An interactive essay in ten scenes & a coda
Quantum chemistry
a visual guide to what we actually compute
The problem
02 · The problem
A molecule is nuclei dressed in electrons.
The electrons define an energy for every geometry R. Both live in one quantum state — the wavefunction.
02 · The problem
Geometry becomes landscape.
Solve the electrons at each R: the energy traces the potential-energy surface.
02 · Eigenstates
Nuclear motion is quantized.
Levels ride between classical turning points; ψₙ has n nodes. The ground state has a sharp energy — and a finite zero-point spread ΔR.
02 · Excitation
Absorb the right hω and the state jumps vertically (Franck–Condon) to the excited surface. Energy rises by exactly hω.
02 · Dynamics
ψ(t) = e⁻ⁱĤt/ħ ψ(0).
Every eigencomponent rotates at its own Eₙ/ħ — their interference, not any rotation of the whole, moves the packet.
02 · Observation
Emission carries the energy back as light. The recorded spectrum — line positions and intensities — is the fingerprint of the calculation.
03 · Zoom
One equation, twelve decades of scale.
03 · Zoom
Each layer borrows its physics from the one below — and hands upward a simpler, effective model.
03 · Zoom
Batteries, catalysts, proteins, weather: the visible tip of a quantum stack, solved layer by layer.
Formulation
04 · Formulation
One object encodes the physics.
04 · The terms
Five pieces, one at a time. Nuclei jiggle (T̂ₙ), electrons stream (T̂ₑ), attractions and repulsions bind the dance (Vₑₙ, Vₑₑ, Vₙₙ).
04 · Born–Oppenheimer
Nuclei are ≥1836× heavier: freeze their positions. R becomes a parameter, T̂ₙ drops out, Vₙₙ becomes the constant Eₙₙ(R) — the electronic Hamiltonian remains.
05 · Representations
One molecule. Many descriptions.
A finite basis turns calculus into algebra: orbitals become coefficient vectors.
The correlated state is a sum over determinants — the CI expansion. A single determinant is one occupation pattern.
Creation operators build and reorder occupations; the sign is the permutation.
Jordan–Wigner carries it onto qubits — same numbers, new hardware.
The physics never changed. Only the bookkeeping — choose the form that makes your question cheap.
06 · The space
Choose your battles.
Full freedom: C(12,4) = 495 determinants. Freeze the core orbital: the fight shrinks to C(10,2) = 45.
Molecular symmetry partitions the space into blocks that never mix — diagonalize each sector on its own.
Every dot is one configuration; the blocks are its symmetry sectors. Spend freedom only where the answer needs it — then verify.
Variational
Make a guess; make it better. Energy is the compass — minimize until ‖∇E‖ → 0.
Projective
Impose the equation itself. Residuals vanish as the ansatz satisfies the eigenproblem.
Dynamical
Don't optimize — propagate. e⁻ⁱᴴᵗ is exact; representation is the challenge.
07 · Paradigms
The workhorse for ground states: Hartree–Fock, CI, MCSCF, DMRG, VQE. (DFT caveat: approximate functionals are variational in KS form only.)
07 · Paradigms
How quantum chemistry actually computes excited states and spectra: Krylov spaces, coupled cluster, DMRG — solve conditions, not gradients.
07 · Paradigms
Spectroscopy, reactions, materials in time — no energy minimum to chase, just unitary evolution (TDVP variants aside).
Evidence
08 · Comparison
Compare like with like.
Five conditions before any classical-vs-quantum claim means anything.
08 · The axes
Regimes, not winners. Scroll steps through the quantities; click a chip to pin it. Values are indicative ranges, not benchmarks.
09 · Complexity, honestly
The bound is not the experience.
Worst case spans all conceivable instances. Chemistry lives in a thin, structured valley — physical, smooth, low-dimensional.
What matters is empirical scaling on real instances — measured, not asserted.
Asymptotics guide; benchmarks decide. The gap between them is where research lives.
10 · Trust
Accuracy is purchased. Never free.
Eight nested knobs, spent in order — scroll to pay. Variational energies come down from above, never up.
Cost climbs faster than error falls (CCSD(T)-like N⁷, basis⁴). The overlap is the budget.
A number without convergence evidence is an anecdote — and converged still differs from experiment. Report both.
Coda
The rest is engineering.
Everything in this essay reduces to one line — and to the discipline of knowing what it cost you, how close you are, and who checked.
Set in
Instrument Serif · Instrument Sans · IBM Plex Mono
Built with
Next.js · Canvas 2D · KaTeX · zero runtime dependencies on the edge
Rendered
locally, in your browser, at 60 fps — no analytics, no cookies