Molecular Orbital Theory - College Chemistry Guide

What is Molecular Orbital Theory?

Molecular Orbital Theory (MOT) is a method for describing the electronic structure of molecules using quantum mechanics. Unlike Valence Bond Theory which treats bonds as localized between two atoms, MOT considers electrons as delocalized over the entire molecule in molecular orbitals formed by the combination of atomic orbitals.

Historical Background

Molecular Orbital Theory was developed in the early 20th century by scientists including Friedrich Hund, Robert Mulliken, and John Lennard-Jones. Robert Mulliken received the Nobel Prize in Chemistry in 1966 for his fundamental work on chemical bonds and the electronic structure of molecules using the molecular orbital method.

Fundamental Concepts

The Basic Premise

When atoms combine to form molecules, their atomic orbitals overlap and combine to form molecular orbitals (MOs). These molecular orbitals belong to the molecule as a whole, not to individual atoms. Electrons in molecules occupy these molecular orbitals according to the same rules that govern atomic orbitals.

Key Principles of MOT:

Atomic orbitals combine to form molecular orbitals
Number of molecular orbitals formed = Number of atomic orbitals combined
Electrons in molecules occupy molecular orbitals
Molecular orbitals are filled according to Aufbau principle, Pauli exclusion principle, and Hund's rule
Bonding MOs are lower in energy than atomic orbitals; antibonding MOs are higher
Electrons are delocalized over the entire molecule

Linear Combination of Atomic Orbitals (LCAO)

Molecular orbitals are formed by the Linear Combination of Atomic Orbitals (LCAO). When two atomic orbitals combine, they form two molecular orbitals:

Bonding Molecular Orbital

Lower Energy - Constructive Interference

ψ_bonding = ψ_A + ψ_B

Formed by in-phase combination (addition) of atomic orbitals

Electron density increases between nuclei

Stabilizes the molecule

Denoted as σ, π, δ (depending on symmetry)

Antibonding Molecular Orbital

Higher Energy - Destructive Interference

ψ_antibonding = ψ_A - ψ_B

Formed by out-of-phase combination (subtraction) of atomic orbitals

Node between nuclei (zero electron density)

Destabilizes the molecule

Denoted as σ*, π*, δ* (with asterisk)

Important: For effective overlap and MO formation, atomic orbitals must have:

Similar energies
Proper symmetry (same symmetry with respect to the molecular axis)
Sufficient overlap

Types of Molecular Orbitals

Based on Symmetry

Sigma (σ) Orbitals

Cylindrical symmetry around the internuclear axis

Formed by head-on overlap

Examples:

• s + s → σ and σ*

• s + p_z → σ and σ*

• p_z + p_z → σ and σ*

Strongest type of covalent bond

Pi (π) Orbitals

Nodal plane containing the internuclear axis

Formed by side-by-side overlap

Examples:

• p_x + p_x → π and π*

• p_y + p_y → π and π*

Weaker than σ bonds

Present in double and triple bonds

Delta (δ) Orbitals

Two nodal planes containing the internuclear axis

Formed by d-orbital overlap

Examples:

• d_xy + d_xy → δ and δ*

• d_x²-y² + d_x²-y² → δ and δ*

Found in metal-metal multiple bonds

Weakest of the three types

MO Diagrams for Homonuclear Diatomic Molecules

Energy Order of Molecular Orbitals

For O₂, F₂, Ne₂ (Z ≥ 8):

σ1s < σ*1s < σ2s < σ*2s < σ2p_z < π2p_x = π2p_y < π*2p_x = π*2p_y < σ*2p_z

For Li₂, Be₂, B₂, C₂, N₂ (Z < 8):

σ1s < σ*1s < σ2s < σ*2s < π2p_x = π2p_y < σ2p_z < π*2p_x = π*2p_y < σ*2p_z

Note: For these lighter molecules, π2p orbitals are lower in energy than σ2p_z due to s-p mixing.

General MO Energy Level Diagram

When two atoms approach each other:

Atomic Orbital A + Atomic Orbital B → Bonding MO (lower energy) + Antibonding MO (higher energy)

Electrons fill the lowest energy MOs first, following the same rules as atomic orbitals

Bond Order and Stability

Bond Order Formula

Bond Order = ½ [(Number of electrons in bonding MOs) - (Number of electrons in antibonding MOs)]

Interpretation:

Bond Order = 0: No bond (molecule doesn't exist)
Bond Order = 1: Single bond
Bond Order = 2: Double bond
Bond Order = 3: Triple bond
Higher bond order = Stronger bond, shorter bond length
Fractional bond orders are possible (e.g., 1.5, 2.5)

Examples with Bond Order Calculations

Molecule	Total Electrons	Electronic Configuration	Bond Order	Magnetic Behavior
H₂	2	σ1s²	1	Diamagnetic
He₂	4	σ1s² σ*1s²	0 (unstable)	-
Li₂	6	(σ1s² σ*1s²) σ2s²	1	Diamagnetic
B₂	10	(KK) σ2s² σ*2s² π2p²	1	Paramagnetic (2 unpaired)
C₂	12	(KK) σ2s² σ*2s² π2p⁴	2	Diamagnetic
N₂	14	(KK) σ2s² σ*2s² π2p⁴ σ2p²	3	Diamagnetic
O₂	16	(KK) σ2s² σ2s² σ2p² π2p⁴ π2p²	2	Paramagnetic (2 unpaired)
F₂	18	(KK) σ2s² σ2s² σ2p² π2p⁴ π2p⁴	1	Diamagnetic
Ne₂	20	(KK) σ2s² σ2s² σ2p² π2p⁴ π2p⁴ σ*2p²	0 (unstable)	-

Note: (KK) represents the filled K shell: σ1s² σ*1s²

                "Molecular Orbital Theory's prediction that O₂ is paramagnetic was a triumph - VBT incorrectly predicted it would be diamagnetic!"
            

Heteronuclear Diatomic Molecules

For molecules like CO, NO, HF, and HCl where the two atoms are different:

Atomic orbitals have different energies
The more electronegative atom's orbitals are lower in energy
Bonding MOs have more character of the lower-energy atomic orbital
Antibonding MOs have more character of the higher-energy atomic orbital
MOs are not symmetric (polar bonds)

Carbon Monoxide (CO)

Electronic Configuration: (σ2s)² (σ*2s)² (π2p)⁴ (σ2p)²

Bond Order: 3 (triple bond)

Special Feature: Has a lone pair on carbon, making it a good ligand

CO is isoelectronic with N₂

Nitric Oxide (NO)

Electronic Configuration: (σ2s)² (σ*2s)² (π2p)⁴ (σ2p)² (π*2p)¹

Bond Order: 2.5

Magnetic Behavior: Paramagnetic (1 unpaired electron)

Readily loses the antibonding electron to form NO⁺ (bond order 3)

Delocalized π Systems

MOT excels at describing molecules with delocalized electrons, such as benzene, ozone, and conjugated systems:

Benzene (C₆H₆)

Six π electrons delocalized over six carbon atoms

Three bonding π MOs (all filled)

Three antibonding π* MOs (all empty)

Extra stability (aromatic stabilization)

Ozone (O₃)

π electrons delocalized over three oxygen atoms

Bond order = 1.5 for each O-O bond

Explains equivalent bond lengths

Better description than resonance structures

Conjugated Systems

Alternating single and double bonds

π electrons delocalized over entire chain

Explains UV-Vis absorption

Important in dyes and pigments

MOT for Coordination Compounds

Molecular Orbital Theory provides the most complete description of bonding in coordination complexes:

Advantages over VBT and CFT:

Accounts for both σ and π bonding between metal and ligands
Explains the spectrochemical series (why some ligands cause larger splitting)
Describes metal-to-ligand and ligand-to-metal charge transfer
Predicts magnetic properties more accurately
Explains partially covalent character of metal-ligand bonds

σ-Donor Ligands

Donate electron density to metal through σ bonds (e.g., NH₃, H₂O, Cl⁻)

Form bonding MOs from ligand lone pairs and metal d, s, and p orbitals

π-Acceptor Ligands (π acids)

Accept electron density from filled metal d orbitals into empty ligand π* orbitals

Strong field ligands: CO, CN⁻, NO⁺

Increase Δ (crystal field splitting) → low-spin complexes

Back bonding strengthens metal-ligand bond

π-Donor Ligands

Donate electron density from filled ligand π orbitals to empty metal d orbitals

Weak field ligands: I⁻, Br⁻, Cl⁻, OH⁻

Decrease Δ → high-spin complexes

                "MOT explains why CO is a strong-field ligand: π back-bonding from metal d orbitals to CO π* orbitals increases the crystal field splitting."
            

Comparison: MOT vs. VBT

Aspect	Molecular Orbital Theory	Valence Bond Theory
Electron Location	Delocalized over entire molecule	Localized between two atoms
Orbital Formation	Atomic orbitals combine to form MOs	Atomic orbitals overlap directly
Magnetic Properties	Correctly predicts paramagnetism of O₂, B₂	Incorrectly predicts O₂ is diamagnetic
Bond Order	Can have fractional values (e.g., 1.5, 2.5)	Always whole numbers (1, 2, 3)
Resonance	Not needed - delocalization is inherent	Requires resonance structures
Computational Ease	More complex calculations	Simpler, more intuitive
Explaining Spectra	Excellent for UV-Vis, photoelectron spectra	Limited spectroscopic predictions
Best For	Delocalized systems, diatomic molecules, spectroscopy	Simple molecules, qualitative descriptions

Applications of Molecular Orbital Theory

Spectroscopy

UV-Visible spectroscopy: Electronic transitions between MOs

Photoelectron spectroscopy: Ionization energies reveal MO energies

Predicting absorption wavelengths in conjugated systems

Organic Chemistry

Explaining aromatic stability (benzene, naphthalene)

Understanding conjugation and resonance

Predicting reactivity in pericyclic reactions

Frontier Molecular Orbital Theory (HOMO-LUMO)

Inorganic Chemistry

Metal-ligand bonding in coordination complexes

Understanding the spectrochemical series

Metal-metal multiple bonds

Organometallic compound bonding

Materials Science

Band theory of solids (extension of MOT)

Conductors, semiconductors, and insulators

Optical properties of materials

Molecular electronics

Biochemistry

Heme proteins (hemoglobin, myoglobin, cytochromes)

Photosynthetic pigments (chlorophyll)

DNA base stacking interactions

Enzyme catalysis mechanisms

Computational Chemistry

Basis for computational methods (Hartree-Fock, DFT)

Drug design and molecular modeling

Predicting molecular properties

Reaction mechanism studies

Important Concepts and Rules

Aufbau Principle

Molecular orbitals are filled in order of increasing energy, starting with the lowest energy MO.

Pauli Exclusion Principle

Each molecular orbital can hold a maximum of two electrons with opposite spins.

Hund's Rule

When filling degenerate (equal energy) molecular orbitals, electrons occupy them singly with parallel spins before pairing.

HOMO and LUMO

HOMO: Highest Occupied Molecular Orbital

LUMO: Lowest Unoccupied Molecular Orbital

The HOMO-LUMO gap determines many molecular properties including reactivity and color.

Study Tips for MOT

Master drawing MO diagrams for homonuclear diatomic molecules
Memorize the MO energy ordering for atoms with Z < 8 and Z ≥ 8
Practice calculating bond orders and predicting magnetic behavior
Understand the difference between σ, π, and δ orbitals
Know why MOT correctly predicts O₂ paramagnetism while VBT fails
Understand the relationship between bond order, bond length, and bond strength
Be able to compare and contrast MOT with VBT
Learn the concepts of π-donor and π-acceptor ligands for coordination chemistry

Chemistry Guide

What is Molecular Orbital Theory?

Historical Background

Fundamental Concepts

The Basic Premise

Key Principles of MOT:

Linear Combination of Atomic Orbitals (LCAO)

Bonding Molecular Orbital

Antibonding Molecular Orbital

Types of Molecular Orbitals

Based on Symmetry

Sigma (σ) Orbitals

Pi (π) Orbitals

Delta (δ) Orbitals

MO Diagrams for Homonuclear Diatomic Molecules

Energy Order of Molecular Orbitals

For O₂, F₂, Ne₂ (Z ≥ 8):

For Li₂, Be₂, B₂, C₂, N₂ (Z < 8):

General MO Energy Level Diagram

Bond Order and Stability

Bond Order Formula

Examples with Bond Order Calculations

Heteronuclear Diatomic Molecules

Carbon Monoxide (CO)

Nitric Oxide (NO)

Delocalized π Systems

Benzene (C₆H₆)

Ozone (O₃)

Conjugated Systems

MOT for Coordination Compounds

Advantages over VBT and CFT:

σ-Donor Ligands

π-Acceptor Ligands (π acids)

π-Donor Ligands

Comparison: MOT vs. VBT

Applications of Molecular Orbital Theory

Spectroscopy

Organic Chemistry

Inorganic Chemistry

Materials Science

Biochemistry

Computational Chemistry

Important Concepts and Rules

Aufbau Principle

Pauli Exclusion Principle

Hund's Rule

HOMO and LUMO

Study Tips for MOT

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