Linear Combination of Atomic Orbitals (LCAO) Conditions 

1.0What is LCAO?

LCAO is a quantum chemistry method where molecular orbitals (MOs) are expressed as a linear combination of atomic orbitals (AOs). Mathematically, each molecular orbital φᵢ is represented as:

${\varphi }_i={\sum }_r{c}_{ri}{\chi }_r$

where χᵣ are atomic orbitals and cᵣᵢ are coefficients indicating each AO’s contribution.

This approach is used qualitatively in JEE contexts, though quantitatively it forms the basis of Hartree–Fock and other advanced computational methods.

2.0Basic Principles of LCAO

The LCAO method is based on the idea that a molecular orbital (Ψ) is obtained by combining two atomic orbitals (φA and φB):

$\Psi =c_A{\phi }_A+c_B{\phi }_B$

Here,

• ${\phi }_{A}and{\phi }_B$= atomic wave functions of atoms A and B
• $c_{A}and{c}_B$ = coefficients determining the contribution of each orbital

The linear combination can lead to two types of orbitals:

• Bonding Molecular Orbital (Ψ+) – constructive overlap
• Antibonding Molecular Orbital (Ψ–) – destructive overlap

But not every atomic orbital can combine. For the combination to be successful, certain LCAO conditions must be satisfied.

3.0Conditions for Linear Combination of Atomic Orbitals

There are three essential conditions for the effective linear combination of atomic orbitals:

Condition 1: Similar Energy Levels

• For effective overlap, the energy difference between combining atomic orbitals must be small.
• Example: 1s orbital of hydrogen can combine with 1s orbital of another hydrogen to form H₂ molecule.
• On the other hand, 1s orbital of hydrogen cannot effectively combine with 2s orbital of lithium because the energy difference is too large.

Condition 2: Proper Symmetry of Atomic Orbitals

• Atomic orbitals must have the same orientation in space for effective overlap.
• If the symmetry of orbitals is different, destructive interference occurs, and no stable molecular orbital is formed.
• Example:
• • s-s overlap → effective and allowed
  • s-p overlap → possible if aligned along the internuclear axis
  • $p_x–p_x$ overlap → strong head-on overlap, forms σ bonds
  • $p_x–p_y$ overlap → not allowed due to improper symmetry

Condition 3: Effective Overlap of Atomic Orbitals

• The extent of overlap determines the strength of the bond.
• Greater overlap → stronger bonding molecular orbital → more stable molecule.
• Example:
• • H₂ molecule is stable due to strong 1s–1s overlap.
  • He₂ molecule does not exist due to ineffective overlap and bond order = 0.

4.0Importance of Linear Combination of Atomic Orbitals in Molecular Orbital Theory

Molecular Orbital Theory (MOT) is more accurate than Valence Bond Theory (VBT) because it explains phenomena like:

• Paramagnetism of oxygen (O₂), which VBT fails to explain.
• Bond order calculations using the molecular orbital diagram.
• Delocalization of electrons in molecules.

At the core of MOT lies the LCAO concept. Without understanding how atomic orbitals combine, one cannot construct molecular orbital diagrams or predict bond properties.

Thus, LCAO is a high-weightage concept in JEE Chemistry, directly linking to molecular orbital diagrams, bond order, and electronic configuration of diatomic molecules.

5.0Application of LCAO 

Understanding LCAO is essential for solving JEE questions such as:

• Identifying stable vs unstable molecules (H₂ vs He₂).
• Calculating bond order and bond length trends across diatomic molecules.
• Explaining why O₂ is paramagnetic while N₂ is diamagnetic.
• Comparing bond strengths in isoelectronic species (e.g., N₂, O₂⁺, F₂).
• Predicting electronic configuration of molecules in terms of σ and π orbitals.