theoretical chemistry unit 6 study guides

Key Concepts

• Molecular orbital theory describes the behavior of electrons in molecules using quantum mechanics principles
• Atomic orbitals combine to form molecular orbitals when atoms bond together
• Bonding orbitals are lower in energy than the atomic orbitals they are formed from and contribute to the stability of the molecule
• Antibonding orbitals are higher in energy than the atomic orbitals they are formed from and destabilize the molecule
• The number of molecular orbitals formed equals the number of atomic orbitals that combine
• Molecular orbital diagrams illustrate the relative energies and electron occupations of molecular orbitals
• Molecular orbital theory helps explain properties such as bond order, magnetic behavior, and spectroscopic transitions

Atomic Orbitals Review

• Atomic orbitals are mathematical functions that describe the probability distribution of electrons around an atom
• The principal quantum number (n) determines the energy and size of the orbital
• The angular momentum quantum number (l) determines the shape of the orbital (s, p, d, f)
• The magnetic quantum number (m_l) determines the orientation of the orbital in space
• The spin quantum number (m_s) describes the intrinsic angular momentum of the electron (±1/2)
• The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers
• Hund's rule states that electrons occupy orbitals of the same energy singly before pairing up, with their spins aligned

Molecular Orbital Theory Basics

• Molecular orbital theory is a quantum mechanical approach to describing the electronic structure of molecules
• Linear combination of atomic orbitals (LCAO) approximation is used to construct molecular orbitals from atomic orbitals
• Molecular orbitals are delocalized over the entire molecule, unlike localized atomic orbitals
• The wave function of a molecular orbital is represented as a linear combination of atomic orbital wave functions: $\Psi_{MO} = c_1\phi_1 + c_2\phi_2 + ...$
• The coefficients (c_i) determine the contribution of each atomic orbital to the molecular orbital
• Molecular orbitals are classified as sigma (σ) or pi (π) based on their symmetry and orientation
  • Sigma (σ) orbitals are symmetric about the bond axis and have no nodal plane between the nuclei
  • Pi (π) orbitals have a nodal plane that includes the bond axis and are less stable than sigma orbitals

Bonding and Antibonding Orbitals

• Bonding orbitals are formed by the constructive interference of atomic orbitals, resulting in increased electron density between the nuclei
• Antibonding orbitals are formed by the destructive interference of atomic orbitals, resulting in a node (zero electron density) between the nuclei
• Bonding orbitals are lower in energy than the atomic orbitals they are formed from, while antibonding orbitals are higher in energy
• Electrons in bonding orbitals contribute to the stability of the molecule by lowering the overall energy
• Electrons in antibonding orbitals destabilize the molecule by raising the overall energy
• The bond order is calculated as (number of bonding electrons - number of antibonding electrons) / 2
  • A bond order of 1 corresponds to a single bond, 2 to a double bond, and 3 to a triple bond
• Molecules with more bonding electrons than antibonding electrons are stable, while those with more antibonding electrons are unstable

MO Diagrams for Diatomic Molecules

• Molecular orbital diagrams show the relative energies and electron occupations of molecular orbitals in a molecule
• For homonuclear diatomic molecules (e.g., H₂, N₂, O₂), the atomic orbitals of the same type and energy combine to form molecular orbitals
• The 1s atomic orbitals combine to form the σ_1s bonding and σ_1s* antibonding molecular orbitals
• The 2s atomic orbitals combine to form the σ_2s bonding and σ_2s* antibonding molecular orbitals
• The 2p atomic orbitals combine to form the σ_2p and π_2p bonding orbitals, and the σ_2p* and π_2p* antibonding orbitals
• Electrons fill the molecular orbitals according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle
• The electronic configuration and properties of the molecule can be determined from the molecular orbital diagram
  • For example, O₂ has two unpaired electrons in the π_2p* antibonding orbitals, making it paramagnetic

Polyatomic Molecules and Hybridization

• Molecular orbital theory can be extended to polyatomic molecules, although the diagrams become more complex
• Hybridization is a concept that explains the mixing of atomic orbitals to form new hybrid orbitals with specific geometries
• sp hybridization involves the mixing of one s and one p orbital to form two linear sp hybrid orbitals (e.g., BeH₂)
• sp² hybridization involves the mixing of one s and two p orbitals to form three trigonal planar sp² hybrid orbitals (e.g., BH₃)
• sp³ hybridization involves the mixing of one s and three p orbitals to form four tetrahedral sp³ hybrid orbitals (e.g., CH₄)
• Hybrid orbitals form stronger bonds than pure atomic orbitals due to better overlap and directionality
• Unhybridized p orbitals can form pi (π) bonds or participate in delocalized bonding (e.g., benzene)

Applications in Spectroscopy

• Molecular orbital theory helps interpret various spectroscopic techniques that probe the electronic structure of molecules
• UV-visible spectroscopy measures the absorption of light due to electronic transitions between molecular orbitals
  • The energy of the absorbed light corresponds to the energy difference between the orbitals involved in the transition
• Photoelectron spectroscopy (PES) measures the ionization energies of electrons in molecular orbitals
  • The ionization energy is related to the binding energy of the electron in the molecular orbital
• Electron spin resonance (ESR) spectroscopy detects the presence of unpaired electrons in molecules
  • Molecules with unpaired electrons in molecular orbitals (e.g., radicals) give rise to ESR signals
• Spectroscopic data can be used to validate molecular orbital diagrams and computational results

Computational Methods

• Computational methods are used to calculate the energies and properties of molecules based on molecular orbital theory
• The Hartree-Fock (HF) method is a variational approach that solves the Schrödinger equation for a multi-electron system
  • HF assumes that each electron moves in the average field of all other electrons and nuclei
  • The HF equations are solved iteratively until self-consistency is reached (SCF)
• Post-Hartree-Fock methods (e.g., configuration interaction, coupled cluster) include electron correlation effects beyond the HF approximation
• Density functional theory (DFT) is an alternative approach that uses the electron density instead of the wave function
  • DFT methods (e.g., B3LYP) are computationally efficient and widely used for larger molecules
• Basis sets are mathematical functions used to represent atomic orbitals in molecular orbital calculations
  • Larger basis sets (e.g., 6-31G*) provide more accurate results but are computationally more expensive
• Computational results (e.g., orbital energies, electron densities, vibrational frequencies) can be compared with experimental data to validate the molecular orbital description of the molecule