Born-Oppenheimer approximation clamped nuclei

The problem, as Woolley addressed it, is that quantum mechanical calculations employ the fixed, or "clamped," nucleus approximation (the Born-Oppenheimer approximation) in which nuclei are treated as classical particles confined to "equilibrium" positions. Woolley claims that a quantum mechanical calculation carried out completely from first principles, without such an approximation, yields no recognizable molecular structure and that the maintenance of "molecular structure" must therefore be a product not of an isolated molecule but of the action of the molecule functioning over time in its environment.47... 

We restrict ourselves to the clamped-nucleus or Born-Oppenheimer approximation [30,31] because essentially all the work done to date on electron momentum densities has relied on it. Therefore we focus on purely electronic wavefunctions and the electron densities that they lead to. 

CSFs into the wavefunction expansion. Although unattainable in molecular calculations, the second limiting case, corresponding to full Cl for a complete orbital set, is called the complete Cl expansion s. The eigenvalues of the complete Cl expansion are the exact energies within the clamped-atomic-nucleus Born-Oppenheimer approximation. A correspondence may then be established with the bracketing theorem between the lowest eigenvalues of a limited CSF expansion and those of the exact complete Cl expansion. This is illustrated schematically in Fig. 2. 

In the clamped-nucleus Born-Oppenheimer approximation, with neglect of relativistic effects, the molecular Hamiltonian operator in atomic units takes the form... 

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 )... 

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... 

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