Born-Oppenheimer approximation 566 INDEX

A global property function is usually expressed as the expectation value of an operator or as the derivative of such an expectation value with respect to an internal or external parameter of the system. In the Born-Oppenheimer approximation, the electronic wave function depends parametrically upon the coordinates of the n nuclei, and therefore a set of the 3 -6 linearly independent nuclear coordinates constitutes the natural variables for such a choice of the potential function. However, the manifold M on which the gradient vector field is bound can be defined on a subset of 1R provided q < 3n-6, for example the intrinsic reaction coordinate (unstable manifold of a saddle point of index 1 of the Born-Oppenheimer energy hypersurface) or the reduced reaction coordinate. 

In the Born-Oppenheimer approximation (omitting, for simplicity, the index e of the electronic eigenfunction), the density matrices relative to the n electrons of the molecule are ... 

Suppose we wish to know the dipole moment of, say, the HCl molecule, the quantity that tells us important information about the charge distribution. We look up the output and we do not find anything about dipole moment. The reason is that all molecules have the same dipole moment in any of their stationary state y, and this dipole moment equals to zero, see, e.g., Piela (2007) p. 630. Indeed, the dipole moment is calculated as the mean value of the dipole moment operator i.e., ft = (T l/i l ) = ( F (2, q/r,) T), index i runs over all electrons and nuclei. This integral can be calculated very easily the integrand is antisymmetric with respect to inversion and therefore ft = 0. Let us stress that our conclusion pertains to the total wave function, which has to reflect the space isotropy leading to the zero dipole moment, because all orientations in space are equally probable. If one applied the transformation r -r only to some particles in the molecule (e.g., electrons), and not to the other ones (e.g., the nuclei), then the wave function will show no parity (it would be neither symmetric nor antisymmetric). We do this in the adiabatic or Born-Oppenheimer approximation, where the electronic wave function depends on the electronic coordinates only. This explains why the integral ft = ( F F) (the integration is over electronic coordinates only) does not equal zero for some molecules (which we call polar). Thus, to calculate the dipole moment we have to use the adiabatic or the Born-Oppenheimer approximation. 

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