⚛️Molecular Electronics Unit 3 – Molecular Orbital Theory

Key Concepts and Foundations

• Molecular Orbital Theory (MOT) is a quantum mechanical approach to describing the behavior of electrons in molecules
• MOT considers the wave-like properties of electrons and treats them as delocalized over the entire molecule
• Molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
• The number of MOs formed is equal to the number of atomic orbitals combined
• MOs are characterized by their energy, shape, and symmetry
• The Pauli Exclusion Principle states that no two electrons in a molecule can have the same set of quantum numbers
• Hund's Rule suggests that electrons occupy orbitals of the same energy singly before pairing up, minimizing electron-electron repulsion

Atomic Orbitals vs. Molecular Orbitals

• Atomic orbitals (AOs) describe the probability distribution of electrons around a single atom
• AOs are characterized by their principal quantum number (n), angular momentum quantum number (l), and magnetic quantum number (m)
• Molecular orbitals (MOs) describe the probability distribution of electrons around a molecule
• MOs are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
• MOs are delocalized over the entire molecule, while AOs are localized around a single atom
• The energy and shape of MOs depend on the type and orientation of the AOs that combine to form them
• MOs can be classified as bonding, antibonding, or non-bonding, depending on their effect on the stability of the molecule

Molecular Orbital Formation

• Molecular orbitals (MOs) are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
• The LCAO method involves adding or subtracting the wavefunctions of the atomic orbitals involved
• Constructive interference of atomic orbital wavefunctions leads to the formation of bonding MOs, which have lower energy than the original AOs
• Destructive interference of atomic orbital wavefunctions results in the formation of antibonding MOs, which have higher energy than the original AOs
• The formation of MOs depends on the symmetry and overlap of the atomic orbitals involved
• Sigma (σ) MOs are formed by the head-on overlap of atomic orbitals along the internuclear axis (s-s, s-p, or p-p)
• Pi (π) MOs are formed by the sideways overlap of p atomic orbitals perpendicular to the internuclear axis

Bonding and Antibonding Orbitals

• Bonding molecular orbitals (MOs) are formed by the constructive interference of atomic orbital wavefunctions
• Electrons in bonding MOs have lower energy than those in the original atomic orbitals, leading to a stabilization of the molecule
• Bonding MOs have an increased electron density between the nuclei, resulting in a net attractive force and a stable chemical bond
• Antibonding MOs are formed by the destructive interference of atomic orbital wavefunctions
• Electrons in antibonding MOs have higher energy than those in the original atomic orbitals, leading to a destabilization of the molecule
• Antibonding MOs have a nodal plane between the nuclei, resulting in a net repulsive force and a weakening of the chemical bond
• The occupation of bonding and antibonding MOs determines the overall stability and bond order of a molecule
  • Bond order = (number of bonding electrons - number of antibonding electrons) / 2

Energy Level Diagrams

• Energy level diagrams are used to visualize the relative energies of molecular orbitals (MOs) in a molecule
• The x-axis represents the internuclear distance, while the y-axis represents the energy of the MOs
• Atomic orbitals (AOs) of the constituent atoms are shown on the left and right sides of the diagram
• MOs are represented by horizontal lines, with bonding MOs having lower energy than the original AOs and antibonding MOs having higher energy
• The energy difference between bonding and antibonding MOs is related to the strength of the chemical bond
• Electrons are filled into the MOs according to the Aufbau principle, starting from the lowest energy MO and obeying the Pauli Exclusion Principle and Hund's Rule
• The electron configuration of a molecule can be determined from its energy level diagram, providing insights into its stability, magnetism, and spectroscopic properties

Molecular Orbital Shapes

• The shape of a molecular orbital (MO) depends on the type and orientation of the atomic orbitals (AOs) that combine to form it
• Sigma (σ) MOs have cylindrical symmetry around the internuclear axis and can be formed by the overlap of s-s, s-p, or p-p AOs
  • Bonding σ MOs have no nodal plane between the nuclei, while antibonding σ* MOs have one nodal plane
• Pi (π) MOs have a nodal plane along the internuclear axis and are formed by the sideways overlap of p AOs
  • Bonding π MOs have no nodal plane perpendicular to the internuclear axis, while antibonding π* MOs have one nodal plane
• Delta (δ) MOs have two nodal planes along the internuclear axis and are formed by the overlap of d AOs
• The shape and symmetry of MOs have important consequences for the reactivity and spectroscopic properties of molecules
• Frontier molecular orbitals (FMOs), namely the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO), play a crucial role in chemical reactions and charge transport in molecular electronics

Applications in Molecular Electronics

• Molecular Orbital Theory (MOT) provides a foundation for understanding charge transport in molecular electronic devices
• The frontier molecular orbitals (FMOs), HOMO and LUMO, are particularly important in determining the electronic properties of molecules
• The HOMO energy level is related to the ionization potential (IP) of a molecule, while the LUMO energy level is related to its electron affinity (EA)
• The HOMO-LUMO energy gap determines the conductivity and optical properties of molecular materials
  • A smaller HOMO-LUMO gap leads to better conductivity and lower-energy optical transitions
• Molecular orbital shapes and symmetry influence the coupling between molecules and electrodes in molecular junctions
• Quantum interference effects, arising from the constructive or destructive interference of electron wavefunctions, can modulate the conductance of molecular junctions
• Molecular orbital engineering, such as the incorporation of electron-donating or electron-withdrawing groups, can be used to tune the electronic properties of molecules for specific applications
• Charge transfer complexes, formed by the partial transfer of electrons between molecules with different electron affinities, can exhibit unique electronic and optical properties

Problem-Solving and Calculations

• Constructing molecular orbital diagrams:
  1. Determine the number and type of atomic orbitals (AOs) involved in the molecule
  2. Arrange the AOs in order of increasing energy
  3. Combine the AOs to form molecular orbitals (MOs) based on their symmetry and overlap
  4. Fill electrons into the MOs according to the Aufbau principle, Pauli Exclusion Principle, and Hund's Rule
• Calculating bond order:
  • Bond order = (number of bonding electrons - number of antibonding electrons) / 2
  • Higher bond orders indicate stronger and shorter bonds
• Determining molecular stability:
  • Compare the total energy of the electrons in the MOs to the total energy of the electrons in the original AOs
  • A net decrease in energy suggests a stable molecule, while a net increase indicates an unstable molecule
• Predicting magnetic properties:
  • Molecules with unpaired electrons in degenerate MOs are paramagnetic, while those with all electrons paired are diamagnetic
• Calculating the HOMO-LUMO energy gap:
  • The HOMO-LUMO gap can be estimated from spectroscopic data or computational methods (Koopman's theorem)
  • $E_\text{HOMO-LUMO} = E_\text{LUMO} - E_\text{HOMO}$
• Interpreting molecular orbital shapes and symmetry:
  • Use group theory to determine the symmetry of MOs and predict allowed electronic transitions
  • Analyze the nodal planes and electron density distribution of MOs to understand chemical reactivity and bonding properties