Beyond two atoms: orbitals that span the molecule
In the diatomic guides you built a molecular orbital diagram from just two atomic orbitals: combine them in phase and you get a bonding orbital that pools electron density between the nuclei; combine them out of phase and you get an antibonding orbital with a node where the bond should be. The whole trick was the linear combination of atomic orbitals — add the wavefunctions, square the result, see where the electrons pile up. Nothing in that recipe says you may only use two atoms. The honest, more general statement is this: every valence orbital on every atom in a molecule throws its hat into the ring, and the molecular orbitals you get are spread over all of them at once.
Here is a counting rule that survives the jump to many atoms unchanged, and it is your anchor: orbitals in equal orbitals out. If you feed N atomic orbitals into the blender, you get exactly N molecular orbitals back — no more, no fewer. Two 1s orbitals gave you one bonding and one antibonding level. Three p orbitals lined up sideways will give you three pi levels — one bonding, one roughly nonbonding, one antibonding. Six orbitals would give six. The orbitals simply stretch over more nuclei, and they range in energy from a most-bonding combination at the bottom to a most-antibonding one at the top.
Sigma and pi, stretched out — and why delocalization lowers the energy
The labels you learned for diatomics still apply, just over a longer stretch. A sigma molecular orbital is symmetric around the line joining the atoms — it has electron density piled up directly along the bonds, the way an s-s or end-on p-p combination did. A pi molecular orbital is built from p orbitals standing up perpendicular to the molecular plane and overlapping sideways; its density sits in lobes above and below that plane, with a node in the plane itself. In a chain of three or more atoms, the same sideways p overlap simply keeps going, so a pi cloud can run continuously down the whole backbone instead of being pinned between one pair of atoms — this is what we mean by a delocalized pi system.
Now the central question: why should letting an electron pair roam over three or four atoms be cheaper, energetically, than keeping it bottled up in one bond? The answer is the same uncertainty-principle logic that made any bond form in the first place. Confine an electron to a small box and quantum mechanics charges you a high kinetic-energy rent; give it more room to move, and that rent drops. The lowest, fully bonding molecular orbital of a delocalized system has the same phase everywhere — no nodes between the atoms — so an electron in it glides smoothly across the whole backbone with no costly wiggling. Spreading out literally lowers the energy of the bottom orbital. Pile your electrons into those low, wide-roaming levels and the molecule sits more stable than any single localised Lewis drawing would suggest.
This is the MO theory translation of something you already met: resonance. Back with Lewis structures, you drew the nitrate or ozone molecule as a blend of several pictures because no single one was right. MO theory says it more cleanly — there is just one true set of orbitals, and they happen to be delocalized. The 'resonance stabilisation' that the Lewis picture had to hand-wave about is, in the orbital picture, simply the energy you save by letting electrons spread out. Same physics, honester bookkeeping.
The frontier: HOMO and LUMO
Once a molecule has many orbitals, you rarely care about all of them at once. Fill them from the bottom up, two electrons per orbital, just as you filled atomic orbitals. The highest one that still has electrons in it is the [[homo-and-lumo|HOMO]], the highest occupied molecular orbital; the lowest one that is still empty, sitting just above it, is the LUMO, the lowest unoccupied molecular orbital. Together these two are the [[frontier-molecular-orbitals|frontier orbitals]] — the border between the molecule's full and empty levels.
Why single out just these two? Because chemistry is electrons changing hands, and the electrons most willing to move are the loosely held ones at the very top — the HOMO's — while the easiest place to put an incoming electron is the lowest vacancy, the LUMO. When a molecule acts as an electron donor (a base, a reducing agent, a nucleophile), it offers its HOMO; when it acts as an acceptor (an acid, an oxidant, an electrophile), it offers its LUMO. A reaction is, to a first approximation, a HOMO of one partner overlapping the LUMO of the other. A small HOMO-LUMO gap means a soft, reactive, easily-polarised molecule; a large gap means an inert, hard one — which, you may notice, is the orbital backbone underneath the hard-soft acid-base ideas waiting for you in a later rung.
A worked example: the pi system of ozone, O3
Let us scale the idea up on a real, bent triatomic: ozone, O3. The Lewis picture forced you into two resonance structures, one with the double bond on the left, one on the right, and you had to average them to get equal bond lengths. The MO picture handles it in one stroke. Set the sigma framework aside — three O-O sigma bonds and the in-plane lone pairs do their familiar localised job — and focus only on the one p orbital on each oxygen that points perpendicular to the molecular plane. Three such p orbitals, three pi molecular orbitals out. That is the whole delocalized pi system of ozone.
Three p(perp) AOs on O(left)-O(center)-O(right) --> three pi MOs: pi* (antibonding) + - + two nodes highest energy EMPTY --------------------------------------------------------------- pi(n) (nonbonding) + 0 - one node middle filled (lone-pair-like) --------------------------------------------------------------- pi (bonding) + + + no node lowest energy filled 4 pi electrons: fill bonding (2) + nonbonding (2); pi* stays empty pi bond order over the WHOLE molecule = 1 bonding pair / 2 O-O links = 1/2 per bond
Read the diagram from the bottom. The lowest pi MO has the same phase on all three oxygens — no node, fully bonding, density running unbroken across both O-O links. The middle one has a node right on the central atom: it is positive on the left oxygen, negative on the right, and zero in the middle, so it neither helps nor hurts the bonding — a genuine nonbonding orbital that behaves like a delocalized lone pair. The top one alternates sign at every step, two nodes, strongly antibonding. The pi system carries four electrons (one from each oxygen's p, plus the extra pair the molecule's charge balance demands), so they fill the bonding and the nonbonding levels and leave the antibonding pi* empty.
Look what falls out for free. Only the bottom orbital actually bonds the oxygens through the pi system, and it is one pair shared across two O-O links — so the pi bond order is one-half per link, on top of the one sigma bond each link already has. That predicts both O-O bonds to be identical and intermediate between a single and a double bond, which is exactly what is measured. No resonance arrows, no averaging by hand — the equal bond lengths simply are the answer the delocalized orbitals give. The nonbonding MO, meanwhile, is the HOMO, and ozone's modest HOMO-LUMO gap (the jump up to that empty pi*) is why it absorbs ultraviolet light so usefully in the stratosphere.
Scaling up further — and the honest limits
Ozone used three orbitals; nothing stops you adding more. Stack the same sideways p overlap down a longer chain and the pi levels multiply and bunch closer together, the lowest always node-free and the highest always full of nodes. Push that to thousands of atoms in a crystal and the levels merge into continuous bands — the very band theory that decides whether a solid is a metal, a semiconductor or an insulator, which you will meet in the solids rung. Delocalization is not a special trick for a few odd molecules; it is the same idea sliding smoothly from a triatomic all the way up to a chunk of silicon.
Two honest caveats keep this from turning into a religion. First, building a full delocalized diagram for a many-atom molecule by hand quickly becomes impractical; the clean three-orbital ozone story is the friendly end. For anything bigger, chemists lean on symmetry — grouping atomic orbitals into matching-symmetry combinations before they combine — which is exactly the symmetry-matching machinery a later rung formalises with character tables. Second, delocalized MOs and localised Lewis-type bonds are two valid languages for the same molecule, not a right and a wrong one. A chemist freely switches: localised pictures for naming bonds and electron counting, delocalized ones for magnetism, colour, and electron transfer. Knowing which lens to pick up when is the real skill.