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05 - Chemistry 2000 Lecture 5 Valence bond theory Marc R...

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Chemistry 2000 Lecture 5: Valence bond theory Marc R. Roussel
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Multi-electron wavefunctions from MOs I The solution to Schr¨ odinger’s equation for a multi-electron molecule (or atom) is a wavefunction involving the coordinates of all the electrons. I A molecular orbital is a one-electron wavefunction which, in principle, extends over the whole molecule. I The multi-electron wavefunction is, roughly speaking, a product of one-electron wavefunctions in MO theory. I Each MO has a geometry consistent with the symmetry of the molecule. This arises naturally in the calculation. Nothing special needs to be done.
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Valence-bond theory I Valence-bond (VB) theory takes a different approach, designed to agree with the chemist’s idea of a chemical bond as a shared pair of electrons. I If φ A ( r ) is an atomic orbital from atom A and φ B ( r ) is an atomic orbital from atom B, with r = ( x , y , z ) the coordinates of an electron, then we construct a two-electron bond wavefunction as follows: ψ ( r 1 , r 2 ) = φ A ( r 1 ) φ B ( r 2 ) ± φ A ( r 2 ) φ B ( r 1 ) I We take the + sign for a bonding interaction. The - sign represents an antibonding interaction. I This is pure covalency : The electrons are equally shared between the two atoms and neither atom has both electrons at the same time.
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Example: H 2 I The VB wavefunction for the ground state of H 2 is ψ ( r 1 , r 2 ) = 1s A ( r 1 )1s B ( r 2 ) + 1s A ( r 2 )1s B ( r 1 ) Contrast: In LCAO-MO theory, the σ molecular orbital is written σ ( r ) = 1s A ( r ) + 1s B ( r ) The two-electron ground-state wavefunction is therefore ψ ( r 1 , r 2 ) = σ ( r 1 ) σ ( r 2 ) = 1s A ( r 1 )1s B ( r 2 ) + 1s A ( r 2 )1s B ( r 1 ) + 1s A ( r 1 )1s A ( r 2 ) + 1s B ( r 1 )1s B ( r 2 )
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Example: H 2 (continued) ψ ( r 1 , r 2 ) = 1s A ( r 1 )1s B ( r 2 ) + 1s A ( r 2 )1s B (
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