LCAO Principle
Molecular Orbital Theory (MOT) offers a powerful quantum-mechanical description of chemical bonding that goes beyond the simple shared-pair picture of Lewis structures. At its heart lies the Linear Combination of Atomic Orbitals (LCAO) principle, which states that when atoms bond together, their atomic orbitals mathematically combine to produce a new set of orbitals that belong to the entire molecule — called molecular orbitals (MOs). Just as atomic orbitals describe where electrons are likely to be found around a single atom, molecular orbitals describe where electrons are likely to be found across a whole molecule. The word 'linear combination' is borrowed from mathematics and simply means that we add or subtract the wave functions (ψ) of atomic orbitals to generate new wave functions for molecular orbitals. If two atomic orbitals combine, they must produce exactly two molecular orbitals — one formed by addition (constructive interference of electron waves) and one formed by subtraction (destructive interference). These two types are called the bonding molecular orbital (BMO) and the antibonding molecular orbital (ABMO), and they differ profoundly in energy, shape, and their effect on molecular stability. LCAO-MOT successfully explains properties that earlier bonding models struggled with, such as the paramagnetism of oxygen (O₂), the bond order of species like He₂⁺, and the existence of three-centre two-electron bonds in molecules like diborane. Understanding the LCAO principle is therefore essential for any student who wants to go beyond empirical rules and grasp the quantum-mechanical reality underlying all of chemistry.
- Start with Atomic Wave Functions
- Constructive Combination — The Bonding MO (BMO)
- Destructive Combination — The Antibonding MO (ABMO)
- Conditions for Effective Combination
- Filling MOs and Calculating Bond Order
Start with Atomic Wave Functions
Every atomic orbital is described by a wave function ψ (psi). For two hydrogen atoms A and B approaching each other, we begin with their 1s wave functions ψ_A and ψ_B. These wave functions have both positive and negative lobes (phases), shown with + and − signs. Phase does NOT mean electric charge — it refers to the mathematical sign of the wave function in a region of space.
Constructive Combination — The Bonding MO (BMO)
When we add the two wave functions (ψ_A + ψ_B), regions of the same phase overlap constructively, increasing electron density between the two nuclei. This produces the bonding molecular orbital, denoted σ1s (sigma 1s) for s-orbital combinations or σ_b. Electrons in a BMO are pulled toward both nuclei simultaneously, which lowers the total energy of the system. A BMO is therefore lower in energy than the original atomic orbitals, and its occupation stabilises the molecule. The electron density (probability) between the nuclei is proportional to (ψ_A + ψ_B)², which is always positive and has a large value in the internuclear region.
Destructive Combination — The Antibonding MO (ABMO)
When we subtract the two wave functions (ψ_A − ψ_B), regions of opposite phase cancel each other. This creates a nodal plane — a flat surface between the two nuclei where the electron density falls to exactly zero. The orbital formed is called the antibonding molecular orbital, denoted σ*1s (sigma-star 1s). Electrons in an ABMO are excluded from the internuclear region and are found mainly outside the two nuclei, so they actually destabilise the bond. An ABMO is higher in energy than the original atomic orbitals, and its occupation weakens or breaks the molecule.
Conditions for Effective Combination
Not all atomic orbitals combine equally well. Three criteria must be satisfied: (a) Symmetry — the orbitals must have compatible symmetry with respect to the bond axis (e.g., two s orbitals or two p_z orbitals can form σ MOs; p_x and p_y form π MOs). (b) Energy similarity — the energies of the combining atomic orbitals must be close (within ~13 eV of each other). (c) Orbital size overlap — the orbitals must overlap sufficiently in space. If any criterion is poorly met, the resulting MOs remain largely 'atomic' in character and the bond is weak.
Filling MOs and Calculating Bond Order
Molecular orbitals are filled with electrons following the same rules as atomic orbitals: Aufbau principle (lowest energy first), Pauli exclusion principle (maximum 2 electrons per MO, with opposite spins), and Hund's rule (degenerate MOs filled singly before pairing). Bond order (BO) is then calculated as: BO = (Bonding electrons − Antibonding electrons) / 2. A bond order of 1 means a single bond, 2 a double bond, etc. If BO = 0, the molecule does not exist. A fractional bond order (e.g., 0.5 for He₂⁺) is physically meaningful and corresponds to a real, albeit weak, bond.
The Key Equation
- \psi_{BMO} — Bonding MO wave function (dimensionless (normalised)): Mathematical description of the bonding molecular orbital; its square gives the probability density of finding an electron in that orbital.
- \psi_{ABMO} — Antibonding MO wave function (dimensionless (normalised)): Mathematical description of the antibonding molecular orbital; contains a nodal plane between the two nuclei where electron density = 0.
- \psi_A,\, \psi_B — Atomic orbital wave functions of atoms A and B (dimensionless (normalised)): The original wave functions of the contributing atomic orbitals on atoms A and B before combination.
Worked Example
Problem. Using LCAO-MOT, determine the molecular orbital electron configuration, bond order, and magnetic properties of the oxygen molecule (O₂). Explain why Lewis structure theory fails here.
Answer. O₂ has bond order = 2 (double bond), and is paramagnetic due to two unpaired electrons in degenerate π*2p antibonding orbitals.
Origins · 1 minRobert Sanderson Mulliken and how this was discovered
Robert Mulliken is the primary architect of Molecular Orbital Theory and the LCAO approach. In the 1920s and 1930s, he developed the concept of molecular orbitals as wave functions that span entire molecules rather than individual atoms, and introduced much of the notation still used today (σ, π, bonding, antibonding). His systematic application of quantum mechanics to diatomic molecules explained spectroscopic data that valence bond theory could not account for. He was awarded the Nobel Prize in Chemistry in 1966 'for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method.' Mulliken was notoriously difficult to understand in lectures — his colleague and rival Linus Pauling reportedly joked that Mulliken's papers were written so densely that only Mulliken himself could truly follow them.
Related Concepts
- Valence Bond Theory (VBT) — VBT is the competing bonding model to MOT. Understanding both theories together — and knowing where each succeeds or fails (e.g., VBT fails for O₂ paramagnetism) — gives a complete quantum-mechanical picture of chemical bonding.
- Bond Order and Molecular Stability — Bond order is the key quantity derived from LCAO-MOT electron configurations. It directly connects the abstract wave function mathematics to measurable properties like bond length, bond energy, and whether a molecule can exist.
- MOT of Homonuclear Diatomic Molecules — Applying LCAO to H₂ through Ne₂ is the direct extension of this concept, where you fill MO diagrams and predict magnetic properties and bond orders for real molecules.
- Atomic Orbitals and Quantum Numbers — LCAO begins with atomic orbitals (1s, 2s, 2p etc.). A thorough understanding of their shapes, energies, and phases (signs of wave functions) is prerequisite knowledge for LCAO-MOT.
- Paramagnetism and Diamagnetism — One of MOT's greatest successes is explaining why O₂ is paramagnetic. Understanding magnetic behaviour of molecules requires knowing how unpaired electrons arise from the LCAO filling of degenerate antibonding orbitals.