Part 2.7: Orbital Diagrams
1
Orbital Diagrams
•
•
•
•
Orbital Interactions
Molecular Orbital Theory
Orbital Energies
MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC
• Hybridization
• Symmetry and Reactivity
2
Atomic Orbitals
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.
1s orbital
2p orbital
3
Atomic Orbitals
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.
1s orbital
2p orbital
4
Atomic Orbitals
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.
Waves can interactconstructively = bonding
destructively = antibonding
not at all = non-bonding
5
Molecular Orbital Theory
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)
LCAO
Y = caya + cbyb
(for diatomic molecules)
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
6
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) ]
7
Combining Atomic Orbitals (H2)
Antibonding
Bonding
8
Combining Atomic Orbitals
Fe(C5H5)2
H2
•
•
•
•
•
2 atoms
Only s orbitals
Linear interaction
Same energy
Uniform symmetry
•
•
•
•
11 relevant atoms
s, p, and d orbitals
various interactions
different energies
9
Combining Atomic Orbitals
Y = caya + cbyb … cnyn
Degree of orbital overlap/mixing depends on:
1) Energy of the orbitals
The closer the energy, the more mixing.
2) Spatial proximity
The atoms must be close enough that there is reasonable
orbital overlap.
3) Symmetry
Atomic orbitals mix if they have similar symmetries.
10
Strength of the bond depends upon the degree of orbital overlap.
Energy of the Orbitals
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).
How do we determine orbital energies?
11
Energy of Orbitals
1) Theoretical calculations
2) Photoelectron spectroscopy
3) Tabulated data
Other peoples UPS/XPS data
12
Photoelectron Spectroscopy
Ionization occurs when matter interacts with light of sufficient
energy (Heinrich Hertz, 1886) (Einstein, A. Ann. Phys. Leipzig
1905, 17, 132-148.)
13
Photoelectron Spectroscopy
Photo-ionization and energy-dispersive analysis of the emitted photoelectrons to
study the composition and electronic state of the sample.
hνo = I(BE) + Ekinetic
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.
14
Photoelectron Spectrometer
X-Ray source
Ion source
Detector
SIMS
Analyzer
Axial Electron Gun
Sample introduction Chamber
Sample
Holder
sample
Roughing Pump
CMA
Slits
Ion Pump
15
Photoelectron Spectrometer
16
Photoelectron Spectroscopy
Counts
17
Photoelectron Spectroscopy
Counts
18
19
Tabulated Data
Miessler and Tarr,
Inorganic Chemistry
Diagram for
methane
(CH4)?
20
Tabulated Data
http://en.wikipedia.org/wiki/Ionization_energy
21
Combining Atomic Orbitals
Y = caya + cbyb … cnyn
Degree of orbital overlap/mixing depends on:
1) Energy of the orbitals
The closer the energy, the more mixing.
2) Spatial proximity
The atoms must be close enough that there is reasonable
orbital overlap.
3) Symmetry
Atomic orbitals mix if they have similar symmetries.
22
Strength of the bond depends upon the degree of orbital overlap.
Symmetry and Orbital Diagrams
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.
J. Chem. Edu. 2004, 81, 997.
23
Constructing MOs
• From inspection
• From Group Theory
24
Constructing MOs
p bond (p and d)
s bond (s, p and d)
p
p
p
d
d bond (d)
d
d
25
Constructing MOs (s-s)
26
Constructing MOs (p-p)
27
Constructing MOs (d-d)
28
Simple Diatomics
29
MO Diagrams from Group Theory
1. 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
4.
5.
6.
7.
8.
9.
Generate a reducible representation
Reduce to Irreducible Representation
Combine central and peripheral orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
30
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
31
HF Orbital Diagram
1. Assign a point group
H-F
C2v
C∞v
C2v
32
HF Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
H-F
C2v
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
H s orbital A1
F s, px, py and pz orbitals
z
y
F
x
H
GH 1 1
1
1
33
HF Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
H-F
C2v
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
H s orbital A1
F s, px, py and pz orbitals
A1
z
y
F
x
H
GFs 1 1
1
1
34
HF Orbital Diagram
1. 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
H-F
C2v
H s orbital A1
F s, px, py and pz orbitals
A1
A1
z
y
F
x
H
GFpz 1 1
1
1
35
HF Orbital Diagram
1. 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
H-F
C2v
H s orbital A1
F s, px, py and pz orbitals
A1 B 1
A1
z
y
F
x
H
GFpx 1 -1
1
-1
36
HF Orbital Diagram
1. 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
H-F
C2v
H s orbital A1
F s, px, py and pz orbitals
A1 B 1 B 2
A1
z
y
F
x
H
GFpy 1 -1 -1
1
37
HF Orbital Diagram
1. 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
H-F
C2v
H s orbital A1
F s, px, py and pz orbitals
A1 B 1 B 2
A1
4. Generate a reducible representation
5. Reduce to irreducible representation
6. Combine orbitals by their symmetry
38
HF Orbital Diagram
6. Combine orbitals by their symmetry
H s orbital
F s, px, py and pz orbitals
39
HF Orbital Diagram
6. Combine orbitals by their symmetry
1s (A1)
pz (A1) px (B1)
py (B2)
2s (A1)
H
H-F
F
H s orbital A1
F s, px, py and pz orbitals
A1 B1 B2
A1
40
HF Orbital Diagram
6. Combine orbitals by their symmetry
A1
1s (A1)
py (B2) px (B1)
pz (A1) px (B1)
py (B2)
A1
H
A1
2s (A1)
H-F
F
41
HF Orbital Diagram
7. Fill MOs with eA1
1s (A1)
py (B2) px (B1)
pz (A1) px (B1)
py (B2)
A1
1 e-
H
A1
2s (A1)
H-F
F
7 e42
e- in MOs
1. 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)
43
HF Orbital Diagram
7. Fill MOs with eA1
1s (A1)
py (B2) px (B1)
pz (A1) px (B1)
py (B2)
A1
H
A1
2s (A1)
H-F
F
44
HF Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
A1
1s (A1)
px (B1)
py (B2)
pz (A1)
px (B1)
py (B2)
A1
Generate a reducible representation
A
Reduce to irreducible representation
H
H-F
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
1
2s (A1)
F
45
HF Orbital Diagram
8. Draw orbitals
F
A1
F
F
1s (A1)
H
B2 B 1
H
A1
F
H
F
F
H
px
F
F
F
z
y
pz py
x
A1
2s (A1)
H-F
F
46
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
47
H2O Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
H2O
C2v
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
H s orbitals A1 + B1
O s, px, py and pz orbitals
z
y
H
O
x
H
GH 2 0 2 0
GH A1 + B1
48
H2O Orbital Diagram
1. 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
H2O
C2v
H s orbitals A1 + B1
O s, px, py and pz orbitals
A1 B 1 B 2
A1
z
y
H
O
x
H
49
H2O Orbital Diagram
1. 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
H2O
C2v
H s orbitals A1 + B1
O s, px, py and pz orbitals
4. Generate a reducible representation
5. Reduce to irreducible representation
6. Combine orbitals by their symmetry
50
HF Orbital Diagram
6. Combine orbitals by their symmetry
H s orbital
O s, px, py and pz orbitals
51
H2O Orbital Diagram
6. Combine orbitals by their symmetry
A1 B 1
pz (A1)
px (B1)
py (B2)
2s (A1)
2xH
O
H s orbital A1 + B1
O s, px, py and pz orbitals
A1 B1 B2
A1
52
H2O Orbital Diagram
6. Combine orbitals by their symmetry
B1
A1
A1 B 1
py (B2)
pz (A1)
px (B1)
py (B2)
A1
B1
2xH
A1
H2O
2s (A1)
O
53
H2O Orbital Diagram
7. Fill MOs with e-
B1
A1
A1 B 1
py (B2)
pz (A1)
px (B1)
py (B2)
A1
B1
2 e-
2xH
A1
H2O
2s (A1)
O
6 e-
54
H2O Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
B1
A1
A1 B1
py (B1)
pz (A1)
px (B1)
py (B2)
A1
B1
Generate a reducible representation
A
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
2s (A1)
1
55
H2O Orbital Diagram
8. Generate SALCs of peripheral atoms
Symmetry adapted linear combination of atomic orbitals (SALC)
• 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.
Pi is the projection operator
li is the dimension of Gi
h is the order of the group
i is an irreducible representation of the group
R is an operation of the group
χi (R) is the character of R in the ith irreducible representation
(R) non-symmetry-adapted basis
56
H2O Orbital Diagram
8. 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
57
H2O Orbital Diagram
8. 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
z
y
f1
H
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
58
H2O Orbital Diagram
8. 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
z
y
f1
H
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
59
H2O Orbital Diagram
8. 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
z
y
f1
H
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
60
H2O Orbital Diagram
8. 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
z
y
f1
H
PA1 = 1/4 [((1) E f1 ) + ((1) C2 f1 ) + ((1) sxz f1 ) + ((1) syz f1 )]
x
O
f1
Hf
GH = A1 + B1
2
f2
f1
f2
PA1 = 1/4 [f1 + f2 + f1 + f2]
PA1 = 1/4 [2f1 + 2f2]
PA1 = 1/2 [f1 + f2]
A1 H1s orbitals
61
H2O Orbital Diagram
8. 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
z
y
f1
H
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
62
H2O Orbital Diagram
8. 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
z
y
f1
H
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
63
H2O Orbital Diagram
8. 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
z
y
f1
H
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
x
O
Hf
2
GH = A1 + B1
64
H2O Orbital Diagram
8. 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
z
y
f1
H
PB1 = 1/4 [((1) E f1 ) + ((-1) C2 f1 ) + ((1) sxz f1 ) + ((-1) syz f1 )]
x
O
f1
Hf
GH = A1 + B1
2
f2
f1
f2
PB1 = 1/4 [f1 - f2 + f1 - f2]
PB1 = 1/4 [2f1 - 2f2]
PB1 = 1/2 [f1 - f2]
B1 H1s orbitals
65
H2O Orbital Diagram
9. Draw SALC with central atom.
B1
A1
A1 B 1
py (B2)
pz (A1)
px (B1)
py (B2)
A1
z
y
H
O
B1
x
H
A1
H2O
2s (A1)
66
H2O Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
B1
A1
A1 B1
py (B2)
pz (A1)
px (B1)
py (B2)
A1
B1
Generate a reducible representation
2s (A )
Reduce to irreducible representation
A
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
1
1
67
Sidenote: Many Electron States
A1
• Determining the symmetry of many electron states
from the symmetry of the individual one electron
wavefunctions.
B2
• Important for formulating spectroscopic selection
rules between orbitals or electronic states.
A1
• State symmetry found from the direct product of all
electron symmetries.
B1
B1
A1
H2O
68
Sidenote: Many Electron States
A1
• Determining the symmetry of many electron states
from the symmetry of the individual one electron
wavefunctions.
B2
• Important for formulating spectroscopic selection
rules between orbitals or electronic states.
A1
• State symmetry found from the direct product of all
electron symmetries.
B1
B1
A1
H2O
69
Sidenote: Many Electron States
B1
H2O: A1
A1
A1 B 2 B 2 A1 A1 B 2
A1 B 2
B1
A1
A 1 B2
B 2 = A1
B2 …etc.
B2
A1
B2
B 2 = A1
or closed shell configurations cancel!
H2O: A1
A1 B2 B2 A1 A1 B2
A1
A1
A1
B2 = A1
A1
A1
H2O
70
Sidenote: Many Electron States
B1
H2O: A1
A1 B 2 B 2 A1 A1 B 2
B 2 = A1
A1
B2
A1
B1
A1
H2O
71
Sidenote: Many Electron States
B1
H2O: A1
A1 B 2 B 2 A1 A1 B 2
H2O+: A1
A1 B 2 B 2 A 1 A1 B 2 = B 2
B 2 = A1
A1
B2
A1
B1
A1
H2O+
72
Sidenote: Many Electron States
H2O: A1
A1 B 2 B 2 A1 A1 B 2
B2
H2O+: A1
A1 B 2 B 2 A 1 A1 B 2 = B 2
A1
H2O- = A1
B1
B 2 = A1
A1
B1
A1
H2O-
73
Sidenote: Many Electron States
H2O: A1
A1 B 2 B 2 A1 A1 B 2
B2
H2O+: A1
A1 B 2 B 2 A 1 A1 B 2 = B 2
A1
H2O- = A1
B1
B 2 = A1
A1
B1
H2O* = B2
A1
H2O*
74
Sidenote: Spin Multiplicity
H2O
H2O+
H2O-
A1
A1
A1
A1
B2
B2
B2
B2
B2
A1
A1
A1
A1
A1
A1
B2
A1
H2O*
or
A1
B2
Spin Multiplicity: (2s+1)G1 or 2
s=0
1A
1
s = 1/2
2B
2
s = 1/2
s=0
s=1
2A
1B
3B
1
2
2
75
H2O Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
B1
A1
A1 B1
py (B2)
pz (A1)
px (B1)
py (B2)
A1
B1
Generate a reducible representation
2s (A )
Reduce to irreducible representation
A
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
1
1
Ground State Symmetry of H2O is 1A1
76
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
77
CO2 Orbital Diagram
1. Assign a point group
CO2
D2h
D∞h
D2h
78
CO2 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
CO2
D2h
C s, px, py and pz orbitals
O px, py and pz orbitals
79
CO2 Orbital Diagram
1. 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
CO2
D2h
C s, px, py and pz orbitals
O px, py and pz orbitals
z
80
CO2 Orbital Diagram
1. 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
CO2
D2h
Ag B3u B2u B1u
C s, px, py and pz orbitals
O px, py and pz orbitals
z
81
CO2 Orbital Diagram
1. 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
CO2
D2h
Ag B3u B2u B1u
C s, px, py and pz orbitals
O px, py and pz orbitals
z
GOpz 2 2
GOpz
0
Ag + B1u
0 0 0
2
2
82
CO2 Orbital Diagram
1. 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
CO2
D2h
Ag B3u B2u B1u
C s, px, py and pz orbitals
O px, py and pz orbitals
z
GOpx 2 -2
GOpx
0
0 0 0
B3u + B2g
2
-2
83
CO2 Orbital Diagram
1. 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
CO2
D2h
Ag B3u B2u B1u
C s, px, py and pz orbitals
O px, py and pz orbitals
z
GOpy 2 -2
GOpy
0
0 0 0
B2u + B3g
-2
2
84
CO2 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
CO2
D2h
Ag B3u B2u B1u
C s, px, py and pz orbitals
O px B3u + B2g
4. Generate a reducible representation py B2u + B3g
5. Reduce to irreducible representation pz Ag + B1u
6. Combine orbitals by their symmetry
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
85
CO2 Orbital Diagram
6. Combine orbitals by their symmetry
C s, px, py and pz orbitals
O px, py and pz orbitals
86
CO2 Orbital Diagram
6. Combine orbitals by their symmetry
B1u
Ag
px
2p
py
pz
B3u B2u
B3u B2u B1u
px
B2g B3g
B3u B2u
2s
Ag
C
py
pz
B2g B3g B1u
B3u B2u
Ag
2p
B1u
Ag
OCO
2xO
87
CO2 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
2p
3. Apply operations
u
Ag
px
py
pz
B3
B2
B1
u
u
u
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
4.
5.
6.
7.
B1
Generate a reducible representation
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with e-
B3u
B2u
B2g
B3g
B3u
2
s
Ag
px
py
pz
B2g
B3g
B1u
B3u
B2u
Ag
2p
B2u
B1
u
Ag
C
OCO
2xO
88
CO2 Orbital Diagram
7. Fill MOs with e-
B1u
Ag
px
2p
py
pz
B3u B2u
B3u B2u B1u
px
B2g B3g
B3u B2u
2s
4 e-
Ag
C
py
pz
B2g B3g B1u
B3u B2u
Ag
2p
B1u
Ag
OCO
2xO
8 e-
89
CO2 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
2p
3. Apply operations
u
Ag
px
py
pz
B3
B2
B1
u
u
u
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
4.
5.
6.
7.
8.
9.
B1
B3u
B2u
B2g
B3g
B3u
2
s
py
pz
B2g
B3g
B1u
B3u
B2u
Ag
2p
B2u
B1
Generate a reducible representation
A
Reduce to irreducible representation C
OCO
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
Ag
px
u
g
2xO
90
CO2 Orbital Diagram
8. Generate SALCs of peripheral atoms
f2
f1
z
GOpz
Ag + B1u
PAg = 1/8 [((1) E f1 ) + ((1) C2 f1 ) + ((1) C2 f1 ) … etc.]
PAg = 1/8 [4f1 + 4f2]
91
CO2 Orbital Diagram
8. Generate SALCs of peripheral atoms
z
92
CO2 Orbital Diagram
9. Draw SALC with central atom.
C
OCO
2xO
93
CO2 Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
Generate a reducible representation
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
94
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
95
Ethene Orbital Diagram
x
z
y
Two different approaches (D2h)
C1 + C 2
H1-4
then combine
CH2
then combine
J. Chem. Edu. 2004, 81, 997
96
Ethene Orbital Diagram
97
Ethene Orbital Diagram
98
Ethene Orbital Diagram
99
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
100
NH3 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
NH2
C3v
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
H s orbitals A1 + E
N s, px, py and pz orbitals
z
x
y
GH 3
0
GH A1 + E
1
101
NH3 Orbital Diagram
1. 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
NH2
C3v
H s orbitals A1 + E
N s, px, py and pz orbitals
A1 E
A1
z
x
y
102
NH3 Orbital Diagram
6. Combine orbitals by their symmetry
H s orbital
N s, px, py and pz orbitals
103
NH3 Orbital Diagram
6. Combine orbitals by their symmetry
A1
E
A1
pz (A1) py, px (E)
E
A1
E
3xH
A1
NH3
s (A1)
N
104
NH3 Orbital Diagram
7. Fill MOs with eA1
E
A1
pz (A1) py, px (E)
E
A1
E
3 e-
3xH
A1
NH3
s (A1)
N
5 e-
105
NH3 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
-if the basis stays the same = +1 A1
-if the basis is reversed = -1
-if it is a more complicated change = 0
4.
5.
6.
7.
8.
9.
A1
E
E
A1
pz py, px (E)
(A1)
s (A1)
E
Generate a reducible representation
A
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
1
106
NH3 Orbital Diagram
8. 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
z
2) generate the rep. then irr. rep. for each set
3) Use projection operator for one basis
x
f3
f2
y
G H = A1 + E
f1
107
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
Separate
classes
108
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
z
x
f3
f2
y
f1
109
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
PA1 ≈ ((1) E f1 ) + ((1) C3+f1 ) + ((1) C3-f1 ) + ((1) s1 f1 ) + ((1) s2 f1 ) + ((1) s2 f1 )
f1
f2
PA1 ≈ [f1 + f2 + f3 + f1 + f3 + f2 ]
PA1 ≈ [ 2f1 + 2f2 + 2f3]
PA1 ≈ [ f1 + f2 + f3]
f1
f3
f3
f2
f3
f2
A1 H1s orbitals
f1
110
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
PE ≈ ((2) E f1 ) + ((-1) C3+f1 ) + ((-1) C3-f1 ) + ((0) s1 f1 ) + ((0) s2 f1 ) + ((0) s2 f1 )
f1
f2
PA1 ≈ [2f1 - f2 - f3]
What about the other E
orbital?
0
f3
f3
f2
0
0
One of the E
orbitals
f1
111
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
112
NH3 Orbital Diagram
8. Generate SALCs of peripheral atoms
3 different E SALCS have been generated but they are all similar.
Use subtraction or addition to generate new SALC.
f3
f2
f3
f1
f2
f1
113
NH3 Orbital Diagram
9. Draw SALC with central atom.
A1
E
A1
pz (A1) py, px (E)
E
A1
E
3xH
A1
NH3
s (A1)
N
114
NH3 Orbital Diagram
9. Draw SALC with central atom.
N
3xH
115
NH3 Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
Generate a reducible representation
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
116
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
117
Benzene MOs and SALC
3 nodes
2 nodes
1 node
0 nodes
118
C6H6 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
C6H6
D6h
only p bonding
C pz orbitals
119
C6H6 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
C6H6
D6h
only p bonding
C pz orbitals
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
C6
C″2
C′2
z axis
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
120
C6H6 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
C6H6
D6h
only p bonding
C pz orbitals
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
C6
4. Generate a reducible representation C″2
5. Reduce to irreducible representation
C′2
z axis
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
Gp: B2g + E1g + A2u + E2u
0
-6
0
2
121
C6H6 Orbital Diagram
1. Assign a point group
2. Choose basis function (orbitals)
3. Apply operations
C6H6
D6h
only p bonding
C pz orbitals
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
C6
4. Generate a reducible representation C″2
5. Reduce to irreducible representation
6. Combine orbitals by their symmetry
C′2
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
Gp: B2g + E1g + A2u + E2u
0
-6
0
2
122
C6H6 Orbital Diagram
6. Combine orbitals by their symmetry
B2g
E2u
E1g
A2u
123
C6H6 Orbital Diagram
7. Fill MOs with eB2g
E2u
E1g
A2u
6 pz orbitals = 6 e-
124
C6H6 Orbital Diagram
1. 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
4.
5.
6.
7.
8.
9.
C6H6
D6h
only p bonding
C pz orbitals
G : B + E + A2u + E2u
p
2g
1g
Generate a reducible representation
Reduce to irreducible representation
Combine orbitals by their symmetry
Fill MOs with eGenerate SALCs of peripheral atoms
Draw peripheral atoms SALC with central atom orbital to
generate bonding/antibonding MOs.
125
C6H6 Orbital Diagram
8. Generate SALCs of peripheral atoms
Simplify usingC6!
D6h
C6
126
C6H6 Orbital Diagram
8. Generate SALCs of peripheral atoms
Simplify usingC6!
D6h
C6
127
C6H6 Orbital Diagram
8. 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
128
C6H6 Orbital Diagram
8. 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
129
C6H6 Orbital Diagram
8. 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
130
C6H6 Orbital Diagram
8. Generate SALCs of peripheral atoms
B ≈ B2g
B
E2
E1
A
A ≈ A2u
131
C6H6 Orbital Diagram
8. 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?
132
C6H6 Orbital Diagram
Contain imaginary components; the real component of the linear
combination may be realized by taking ± linear combinations:
133
C6H6 Orbital Diagram
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
134
C6H6 Orbital Diagram
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)
135
C6H6 Orbital Diagram
What are the pictorial representation of the SALC’s?
136
C6H6 Orbital Diagram
What are the pictorial representation of the SALC’s?
137
Projection Operator: Benzene
What are the pictorial representation of the SALC’s?
3 nodes
2 nodes
1 node
0 nodes
138
Orbital Diagrams
•
•
•
•
Orbital Interactions
Molecular Orbital Theory
Orbital Energies
MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC
• Hybridization
• Symmetry and Reactivity
139
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
140
Side Note: Orbital Hybridization
In 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.
H
C
141
s + p Hybrid Orbitals
Miessler and Tarr,
Inorganic Chemistry
142
s + p + d Hybrid Orbitals
143
BF3 Hybridization
Steps to determine the hybridization of a bond.
BF3
1. Assign a point group
2. Choose basis function (s bonds)
3. Apply operations
D3h
s bonds
-if the basis stays the same = +1
-if the basis is reversed = -1
-if it is a more complicated change = 0
D3h
Гs 3
0
1
3
0
1
144
BF3 Hybridization
Steps to determine the hybridization of a bond.
4. Reduce to irreducible representation
BF3
D3h
s bonds
Гs 3 0 1
3 0
1
Gs: A1’ + E’
145
BF3 Hybridization
Steps to determine the hybridization of a bond.
6. Compare symmetry of irr. rep. to central atom MOs
BF3
D3h
Gs: A1’ + E’
B (s) =
B (px)=
B (py)=
B (pz)=
A1’
E’
E’
A2”
146
BF3 Hybridization
Steps to determine the hybridization of a bond.
6. Compare symmetry of irr. rep. to central atom MOs
z
z
z
y
x
Gs:
y
x
A1’
+
E’
y
x
s = A1’
z
y
x
px = E’
z
y
y
x
py = E’
sp2 hybridization (s, px, py)
pz = A2”
147
Orbital Diagrams
•
•
•
•
Orbital Interactions
Molecular Orbital Theory
Orbital Energies
MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC
• Hybridization
• Symmetry and Reactivity
148
Hybridization
Steps to determine the hybridization of a bond.
1. Assign a point group
2. 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 representation
5. Reduce to irreducible representation
6. Compare symmetry of irr. rep. to central atom MOs
149
Symmetry and Reactivity
(2 + 2) cycloaddition
2 x ethylene
p orbitals
cyclobutane
s bonds
Orbital symmetry is retained during the reaction!
150
Symmetry and Reactivity
(2 + 2) cycloaddition
151
Symmetry and Reactivity
Photo
Reaction
Thermal
Reaction
2 bonding + 2 antibonding eThermally Forbidden
(~115 kcal/mol)
3 bonding + 1 antibonding ePhotochemically Allowed
Orbital Diagrams
•
•
•
•
Orbital Interactions
Molecular Orbital Theory
Orbital Energies
MO Diagrams
– HF, H2O, CO2, C2H4, NH3, Benzene
• SALC
• Hybridization
• Symmetry and Reactivity
153