H2+ionMO

H2 ionMO - rA Hydrogen Molecule Ion Variation Theory 2 h e2 1 1 1 2 r = 2m 4 o rA rB AB rB r AB EE EA EB E MO = cA A cB B E= E= MO H MO d 2 MO d(cA

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Hydrogen Molecule Ion- Variation Theory r AB + r A r B + - h 2 2m 2 Ψ + e 2 4 πε o 1 r A 1 r B + 1 r AB Ψ = ΕΨ E E E E E A B + - Ψ MO = c A Ψ A + c B Ψ B ground state: Ψ A = 1s A , Ψ B = 1s B E = Ψ MO * H Ψ MO d τ Ψ 2 MO d τ E = (c A Ψ A + c B Ψ B ) * H (c A Ψ A + c B Ψ B ) d τ (c A Ψ A + c B Ψ B ) 2 d τ E = c 2 A Ψ A * H Ψ A d τ + c 2 B Ψ B * H Ψ B d τ +2 c A c B Ψ A * H Ψ B d τ c 2 A Ψ 2 A d τ + c 2 B Ψ 2 B d τ + 2 c A c B Ψ A Ψ Β d τ H AA = Ψ A * H Ψ A d τ Coulomb Integral E A (single electron atom A) H AB = Ψ A * H Ψ B d τ Resonance Integral S AA = Ψ 2 A d τ Atomic Normalization S AB = Ψ A Ψ Β d τ Overlap Integral E = c 2 A H AA + c 2 B H BB +2 c A c B H AB c 2 A S AA + c 2 B S BB + 2 c A c B S AB = N D ( E C A ) C B = 0 = [2c A H AA + 2c B H AB ]D - [2c A S AA + 2c B S AB ]N 0 = 2c A H AA + 2c B H AB - E [2c A S AA + 2c B S AB ] 0 = c A H AA + c B H AB - E [c A S AA + c B S AB
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( E C A ) C B = 0 = c A H AA + c B H AB - E [c A S AA + c B S AB ] ( E C B ) C A = 0 = c B H BB + c A H AB - E [c B S BB + c A S AB ] c A (H AA - E S AA ) + c B (H AB - E S AB ) = 0 c A (H AB - E S AB ) + c B (H BB - E S BB ) = 0 H AA - E S AA H AB - E S AB H AB - E S AB H BB - E S BB = 0 Ax + By = 0 Cx + Dy = 0 A B C D = AD - BC = 0
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H2 ionMO - rA Hydrogen Molecule Ion Variation Theory 2 h e2 1 1 1 2 r = 2m 4 o rA rB AB rB r AB EE EA EB E MO = cA A cB B E= E= MO H MO d 2 MO d(cA

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