Molecular Orbital Theory (MOT)

Molecular Orbital Theory (MOT)

• Developed by F. Hund and R.S. Mulliken in 1932.
• Electrons in molecules reside in various molecular orbitals.
• Molecular orbitals form through the combination of atomic orbitals with comparable energies and proper symmetry.
• Atomic orbitals are monocentric, influenced by one nucleus
• Molecular orbitals are polycentric, influenced by two or more nuclei.
• Number of molecular orbitals formed equals the number of combining atomic orbitals.
• Atomic orbital combination yields bonding and anti-bonding molecular orbitals.
• Bonding molecular orbitals have lower energy and greater stability than antibonding ones.
• Electron probability distribution around nuclei in a molecule is given by molecular orbitals.
• Molecular orbitals fill according to the Aufbau principle, Pauli Exclusion principle, and Hund's Rule.
  • Filling order is experimentally determined, without a strict (n + l) rule

Formation of Molecular Orbitals: Linear Combination of Atomic Orbitals (LCAO)

• Focuses on molecular hydrogen (H₂) as an example.
• Molecular orbitals in H₂ can be approximated through linear combinations of atomic orbitals.
• For nuclei A and B, the lowest energy orbital is the 1s orbital, represented by wave functions ΨA or ΨB.
• Molecular orbitals are represented by a linear combination of atomic orbitals.
• For identical atoms like A and B, atomic orbitals contribute equally.
• Two molecular orbitals from combining 1s atomic orbitals:
  • Ψm = [ΨA + ΨB] (bonding)
  • Ψ*m = [ΨA - ΨB] (antibonding)

Bonding and Antibonding Orbitals

• Molecular wave functions Ψm and Ψ*m represent bonding and antibonding molecular orbitals.
• Orbitals with symmetry along the line joining two nuclei become σ orbitals (bonding) or σ* orbitals (antibonding), denoted as σ1s and σ*1s.
• Adding 1s wave functions reinforces electron density, especially between nuclei, reducing internuclear repulsion and resulting in a strong bond.
• Subtracting 1s wave functions cancels electron density in a plane between nuclei, increasing internuclear repulsion and resulting in an antibonding orbital.

Further Considerations for Hydrogen Molecule (H₂)

• Ground state: both electrons occupy the σ1s orbital.
• Hydrogen molecule ion (H+2): one electron occupies the σ1s orbital.
• Total bonding energy: 269 kJ/mol for H+2, compared to 458 kJ/mol for H₂.
• σ and σ* orbitals are centrosymmetric and non-centrosymmetric, denoted as σg and σu.
• Molecular orbital wave functions are designated as Ψg and Ψu.
• g and u refer to the symmetry of the orbital about its center
• g (gerade): wave function has the same sign at equal distances from the center.
• u (ungerade): wave function changes sign upon inversion about the center.

Conditions for Atomic Orbital Combination

• Combining atomic orbitals must have similar energy levels, so a 1s orbital can combine with another 1s orbital, but not with a 2s orbital.
• Combining atomic orbitals must have the same symmetry about the molecular axis (z-axis).
• Atomic orbitals with similar energy won't combine without compatible symmetry.
• Combining atomic orbitals must overlap to the maximum extent for greater electron density between nuclei.

Types of Molecular Orbitals

• Diatomic molecules are designated as σ(sigma), π(pi), δ(delta), etc.
• Sigma (σ) molecular orbitals are symmetrical around the bond axis.
• Pi (π) molecular orbitals are not symmetrical around the bond axis.
• Combination of 1s orbitals centered on two nuclei produces σ-type molecular orbitals, designated as σ1s and σ*1s.
• Linear combination of 2pz orbitals yields sigma molecular orbitals (σ2pz and σ*2pz).
• Molecular orbitals from 2px and 2py orbitals are not symmetrical around the bond axis but have positive lobe planes and are labeled π and π*.
• A π-bonding MO has high electron density above and below the internuclear axis
• A π*-antibonding MO has a node between the nuclei.
• δ-type molecular orbitals involve d-orbitals in bonding.

Differences Between π and σ Molecular Orbitals

• π overlap occurs when the lobes of atomic orbitals are perpendicular to the line joining the nuclei. σ overlap occurs when lobes point along that line.
• For a π molecular orbital, Ψ is zero along the internuclear line, so electron density, Ψ², is also zero, unlike in σ orbitals.
• π molecular orbitals exhibit different symmetry than σ orbitals.
• π bonding MOs are ungerade (sign changes upon rotation)
• σ bonding MOs are gerade (sign remains the same).
• Antibonding π MOs are gerade.
• Antibonding σ MOs are ungerade.

Energy Level Diagrams for Molecular Orbitals

• Molecular orbital energy levels are experimentally determined via spectroscopy for homonuclear diatomic molecules.
• Energy order for O₂ and F₂: σ1s < σ1s < σ2s < σ2s < σ2pz < (π2px = π2py) < (π2px = π2py) < σ*2pz
• Key feature: σ2pz molecular orbital energy is higher than π2px and π2py molecular orbitals.
• Energy order for Be₂, B₂, C₂, N₂: σ1s < σ1s < σ2s < σ2s < (π2px = π2py) < σ2pz < (π2px = π2py) < σ*2pz

Electronic Configuration and Molecular Behavior

• Electron distribution among molecular orbitals defines the electronic configuration of a molecule.
• Electronic configuration provides information about the molecule.
• If the number of electrons in bonding orbitals (Nb) is greater than in antibonding orbitals (Na), the molecule is stable.
• If Nb is less than Na, the molecule is unstable.
• More occupied bonding orbitals lead to stronger bonding and a stable molecule.
• Stronger antibonding influence makes the molecule unstable.

Bond Order

• Bond order (B.O.) = ½ (Nb - Na)
• Positive B.O. (Nb > Na) means a stable molecule.
• Negative or zero B.O. (Nb ≤ Na) means an unstable molecule.
• Integral bond order values of 1, 2, and 3 correspond to single, double, and triple bonds, respectively.
• Approximate measure of bond length comes from bond order between two atoms in a molecule.
• Bond length decreases with increasing bond order.

Magnetic Nature

• Diamagnetic: all molecular orbitals are doubly occupied (repelled by a magnetic field), e.g., N₂.
• Paramagnetic: one or more molecular orbitals are singly occupied (attracted by a magnetic field), e.g., O₂.

Bonding in Homonuclear Diatomic Molecules

Hydrogen

• H₂: (σ1s)²
• Bond order = (2-0)/2 = 1 (single covalent bond)
• Bond dissociation energy: 438 kJ/mol.
• Bond length: 74 pm.
• Diamagnetic (no unpaired electrons).

Helium

• He₂: (σ1s)² (σ*1s)²
• Bond order = ½ (2-2) = 0 (no bond)
• Helium does not form diatomic molecules and exists as free atoms.
• Very low binding energy (0.01J/mol).
• H₂ has a bond energy of 436 kJ/mol.

Lithium

• Li₂: (σ1s)² (σ*1s)² (σ2s)²
• Bond order = ½ (4-2) = 1.
• Stable and diamagnetic.
• Exists in the vapor phase.
• It has a single Li-Li bond.

Beryllium

• Be₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)²
• Has an equal number of bonding and antibonding electrons.
• Bond order is zero, so it's not stable.

Boron

• B₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (π2p1x = π2p1y)
• Advantage of the Molecular orbital model over the Lewis dot picture.
• Exists in the gas phase (paramagnetic).
• Shift in energy levels is caused by mixing s and p orbitals.
• Without mixing, σg(2p) would be lower in energy than π(2p).
• Mixing lowers the energy of σg(2s) and increases that of σg(2p).
• The last two electrons are unpaired in degenerate π orbitals, so the molecule is paramagnetic.
• B.O. is 1/2 (6-4) = 1.

Carbon

• C₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (π2p2x = π2p2y).
• Doubly bonded (predicted from simple MO picture with paired electrons).
• It has two π bonds and no σ bond.
• C₂ is not commonly found as a species (carbon is more stable).
• Acetylide ion (C22-) exists in compounds with alkali metals, alkaline earths, and lanthanides.
• C22- B.O. is 3 [configuration (ππ2σσ2)].
• B.O. of C₂ is ½ (8-4) = 2 and should be diamagnetic, and double bond in C2 consists of both pi bonds
• A double bond is made up of one sigma and one pi bond in most other molecules.

Nitrogen

• N₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (π2p2x = π2p2y) (σ2pz)²
• Consists of a triple bond (both Lewis and MO models)
• Short N-N distance (109.8 pm) and high bonding dissociation energy (942 KJ/mol)
• The effect of the nuclear charge causes shielding and interactions, causing an increase in the 2s and 2p orbital energies.
• B.O. of N₂ is ½ (10-4) = 3, containing one sigma and two pi bonds.

Anionic Nitrogen

• N₂-: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (π2p2x = π2p2y) (σ2pz)² (π*2px)¹
• B.O. of N₂ is ½ (10-5) = 2.5 (paramagnetic).
• Less stable than N₂+
• N₂- has a greater number of antibonding electrons.

Oxygen

• O₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (σ2pz)² (π2p2x = π2p2y) (π2p1x = π2p1y)
• O₂ is paramagnetic.
• B.O. for O2 is ½ [10-6] = 2, so in the oxygen molecules, atoms are held by a double bond.
• Molecular orbitals have two unpaired electrons (π2px, π2py)

Bond Order and Oxygen Distance Correlation

• As related to the molecular orbital model.
  • O2+ (dioxygenyl) has a bond order of 2.5 and an internuclear distance of 112.3 pm.
  • O2 (dioxygen) has a bond order of 2.0 and an internuclear distance of 120.07 pm.
  • O2- (superoxide) has a bond order of 1.5 and an internuclear distance of 128 pm.
  • O22- (peroxide) has a bond order of 1.0 and an internuclear distance of 149 pm.
• The mixing percentage change is not sufficient enough in O2 to direct the 𝜎(2p) orbitals to higher energy than the 𝜋(2𝑝) orbitals.
• The order of molecular orbitals is consistent with the photoelectron spectrum.

Fluorine

• F₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (σ2pz)² (π2p2x = π2p2y) (π2p2x = π2p2y)
• It shows a diamagnetic molecule.
• The net bond order (N₂, O₂, F₂) remains irrespective of the mixing is considered (or not).
• Minor switches can occur due to the energy levels of the σ(2p) and π(2p) since they are so close.
• Returns the molecule in CO and F2 to show higher sigma levels because energy difference raises, higher orbitals are seen again.

Neon

• Ne₂: (σ1s)² (σ1s)² (σ2s)² (σ2s)² (σ2pz)² (π2p2x = π2p2y) (π2p2x = π2p2y) (σ*2pz)²
• It is equal in numbers where all molecular orbitals are filled.
• Transient species with bond order (therefore) zero if it exists at all.