antibindingregionbindingregionantibindingregionQuantum theory of bonding: preliminary considerationsPreliminary considerations: delocalizationX Coordinate (Angstroms)-4 -3 -2 -1 0 1 2 3 4Energy (eV)-60-50-40-30-20-10010X coordinate (Angstroms)-4 -3 -2 -1 0 1 2 3 4Energy (eV)-60-50-40-30-20-10010(a)(b)V = -e2/rA - e2/rB + e2/ReqV = -e2/r2.77 eVH atomH2+moleculezzzxyABrABerAerBeABBArerereV222+−−=EψψV2m2m2m2ee2BB2AA=⎥⎦⎤⎢⎣⎡+∇−∇−∇−hhhQuantum theory of the H2+moleculer (Angstroms)01234Ψ(r) (Angstroms-3/2)0.00.20.40.60.81.01.21.41.6Molecular orbital methodObjective: construct approximate wave functions (molecular orbitals) by combining valence atomic wavefunctions of the atoms forming a bond.The LCAO-MO method: Linear Combination of Atomic Orbitals to forma Molecular Orbital.1s wavefunction0/2/13011arsea−⎟⎟⎠⎞⎜⎜⎝⎛=πϕz/a0-8-6-4-202468ΨgABz/a0-8-6-4-202468ϕ1sABFormation of a sigma bonding molecular orbitalψA= φ1s(rA)ψB= φ1s(rB)ψg= Ng[φ1s(rA) + φ1s(rB)]R/a0R/a0z/a0-8 -6 -4 -2 0 2 4 6 8ΨuABz/a0-8 -6 -4 -2 0 2 4 6 8ϕ1sABFormation of an antibonding sigma molecular orbitalψA= φ1s(rA)ψB= −φ1s(rB)ψu= Nu[φ1s(rA)−φ1s(rB)]R/a0R/a0Exercise: Obtain an expression for the electron probability densitiesψg2and ψu2to show explicitly that ψg2contains a term that increases the electron probability density between the two nuclei.Representing MO’s using the angular wavefunctions1s++1s++σ1s(σg)1s+ −1s++σ1s*(σu)−Correlation diagram for the H2+molecule*(The red curves show the energy without the nuclear repulsionterm e2/rAB).ReMolecular orbital energy level diagram for H2+φ1sAφ1sBσg= Ng(φ1sA + φ1sB)σu= Nu(φ1sA −φ1sB)++++−Molecular orbitals formed from 2s and 2p atomic orbitalsMolecular orbital energy level diagrams for second row homonuclear diatomic moleculesLi2through N2O2through Ne2Exercise: Construct a molecular orbital energy level diagram for N2and O2.Bond order = net number of bonding electron pairsMolecular orbital energy level diagram for O2H2Molecular orbital energy level diagram for O2Liquid oxygen sticks to the poles of a magnet because it is paramagneticMolecular orbital configurations and properties of the second row homonuclear diatomic moleculesMolecular orbital energy level diagram for HFE1s= −685 eVE2s= −34 eVE2p= −8.6 eVFluorineHydrogenE1s= −13.6 eVσσ*HFσ = c1aφ1s(H) + c2aφ2pz(F)σ*= c1bφ1s(H) − c2bφ2pz(F)CarbonOxygenCOσ2sσ*2sσ*2pσ2pπ2pπ*2p2s2s2p2pMolecular orbital energy level diagram for COMolecular orbitals for linear methyleneH C HCH22s2p1sCCH2HHσsσppxpyσs*σp*2s2pC1sHHa13a11b24a11b12b2CH2a). linearb). bentMolecular orbital energy level diagrams for methyleneWalsh diagram for methylenebond angleenergy90o180o2a13a11b21b14a12b2σsσpzpx, pyσs*σp*dz2dxz,yzdz2dxz,yzyzdxydxyxσπδδ∗π∗MMσ∗Metal-Metal Bonding(molecular orbitals constucted from d atomic orbitals)2s2p2s + 2p2s + 2p +2p +2p2s + 2p +2p2pspσspπsp22pπσsp2sp3σsp3Another way to construct molecular wavefunctions:valence-bond hybridized orbitals for the carbon atomsp22pπσsp2sp22pπσsp2sp22pπσsp2lineartrigonal planarsp hybrid orbitalssp2hybrid orbitalstetrahedraltrigonalbipyramidaloctahedralsp3hybrid orbitalsdsp3hybrid orbitalsd2sp3hybrid orbitalssp3hybrid orbitalsdsp3hybrid orbitalsd2sp3hybrid orbitalsdsp3hybrid orbitalsDirectional characteristics of hybridized orbitalsHybrid type Coord. No. Geometry Bond angleBonding in the ethene molecule, C2H4Bonding in the ethyne molecule, C2H2sp hybridizationpypxpypxBonding in conjugated hydrocarbonsExamples: 1,3 butadiene CH2= CH − CH = CH2benzene C6H6Construct sigma bonds using valence-bond hybridization.Construct delocalized molecular orbitals for the pi electrons.Energy levels and molecular orbitals for the π electrons of
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