Molecular Orbitals: Introduction
Molecular Orbitals: How to Build an MO Diagram
MOs are made up of linear combinations of AOs.
The two 1s AOs in H2 Φ1 and Φ2 interact to form MOs Ψ+ and Ψ– .
How to Build an MO Diagram: Diatomic
1) Determine point group (either D∞h or C∞v)
2) Use character table to find symmetry labels on SALCAOs for each pair of AOs
3) SALCAOs are the MOs: in-phase = bonding, out-of-phase = antibonding
 (usual) MO energy order: MOs from low-E AOs < MOs from high-E AOs
then b < nb < ab
4) Count orbitals and electrons, fill orbitals with electrons:
 # AOs in = # MOs out, # MOs filled = # electrons / 2
 fill MOs low-E to high-E, (usually) b < nb < ab
 (usually) b MOs are mostly higher  atom character, ab are lower 
Ψ+
How to Build an MO Diagram: Polyatomic with Central Atom
Lewis  VSPER  point group
Use character table to find symmetry labels on centre atom AOs
Use character table to find symmetry labels on each set of SALCAOs
Match symmetries of centre AOs to SALCAOs:
 2 with same symmetry label: bonding + antibonding MOs
 3 with same symmetry label: (usually) bonding, non-bonding, antibonding MOs
 (usual) MO energy order: b < nb < ab
5) Count orbitals and electrons, fill orbitals with electrons:
 # AOs in = # MOs out, # MOs filled = # electrons / 2
 fill MOs low-E to high-E, (usually) b < nb < ab
 (usually) b MOs are mostly higher  atom character, ab are lower 
1)
2)
3)
4)
Φ1 + Φ2 = Ψ+
Φ1 – Φ2 = Ψ–
const. interference of e– waves
results from in-phase overlap
e– density between atoms ↑
lower E than AOs
e– in Ψ+ pull atoms together
“bonding” MO
single phase connects two atoms
- 2e in bonding MO = +1 bond
Ψ+ : -
deconst. interference of e– waves
results from out-of-phase overlap
e– density between atoms ↓ (a node!)
higher E than AOs
e– in Ψ– repel atoms
- “antibonding” MO (denoted with * )
- phase change between two atoms
- 2e in antibonding MO = –1 bond
Ψ– : -
In MO theory, bond order = ½(e– in bonding MOs – e– in antibonding MOs)
s orbital overlap: σ
p orbital overlap: σ
p orbital overlap: π
How to Build an MO Diagram: Polyatomic without Central Atom
1) Lewis  VSPER  point group
2) Use character table to find symmetry labels on SALCAOs for each set of AOs
3) SALCAOs are the MOs
 MO energy order: fewer nodes < more nodes
4) Count orbitals and electrons, fill orbitals with electrons:
 # AOs in = # MOs out, # MOs filled = # electrons / 2
 fill MOs low-E to high-E, fewer nodes < more nodes
 (usually) b MOs are mostly higher  atom character, ab are lower 
How to Find SALCAOs and SALCAO symmetries
σ or σg
σ* or σu
σ or σg
σ* or σu
π or πu
π* or πg
Licensed by WSM under a Creative Commons Attribution‐NonCommercial‐ShareAlike 2.5 Canada Licence. 1) Apply each symmetry operation to each individual AO in a set, determine resulting character:
 the AO superimposes and retains phase: +1
 the AO stays in position but changes phase: –1
 the AO changes position and does not superimpose: 0
2) for each operation, sum the characters for all AOs to generate the reducible representation (RR) for
that set of AOs, a row of characters for each operation
3) determine the irreducible representations – individual rows from the character table – that sum to the
RR: symmetry labels of the summing IRs are symmetry labels of the SALCAOs
4) common patterns of SALCAOs:
two AOs  two SALCAOs, + + / + –
three AOs  three SALCAOs: + + + / + · – / + – +
Licensed by WSM under a Creative Commons Attribution‐NonCommercial‐ShareAlike 2.5 Canada Licence. Information in a Molecular Orbital Diagram
A molecular orbital diagram shows:
 energies and degeneracies of all valence orbitals in a molecule
 occupancy of all orbitals, filled or unfilled
 labels of each (set of) MO(s), determined by symmetry
 HOMO = highest occupied molecular orbital (or SOMO: singly)
LUMO = lowest unoccupied molecular orbital
For simple compounds, an MO diagram may additionally include:
 to sides of MOs: energy, number, symmetry of contributing AOs
 dashed lines connecting MOs to AOs from which they are formed
 clues to MO binding properties:
o MO is bonding if lower-E than contributing (SALC)AOs
o MO is antibonding if higher-E than contributing (SALC)AOs
o MO is non-bonding if identical E to contributing (SALC)AO
Molecular Orbitals: Row 2 Homonuclear Diatomics
2u
1g
2p
2g
1u
2s
1u
1g
Li2 to N2
2u
Molecular Orbitals: Row 1 Diatomics
1g
In H2, these MOs can be labeled as:
u
2p
σu or σ*: antibonding overlap of 1s AOs
E
1u
2g
2s
1u
antibonding
1g
g
MO Diagram of H2
σg or σ: bonding overlap of 1s AOs
bonding
The two valence e– fill the lowest-E
orbital: H2 has a (σg)2 configuration.
If neighbouring atoms have e– density overlapping in-phase: bonding MO
e– density overlapping in-phase: antibonding MO
no overlapping e– density: non-bonding MO
In MO theory, bond order = ½ (e– in bonding MOs – e– in antibonding MOs)
i.e. 2e in b MO = +1 bond
2e in ab MO = –1 bond
H2 has a BO = ½(2 – 0) = 1,
as expected!
The same qualitative H2 MO
diagram may be used for any
diatomic molecule using 1s
AOs.
Licensed by WSM under a Creative Commons Attribution‐NonCommercial‐ShareAlike 2.5 Canada Licence. O2 and F2
Molecular Orbitals: Heteronuclear Overlap
2
2
2
A
A
A
B
1
1
B
1
B
covalent
A = B
polar
A < B
ionic
A << B
E(A) = E(B)
strong overlap
1 is 1:1 A and B
2 is 1:1 A and B
electrons shared equally
e.g. N2
E(A) > E(B)
weaker overlap
1 is mostly B
2 is mostly A
electrons mostly on B
e.g. CO
E(A) >> E(B)
almost no overlap
1 is nearly exclusively  B
2 is nearly exclusively  A
electrons localized on B
e.g. LiF
Licensed by WSM under a Creative Commons Attribution‐NonCommercial‐ShareAlike 2.5 Canada Licence.