Clamped-Nucleus Treatment

In order to evaluate the vibrational polarizability, Eq. (8.8), one needs the energies of all vibrational states of the electronic ground state and the corresponding vibrational dipole transition moments, which requires knowledge of the potential energy and electric dipole moment surface of this single electronic state. For the electronic-vibrational polarizability, Eq. (8.7), however, one would need to know not only all excited electronic states, and the electronic dipole transition moments to them but also all the vibrational states, of these excited states, which makes this 

However, we can make the approximation in Eq. (8.7) that the differences between the vibrational energies are much smaller than the differences between the electronic energies, i.e. 

In the second approach, the so-called clamped-nucleus treatment, the effect of the perturbation on the electronic and nuclear motion is treated sequentially. First, the Born-Oppenheimer approximation is applied to the vibronic wavefunction of the ground state, ov ri, Rk ), which is therefore expressed as a product of an electronic wavefunction o( ri Rk ) vibrational wavefunction (. R/f ) 

Secondly, in the presence of an external electric field with component js, the field gives rise to a first-order perturbation Hamiltonian, Eq. (4.29), and the electronic wavefunction can be expanded in a perturbation series Eq. (3.16). To first order the electronic wavefunction, Eq. (3.27), and the electronic energy including the nuclear repulsion, Eq. (3.29), are then given as 

With this energy as potential energy for the nuclear motion the nuclear Schrodinger equation in the Born-Oppenheimer approximation, Eq. (2.12), becomes 

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For polyatomic molecules the electronic polarizability in the clamped-nucleus treatment is frequently expressed as the polarizability evaluated at an equilibrium geometry Rk,s plus a vibrational correction Aa ... 

At the same time that Heisenberg was formulating his approach to the helium system, Born and Oppenheimer indicated how to formulate a quantum mechanical description of molecules that justified approximations already in use in treatment of band spectra. The theory was worked out while Oppenheimer was resident in Gottingen and constituted his doctoral dissertation. Born and Oppenheimer justified why molecules could be regarded as essentially fixed particles insofar as the electronic motion was concerned, and they derived the "potential" energy function for the nuclear motion. This approximation was to become the "clamped-nucleus" approximation among quantum chemists in decades to come.36... 

It will turn out in the following treatment that these equations need not be made more explicit since various approximations (clamped nucleus, approxi-... 

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