Fix the rules: in atomic units, a molecule’s energy operator has four terms — electron kinetic energy, electron–nucleus attraction, electron–electron repulsion, and (once nuclei are clamped) a constant nuclear repulsion. Find its lowest eigenvalue and you have the ground state.
The kinetic and attraction terms act on one electron at a time — if that were all, you could solve each electron separately and multiply. The repulsion sum is what ruins that: it ties every electron to every other.
Solve this eigenvalue problem and bonding, geometries, barriers, and spectra all follow. The entire field is a sequence of controlled approximations to a single equation we cannot solve exactly past one electron.
Because each electron repels all the others, you can’t place them one at a time. The wavefunction depends on all electron coordinates at once — it lives in dimensions. Add one electron and the cost doesn’t add, it multiplies.
Grid each coordinate into points and the exact wavefunction needs about values. For a handful of electrons that already exceeds the number of atoms in the universe. Brute force is off the table — forever.
Nuclei are thousands of times heavier than electrons, so they crawl while electrons fly. Clamp the nuclei; solve the electrons in that fixed frame. The nuclear repulsion becomes a constant you add at the end.
Solve the electrons for each nuclear geometry and the energies trace a potential energy surface. Its minima are stable molecules, its passes are reaction barriers. Chemistry is the shape of that surface.
Think before you swipe. Of the four terms in the Hamiltonian, which single one blocks solving for each electron independently — and what would happen if you simply deleted it?
Drop the electron–electron repulsion and the Hamiltonian separates into independent one-electron problems you could solve and multiply together. Keeping it (but replacing it with an average) is exactly the mean-field idea coming next.
Build the many-electron wavefunction from one-electron pieces. A spin-orbital is a spatial orbital — where the electron is — times a spin function, up or down. It’s the smallest brick we’ll stack.
The combined coordinate x bundles position r and spin ω. Two electrons can share the same spatial orbital only if their spins differ — the seed of the Pauli principle, made precise in three cards.
The obvious many-electron guess: put electron 1 in orbital , electron 2 in , and multiply. This Hartree product treats electrons as independent and labeled — and it’s wrong for a deep reason.
A Hartree product says “electron 1 is here, electron 2 is there.” But electrons are identical and indistinguishable — there is no fact about which is which. The math must not let you tell them apart.
Electrons are fermions, so the wavefunction must change sign whenever you exchange any two of them. Not “stay the same” — flip. This single rule carries all of Pauli exclusion inside it.
Put two electrons in the very same spin-orbital and swapping them must both leave the function unchanged (same state) and flip its sign (antisymmetry). The only number equal to its own negative is zero. The state is forbidden.
A determinant flips sign when you swap two rows and vanishes when two columns match. Put electrons in the rows and spin-orbitals in the columns and antisymmetry is automatic — for free.
Expand it: . Both ways of assigning the two electrons appear, with a minus sign — so you can’t say which electron is in which orbital. That’s the fix.
Hartree–Fock’s entire ansatz is: approximate the true wavefunction by a single Slater determinant, then choose the best possible orbitals to fill it. Everything that follows is making “best” precise.
We never added “no two electrons in the same state” as a separate rule. Swipe only after you can say which property of the Slater determinant produces it.
Two electrons in the same spin-orbital means two identical columns, and a determinant with identical columns is exactly zero. Pauli exclusion isn’t an extra axiom — it’s a theorem about determinants.
Any normalized trial wavefunction has an energy that is never below the true ground state. So we don’t solve the equation — we minimize a number. Lower energy is provably a better wavefunction.
Our trial function is one Slater determinant; the adjustable knobs are the orbitals inside it. Minimize the energy over all choices of orbitals and you’ve found the best single-determinant description allowed.
Plug the determinant into the energy and it splits cleanly: a sum of one-electron energies (kinetic + nuclear attraction) plus, for every pair of electrons, a Coulomb term minus an exchange term.
The Coulomb part is what you’d expect from charge clouds repelling. The subtracted exchange part has no classical analog — it’s the price the determinant’s minus sign extracts. Two cards from now it does real work.
The Coulomb integral is the everyday electrostatic repulsion between two smeared-out electron densities. Two clouds of negative charge push apart; J is the energy cost of that. Nothing quantum here — yet.
Exchange has no classical picture. It lowers the energy of electrons with the same spin, as if antisymmetry carves a little “hole” around each electron that keeps like-spin neighbors away. It only acts between parallel spins.
Because the determinant forbids two same-spin electrons from coinciding, they naturally stay apart — the Fermi hole. Exchange is the energy bonus of that built-in avoidance. Opposite-spin avoidance is what HF will miss.
Two electrons sit in a molecule. Predict: does the exchange term act between them if their spins are parallel? Antiparallel? Swipe when you’ve committed to an answer for both.
Exchange couples only same-spin electrons — antiparallel pairs have zero exchange. That’s why it’s tied to the Fermi hole: antisymmetry keeps like spins apart, and exchange is the energy of that avoidance.
Demand that the energy be stationary as you vary the orbitals (keeping them orthonormal). The condition collapses into an eigenvalue equation for an effective one-electron operator — the Fock operator.
Each electron feels its own kinetic + nuclear core term, plus the summed Coulomb and exchange fields of all the other occupied orbitals. The many-body problem became a one-body problem — in an averaged field.
The best orbitals are the ones that diagonalize the Fock operator; their eigenvalues are orbital energies. Fill the lowest for the ground state. This is one tidy one-electron eigenvalue problem.
Look back: the Fock operator is built from the occupied orbitals — the very things you’re solving for. The equation defines the orbitals in terms of themselves. That circularity is the whole story of the SCF loop.
The orbital energies aren’t just bookkeeping. Minus the highest occupied one approximates the energy to rip an electron out — the ionization potential. A real, measurable number drops out of the math for free.
Searching over “all functions” is infinite freedom. Instead, fix a small set of atom-centered basis functions and write each molecular orbital as a weighted sum. Now an orbital is just a column of coefficients.
The unknowns went from whole functions to a finite matrix of coefficients C. Solving Hartree–Fock now means solving for numbers — something a computer can actually do.
True atomic orbitals decay like with a sharp cusp at the nucleus. We use Gaussians, , which have neither — because the integrals between Gaussians are fast to compute. STO-3G fakes each true orbital with three Gaussians.
More basis functions hug the true orbitals better and, by the variational principle, push the energy down — but cost climbs steeply. Choosing a basis is buying accuracy by the pound.
Insert the LCAO expansion and the Fock equation turns into a matrix eigenvalue problem: the Fock matrix times the coefficients equals the overlap matrix times the coefficients times the orbital energies.
is the overlap matrix — atom-centered functions aren’t orthogonal, so it isn’t the identity. Orthogonalize the basis and this collapses to an ordinary eigenproblem you diagonalize directly.
The field each electron feels depends only on where all the electrons are — the density. Collect the occupied coefficients into a density matrix . It’s the single object the Fock matrix actually needs.
The Fock matrix is the core part plus a contraction of against the two-electron integrals. So depends on , and comes from diagonalizing . There it is again — the circle.
Every Coulomb and exchange contribution is a two-electron integral over four basis functions. With basis functions there are on the order of of them — the dominant cost and memory of the whole calculation.
Guess a density . Build . Solve for new coefficients. Form a new . Repeat. When stops changing, the field is consistent with the orbitals that made it — self-consistent.
It’s a fixed-point iteration: feed the field back into itself until it reproduces itself. Like a seating chart where everyone wants to sit near everyone’s final seat — reshuffle until nobody moves.
Take H at its bond length, about bohr, in STO-3G: one function on each atom, so matrices throughout. Two electrons, opposite spins, both in the bonding combination of the two functions.
The two functions overlap strongly — off-diagonal near , far from orthogonal. Start from a guess density, run the loop a handful of times, and the energy settles to about hartree — the textbook minimal-basis result.
That one number used all of it: the determinant, Coulomb minus exchange, the Fock matrix built from , , and the loop. The abstract machine produced a concrete energy you could compare to experiment.
You can diagonalize a matrix in one shot. So why does Hartree–Fock need a loop instead of a single solve? Swipe once you can name the dependency that forces it.
The Fock matrix is built from the density , but comes from diagonalizing . You can’t build the matrix until you know the answer — so you guess, solve, rebuild, and repeat until it’s consistent.
Each electron felt only the average of the others, so opposite-spin electrons never learned to dodge in real time. The exact energy minus the Hartree–Fock energy is, by definition, the correlation energy.
Correlation is a percent or two of the total energy but a large share of the differences chemistry cares about — bond strengths, barriers. Stretch H₂ apart and restricted HF fails badly: it can’t break the bond correctly.
Once you can name what HF misses, you can chase it. MP2 adds correlation by perturbation; configuration interaction and coupled cluster mix in other determinants; full CI is exact in the basis. Hartree–Fock is the floor they all build on.
Almost every method in quantum chemistry starts from a Hartree–Fock reference. Understand this one determinant — its orbitals, its Fock operator, its loop — and the rest of the field has a foundation to stand on.
From the molecular Hamiltonian to a converged H₂ energy — the whole chain, no gaps.
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