Polyatomic Molecular Orbital Theory

1 1 Polyatomic Molecular Orbital TheoryTransformational properties of atomic orbitalsAtomic orbitalTransforms assx2+y2+z2pxxpyypzzdz2z2, 2z2-x2-y2dx2-y2x2-y2dxyxydxzxzdyzyzSpy When bonds are formed, atomic orbitals combine according to their symmetry. Symmetry properties and degeneracy of orbitals and bonds can be learned from corresponding character tables by their inspection holding in mind the following transformational properties:dz22 Examples of atomic orbitals symmetry analysisC2vA1zx2, y2, z2A2 RzxyB1x, RyxzB2y, RxyzAtomic orbitalMulliken labelsC2vD3hD4hTdOhspxpypzdz2dx2-y2dxydx zdyzD3hA1 x2+y2, z2A2 RzE (x,y)(x2-y2, xy)A1 A2 zE (Rx,Ry)(xz, yz)TdA1x2+y2+z2A2E(2z2-x2-y2, x2-y2)T1(Rx,Ry,Rz)T2(x,y,z)(xz, yz, xy)OhA1gx2+y2+z2Eg(2z2-x2-y2, x2-y2)T1g(Rx,Ry,Rz)T2g(xz, yz, xy)T1u(x,y,z).

2 D4hA1gx2+y2, z2B1gx2-y2B2gxyEg(Rx,Ry)(xz, yz)A2uzEu(x, y)b2b1a1a1a1a1a2b1b2a1'e'e'a2"a1'e'e'e"e "a1geueua2ua1gb1gb2gegega1t2t2t2eet2t2t2 a1gt1ut1ut1uegegt2gt2gt2gMO diagram of homonuclear diatomic molecules Filling the resulting MO s of homonuclear diatomic molecules with electrons leads to the following results:Bond order = (#Bonding e s - #Antibonding e s)AB1 g2 u1s1s2s2s2p2p3 g4 u1 u5 g2 g6 uAB# of e sBond order # sBond energy, levels in N2molecule Photoelectron spectroscopy of simple molecules is an invaluable source of the information about their electronic structure.

3 The He-I photoelectron spectrum of gaseous N2below proves that there is the - level inversion in this molecule. It also allows identify bonding (peaks with fine vibronic structure) and non-bonding MO (simple peaks) in it. NN1 g2 u1s1s2s2s2p2p3 g4 u1 u5 g2 g6 uE, s Play with some MO to Deal with more than two atoms?You use some Acronyms! Sounds Good! SALCs Symmetry Adapted Linear Combinations LGOs Ligand Group OrbitalsLets Separate the O for the H HOxygenHydrogenHaHbTo the Character Tables5O (py)O (px)O (pz)O (s)A1 OxygenB1B2 OHHzxyHydrogenHa- Hb(s)Ha+ Hb(s)

4 A1B1 OHHzxy6 NOW Let s Make Some MOsB1A1A1 LGOsHydrogensOxygenNOW Let s Make Some MOsB2A1A12 A13 A14 A1 LGOsHydrogensOxygen7 NOW Let s Make Some MOsB2A1A12 A13 A14 A1 LGOsHydrogensOxygenNOW Let s Make Some MOs2 B1B1A1A11B21B1 LGOsHydrogensOxygen8 NOW Let s Make Some MOs2 B1B1A1A11B21B1 LGOsHydrogensOxygenNOW Let s Make Some MOs2 B1B1A1A12 A13 A14 A11B21B1 LGOsHydrogensOxygen9 NOW Let s Make Some MOs2 B1B1A1A12 A13 A14 A11B21B1 LGOsHydrogensOxygenNow for the Million Dollar QuestionWhy is Water Bent???10 Molecular Orbital Theory linear XH2moleculesMolecular orbitals of BeH2 BeH2(D h)1 g22 g23 u2 Group orbitals of 2H's: s + s s - sBeHHzy2 HBe2px2py2pz2s1ss + ss - s-128 eVD h vi g+11x2+y2, z2 u+1-1z u0-2(x,y) g+ u+ g+ u+ u1 g u2 g4 g3 u5 u11 Molecular Orbital Theory Walsh diagramThe Walsh diagram shows what happens to the Molecular orbitals for a set of molecules which are related in structure.

5 In this case, the difference is the H-X-H bond angle which decreases from 180oto 90oMolecular Orbital Theory Walsh diagramWater XHHHOH124) MO Theory and Molecular geometry (Walsh diagrams) Correlate changes in energy of MO s between species ABxof high and lower symmetry, such as BeH2and H2O. BeH22 g23 u2linearBH22a121b223a11, 131oCH2 2a121b223a12, 102oNH22a121b223a121b11, 103oOH22a121b223a121b12, 105oFH2+2a121b223a121b12, 113o2a13a11b2b14a12b22 g3 u4 g u5 uzyOHHxMolecular Orbital Theory BH3BH3has a C3principal axis of symmetry, 3 C2axes ( C3), 3 v, and h it is in a D3hpoint group13 Molecular Orbital Theory BH3 BHHHzyxThe BH3molecule exists in the gas phase, but dimerizes to B2H6(which we will look at a bit later)

6 2 BH3B2H6 The BH3molecule is trigonal planar and we will make the C3principal axis of symmetry the z axis, with the x and y axes in the plane of the molecule. The y axis (arbitrary) will be along one of the B-H Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2 11-1-1-11E 2-10-21014 Molecular Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2 11-1-1-11E 2-10-210 Molecular Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2 11-1-1-11E 2-10-2102s15 Molecular Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2

7 11-1-1-11E 2-10-2102pzMolecular Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2 11-1-1-11E 2-10-2102px 2pydoubly degenerate16 Molecular Orbital Theory D3hCharacter TableD3hE2C33C2 h2S33 vA1 111111A2 11-111-1E 2-102-10A1 111-1-1-1A2 11-1-1-11E 2-10-210 Molecular Orbital Theory LGOs on H atomsHHHIn BH3we need three LGOs, formed from linear combinations of the H 1s orbitalsEC3C2 hS3 v301301 What happens if we carry out the D3hsymmetry operations on this group of H 1s orbitals ?

8 How many remain unchanged?17 Molecular Orbital Theory LGOs on H atomsEC3C2 hS3 v301301D3hE2C33C2 h2S33 vA1 111111E 2-102-10 LGOs301301 The resulting row of characters is also obtained by adding the characters of the A1 and E representationsMolecular Orbital Theory LGOs for BH3 (a1 ) = (1/ 3)( 1 + 2+ 3) (e )1= (1/ 6)(2 1 2 3) (e )2= (1/ 2)( 2 3) 18 Molecular Orbital Theory LGOs for BH3nodal planese a1 Molecular Orbital Theory19 Molecular Orbital Theory BH3 Three bonding MOs are filled, accounting for the three B-H bondsEmpty anti-bonding MOs Empty non-bonding MO Molecular Orbital Theory20 Molecular Orbital Theory NH3 The ammonia molecule, NH3, has C3vsymmetry.

9 With a C3principal axis of symmetry and 3 vertical planes of symmetryzyxNHHHM olecular Orbital Theory NH3C3vE2C33 vA1111A211-1E2-10 Part of the character table of C3v21 Molecular Orbital Theory NH3C3vE2C33 vA1111A211-1E2-10 Part of the character table of C3v2s and 2pzorbitals on NMolecular Orbital Theory NH3C3vE2C33 vA1111A211-1E2-10 Part of the character table of C3v2pxand 2pyorbitals on N22 Molecular Orbital Theory NH3C3vE2C33 vA1111A211-1E2-10 LGOs301 Part of the character table of C3va1+ e2) Molecular orbitals of NH3(C3v)C3vE 2C33 vA11 11zx2+y2, z2A21 1-1E2 -10(x,y) NH3(C3v: E, 2C3, 3 v)The symmetry of 3H s group orbitals : r= 3E+0C3+ v = A1+ EzxNH3H(1)H(2)H(3)y3 HNa1 (2pz)a1 (2s)e (2px, 2py)ea12s1 - s2 - s3- s2 + eVs1 + s2 + s3symmetry adapted linear combinations (SALC) of three 1s orbitals can be found with help of the "projection operator" technique ( Cotton, p.

10 114) eV4a123 Molecular Orbital TheoryMolecular Orbital Theory NH3N-H bondingorbitalslone pair on N24 Molecular Orbital Theory Methane TdMethane has Tdsymmetry, a cubic point groupThe C3axes in CH4coincide with the C-H bondsThe C2and S4axes coincide with the x, y, and z axesC3zxyMolecular Orbital Theory Tdcharacter tableTdE8C33C26S46sdA111111A2111-1-1E2-1 200T130-11-1T230-1-11 Part of the TdCharacter Table25 Molecular Orbital Theory Tdcharacter tableTdE8C33C26S46sdA111111A2111-1-1E2-1 200T130-11-1T230-1-11 Part of the TdCharacter Table2s of CMolecular Orbital Theory Tdcharacter tableTdE8C33C26S46sdA111111A2111-1-1E2-1 200T130-11-1T230-1-11 Part of the TdCharacter Table2px, 2py.