If we look again at the Hamiltonian, we would notice right away that the term would prevent us from applying this separation of variables. Is an electronic wave function that depends parametrically on the nuclear positions. We know that if a Hamiltonian is separable into two or more terms, then the total eigenfunctions are products of the individual eigenfunctions of the separated Hamiltonian terms, and the total eigenvalues are sums of individual eigenvalues of the separated Hamiltonian terms.Ĭonsider, for example, a Hamiltonian which is separable into two terms, one involving coordinate and the other involving coordinate. The nuclear motion (e.g., rotation, vibration) sees a smeared out potential from the speedy electrons. The electronic wavefunction depends upon the nuclear positions but not upon their velocities, i.e., the nuclear motion is so much slower than electron motion that they can be considered to be fixed. It leads to a molecular wave function in terms of electron positions and nuclear positions. The Born-Oppenheimer Approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. Next: The helium atom Up: The Hartree-Fock method Previous: The Hartree-Fock method
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