| a KK is the electron configuration for He2 given as (σ1s)2(σ*1s)2 |
Molecular Orbital Theory
Chapter 09
Molecular orbital (MO) theory is a method for describing the electronic structure of molecules using quantum mechanics. MO theory uses molecular orbitals unlike valence bond theory (VBT) which uses hybrid orbitals. MO theory is able to describe properties about a molecule that cannot such as the paramagnetic nature of O2 or the delocalized nature of electrons across the entire molecule.
Molecular Orbitals
Molecular orbitals (MO) constructed from the combination of atomic orbitals (AOs) to give bonding MOs and antibonding MOs. Orbitals of similar energy and symmetry can combine.
Types of MOs
Bonding MOs (σ, π) are lower in energy where the associated electrons stabilize the bond. These MOs form due to the constructive combination of AOs. Electron density exists between the atoms involved in the bond. Antibonding MOs (σ*, π*)are higher in energy and the associated electrons destabilize the bond. These MOs for due to the destructive combination of AOs. Electron density does not exist between the atoms involved in the bond.
σ MOs form due to the overlap of
- two s atomic orbitals
- one s and one p atomic orbitals
- the head-to-head overlap of two p orbitals
π MOs form due to the overlap of side-by-side p orbitals.
MOs span the molecule they form in and the electrons in MOs are said to be delocalized across the entire molecule.
MO diagrams show the combination of AOs (listed on the outside of the diagram) to give MOs (listed in the middle of the diagram) and orbitals are labeled.
Electrons fill molecular orbitals from the lowest to highest energy (similar to Aufbau) and follow the Pauli exclusion principle as well as Hund’s rule.
The highest occupied molecular orbital (HOMO) is the highest energy molecular orbital that contains at least one electron. The lowest unoccupied molecular orbital (LUMO) is the lowest energy molecular orbital that does not contain any electrons.
A molecule is more ‘stable’ when
- the number of electrons in bonding MOs increases
- the number of electrons in antibonding MOs decreases
A molecule does not exist when the number of electrons in bonding MOs equals the number of electrons in antibonding MOs.
Bond order
The bond order of the molecule can be determined by considering the number of electrons in bonding and antibonding MOs such that
\[\mathrm{bond~order} = \dfrac{e^-~\mathrm{in~bonding~MOs} ~-~ e^-~\mathrm{in~antibonding~MOs}}{2}\]
Larger bond order results in shorter bond lengths. A bond order of zero means the molecule does not exist (it is not stable).
Note: Electrons in bonding MOs are not always bonding electrons. Remember that only valence electrons are involved in bonding.
Electron configuration
The electron configuration of electrons in MOs are written similar to electron configurations of AOs. The MOs are listed from lowest to highest energy and the number of electrons in each MO are included. MOs with core electrons can be denoted using X-ray notation and is dependent on the shell/quantum number (n) they are in.
- n = 1: K-shell → K
- n = 2: L-shell → L
- n = 3: M-shell → M
- n = 4: N-shell → N
For example, when abbreviating two core MOs in the n = 1 shell, one can write “KK” before the valence MOs.
Homonuclear diatomics
Presented below are the MO diagrams for various homonuclear diatomic molecules from the first two rows of the periodic table. Take note of the following:
- Only valence MOs are shown
- The energy ordering of the MOs for row 2 elements Li → N are the same. The order changes when reaching O and remains the same from O→F.
s-p mixing
The MO energy ordering changes when going from N2 to O2 due to a phenomenon called s-p orbital mixing. This occurs because of the energies of a 2s and a 2p orbital are close enough to interact significantly. When these orbitals mix, the resulting σ2s MO is lower in energy than the π2p MO. The effects of sp-mixing disappears upon reaching the heavier oxygen atom where the σ2s is now higher in energy than the π2p.
H2
Key features
- Bond order: 1
- Magnetism: diamagnetic
- Electron configuration: (σ1s)2
- HOMO: σ1s
- LUMO: σ*1s
H2+
The bond order in H2+ is lower than that of H2 making the bond in H2+ weaker and longer. H2+ is more stable than H2– because no antibonding electron is present.
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: (σ1s)1
- HOMO: σ1s
- LUMO: σ*1s
- Less stable than H2 and more stable than H2–
H2–
The bond order in H2– is lower than that of H2 making the bond in H2– weaker and longer. Due to the antibonding electron, the bond in H2– is also weaker and longer than seen in H2+.
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: (σ1s)2(σ*1s)1
- HOMO: σ*1s
- LUMO: σ2s
- Less stable than H2 and H2+
Bond Length: H2 < H2+ < H2–
Stability: H2– < H2+ < H2
He2
Key features
- Bond order: 0 (does not exist; not stable)
- Magnetism: hypothetically diamagnetic
- Electron configuration: (σ1s)2(σ*1s)2
- HOMO: σ*1s
- LUMO: σ2s
Li2
Key features
- Bond order: 1
- Magnetism: diamagnetic
- Electron configuration: KK(σ2s)2
- HOMO: σ2s
- LUMO: σ*2s
Li2+
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)1
- HOMO: σ2s
- LUMO: σ*2s
Li2–
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)2(σ*2s)1
- HOMO: σ*2s
- LUMO: π2s
Bond Length: Li2 < Li2+ < Li2–
Stability: Li2– < Li2+ < Li2
Be2
Key features
- Bond order: 0 (does not exist; not stable)
- Magnetism: hypothetically diamagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2
- HOMO: σ*2s
- LUMO: π2p
Be2+
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)2(σ*2s)1
- HOMO: σ*2s
- LUMO: π2p
Be2–
Key features
- Bond order: 0.5
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(π2p)1
- HOMO: π2p
- LUMO: σ2p
Bond Length: Be2 < Be2+ < Be2–
Stability: Be2– < Be2+ < Be2
B2
Key features
- Bond order: 1
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(π2p)2
- HOMO: π2p
- LUMO: σ2p
C2
Key features
- Bond order: 2
- Magnetism: diamagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(π2p)4
- HOMO: π2p
- LUMO: σ2p
N2
Key features
- Bond order: 3
- Magnetism: diamagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(π2p)4(σ2p)2
- HOMO: σ2p
- LUMO: π*2p
O2
Key features
- Bond order: 2
- Magnetism: paramagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(σ2p)2(π2p)4(π*2p)2
- HOMO: π*2p
- LUMO: σ*2p
F2
Key features
- Bond order: 1
- Magnetism: diamagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(σ2p)2(π2p)4(π*2p)4
- HOMO: π*2p
- LUMO: σ*2p
Ne2
Key features
- Bond order: 0 (does not exist; not stable)
- Magnetism: hypothetically diamagnetic
- Electron configuration: KK(σ2s)2(σ*2s)2(σ2p)2(π2p)4(π*2p)4
- HOMO: σ*2p
- LUMO: σ3s
Heteronuclear diatomics
When writing MO diagrams for heteronuclear diatomic molecules, the AOs on the more electronegative atom are lower in energy than the AOs on the less electronegative atom. The shared electrons (the electrons in the MOs) are not shared equally. More electron density will reside closer to the more electronegative atom as the bonding MO will have character that more closely matches the more electronegative atom.
sp-mixing in heteronuclear diatomics can be predicted by
- valence electron count
- valence electrons is < 14, sp-mixing likely - Examples: NO (11 electrons), CO (10 electrons)
- valence electrons is > 14, sp-mixing unlikely
- electronegativity difference
- small difference, sp-mixing likely - Example: CO
- large difference, sp-mixing unlikely - Example: HF
NO
Notice that s-p orbital mixing occurs here (i.e. π2p is lower in energy than the σ2p). The bonding MOs have more oxygen character as these MOs are closer in energy to oxygen’s atomic orbitals. Therefore, more electron density resides on oxygen than nitrogen, demonstrating the polarity of the N–O single bond.
Key features
- Bond order: 2.5
- Magnetism: paramagnetic
- Electron configuration:
- HOMO: π*2p
- LUMO: σ*2p
- Polar bond
HF
Sometimes electrons remain localized on one of the atoms such as the case with HF where fluorine has three lone electron pairs that are not involved in bonding (i.e. nonbonding; nb). These electrons remain in orbitals that do not combine with an AO (such as the 1s orbital in hydrogen).
Here, the 2p orbitals in fluorine are closer in energy to the 1s orbitals in hydrogen. These AOs form the bonding (and antibonding) MOs. Notice how fluorine’s 2s atomic orbitals are much lower in energy than hydrogen’s 1s, and therefore does not form the MO.
The bonding MOs have more fluorine character as these MOs are closer in energy to fluorine’s atomic orbitals. Therefore, more electron density resides on fluorine than hydrogen, demonstrating the polarity of the H–F single bond.
Key features
- Bond order: 1
- Magnetism: diamagnetic
- Polar bond
Summary
MO theory, unlike VB theory, can explain:
- Electron delocalization (resonance and conjugation)
- Aromaticity and bond equalization in cyclic systems
- Magnetism
- Fractional bond orders
- Excited states
- Bond polarity
MO Diagram for Li2 → N2
π is lower in energy than σ due to sp-mixing.
MO Diagram for O2 → Ne2
π is higher in energy than σ due to no sp-mixing.
MO Diagram for some period 2 heteronuclear diatomics
π is lower in energy than σ due to sp-mixing.
π is higher in energy than σ due to no sp-mixing.
