linear combination of atomic orbitals (LCAO)
/ L-C-A-O, 'el-see-ay-oh' /
If molecular orbitals belong to the whole molecule, where do we get them from? We do not invent them out of thin air. Instead, we build each one as a recipe — a weighted mix of the atomic orbitals we already understand. Adding and subtracting those familiar atomic clouds in different proportions gives us the new molecular ones. That building-by-mixing recipe is the LCAO approximation.
Concretely, take two hydrogen 1s orbitals, call them A and B. Adding them in phase, (A + B), makes a molecular orbital with extra electron density piled up between the nuclei — a bonding orbital. Subtracting them, (A - B), makes one with a node, a flat sheet of zero density, right between the nuclei — an antibonding orbital. Two atomic orbitals in always give two molecular orbitals out, and the squared sizes of the coefficients on A and B tell you which atom the electrons lean toward. The word 'linear' just means we are scaling and summing, not multiplying the orbitals together.
LCAO is the everyday workhorse of molecular orbital theory because it is honest about being an approximation while still being good enough to predict bond order, magnetism, and the rough energy ladder of a molecule. The true molecular orbital is more complicated than any finite sum of atomic orbitals, but the atomic-orbital mix captures the chemistry that matters — which is exactly why almost every MO diagram you will ever draw is secretly an LCAO.
For H2: the bonding sigma orbital is roughly (1sA + 1sB) and the antibonding sigma-star orbital is roughly (1sA - 1sB). Both come from the same two atomic orbitals — one by adding, one by subtracting.
Add the atomic orbitals in phase for the bonding MO; subtract them for the antibonding MO.
LCAO is an approximation, not the exact truth — real molecular orbitals are not literally sums of isolated atomic orbitals, but the mix is accurate enough to get bonding right for the cases chemists care about.