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Unformatted text preview: Diatomic Molecules 7th May 2009 1 Hydrogen Molecule: BornOppenheimer 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 4body 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 π² r 1 A Z A e 2 4 π² r 2 A Z B e 2 4 π² r 1 B Z B e 2 4 π² r 2 B + e 2 4 π² r 12 + Z A Z B e 2 4 π² 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 π² r 1 A Z A e 2 4 π² r 2 A Z B e 2 4 π² r 1 B Z B e 2 4 π² r 2 B + e 2 4 π² r 12 + Z A Z B e 2 4 π² R AB 1 • 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 nonradially symmetric interactions between electrons, nuclei, and electronsnuclei. The first approximation we make is the BornOppenheimer • 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...
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This note was uploaded on 02/02/2012 for the course CHEM 444 taught by Professor Dybowski,c during the Fall '08 term at University of Delaware.
 Fall '08
 Dybowski,C
 Physical chemistry, Atom, Mole, pH

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