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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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...
View Full Document

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.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online