Core Principles of Molecular Orbital Theory

1. Linear Combination of Atomic Orbitals (LCAO)

Molecular orbitals are formed by combining atomic orbitals. This is mathematically represented as:

$\Psi_{MO} = c_1\phi_1 + c_2\phi_2 + ... + c_n\phi_n$

where $\Psi_{MO}$ is the molecular orbital, $\phi_i$ are the atomic orbitals, and $c_i$ are coefficients (Bussy & Hutter, 2021)

Bonding and Antibonding Orbitals

• Bonding orbitals: Lower energy, increased electron density between nuclei
• Antibonding orbitals: Higher energy, node in electron density between nuclei

2. Orbital Symmetry and Overlap

The formation of molecular orbitals depends on the symmetry and effective overlap of atomic orbitals. This principle determines which combinations are allowed and their relative energies.

Types of Orbital Overlap

1. Sigma (σ) bonds: Head-on overlap
2. Pi (π) bonds: Side-by-side overlap
3. Delta (δ) bonds: Overlap of d orbitals

3. Aufbau Principle and Orbital Filling

Electrons fill molecular orbitals from lowest to highest energy, following Hund's rule and the Pauli exclusion principle (Hauck et al., 2023)

Molecular Orbital Diagrams

Visual representations of orbital energies and electron configurations in molecules:

     σ*2s
  ↑↓  ↑↓
σ2s   π2p
↑↓    ↑↓
     σ2p
↑↓
σ1s

Example: MO diagram for N2 molecule

4. Hybridization

Mixing of atomic orbitals to form new hybrid orbitals, explaining molecular geometries and bonding patterns (Narsaria et al., 2018)

Common Hybridizations

1. sp³: Tetrahedral geometry (e.g., CH4)
2. sp²: Trigonal planar geometry (e.g., BF3)
3. sp: Linear geometry (e.g., CO2)

5. Molecular Orbital Energy Levels

Understanding the relative energies of molecular orbitals is crucial for predicting molecular properties and reactivity (Ikeno et al., 2004)

HOMO-LUMO Gap

The energy difference between the Highest Occupied Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO) is a key factor in determining molecular reactivity and spectroscopic properties (Narsaria et al., 2018)

6. Delocalization and Aromaticity

Molecular orbital theory explains the concept of electron delocalization in conjugated systems and aromatic compounds, leading to enhanced stability

Hückel's Rule

Aromatic compounds have (4n+2) π electrons in a planar, cyclic conjugated system, where n is a non-negative integer

7. Computational Methods

Various computational approaches are used to calculate molecular orbitals and their properties (Bussy & Hutter, 2021)

Density Functional Theory (DFT)

A popular method for calculating electronic structures, using electron density functionals to approximate exchange-correlation energies (Coretti et al., 2022)

Time-Dependent DFT (TD-DFT)

Used for calculating excited states and spectroscopic properties of molecules (Bussy & Hutter, 2021)

8. Applications in Spectroscopy

Molecular orbital theory is crucial for interpreting various spectroscopic techniques (DiMucci et al., 2023)

X-ray Absorption Spectroscopy (XAS)

Probes core-level electronic transitions, providing information about oxidation states and local electronic structure (DiMucci et al., 2023)

UV-Visible Spectroscopy

Analyzes electronic transitions between molecular orbitals, useful for studying conjugated systems and charge transfer complexes