Ch 5 Lecture 3 Group Theory and MOs

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Ch 5 Lecture 3 Group Theory and MOs

• The Group Theory Approach to Molecular Orbitals
• Steps of the Group Theory Approach
• Find the point group (substitute D2h for D8h, and
  C2h for C8h)
• Assign xyz coordinates (Cn z, y points toward
  central atom)
• Find characters and representations for groups of
  outer atoms (2s, 2p)
• An orbital that changes position 0
• An orbital in the same position 1
• An orbital with reversed sign -1
• Reduce each representation to its irreducible
  representations
• This identifies the symmetries of the various
  group orbitals
• Group orbitals combinations of AOs with
  matching symmetry
• Find atomic orbitals of the central atom with the
  same symmetry as the group orbitals (ie, same
  irreducible representations) and combine those of
  similar energies to form MOs
• The total number of MOs formed number of AOs
  used from all atoms

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• H2O
• Water is in the C2v point group
• The C2 axis z-axis xz plane is the plane of
  the molecule
• Since the molecule is planar, we dont need to
  assign axes to Hs
• H(1s) orbitals have no directionality
• Use the H(1s) orbitals to find the reducible
  representation
• Use the reduction formula to identify the
  irreducible representations
• A1 caH(1s)a cb(H1s)b
• B1 caH(1s)a - cb(H1s)b

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• Find O AOs with the same symmetry
• a) 1s, 2s orbitals are of A1 symmetry (s is
  always most symmetric)
• b) 2pz orbital is of A1 symmetry (use z in
  character table)
• c) 2px orbital is of B1 symmetry (use x in
  character table)

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• There are 6 total MOs
• The A1 group O(2s) O(pz)
• Y1 Bonding
• Y3 Nonbonding
• Y5 Antibonding
• b. The B1 group O(px)
• Y2 Bonding
• Y6 Antibonding
• The oxygen py Y4
• 8 valence electrons fill Y1-Y4
• BO 2
• Lone pair 2
• Compare to Lewis Structure
• 1 lone pair nonbonding HOMO (2py)
• 1 lone pair LOMO (Y3 2s)
• 2 bonding pairs in the middle
• All four orbitals are different

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• NH3
• Ammonia is in C3v point group
• C3 axis is assigned as the z-axis, the yz plane
  passes through N and one H
• Use character table to find irreducible
  representations
• Identify orbital and group orbital symmetries
• A1 Ha Hb Hc
• E Hb Hc
• 2Ha Hb Hc
• A1 s, pz
• E px, py

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Ha
A1
Hb
Hc
NH3, there are 3 hydrogen atoms. ? 3 1s orbitals
2Ha
Hb
Hc
E
Hc
Hc
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Bonding molecular orbitals
A1
s
py
E
px
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• Combine Orbitals to give 7 MOs
• A1 group 2s 2pz a1
• Bonding 2a1
• Antibonding 4a1
• Nonbonding 3a1
• E group 2px 2py e
• 2 bonding (degenerate) 1e
• 2 antibonding (degenerate) 2e
• Put in 8 valence electrons
• BO 3 (2a1, 1e)
• Lone pair 3a1

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• D. BF3 (Lewis Acid)
• Point Group D3h
• C3 axis z-axis, F y-axis points to B, F x-axis
  is in the plane of the molecule
• Find Reducible, Irreducible Representations
• Find suitable orbitals

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• Results
• BF should have some double bond character
• 4a1, 3e are bonding BO 3
• 1a2 is somewhat bonding too
• The LUMO is an empty p orbital
• 2a2 is LUMO
• Lewis acidity explained
• More complex molecules by computer program

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• Hybrid Orbitals
• Utility
• Developed in Valence Bond Theory
• Not needed anymore, but emphasizes symmetry
• Hybrid Orbitals
• Orbitals on a central atom can be combined 1s
  3p orbitals 4sp3 orbitals
• Equivalent, directional orbitals result
• Methane
• Reducible representation found from what happens
  to four vectors
• Irreducible representations found as usual
• A1 2s, T2 px, py, pz (or dxy, dxz, dyz) sp3
  hybrid orbitals
• Example Find hybridization and orbital involved
  for BF3

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Molecules we learn

• Homonuclear diatomic molecules (nonpolar)
• H2 ? Ne2
• Heteronuclear diatomic molecules (polar)
• C?O, LiF-ionic
• Linear large molecules (group orbitals group
  theory approach)
• FHF- (central atom ? s orbital only)
• OCO (central atom ? s, p orbitals)
• Non-linear large molecules (group orbitals
  group theory approach)
• H2O (outer atoms ? s orbital only)
• NH3 (outer atoms ? s orbital only)
• BF3 (the most difficult) (outer atoms ? s,p
  orbital only)

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We need

• Able to draw molecular orbital diagrams of all
  these molecules.
• Energy level MO sketch
• Bond order prediction