6.2 More about Energy Levels and Molecular OrbitalsThe BOUND and SUM of energy levels（能级的 “和” 与 “界”）Alternant Hydrocarbons（交替烃）Bound of the Energy LevelsTheoremThe energy levels of the conjugated hydrocarbons are bounded, satisfying |x|3 |x| FbFg FaFbAnthraceneNaphthalene4. Molecular Graph0.840.890.450.390.520.550.730.800.100.450.406.4 Aromaticity Aromaticity, Anti-aromaticity, 4n+2 rule Resonance Energy （共振能） The Eight-parameter Scheme The Five-parameter Scheme1. Aromaticity, anti-aromaticity and 4n+2 ruleCyclic conjugated hydrocarbons are, either,Aromatic Having 4n+2 pi electrons, extra stabilities is originated from the -electron delocalizationAnti-aromatic Having 4n pi electrons, destabled due to the -electron delocalization2. Resonance Energy, a Criteria of AromaticityRE = E ERER? 0 Aromatic, benzene, etc. = 0 non-aromatic, acyclic hydrocarbons= (13.128+2*13.1135)/3= 13.124 E = 13.68 RE = 13.68-13.124 = 0.56 (b)The Eight-parameter schemeDefine: REPE = RE/NResonance Energies per ElectronFor Benzene REPE = 0.39/6 =0.065For Naphthalene REPE = 0.55/10 = 0.055The Eight-parameter schemeMoleculeRE(b)REPE(H-S) benzene2.000.065naphthalene3.680.056anthracene3.360.020biphenylene4.510.027pentalene2.46*-0.018haptalene3.62*-0.0044. Five-parameter SchemeJ-T-H five-parameter SchemeYuansheng Jiang (江元生） Au-Chin Tang （唐敖庆） Roald HoffmannProblem Set:7，8，10，14Charge DensitiesFor neutral A.Hs, the electron charges on all of the carbon atoms are equal to 1.TheoremProofFor A.H.s, the energy levels occurs in pairs (x). The molecular orbitals of each pair should have the property:Charge DensitiesProofThe probability of finding the rth AO in all of the MOs is equal to 1m: the MO index for bonding MOs m: the MO index for anti-bonding MOs