lecture05_2011

lecture05_2011 - 12Jan2011 Chemistry 21b Spectroscopy...

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12Jan2011 Chemistry 21b – Spectroscopy Lecture # 5 – MO Theory for & H 2 , Electronic Nomenclature for Molecules 1. The Chemical Bond & Molecular Hydrogen Recall that the polyatomic Hamiltonian is: ˆ H = ˆ T N + ˆ T e + ˆ V Ne + ˆ V ee + ˆ V NN (5 . 1) ˆ T N + ˆ H el where ˆ T N = summationdisplay α (1 / 2) M α 2 α , ˆ T e = summationdisplay i (1 / 2) 2 i (5 . 2 3) ˆ V Ne = summationdisplay α,i Z α | R α r i | , ˆ V ee = summationdisplay i>j 1 | r i r j | , ˆ V NN = summationdisplay α>β Z α Z β | R α R β | (5 . 4 6) Atomic units have been used, in which ¯ h = m e = e = 1 . The simplest molecule in which electron-electron repulsion (and thus electron-electron correlation) is present is H 2 . The electronic Schr¨ odinger equation for even this simplest of multi-electron molecules cannot be solved analytically. As a first approximation, it is typical to follow the Hartree-Fock approach and use Slater determinants to create what are now properly anti-symmetrized molecular orbitals φ el ± for the H + 2 molecule as a starting point, and form an H 2 wavefunction from these to be variationally optimized. In the ground state of H 2 , both electrons will be in the energetically more favorable orbital φ el + . Because of the Pauli principle, the two electrons must then be paired as a singlet, which for this simple two electron case we can write as a product of a spatial wavefunction and a spin wavefunction, just as for the He atom: Φ 1 = radicalbigg 1 2 φ + (1) φ + (2) { α 1 β 2 β 1 α 2 } , (5 . 7) with φ + given by (4.24). This wavefunction works well near R e , but as McQuarrie describes it fails terribly at large R , where it partly represents an H - and an H + ion pair rather than two ground-state H atoms. This can be seen by expanding (5.7) into the atomic orbitals φ A and φ B : Φ 1 = ( φ A + φ B ) 1 (2 + 2 S ) 1 / 2 ( φ A + φ B ) 2 (2 + 2 S ) 1 / 2 = 1 2 + 2 S { φ A φ A + φ A φ B + φ B φ A + φ B φ B } = 1 2 + 2 S { ( φ B φ A + φ A φ B ) + ( φ A φ A + φ B φ B ) } . (5 . 8) The first two terms correspond to one electron/H nucleus, which is appropriate, but the second pair of terms correspond to both electrons being on one nucleus or another (Why are two terms appropriate for each of these limits?). Recall that the ionization energy of a hydrogen atom is 13.6 eV, and that it’s electron affinity is 0.75 eV, and so an equal 42
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contribution of these so-called atomic (or valance bond) and ionic components of the simplest molecular orbital wavefunction has an asymptotic limit for large R that is in error by considerably more than a chemical bond! This problem can be remedied by adding a second configuration made up of the φ - combination: Φ 2 = radicalbigg 1 2 φ - (1) φ - (2) { α 1 β 2 β 1 α 2 } (5 . 9) so that the CI wavefunction becomes 1 Ψ el g = c 1 Φ 1 + c 2 Φ 2 (5 . 10) and the coefficients c i can be obtained by diagonalizing the 2 × 2 interaction (secular) matrix (for each/all separations R ). There are actually six possible spatial-spin wavefunction combinations, but as McQuarrie nicely outlines (pp. 531-534) most terms are zero by symmetry, leading to eq. (5.10). As R → ∞ , c 1 , c 2 1 /
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