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05072009(1) - Diatomic Molecules 7th May 2009 1 Hydrogen...

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Diatomic Molecules 7th May 2009 1 Hydrogen Molecule: Born-Oppenheimer Approx- imation In this discussion, we consider the formulation of the Schrodinger equation for diatomic molecules; this can be extended to larger molecules. First we will consider the separation of the total Hamiltonian for a 4-body prob- lem into a more tractable form. We will afterward discuss the molecular wavefunctions. For the hydrogen molecule, we are concerned with 2 nuclei and 2 elec- trons. The total Hamiltonian, representing the total energy operator, is: ˆ H ( ~ r, ~ R ) = - ¯ h 2 2 M 2 A + 2 B - ¯ h 2 2 m e 2 1 + 2 2 - Z A e 2 4 π 0 r 1 A - Z A e 2 4 π 0 r 2 A - Z B e 2 4 π 0 r 1 B - Z B e 2 4 π 0 r 2 B + e 2 4 π 0 r 12 + Z A Z B e 2 4 π 0 R AB Let’s define: ˆ H N ( ~ R ) = - ¯ h 2 2 M 2 A + 2 B ˆ H electronic ( ~ r, ~ R ) = - ¯ h 2 2 m e 2 1 + 2 2 - Z A e 2 4 π 0 r 1 A - Z A e 2 4 π 0 r 2 A - Z B e 2 4 π 0 r 1 B - Z B e 2 4 π 0 r 2 B + e 2 4 π 0 r 12 + Z A Z B e 2 4 π 0 R AB 1
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NOTE: For the present purposes, ˆ H N is only a function of ~ R and only depends on the coordinates of the nuclei. It is the bf kinetic energy operator of the nuclei. ˆ H electronic ( ~ r, ~ R ) is the electronic Hamiltonian. Thus, ˆ H ( ~ r, ~ R ) = ˆ H N ( ~ R ) + ˆ H electronic ( ~ r, ~ R ) To solve the full Schrodinger equation for electrons and nuclei, one has to make approximations. This is because, as in the hydrogen atom case, there are non-radially symmetric interactions between electrons, nuclei, and electrons-nuclei. The first approximation we make is the Born-Oppenheimer Due to the large relative difference in electronic and nuclear masses, a first approximation is to assume that the time scales of motion of electrons and nuclei are separable . Effectively, the nuclei are at rest relative to the electrons; as the nuclear configuration changes, the electronic degrees of freedom “relax instantaneously”.
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