Molecules and the Born-Oppenheimer Approximation

The electronic structure of molecules is a more complicated problem than the electronic structure of atoms because instead of one central nuclear potential there are several positively charged nuclei distributed in space. Analyzing molecular electronic structure usually begins with an approximate separation of the electronic problem from the problem of nuclear motion. This separation was implicit in the treatment of vibration and rotation in Chapter 9 and in the discussion of potential energy surfaces in Chapter 6. 

The general molecular Schrodinger equation, apart from electron spin interactions, is 

There is a repulsive Coulombic interaction among the nuclear charges and a repulsive charge-charge interaction among the electrons. However, the interaction potential between electrons and nuclei is attractive since the particles are oppositely charged. This particular interaction couples the motions of the electrons with the motions of the nuclei. 

The wavefunctions that satisfy Equation 10.28 must be functions of both the electron position coordinates and the nuclear position coordinates, and this differential equation is not separable. In principle, true solutions could be found, but the task is difficult as this is a formidable differential equation for even simple molecules. An alternative is an approximate separation of the differential equation based on the sharp difference between the mass of an electron and the masses of the nuclei. The difference suggests that the nuclei motions are sluggish relative to the electron motions. Over a brief period of time, the electrons see the nuclei as if fixed in space since during such a period the nuclear motions are relatively slight. The nuclei, on the other hand, see the electrons as something of a blur, given their relatively faster motions. 

We can exploit the distinction between sluggish and fast particles to achieve an approximate separation of variables. First, the wavefunction is taken to be (approximately) a product of a function of nuclear coordinates only, (p, and a function of electron coordinates, /, at a specific nuclear geometry. At the specific nuclear geometry, R = Rq R collectively stands for all nuclear position coordinates), this is expressed as 

---