Validity of the BO approximation

There is no doubt that the Born-Oppenheimer approximation captures the essential feature of the molecular properties of stable molecules. (So many papers have been published on the mathematical and numerical analyses of the Born-Oppenheimer theory, but it is beyond the scope of this book. See Refs. [181, 405, 409] for relevant recent literature.) The validity or the level of accuracy of the Born-Oppenheimer separation is roughly assessed as follows. Let m be the mass of an electron and let M be the mass of the /-th nucleus. The perturbation parameter n is taken as 

They expanded the total energy according to the order of the perturbation interaction such that 

They showed that is the electronic energy, corresponds to the 

The energy terms up to constitute the so-called Born-Oppenheimer 

The critical question is how accurate (or inaccurate) the Born-Oppenheimer energy of a molecule defined as above is. Recently Takahashi and Takatsuka have addressed this question and reported a semiclassical analysis of this matter [413]. They explicitly showed that the fifth order term is also exactly zero, and therefore the lowest-order correction to the Born-Oppenheimer energy must be of the order of 

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Assuming the validity of the BO approximation, we deal with vibronic states that can be written as a product 4, ) = (gi where d>, ) is the electronic state and... 

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