Linear combination of atomic orbitals

Linear combination of atomic orbitals (LCAO) approximates a molecular orbital as a weighted sum of atomic orbitals centered on constituent atoms. The method underlies molecular-orbital calculations and explains how atomic orbitals combine into bonding and antibonding states.

Molecular orbital: A wavefunction describing an electron’s state in a molecule, typically spread over multiple nuclei. LCAO is a method for constructing these molecule-wide electron states.
Schrödinger equation: The equation governing how quantum-mechanical wavefunctions relate to energy and interactions. Molecular orbitals constructed by LCAO are approximate solutions to this equation.
Hückel method: A simplified molecular-orbital method for estimating π-electron energies in conjugated molecules. It applies LCAO to π orbitals with a particularly simple set of matrix assumptions.
Valence bond theory: A theory of chemical bonding that describes bonds through localized electron pairs and overlapping atomic orbitals. It builds bonds from localized structures rather than primarily delocalized molecular orbitals.
Atomic orbital: A one-electron wavefunction associated with an atom, characterized by its shape, energy, and angular momentum. These are the basis functions whose weighted sum forms an LCAO molecular orbital.
Basis set (chemistry): A finite collection of functions used to represent orbitals in an electronic-structure calculation. LCAO calculations use selected atomic-like functions as their basis set.
Extended Hückel method: A semiempirical molecular-orbital method that includes valence orbitals and approximate overlap effects. It extends the LCAO treatment beyond the π-electron simplifications of Hückel theory.
Molecular orbital theory: A framework that describes electrons in molecules using orbitals extending over multiple atoms. LCAO is a common construction method within this broader theory.
Linear combination: A sum of vectors or functions multiplied by scalar coefficients. LCAO applies this mathematical operation to atomic-orbital wavefunctions.
Variational method: A method that estimates a system’s lowest energy by optimizing a trial wavefunction. Optimizing LCAO coefficients can apply the variational method to molecular states.

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