# 05072009 - Diatomic Molecules 12th May 2009 1 Hydrogen...

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Unformatted text preview: Diatomic Molecules 12th 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 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 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”. This is also referred to the adiabatic approximation. This is a good assumption for most cases. • Because we consider the separation in time scales of nuclear and elec- tronic degrees of freedom, we assume a separable ansatz of the form: Ψ( ~ r, ~ R ) = ψ el ( ~ r, ~ R ) ψ N ( ~ R ) Thus, if we consider the usual approach to setting up the Schrodinger equa- tion: h ˆ H N ( ~ R ) + ˆ H electronic ( ~ r, ~ R ) i ψ el ( ~ r, ~ R ) ψ N ( ~ R ) = Eψ el ( ~ r, ~ R ) ψ N ( ~ R ) ˆ H N ( ~ R ) ψ el ( ~ r, ~ R ) ψ N ( ~ R ) + ˆ H electronic ( ~ r, ~ R ) ψ el ( ~ r, ~ R ) ψ N ( ~ R ) = Eψ el ( ~ r, ~ R ) ψ N ( ~ R ) ψ el ( ~ r, ~ R ) ˆ H N ( ~ R ) ψ N ( ~ R ) + ψ N ( ~ R ) ˆ H electronic ( ~ r, ~ R ) ψ el ( ~ r, ~ R ) = Eψ el ( ~ r, ~ R ) ψ N ( ~ R ) ψ el ( ~ r, ~ R ) ˆ H N ( ~ R )...
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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.

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05072009 - Diatomic Molecules 12th May 2009 1 Hydrogen...

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