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§ Phase viii · Molecules · Chapter 01

LCAO bonding

By 1927 quantum mechanics had cracked the hydrogen atom. The natural next question was what happens when you bring two of them together. The answer, worked out by a small group of theorists in Göttingen and Chicago, was that the atomic orbitals do not just sit beside each other. They mix. They add and subtract. And the sums and differences have totally different fates.

In the summer of 1927 the hydrogen atom was already solved. Schrödinger had published his wave equation eighteen months earlier; Pauli had explained the periodic table; Heisenberg’s matrix mechanics had merged with Schrödinger’s waves into a single framework. A young theorist could open a textbook, look up the 1s orbital of hydrogen, and compute its energy to four decimal places. What no one had done yet was put two hydrogen atoms in the same room and explain, from those same equations, why they stuck together.

That problem turned out to be harder than it looked. The hydrogen molecule has two protons and two electrons, which means the Schrödinger equation already has six spatial coordinates and a Coulomb potential with four singular points. There is no clean closed-form answer. To make progress, theorists had to compromise: keep the spirit of the atomic-orbital picture but allow it to deform when the atoms come close. The first serious attempt, by Walter Heitler and Fritz London in Zurich that summer, kept the electrons paired up with their home atoms and let the wave function trade them. It worked, more or less. It got a bond.

The second attempt came from

The cleanest place to see LCAO at work is not even a full molecule. It is the hydrogen molecular ion, H₂⁺: two protons and one electron. There is nothing simpler in chemistry. There is no electron-electron repulsion to worry about, no Pauli exclusion, no spin pairing. Just one electron and two attracting protons. If LCAO does not work here, it does not work anywhere.

Call the two protons A and B. When the electron is near A and far from B, the system looks exactly like a hydrogen atom centered on A, and the electron sits in its 1s orbital. We will call that wave function 1s_A. When the electron is near B and far from A, it sits in 1s_B. Both of these are real, positive, spherically symmetric blobs centered on their nucleus. They are tabulated in every quantum-mechanics textbook. They are not the answer to the H₂⁺ problem (the electron is not glued to either proton), but they are an excellent starting point. LCAO says: build a trial wave function out of those two atomic pieces. Write ψ as a sum.

ψ = c_A · 1s_A + c_B · 1s_B

That is the entire LCAO ansatz for this problem. Two atomic orbitals, two coefficients to determine. Symmetry does the rest of the work for us. The molecule is identical if you swap A and B (two protons, one electron, no distinction between them), so the wave function must be either unchanged or simply flipped in sign under that swap. Unchanged means c_A = c_B. Flipped means c_A = -c_B. Those are the only two solutions, and after normalization they collapse to:

ψ_+  ∝  1s_A + 1s_B
ψ_−  ∝  1s_A − 1s_B

The plus combination is called the bonding orbital. The minus combination is called the antibonding orbital. The names are not decoration. They mean exactly what they say.

Why “bonding” and “antibonding”? The Born rule from earlier in this book tells us that the probability of finding the electron at a point is |ψ|² at that point. So square the two combinations and see where the charge lives.

For the plus combination, |ψ_+|² ∝ (1s_A + 1s_B)² = (1s_A)² + (1s_B)² + 2(1s_A)(1s_B). The first two terms are just the atomic densities (one blob on A, one on B), but the third is an extra contribution that exists only where both atomic orbitals overlap, which is the region between the nuclei. Adding wave functions in phase piles up extra probability where they meet. The electron, in the bonding orbital, spends a disproportionate amount of its time directly between the two protons. From there it pulls on both at once. The electrostatic attraction is double-counted. The molecule binds.

For the minus combination, |ψ_−|² ∝ (1s_A − 1s_B)² = (1s_A)² + (1s_B)² − 2(1s_A)(1s_B). The cross term has flipped sign. The electron is less likely to be found between the nuclei than the bare sum of the two atomic densities would predict. Exactly on the midplane between A and B, where 1s_A = 1s_B by symmetry, the antibonding wave function vanishes. There is a node. The electron is forbidden from the spot where it would have done the most good. Now the protons see each other directly, with no shielding electron in between, and they push apart.

This is the entire intuition of LCAO bonding, and it is worth pausing to feel its strangeness. Nothing changed about the atoms. The 1s orbitals on A and B are the same blobs they always were, the ground state of a single proton plus a single electron. All we did was add them in two different ways. And yet one combination corresponds to a stable chemical bond and the other to a repulsive scattering state. The bonding does not live in the atoms. It lives in the interference between them.

To make this quantitative, we have to put the two combinations back into the Schrödinger equation and compute their energies as a function of the proton-proton distance R. That is the H₂⁺ potential energy curve. The bonding energy E_+(R) drops as the two protons approach, reaches a minimum, and then rises sharply when the bare proton-proton repulsion overwhelms the electronic attraction. The antibonding energy E_−(R) rises monotonically from the dissociation limit. There is no minimum. There is no equilibrium. Push the two protons together in the antibonding state and they push back at every distance.

The shape of those curves is not an accident. It is what the variational principle (the rule that says the best wave function is the one with the lowest energy) hands you when you allow only two basis orbitals. You can crank through the algebra yourself if you want it. The result is sometimes called the secular equation, and it is the first calculation anyone does in molecular-orbital theory.

H_AA = ⟨1s_A | H | 1s_A⟩   (the on-site energy; equal to H_BB by symmetry)
H_AB = ⟨1s_A | H | 1s_B⟩   (the hopping matrix element; negative)
S    = ⟨1s_A | 1s_B⟩       (the overlap integral; positive, less than 1)

| H_AA − E       H_AB − E S | = 0
| H_AB − E S     H_AA − E   |

E_+  =  (H_AA + H_AB) / (1 + S)        (bonding)
E_−  =  (H_AA − H_AB) / (1 − S)        (antibonding)

The reason all of this matters, and the reason it has occupied a chapter of its own, is that the bonding-and-antibonding split is not a fact about hydrogen. It is a fact about superposition. Whenever you bring two atomic orbitals into overlap range and they are close enough in energy to mix, the same algebra runs: the in-phase combination sinks, the out-of-phase combination rises, the in-between region either fills with charge or develops a node. Two carbon 2p orbitals overlapping side-on give you the π and π* of ethylene. A sodium 3s and a chlorine 3p form the σ and σ* of NaCl. The d orbitals of a metal atom and the p orbitals of a ligand mix into the bonding and antibonding combinations that explain transition-metal complexes. Every covalent bond in every molecule you have ever heard of is a constructive interference of atomic orbitals, and every antibonding state is the destructive partner that has to stay empty for the molecule to hold together.

That is the structural-chemistry payoff, and it is enormous. The valence-bond picture from Heitler and London survives in the language of organic chemistry (the bond between two atoms, the lone pair, the resonance structure), but when chemists want to compute anything (a UV absorption spectrum, an oxidation potential, the geometry of a transition state, the photophysics of a chromophore) they reach for the molecular-orbital picture and the LCAO ansatz that underlies it. Modern density-functional theory codes, which now run on every supercomputer in every chemistry department in the world, still build their initial guesses out of linear combinations of atomic orbitals. The acronym in their input files is literally LCAO. The intuition you just absorbed from squaring a sum of two 1s functions is the same intuition that drives a billion CPU-hours a year of computational chemistry.

The deeper lesson is one we have already met several times in this book and will meet again. Quantum mechanics is a theory about superposition. Whenever two states with similar energies are coupled by some interaction (an electron hopping from atom to atom, a photon nudging a spin, a vibration exchanging quanta with a nearby mode), the eigenstates of the joint system are not either of the originals. They are sums and differences, sometimes equal-weight and sometimes lopsided, but always paired. One combination lies below the average. The other lies above. Wherever you see a splitting in physics (the symmetric and antisymmetric modes of two coupled pendulums, the even and odd parities of an ammonia umbrella, the bonding and antibonding orbitals of a molecule) the same algebra runs underneath. LCAO is just the version of that algebra written in the chemist’s accent. It is the same idea that makes a two-level atom oscillate between its ground and excited states under a driving field, the same idea that lets a Cooper pair condense, the same idea that lets a graphene electron tunnel sideways from one carbon to the next. Wave functions in close contact mix, and mixing always splits.

It is also the intuition that explains why the periodic table makes molecules. Two helium atoms, with their full 1s shells, would have to put four electrons into the σ_g and σ_u* combinations. Two go into bonding, two into antibonding, the antibonding rise outweighs the bonding drop, and He₂ does not exist in any room-temperature gas. Two hydrogen atoms have only two electrons total: both fit into σ_g, antibonding stays empty, and H₂ is the most abundant molecule in the universe. The whole question “does a covalent bond form?” reduces, in the simplest models, to “is the bonding orbital filled and the antibonding orbital empty?” That rule of thumb (the molecular-orbital bond order) is the LCAO picture compressed to a single number.