The laws are known. The equations are not soluble.

In 1929, Paul Dirac wrote what may be the most consequential single sentence in the history of theoretical physics:

„The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.“

— P.A.M. Dirac, Quantum Mechanics of Many-Electron Systems, Proc. R. Soc. London A 123 (1929) 714–733

The diagnosis was complete. The prescription followed immediately:

„It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.“

Dirac was not apologising for a limitation of human ingenuity. He was naming the actual scientific programme. The work of atomic and molecular physics — its entire theoretical architecture — is the discipline of controlled approximation: the art of deciding which features of reality to keep and which to set aside, while retaining the capacity to account for what is observed.

This page is an orientation to that programme.

The many-body problem

The Schrödinger equation for a single hydrogen atom is exactly soluble. For helium — two electrons — it is not. For any system with more than one electron interacting with more than one nucleus, exact analytic solutions cease to exist. The obstacle is the Coulomb repulsion between electrons: each electron’s motion depends on every other electron’s position, making the system irreducibly coupled. This is the many-body problem.

It is not a temporary gap in technique. The Hilbert space of N interacting quantum particles grows exponentially in N. Even for a modest molecule of ten atoms, a full many-body wavefunction is not just computationally inconvenient — it is a mathematical object of astronomic dimensionality that no conceivable future computer could store, let alone evolve.

Dirac understood this in 1929. The field has spent the decades since building the hierarchy of approximations that make the problem tractable.

A hierarchy of approximations

Each step in the following sequence gives up something in exchange for tractability. Each is a legitimate scientific instrument, not a compromise.

Exact quantum mechanics (theoretical baseline)
The full many-body Schrödinger equation. Defines the target. Directly applicable only to systems of two to three particles.

Born–Oppenheimer approximation
Separates nuclear and electronic motion on the basis of their mass ratio. Nuclei, being ~1836× heavier than electrons, move on timescales long enough that electrons equilibrate instantaneously to any nuclear configuration. This decouples the nuclear and electronic problems: solve the electronic structure for fixed nuclei, then construct an effective potential energy surface on which nuclei move. The foundational approximation of all molecular science.
Breaks down when: two electronic states approach degeneracy (conical intersections, photochemical dynamics, tunnelling processes).

Variational methods
The variational principle states that for any trial wavefunction, the expectation value of the Hamiltonian is an upper bound on the true ground state energy. This converts the eigenvalue problem into an optimisation: improve the trial wavefunction, lower the energy. Systematic and controllable.

Hartree–Fock (mean-field)
Each electron moves in the average field of all others. The many-body wavefunction is approximated as an antisymmetrised product of single-particle states (a Slater determinant). Computationally tractable for large systems; captures ~99% of the total electronic energy. The missing 1% — electron correlation — is where the interesting physics of strongly correlated systems lives.

Post–Hartree–Fock methods (CI, MP, CC)
Systematic corrections to HF: configuration interaction (CI), Møller–Plesset perturbation theory (MP2, MP4), coupled-cluster theory (CCSD, CCSD(T)). The last is widely called the „gold standard“ of quantum chemistry for small and medium molecules.
Reference: Szabo & Ostlund, Modern Quantum Chemistry (1982/1996 Dover) — the standard text connecting Dirac’s programme to computational practice.

Density Functional Theory (DFT)
Hohenberg, Kohn, and Sham (1964–1965) showed that the ground-state energy is a functional of the electron density ρ(r) alone — not the full wavefunction. The density is a function of three spatial coordinates regardless of the number of electrons, making DFT scalable to systems of thousands of atoms. The approximation lies in the exchange-correlation functional, which is known exactly in form but must be approximated in practice. DFT is the workhorse of condensed matter physics and materials science.

Dynamical Mean-Field Theory (DMFT)
For systems where electron-electron correlations are local and strong — where the Hubbard-U (on-site repulsion) dominates the kinetic energy — DFT fails qualitatively. DMFT maps the lattice problem onto a self-consistently determined quantum impurity problem: a single correlated site embedded in an effective bath representing the rest of the lattice. Non-perturbative in the local interaction. The method that finally made strongly correlated electron systems computationally accessible.

Quantum Monte Carlo (QMC)
Stochastic sampling of the many-body wavefunction space. Variants: variational (VMC), diffusion (DMC), determinant (DQMC). Among the most accurate methods available for correlated systems; computationally expensive; sign problem limits applicability to certain Hamiltonians.

Where this field connects to the frontiers

The programme Dirac outlined in 1929 runs directly into the two classes of material systems that have defined the frontier of condensed matter physics for the past four decades: heavy fermionic systems and high-temperature superconductors. Both are defined by precisely what the intermediate levels of the hierarchy above cannot handle — strong, non-local, dynamical electron-electron correlations.

→ Section A: Heavy Fermionic Systems
(Uranium-Platinum, Plutonium composites, Kondo lattice physics, NRG and DMFT numerics)
The f-electron materials — UPt₃, CeAl₃, PuCoGa₅ — where effective electron masses reach hundreds of bare electron masses. Where superconductivity emerges from the same electrons that form a dense Kondo lattice. Where the question of pairing symmetry in UPt₃ remains contested after forty years.

→ Section B: High-Temperature Superconductivity
(Lanthanum-Barium-Copper-Oxide, Yttrium-Barium-Copper-Oxide, iron-based superconductors, t-J and Hubbard model numerics)
The cuprate and iron-based superconductors — where Cooper pairs form at temperatures impossible under the original BCS theory, where the pairing mechanism remains the central open question of theoretical physics, and where the pseudogap phase above Tc continues to resist a definitive theoretical description.

Selected references

  • P.A.M. Dirac, „Quantum Mechanics of Many-Electron Systems,“ Proc. R. Soc. London A 123 (1929) 714–733. DOI: 10.1098/rspa.1929.0094
  • A. Szabo & N.S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover (1996). ISBN 0-486-69186-1
  • W. Kohn & L.J. Sham, „Self-Consistent Equations Including Exchange and Correlation Effects,“ Phys. Rev. 140 (1965) A1133
  • A. Georges, G. Kotliar, W. Krauth & M.J. Rozenberg, „Dynamical mean-field theory of strongly correlated fermion systems,“ Rev. Mod. Phys. 68 (1996) 13