed electron-positron systems immediately recognize some interesting consequences of this transparent observation. For example, (a) the familiar Born-Oppenheimer approximation cannot be used for positrons, but rather positrons must be treated as distinguishable electrons (b) electron-positron correlation is more important, pair by pair, than correlation between leptons of like charge and (c) there are always core electrons (except for the simplest systems), but positrons congregate in the valence region or beyond. 

G.A. Worth, L.S. Cederbaum, Beyond Born-Oppenheimer Molecular dynamics through a conical intersection. Annual Rev. Chem. Phys. 55 (127) (2004) 127-158. 

In the framework of the Born-Oppenheimer approximation, radiationless transitions from one surface to another are impossible. (See, e.g., Michl and BonaCit -Koutecky, 1990.) It is therefore necessary to go beyond the Born-Oppenheimer approximation and to include the interaction between different electronic molecular states through the nuclear motion in order to be able to describe such transitions. Using the time-dependent perturbation theory for the rate constant of a transition between a pair of states one arrives at... 

Many chemical reactivity aspects are beyond the present framework. Since this methodology studies the intrinsic or initial response, properties associated with advanced steps of the reaction, such as those related with the transition state or the reaction product, may be inaccurately described. The same can be said about situations where the Born-Oppenheimer approximation fails, like tunnelling contributions. 

Woolley, R. G. 1991. "Quantum Chemistry Beyond the Born-Oppenheimer Approximation." Journal of Molecular Structure (Theochem), 230 17—46. 

Indeed, for certain crucial cases of very small energy differences in the discrete or the continuous spectrum, it is often necessary to aim at computing highly accurate total energies. This is also necessary when the rovibra-tional spectrum for a particular Born-Oppenheimer (B-O) state must be known with great precision. In fact, in certain cases, going beyond the B-O approximation may be desirable. 

What are the advantages of the Born Oppenheimer approximation and when is it necessary to go beyond [Separation of electronic and vibrational wavefunctions, existence of PES calculation of IC and ISC rate constants, near-degenerate states]... 

---