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HUCKEL THEORY - University of Calgary CHEM 453

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Molecular Orbital Theory While FMO theory allows prediction of reactions (by thermodynamics, regio or stereochemistry), all predictions seen so far have been qualitative We have predicted that HOMO... or LUMO levels raise or lower due to degree of mixing of orbitals and the charge, or electronegativity, of orbitals that are being mixed Ideally we would like a quantitative method to predict reactions In theory this is possible if we can solve the Schrödinger equation ĤΨ = EΨ" Ψ = electronic wave function (Ψ2 describes where electrons are located) Ĥ = Hamiltonian operator E = energy value The Schödinger equation thus means the energy levels for a molecule are quantized 96# Molecular Orbital Theory To solve the Schrödinger equation, first use the Born-Oppenheimer approximation which states that we can fix the location of the nuclei and consider only motion of the electrons (if everything is moving like in reality, then there are far too many unknowns for the number of equations, impossible to solve) ĤΨ = EΨ" Multiple each side by Ψ and then integrate over all space ∫ ΨĤΨ dτ = ∫ ΨEΨ dτ Since E is only a number, it can be brought out of the integral and solve for E E = ∫ ΨĤΨ dτ ∫ ΨΨ dτ If consider a diatomic molecule with LCAO, ΨMOL = c1φ1 + c2φ2 E = ∫ (c1φ1 + c2φ2)Ĥ(c1φ1 + c2φ2) dτ ∫ (c1φ1 + c2φ2)2 dτ 97# Molecular Orbital Theory E = ∫ (c1φ1Ĥc1φ1 + c1φ1Ĥc2φ2 +c2φ2Ĥc1φ1 + c2φ2Ĥc2φ2) dτ ∫ (c1 2φ1 2 + 2c1φ1c2φ2 + c2 2φ2 2) dτ If we rewrite equation after multiplying with the operator term: Some common terms: 1) ∫ φaφb dτ = Sab Overlap Integral (related to overlap of atomic orbitals) 2) ∫ φaĤφa dτ = Haa Coulomb Integral (related to binding energy of electron in an orbital) 3) ∫ φaĤφb dτ = Hab Resonance Integral (related to energy of an electron in the field of 2 or more nuclei) 98# Molecular Orbital Theory E = c1 2H11 + 2c1c2H12 + c2 2H22 c1 2S11 + 2c1c2S12 + c2 2S22 Substitute these common terms into energy solutions This equation will thus hold valid for each valid solution, the different solutions for E correspond to the different MO energy levels To obtain the solutions, need to undertake a partial derivative with respect to each coefficient to minimize the energy with respect to the coefficient δE / δC1 = 0 δE / δC2 = 0 For the partial derivative with respect to C1: E(2C1S11 + 2C2S12) = 2C1H11 + 2C2H12 C1(H11 – ES11) + C2(H12 – ES12) = 0 For the partial derivative with respect to C2: C1(H12 – ES12) + C2(H22 – ES22) = 0 99# Molecular Orbital Theory The solutions of the partial derivatives were written by grouping the terms for each coefficient to write the equation in a secular equation form C1(H11 – ES11) + C2(H12 – ES12) = 0 C1(H12 – ES12) + C2(H22 – ES22) = 0 Secular Equations This allows the equations to be written in a secular determinant H11 – ES11 H12 – ES12 H12 – ES12 H22 – ES22 = 0 Secular Determinant Diagonal elements: Coulomb integrals (terms dealing with each nuclei) Off-Diagonal elements: Resonance integrals (terms dealing with different nuclei) Remember that this secular determinant was formed considering the mixing of two atom [Show More]

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