part 2.7: orbital diagrams 1. orbital diagrams orbital interactions molecular orbital theory orbital...
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Part 2.7: Orbital Diagrams
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Orbital Diagrams• Orbital Interactions• Molecular Orbital Theory• Orbital Energies• MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC• Hybridization• Symmetry and Reactivity
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Atomic Orbital- is a mathematical function (Y) that describes the wave-like behavior of electrons in an atom.Used to calculate the probability (Y2 ) of finding any electron of an atom in any specific region around the atom's nucleus.
Atomic Orbitals
1s orbital
2p orbital
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Atomic Orbital- is a mathematical function (Y) that describes the wave-like behavior of electrons in an atom.Used to calculate the probability (Y2 ) of finding any electron of an atom in any specific region around the atom's nucleus.
Atomic Orbitals
1s orbital
2p orbital
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Atomic Orbitals
constructively = bonding destructively = antibonding
not at all = non-bonding
Waves can interact-
Atomic Orbital- is a mathematical function (Y) that describes the wave-like behavior of electrons in an atom.Used to calculate the probability (Y2 ) of finding any electron of an atom in any specific region around the atom's nucleus.
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1. ATOMIC ORBITALS of different atoms combine to create MOLECULAR ORBITALS 2. The number of ATOMIC ORBITALS = the number of MOLECULAR ORBITALS
3. Electrons in these MOLECULAR ORBITALS are shared by the molecule as whole
4. MOLECULAR ORBITALS can be constructed from Linear Combination of Atomic Orbitals (LCAO)
Molecular Orbital Theory
Y = caya + cbyb (for diatomic molecules)LCAO
BONDING Orbitals have most of the electron density between the two nuclei
ANTI-BONDING Orbitals have a node between the two nuclei
NONBONDING Orbitals are essentially the same as if it was only one nuclei
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Combining Atomic Orbitals
Bonding:Ψ(σ) or Ψ+ = (1/√2 ) [φ(1sa) + φ(1sb) ]
Antibonding:
Ψ(σ*) or Ψ- = (1/√2 ) [φ(1sa) - φ(1sb) ]
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Combining Atomic Orbitals (H2)Antibonding
Bonding
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Combining Atomic OrbitalsH2 Fe(C5H5)2
•2 atoms•Only s orbitals•Linear interaction•Same energy•Uniform symmetry
•11 relevant atoms•s, p, and d orbitals•various interactions•different energies
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Degree of orbital overlap/mixing depends on: 1) Energy of the orbitals
The closer the energy, the more mixing.
2) Spatial proximityThe atoms must be close enough that there is
reasonable orbital overlap.
3) SymmetryAtomic orbitals mix if they have similar symmetries.
Combining Atomic Orbitals
Strength of the bond depends upon the degree of orbital overlap.
Y = caya + cbyb … cnyn
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For heteronuclear molecules:
1. The bonding orbital(s) will reside predominantly on the atom of lower orbital energy (the more electronegative atom).
2. The anti-bonding orbital(s) will reside predominantly on the atom with greater orbital energy (the less electronegative atom).
Energy of the Orbitals
How do we determine orbital energies?
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Energy of Orbitals1) Theoretical calculations
2) Photoelectron spectroscopy
3) Tabulated dataOther peoples UPS/XPS data
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Photoelectron Spectroscopy
Ionization occurs when matter interacts with light of sufficient energy (Heinrich Hertz, 1886) (Einstein, A. Ann. Phys. Leipzig 1905, 17, 132-148.)
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Photo-ionization and energy-dispersive analysis of the emitted photoelectrons to study the composition and electronic state of the sample.
X-ray Photoelectron Spectroscopy(XPS)
- using soft (200-2000 eV) x-ray excitation to examine core-levels.
Ultraviolet Photoelectron Spectroscopy(UPS)
- using vacuum UV (10-45 eV) radiation from discharge lamps to examine valence levels.
Photoelectron Spectroscopy
hνo = I(BE) + Ekinetic
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X-Ray source
Ion source
Axial Electron Gun
Detector
CMAsample
SIMS Analyzer
Sample introduction Chamber
Sample Holder
Ion PumpRoughing Pump Slits
Photoelectron Spectrometer
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Photoelectron Spectrometer
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Photoelectron Spectroscopy
Counts
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Photoelectron Spectroscopy
Counts
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Miessler and Tarr, Inorganic Chemistry
Tabulated Data
Diagram for methane
(CH4)?
21http://en.wikipedia.org/wiki/Ionization_energy
Tabulated Data
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Degree of orbital overlap/mixing depends on: 1) Energy of the orbitals
The closer the energy, the more mixing.
2) Spatial proximityThe atoms must be close enough that there is
reasonable orbital overlap.
3) SymmetryAtomic orbitals mix if they have similar symmetries.
Combining Atomic Orbitals
Strength of the bond depends upon the degree of orbital overlap.
Y = caya + cbyb … cnyn
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Symmetry and Orbital Diagrams
J. Chem. Edu. 2004, 81, 997.
1. Number of MOs = number of incipient orbitals. This rule could be
referred to as “the conservation of orbitals.” 2. Orbitals of the same symmetry mix.
3. Orbital interactions can be bonding, nonbonding or antibonding.
4. There are three basic types of orbital overlap: s (end on interaction), p (side by side approach) and d (off-axis approach).
5. Orbitals with the correct symmetry and most similar energy mix to the greatest extent.
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Constructing MOs
• From inspection
• From Group Theory
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Constructing MOss bond (s, p and d)
d d
p bond (p and d)
p p p d
d bond (d)
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Constructing MOs (s-s)
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Constructing MOs (p-p)
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Constructing MOs (d-d)
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Simple Diatomics
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MO Diagrams from Group Theory1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to Irreducible Representation6. Combine central and peripheral orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
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MO Diagrams from Group Theory• H-F
– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method• Benzene
– Real + Imaginary SALC
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HF Orbital Diagram1. Assign a point group
C2vC∞v
H-FC2v
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HF Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
F
H
z
xy
GH 1 1 1 1
H s orbitalF s, px, py and pz orbitals
H-FC2v
A1
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HF Orbital DiagramH-F1. Assign a point group
2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
C2v
F
H
z
xy
GFs 1 1 1 1
H s orbitalF s, px, py and pz orbitals
A1
A1
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H s orbitalF s, px, py and pz orbitals
HF Orbital DiagramH-F1. Assign a point group
2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
C2v
F
H
z
xy
GFpz 1 1 1 1
A1
A1 A1
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H s orbitalF s, px, py and pz orbitals
HF Orbital DiagramH-F1. Assign a point group
2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
C2v
F
H
z
xy
GFpx 1 -1 1 -1
A1
A1 A1B1
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H s orbitalF s, px, py and pz orbitals
HF Orbital DiagramH-F1. Assign a point group
2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
C2v
F
H
z
xy
GFp
y
1 -1 -1 1
A1
A1 A1B1 B2
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HF Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry
H s orbitalF s, px, py and pz orbitals
H-FC2v
A1
A1 A1B1 B2
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HF Orbital Diagram6. Combine orbitals by their symmetry
H s orbitalF s, px, py and pz orbitals
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HF Orbital Diagram6. Combine orbitals by their symmetry
H F
2s (A1)
H s orbitalF s, px, py and pz orbitals
A1
A1 A1B1 B2
pz (A1)py (B2)
px (B1)1s (A1)
H-F
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HF Orbital Diagram6. Combine orbitals by their symmetry
H F
2s (A1)
pz (A1)py (B2)
px (B1)1s (A1)
A1
A1
A1
py (B2) px (B1)
H-F
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HF Orbital Diagram7. Fill MOs with e-
H F
2s (A1)
pz (A1)py (B2)
px (B1)1s (A1)
A1
A1
py (B2) px (B1)
1 e- 7 e-
A1
H-F
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e- in MOs1. Electrons preferentially occupy molecular
orbitals that are lower in energy. (Aufbau Principle)
2. If two electrons occupy the same molecular orbital, they must be spin paired. (Pauli Exclusion Principle)
3. When occupying degenerate molecular orbitals, electrons occupy separate orbitals with parallel spins before pairing. (Hund’s Rule)
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HF Orbital Diagram7. Fill MOs with e-
H F
2s (A1)
pz (A1)py (B2)
px (B1)1s (A1)
A1
A1
py (B2) px (B1)
H-F
A1
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1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
2s (A1)
HF Orbital Diagram
H F
pz (A1)
py (B2)
px (B1)1s (A1)
A1
A1
A1
py (B2) px (B1)
H-F
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HF Orbital Diagram8. Draw orbitals
2s (A1)
pz py px
1s (A1)
A1
A1
B2 B1
F
H
z
xy
F
F
H
F
F
FF
H F
F
F
H
A1
H-F
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• H-F– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
MO Diagrams from Group Theory
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H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
OH
z
xy
GH 2 0 2 0
H s orbitalsO s, px, py and pz orbitals
H2OC2v
H
A1 + B1GH
A1 + B1
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H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
OH
z
xy
H s orbitalsO s, px, py and pz orbitals
H2OC2v
H
A1 + B1
A1 A1B1 B2
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H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry
H s orbitalsO s, px, py and pz orbitals
H2OC2v
A1 + B1
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HF Orbital Diagram6. Combine orbitals by their symmetry
H s orbitalO s, px, py and pz orbitals
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H2O Orbital Diagram6. Combine orbitals by their symmetry
2 x H O
2s (A1) H s orbitalO s, px, py and pz orbitals
A1 + B1
A1 A1B1 B2
pz (A1)py (B2)
px (B1)A1 B1
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H2O Orbital Diagram6. Combine orbitals by their symmetry
2 x H O
2s (A1)
pz (A1)py (B2)
px (B1)A1 B1
H2OA1
A1
A1
B1
B1
py (B2)
H2O Orbital Diagram7. Fill MOs with e-
2 x H O
2s (A1)
pz (A1)py (B2)
px (B1)A1 B1
H2OA1
A1
A1
B1
B1
py (B2)
2 e- 6 e- 54
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H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
2s (A1)
pz (A1)
py (B2)
px (B1)A1 B1
A1
A1
B1
B1
py (B1)
A1
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H2O Orbital Diagram8. Generate SALCs of peripheral atoms
• Use projection operator to generate SALC.
• Projection operators constitute a method of generating the symmetry allowed combinations.
• Taking one AO and projecting it out using symmetry.
Symmetry adapted linear combination of atomic orbitals (SALC)
Pi is the projection operatorli is the dimension of Gi
h is the order of the groupi is an irreducible representation of the groupR is an operation of the groupχi (R) is the character of R in the ith irreducible representation(R) non-symmetry-adapted basis
57
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the atomic orbitals in the molecule into sets which are equivalent by symmetry
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
58
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
59
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
60
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
61
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
f1 f2 f1 f2
PA1 = 1/4 [f1 + f2 + f1 + f2]
PA1 = 1/4 [2f1 + 2f2]
PA1 = 1/2 [f1 + f2]
HO
H
A1 H1s orbitals
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H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
63
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
64
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
65
H2O Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
OH
z
xy
Hf1 f2
A1 + B1GH =
f1 f2 f1 f2
PB1 = 1/4 [f1 - f2 + f1 - f2]
PB1 = 1/4 [2f1 - 2f2]
PB1 = 1/2 [f1 - f2]
HO
H
B1 H1s orbitals
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
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H2O Orbital Diagram9. Draw SALC with central atom.
2s (A1)
pz (A1)py (B2)
px (B1)A1 B1
H2OA1
A1
A1
B1
B1
py (B2)
OH
z
y
H
x
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H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
2s (A1)
pz (A1)
py (B2)px (B1)
A1 B1
A1
A1
A1
B1
B1
py (B2)
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Sidenote: Many Electron States
H2OA1
A1
A1
B1
B1
B2
• Determining the symmetry of many electron states from the symmetry of the individual one electron wavefunctions.
• Important for formulating spectroscopic selection rules between orbitals or electronic states.
• State symmetry found from the direct product of all electron symmetries.
69H2OA1
A1
A1
B1
B1
B2
• Determining the symmetry of many electron states from the symmetry of the individual one electron wavefunctions.
• Important for formulating spectroscopic selection rules between orbitals or electronic states.
• State symmetry found from the direct product of all electron symmetries.
Sidenote: Many Electron States
70H2OA1
A1
A1
B1
B1
B2
H2O: A1 A1 B2 B2 A1 A1 =
A1 B2 B2 A1 A1
B2 …etc.
or closed shell configurations cancel!
A1 A1 A1
A1 B2 B2
= A1 B2 B2
H2O: A1 A1 B2 B2 A1 A1 = A1 B2 B2
A1
Sidenote: Many Electron States
71H2OA1
A1
A1
B1
B1
B2
H2O: A1 A1 B2 B2 A1 A1 = A1 B2 B2
Sidenote: Many Electron States
72H2O+
A1
A1
A1
B1
B1
B2
H2O: A1 A1 B2 B2 A1 A1 = A1 B2 B2
H2O+: A1 A1 B2 B2 A1 A1 =B2 B2
Sidenote: Many Electron States
73H2O-
A1
A1
A1
B1
B1
B2
H2O: A1 A1 B2 B2 A1 A1 = A1 B2 B2
H2O+: A1 A1 B2 B2 A1 A1 =B2 B2
H2O- = A1
Sidenote: Many Electron States
74H2O*
A1
A1
A1
B1
B1
B2
H2O: A1 A1 B2 B2 A1 A1 = A1 B2 B2
H2O+: A1 A1 B2 B2 A1 A1 =B2 B2
H2O- = A1
Sidenote: Many Electron States
H2O* = B2
75
(2s+1)G1 or 2
A1
A1
B2
H2O
A1
H2O+
B2
H2O-
A1
Sidenote: Spin MultiplicityH2O*
B2
A1
A1
B2
A1
A1
B2
A1
A1
B2
Spin Multiplicity:
s = 0
1A1
s = 1/2
2B2
s = 1/2
2A1
s = 0
1B2
A1
A1
B2
or
s = 1
3B2
76
H2O Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
2s (A1)
pz (A1)
py (B2)px (B1)
A1 B1
A1
A1
A1
B1
B1
py (B2)
Ground State Symmetry of H2O is 1A1
77
MO Diagrams from Group Theory• H-F
– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
78
CO2 Orbital Diagram1. Assign a point group
D2hD∞h
CO2D2h
79
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
80
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
z
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
81
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
Ag
z
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
B3u B2u B1u
82
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
Ag
z
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
B3u B2u B1u
GOpz 2 2 0 00 0 2 2
GOpz Ag + B1u
83
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
Ag
z
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
B3u B2u B1u
GOpx 2 -2 0 00 0 2 -2
GOpx B3u + B2g
84
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
Ag
z
C s, px, py and pz orbitalsO px, py and pz orbitals
CO2D2h
B3u B2u B1u
GOpy 2 -2 0 00 0 -2 2
GOpy B2u + B3g
85
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry
Ag
C s, px, py and pz orbitals
CO2D2h
B3u B2u B1u
O px
py pz
B2u + B3g
B3u + B2g
Ag + B1u
86
CO2 Orbital Diagram6. Combine orbitals by their symmetry
C s, px, py and pz orbitalsO px, py and pz orbitals
87
Ag
CO2 Orbital Diagram6. Combine orbitals by their symmetry
C 2 x O
Ag
B3u B2u B1u
2s
2p
2pB2g
B3u
px py pz
B2u
B3g
Ag
B1u
px py pz
B2g B3g
OCO
Ag
B1u
B1u
B3u B2u
B3u B2u
88
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
C 2 x O
Ag
B3u B2u B1u
2s
2p
2pB2g
B3u
px py pz
B2u
B3g
Ag
B1u
px py pz
B2g B3g
OCO
Ag
Ag
B1u
B1u
B3u B2u
B3u B2u
89
CO2 Orbital Diagram7. Fill MOs with e-
C 2 x O
Ag
B3u B2u B1u
2s
2p
2pB2g
B3u
px py pz
B2u
B3g
Ag
B1u
px py pz
B2g B3g
OCO
Ag
Ag
B1u
B1u
B3u B2u
B3u B2u
4 e- 8 e-
90
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
C 2 x O
Ag
B3u B2u B1u
2s
2p
2pB2g
B3u
px py pz
B2u
B3g
Ag
B1u
px py pz
B2g B3g
OCO
Ag
Ag
B1u
B1u
B3u B2u
B3u B2u
91
CO2 Orbital Diagram8. Generate SALCs of peripheral atoms
z
GOpz Ag + B1u
f1f2
PAg = 1/8 [((1) E f1 ) + ((1) C2 f1 ) + ((1) C2 f1 ) … etc.]
PAg = 1/8 [4f1 + 4f2]
92
CO2 Orbital Diagram8. Generate SALCs of peripheral atoms
z
93
CO2 Orbital Diagram9. Draw SALC with central atom.
C 2 x OOCO
94
CO2 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
95
• H-F– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
MO Diagrams from Group Theory
96
Two different approaches (D2h)
Ethene Orbital Diagram
C1 + C2
H1-4
then combine
CH2
then combine
C CH
HH
H
z
y
x
C CH
HH
HC C
H
HH
H
J. Chem. Edu. 2004, 81, 997
97
C CH
HH
H
Ethene Orbital Diagram
98
Ethene Orbital Diagram
C CH
HH
H
99
C CH
HH
H
Ethene Orbital Diagram
100
• H-F– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
MO Diagrams from Group Theory
101
N
HH
H
NH3 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
z
x
y
GH 3 0 1
H s orbitalsN s, px, py and pz orbitals
NH2C3v
A1 + EGH
A1 + E
102
N
HH
H
NH3 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
z
x
y
H s orbitalsN s, px, py and pz orbitals
NH2C3v
A1 + E
A1 E A1
103
NH3 Orbital Diagram6. Combine orbitals by their symmetry
H s orbitalN s, px, py and pz orbitals
104
E
A1
NH3 Orbital Diagram6. Combine orbitals by their symmetry
3 x H N
s (A1)
pz (A1) py, px (E)EA1
NH3
A1
A1
E
105
E
A1
NH3 Orbital Diagram
3 x H N
s (A1)
pz (A1) py, px (E)EA1
NH3
A1
A1
E
7. Fill MOs with e-
3 e- 5 e-
106
NH3 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
E
A1
s (A1)
pz (A1)
py, px (E)EA1
A1
A1
E
107
NH3 Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the atomic orbitals in the molecule into sets which are equivalent by symmetry
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
f1f2
A1 + EGH =
N
HH
H
z
x
y
f3
108
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
Separate classes
109
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
f1f2
N
HH
H
z
x
y
f3
110
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
PA1 ≈ ((1) E f1 ) + ((1) C3+f1 ) + ((1) C3
-f1 ) + ((1) s1 f1 ) + ((1) s2 f1 ) + ((1) s2 f1 )
f1 f2 f3
PA1 ≈ [f1 + f2 + f3 + f1 + f3 + f2 ]
PA1 ≈ [ 2f1 + 2f2 + 2f3]A1 H1s orbitals
f1 f3 f2
PA1 ≈ [ f1 + f2 + f3] f1f2
HH
Hf3
111
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
PE ≈ ((2) E f1 ) + ((-1) C3+f1 ) + ((-1) C3
-f1 ) + ((0) s1 f1 ) + ((0) s2 f1 ) + ((0) s2 f1 )
f1 f2 f3
PA1 ≈ [2f1 - f2 - f3] One of the E orbitals
0 0 0
f1f2
HH
Hf3
What about the other E orbital?
112
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
113
8. Generate SALCs of peripheral atoms
NH3 Orbital Diagram
f1f2
HH
Hf3
3 different E SALCS have been generated but they are all similar.
Use subtraction or addition to generate new SALC.
f1f2
HH
Hf3
114
E
A1
NH3 Orbital Diagram
3 x H N
s (A1)
pz (A1) py, px (E)EA1
NH3
A1
A1
E
9. Draw SALC with central atom.
HH
HH
HH
HH
H
115
NH3 Orbital Diagram9. Draw SALC with central atom.
3 x HN
HH
HH
HH
HH
H
116
NH3 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
117
• H-F– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
MO Diagrams from Group Theory
118
Benzene MOs and SALC
0 nodes
1 node
2 nodes
3 nodes
119
C6H6 Orbital Diagram1. Assign a point group2. Choose basis function (orbitals)
C6H6D6h
only p bondingC pz orbitals
120
C6H6 Orbital DiagramC6H6D6h
only p bondingC pz orbitals
1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ
z axis
C′2C″2
C6
6 0 0 0 -2 0 0 0 0 -6 0 2
121
C6H6 Orbital DiagramC6H6D6h
only p bondingC pz orbitals
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ
z axis
Gp: B2g + E1g + A2u + E2u
C′2C″2
C6
1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation
6 0 0 0 -2 0 0 0 0 -6 0 2
122
C6H6 Orbital DiagramC6H6D6h
only p bondingC pz orbitals
D6h E 2C6 2C3 C2 3C′2 3C″2 i 2S3 2S6 σh 3 σd 3 σv
Гπ 6 0 0 0 -2 0 0 0 0 -6 0 2
C′2C″2
C6
Gp: B2g + E1g + A2u + E2u
1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry
123
C6H6 Orbital Diagram6. Combine orbitals by their symmetry
E1g
B2g
A2u
E2u
124
C6H6 Orbital Diagram
E1g
B2g
A2u
E2u
7. Fill MOs with e-
6 pz orbitals = 6 e-
125
1. Assign a point group2. Choose basis function (orbitals)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Combine orbitals by their symmetry7. Fill MOs with e-
8. Generate SALCs of peripheral atoms9. Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
C6H6 Orbital DiagramC6H6D6h
only p bondingC pz orbitals
Gp: B2g + E1g + A2u + E2u
126
Simplify usingC6!
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
C6D6h
127
Simplify usingC6!
C6
D6h
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
128
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
A orbital
129
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
130
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
B orbital
131
C6H6 Orbital Diagram
E1
B
A
E2
8. Generate SALCs of peripheral atoms
B ≈ B2g
A ≈ A2u
132
C6H6 Orbital Diagram8. Generate SALCs of peripheral atoms
To generate SALCs, the steps are:
1) group the similar AOs
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
ok
ok
What?
Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations:
C6H6 Orbital Diagram
133
134
Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations:
For C6 point group:
or from Euler’s formula
C6H6 Orbital Diagram
135
Contain imaginary components; the real component of the linear combination may be realized by taking ± linear combinations:
divide out and remove prefactor constant (-i√3)
C6H6 Orbital Diagram
136
What are the pictorial representation of the SALC’s?
C6H6 Orbital Diagram
137
What are the pictorial representation of the SALC’s?
C6H6 Orbital Diagram
138
Projection Operator: BenzeneWhat are the pictorial representation of the SALC’s?
0 nodes
1 node
2 nodes
3 nodes
139
Orbital Diagrams• Orbital Interactions• Molecular Orbital Theory• Orbital Energies• MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC• Hybridization• Symmetry and Reactivity
140
• H-F– diatomic, H = 1s; F = 2s, 3 x 2p
• H2O– triatomic, H = 2 x 1s; O = 2s, 3 x 2p
• CO2
– p bonding, O = 2s, 3 x 2p; C = 2s, 3 x 2p
• C2H4
– Fragmentation method
• NH3/Benzene– Real + Imaginary SALC
MO Diagrams from Group Theory
141
Side Note: Orbital HybridizationIn chemistry, hybridization is the concept of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the component atomic orbitals) suitable for the pairing of electrons to form chemical bonds.
CH
142
Miessler and Tarr, Inorganic Chemistry
s + p Hybrid Orbitals
143
s + p + d Hybrid Orbitals
144
1. Assign a point group2. Choose basis function (s bonds)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
BF3 HybridizationSteps to determine the hybridization of a bond.
BF3D3h
s bonds
D3h
Гs 3 0 1 3 0 1
145
4. Reduce to irreducible representation
BF3 HybridizationSteps to determine the hybridization of a bond.
BF3D3h
s bonds
Гs 3 0 1 3 0 1
Gs: A1’ + E’
146
6. Compare symmetry of irr. rep. to central atom MOs
BF3 HybridizationSteps to determine the hybridization of a bond.
BF3D3h
Gs: A1’ + E’
B (s) = A1’
B (px)= E’
B (py)= E’
B (pz)= A2”
147
6. Compare symmetry of irr. rep. to central atom MOs
BF3 HybridizationSteps to determine the hybridization of a bond.
Gs: A1’ + E’ s = A1
’
z
y
x
z
y
x
z
y
x
px = E’ py = E’
z
y
x
y
z
y
x
pz = A2”
sp2 hybridization (s, px, py)
148
Orbital Diagrams• Orbital Interactions• Molecular Orbital Theory• Orbital Energies• MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC• Hybridization• Symmetry and Reactivity
149
Hybridization
1. Assign a point group2. Choose basis function (s bonds)3. Apply operations
-if the basis stays the same = +1-if the basis is reversed = -1-if it is a more complicated change = 0
4. Generate a reducible representation5. Reduce to irreducible representation6. Compare symmetry of irr. rep. to central atom MOs
Steps to determine the hybridization of a bond.
150
(2 + 2) cycloaddition
Symmetry and Reactivity
p orbitals2 x ethylene cyclobutane
s bonds
Orbital symmetry is retained during the reaction!
151
(2 + 2) cycloaddition
Symmetry and Reactivity
Symmetry and Reactivity
2 bonding + 2 antibonding e-
Thermally Forbidden(~115 kcal/mol)
3 bonding + 1 antibonding e-
Photochemically Allowed
Thermal Reaction
Photo Reaction
153
Orbital Diagrams• Orbital Interactions• Molecular Orbital Theory• Orbital Energies• MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC• Hybridization• Symmetry and Reactivity
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