Roothaan

# Roothaan - Self-Consistent Field Hartree Fock Theory...

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Colby College Self-Consistent Field Hartree Fock Theory: Roothaan Equations LCAO approach: Ψ a = i=1 N c ia φ i Ψ a = one-electron molecular orbital φ i = atomic orbitals for N orbitals (Slater or Gaussian) c ia = orbital coefficients -- variational parameters Molecule wavefunctions are Slater determinants of one-electron MO’s Ψ = 1 / N! Ψ a (1) α (1) Ψ a (1) β (1) Ψ b (1) α (1) Ψ b (1) β (1) Ψ a (2) α (2) Ψ a (2) β (2) Ψ b (2) α (2) Ψ b (2) β (2) Ψ a (3) α (3) Ψ a (3) β (3) Ψ b (3) α (3) Ψ b (3) β (3) : : : : Spin wavefunction: σ (1) = α (1) or β (1) Variation Theorem: vary c ia ’s to find the lowest energy: Hartree-Fock Equations: f 1 Ψ a (1) σ (1) = ε a Ψ a (1) σ (1) ε a = one-electron orbital energy for MO a, b, c, … f 1 = h 1 + j=1 n/2 {2J j (1) - K j (1) } j = all the other filled orbitals for electron 1, 2, 3, … core Hamiltonian: h 1 = h 2 2m 2 1 - k=1 m Z k e 2 4 π ε o r k 1 k = all nuclei electron-electron repulsion: Coulomb operator: J j (1) Ψ a (1) =

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