Molecular Orbital Theory

Chapters 16-17: Molecular Orbital Theory for Covalent Bonding

Overview of Bonding Theories

  1. Lewis Octet Rule

    • Effective for understanding bonding of second-row atoms.

  2. Valence Shell Electron Repulsion Theory (VSEPR)

    • Explains molecular structure through electron pair repulsions.

  3. Valence Bond Theory (VB)

    • Explains bond formation, bond order, and molecular structures, primarily focusing on carbon-containing molecules.

  4. Molecular Orbital Theory (MO)

    • Built on mathematical interactions of quantum mechanically determined neighboring atomic orbitals.

    • More general success in describing molecular properties, including:

      • Bond energy

      • Molecular structure

      • Magnetic properties

      • Molecular interaction with light

Molecular Orbitals and Atomic Orbitals

  1. Formation of Molecular Orbitals

    • In MO theory, electrons from neighboring atomic orbitals combine to form molecular orbitals.

    • Molecular orbitals are the result of linear combinations of atomic orbital wavefunctions, commonly referred to as Linear Combination of Atomic Orbitals (LCAO).

    • Definition: An LCAO-MO is a molecular orbital formed from the linear combination of atomic orbitals.

Hydrogen Molecule Example

  1. Bonding Molecular Orbital of the Hydrogen Molecule

    • The molecular orbital for H2: ext{Ψ} = ext{Ψ}{A1s} + ext{Ψ}{B1s} where

      • Ψ is the wavefunction of the molecular orbital.

      • ΨA1s is a 1s orbital centered on atom A.

      • ΨB1s is a 1s orbital centered on atom B.

    • The linear combination involves constructive interference denoted by the “+” symbol in the equation.

  2. Antibonding Molecular Orbital of the Hydrogen Molecule

    • For every bonding orbital, there exists an antibonding orbital as well.

    • The second molecular orbital for molecular hydrogen is given by:
      ext{Ψ} = ext{Ψ}{A1s} - ext{Ψ}{B1s}

    • Antibonding orbitals show destructive interference, hence the “−” symbol, and are denoted with an asterisk, e.g., σ1s*.

Molecular Orbital Energy Level Diagram

  1. Understanding the Diagram

    • The molecular orbital energy level diagram illustrates the relative energies of the original atomic orbitals alongside the new bonding and antibonding molecular orbitals.

    • Bonding orbital energies are lower than the energies of the atomic orbitals, while antibonding orbital energies are higher.

Filling Electrons in Molecular Orbitals

  1. Basic Principles

    • When filling electrons in molecular orbitals, follow these rules:

      • Electrons start filling the lowest-energy molecular orbital.

      • Pauli Exclusion Principle: Each molecular orbital can accommodate a maximum of two electrons with opposite spins (spin-paired electrons).

      • Hund’s Rule: If multiple molecular orbitals of the same energy are available, electrons will enter them singly and adopt parallel spins until all have one electron.

    • If there are more electrons in bonding MOs than in antibonding MOs, a covalent bond is formed.

Formation of Molecular Orbitals from Atomic Orbitals

  1. Combination of Atomic Orbitals

    • In H2, there are two 1s orbitals that merge to form two molecular orbitals:

      • Bonding orbital is denoted as σ1s.

      • Antibonding orbital is denoted as σ1s*.

      • The notation σ indicates the formation of a "σ-orbital," which is cylindrical in shape.

Bonding in Diatomic Molecules

  1. Understanding Electron Occupation

    • For H2, both electrons occupy the bonding σ1s orbital, which is the lower-energy orbital.

    • A pair of electrons is the maximum allowed per orbital.

    • Unlike Lewis theory, which assumes bond strength based on electron pairing, MO theory allows the validity of a single electron bond, albeit with half the strength of an electron-pair bond.

Homonuclear Diatomic Molecules of Period 2 Elements

  1. Constructing MO Energy-level Diagrams

    • Begin with valence-shell atomic orbitals from each atom.

    • Two atom contributions—both 2s- and 2p-orbitals—yield a total of eight atomic orbitals, resulting in eight molecular orbitals.

    • Only atomic orbitals from different atoms at similar energy levels will interact.

    • Sigma-bonding interactions are stronger than pi-bonding interactions due to the spatial orientation:

      • Sigma-bonding: Orbitals overlap along the internuclear axis.

      • Pi-bonding: Orbitals overlap side-by-side.

Bonding and Antibonding σ Bond Molecular Orbitals

  1. Sigma Bonds from 2p Orbitals

    • When two 2p-orbitals are directed toward each other, they can form two molecular orbitals:

      • One bonding σ-orbital (σ2p).

      • One antibonding σ* orbital (σ2p*).

Bonding and Antibonding π Bond Molecular Orbitals

  1. Formation of π Bonds

    • Perpendicular p-orbitals overlap side by side to form bonding and antibonding π-orbitals.

Comparison of Period 2 MO Energy Diagrams

  1. Energy Differences and Predictions

    • The energy differences between the MO diagrams of homonuclear diatomic molecules Li2 through N2 are attributed to shielding effects.

    • Accurate predictions regarding orbital placements require detailed calculations, especially for O2 and F2.

Bond Orders from MO Diagrams

  1. Connecting Lewis and MO Theories

    • At first glance, the molecular orbital description of N2 may appear different from the Lewis diagram (such as :N≡N:), but they correlate when comparing bond orders.

    • Bond Order Formula:
      ext{Bond Order} = rac{1}{2} imes ( ext{Number of electrons in bonding orbitals} - ext{Number of electrons in antibonding orbitals})

    • For N2:
      BO = rac{1}{2} imes (8 - 2) = 3

    • The calculated bond order matches the prediction from Lewis’s method.

Paramagnetism of Molecular Oxygen

  1. Addressing Electron Configurations

    • While Lewis and valence-bond theories suggest all electrons in O2 are paired, the molecular orbital diagram indicates oxygen is actually paramagnetic due to the presence of unpaired electrons.

    • Oxygen exhibits paramagnetism as a property resulting from unpaired electrons, contrasting with diamagnetism, where all electrons are paired.

Self-test 2G.1B: Bond Order of O2+ Ion

  1. Identify Valence Atomic Orbitals

    • To find the electron configuration for O2+:

    • Calculation of valence electrons:
      6 + 6 - 1 = 11

  2. Constructing the Electron Configuration

    • The resulting molecular orbital configuration:

    • ext{σ}2s^2 ext{σ}2s^2 ext{σ}2p^2 ext{π}2p^4 ext{π}2p^1

    • Bond Order Calculation:
      BO = rac{1}{2} imes (8 - 3) = 2.5

    • Since at least one MO has an unpaired electron, O2+ is considered paramagnetic.

Bonding in Heteronuclear Diatomic Molecules

  1. Formation of Molecular Orbitals

    • In heteronuclear diatomic molecules, atomic orbitals (AOs) with unequal energy share electrons to form molecular orbitals (MOs).

    • For a molecule represented as:
      ext{Ψ} = cA ext{Ψ}A + cB ext{Ψ}B

    • Where:

      • cA and cB represent the weights of the respective AOs in the MO.

    • The more electronegative element will have lower energy, dominating the MOs, which leads to greater electron density near that atom.

    • Example: For nitrogen oxide (NO), determine the bond order and identify which atom is more electronegative.

Self-test 2G.2B: Configuration of the Cyanide Ion (CN−)

  1. Ground State Configuration Steps

    • Count valence electrons within CN−:
      4 + 5 + 1 = 10

  2. Constructing the Electron Configuration

    • Resulting electron configuration:
      1σ^2 2σ^*2 1π^4 3σ^2

    • Bond Order Calculation:
      BO = rac{1}{2} imes (8 - 2) = 3

    • The cyanide ion is characterized as diamagnetic, and it is important to note the position of the negative charge.

Non-bonding Orbitals in Polyatomic Molecules

  1. Introduction to Non-bonding Orbitals

    • In MO theory, there exists a third type of orbital: nonbonding molecular orbitals.

    • Characteristics:

      • Consist of atomic orbitals with minimal or no interaction, forming two electrons in bonding orbitals and two electrons in antibonding orbitals, yielding a bond order of zero; hence, nonbonding orbitals do not contribute to bond strength.

MO Theory of Water

  1. Construction of MOs in H2O

    • Water contains six atomic orbitals (1 O2s, 3 O2p, and 2 H1s) which are employed to construct six molecular orbitals.

    • When the eight valence electrons are arranged, they fill the two bonding orbitals and two nonbonding orbitals.

    • Resultantly, the electrons in bonding orbitals symbolize the two single OH bonds found in the Lewis structure, while the electrons in nonbonding orbitals denote the two lone pairs on H2O.

Summary of Bonding Models

  1. Comparison of Theories

    • Lewis Octet Rule:

      • Electron Location: Visualized as dots around the bonding atom.

      • Bonding Character: Number of bonding electrons determines bond order.

      • Molecular Shape: Predicted by VSEPR theory.

    • Valence Bond Theory:

      • Electron Location: Between neighboring atoms but includes hybridization and resonance considerations.

      • Constructed by overlapping neighboring atomic orbitals to form bonds.

    • Molecular Orbital Theory:

      • Electron Location: Spread out throughout the wavefunction regions of participating atomic orbitals.

      • Built through linear combinations of atomic orbitals, sequentially filling the lowest-energy molecular orbitals.

      • Bonding and antibonding electrons are accounted for when determining the stability and configuration of molecules.