Single-Particle Models

Foundations

Here is something nobody ever sat me down and pointed out, and it took an embarrassing amount of time to notice: every successful many-body theory in quantum chemistry, condensed matter, and nuclear physics quietly does the same thing. They all build a fictitious one-body problem, diagonalize it, fill the levels, and declare that an approximation to the true many-body ground state. The mean field changes, the occupation rule changes, the language changes — but the move is the same. Hartree-Fock, Kohn-Sham DFT, the nuclear shell model, BCS, HFB, tight-binding, band theory: same skeleton, different coat of paint.

This page is about that skeleton. What is the move, exactly? Why does it work so absurdly well when at first glance it shouldn't? Where does it break, and what do you do then? The unifying lens is under-explained in the standard textbooks — each subfield introduces its own single-particle theory and rarely points out that it's an instance of a much more general pattern. So: a tour.

The move

Take an interacting many-body Hamiltonian — the actual, true one, with all the two-body Coulomb (or strong-force, or whatever) interactions:

\hat{H} = i \sum \hat{h}_{0} (r_{i}) + \frac{1}{2} i \neq = j \sum v (r_{i}, r_{j}) .

The one-body part $\hat{h}_{0}$ contains kinetic energy and any external potential (the nuclei in a molecule, the lattice in a solid, the confining well in a nucleus). The two-body part is the interaction — Coulomb between electrons, the nuclear force between nucleons, on-site Hubbard repulsion on a lattice. This Hamiltonian has, for $N$ particles, a configuration space of dimension roughly $(N M)$ where $M$ is the size of your single-particle basis. For any reasonable $N$ and $M$ , that's astronomical. You cannot diagonalize it.

The move: invent a one-body operator $\hat{h}_{eff}$ that does some of the work the two-body interaction would have done. Solve the resulting eigenvalue problem:

\hat{h}_{eff} ψ_{α} (r) = ε_{α} ψ_{α} (r) .

You get a stack of single-particle levels ${ε_{α}}$ with corresponding wavefunctions ${ψ_{α}}$ . Now fill the lowest $N$ of them according to whatever statistics your particles obey (Pauli for fermions: one particle per spin-orbital). The many-body state is the antisymmetric product of the filled levels — a Slater determinant. That Slater determinant is your approximation to the true ground state.

That's the entire move. Vary $\hat{h}_{eff}$ , the occupation rule, or the way you wrap the filled levels back into a many-body state, and you get every single-particle method that exists.

The common skeleton

Every single-particle theory has the same four pieces. Once you see the skeleton it's hard to unsee it.

(1) Effective one-body Hamiltonian $\hat{h}_{eff}$ . Sometimes prescribed (the nuclear shell model puts a harmonic oscillator plus spin-orbit on the table by hand). Sometimes built self-consistently from the density the method produces (HF, KS-DFT). Always one-body: it acts on a single particle's coordinates, even if its construction encodes the average effect of the other particles.

(2) Eigenvalue problem. Solve $\hat{h}_{eff} ψ_{α} = ε_{α} ψ_{α}$ . This is the entire point of the simplification — you reduced a many-body problem of size $\sim e^{N}$ to a single-particle problem of size $M$ , the basis size. Linear algebra you can actually run.

(3) Occupation rule. Some statistics for how the particles populate the levels. For fermions at zero temperature: fill the lowest $N$ levels (Aufbau), no more than one per spin-orbital (Pauli). At finite temperature: Fermi-Dirac. For bosons: Bose-Einstein (all in the ground state at $T = 0$ ). The occupation rule is where the statistics of identical particles enter.

(4) Reconstruction. Wrap the filled levels back into a many-body state. For fermions: a Slater determinant. For pairs: a coherent superposition (BCS state). The reconstruction is what lets you compute observables — energies, densities, correlation functions — from the single-particle pieces. Sometimes the reconstruction is explicit (HF builds the Slater determinant). Sometimes the auxiliary determinant doesn't represent anything physical, but the density it produces is what matters (KS-DFT). Sometimes the levels themselves carry the physics and the determinant is implicit (band theory).

Everything that follows is a tour of methods that put different things in those four slots.

Catalog: Hartree-Fock

The granddaddy. Start with the variational principle: the best Slater determinant is the one that minimizes $⟨ Φ∣ \hat{H} ∣Φ ⟩$ over all single-determinant trial wavefunctions. The minimizer satisfies the Hartree-Fock equation:

\hat{F} ψ_{i} = ε_{i} ψ_{i}, \hat{F} = \hat{h}_{0} + j \sum (\hat{J}_{j} - \hat{K}_{j}) .

The Fock operator $\hat{F}$ contains the one-body piece $\hat{h}_{0}$ plus the Hartree (Coulomb) potential $\hat{J}_{j}$ from each other occupied orbital plus the exchange potential $\hat{K}_{j}$ arising from antisymmetry. The catch: $\hat{F}$ itself depends on the occupied orbitals through $\hat{J}$ and $\hat{K}$ , so you solve by self-consistent iteration: guess orbitals, build $\hat{F}$ , diagonalize to get new orbitals, repeat until convergence.

Slot-by-slot:

Effective Hamiltonian: the Fock operator $\hat{F}$ .
Eigenvalue problem: $\hat{F} ψ_{i} = ε_{i} ψ_{i}$ , self-consistent.
Occupation rule: fill the $N /2$ lowest spatial orbitals with two electrons each (RHF), or use independent spin-up/down orbitals (UHF).
Reconstruction: a single Slater determinant of the occupied spin-orbitals. This determinant IS the HF wavefunction.

What's missing: dynamic correlation. Two electrons in the HF approximation don't avoid each other's instantaneous position the way they should — the mean-field Coulomb potential is everywhere at once, the Coulomb hole isn't there. The error is sometimes called "the correlation energy" and is the target of every post-HF method (MP2, CCSD, FCI, etc.).

The HF page on this site has more, and the from-scratch H2 series implements it end to end.

Catalog: Kohn-Sham DFT

Different philosophy, same skeleton. Hohenberg-Kohn proved that the ground-state density $ρ (r)$ determines the total energy of an interacting electron system. Kohn and Sham then said: imagine a fictitious non-interacting electron system with a single-particle potential $v_{KS} (r)$ chosen so its density exactly matches the interacting density. The fictitious system is a single-particle problem, trivially solvable:

\hat{h}_{KS} ψ_{α} = ε_{α} ψ_{α}, \hat{h}_{KS} = - \frac{1}{2} \nabla^{2} + v_{ext} (r) + v_{H} [ρ] (r) + v_{xc} [ρ] (r) .

The external potential $v_{ext}$ is the nuclear attraction. The Hartree potential $v_{H} [ρ]$ is the classical Coulomb response to the density. The exchange-correlation potential $v_{xc} [ρ]$ is everything the previous two terms miss, packed into a universal functional of the density. The whole game in DFT is approximating $v_{xc}$ — LDA, GGA, hybrids, double hybrids, range-separated functionals. The choice determines whether you get bond energies right, whether band gaps come out close to experiment, whether van der Waals dispersion shows up.

Slot-by-slot:

Effective Hamiltonian: the KS Hamiltonian $\hat{h}_{KS}$ , self-consistent through its density dependence.
Eigenvalue problem: single-particle Schrödinger equation in the KS potential.
Occupation rule: Aufbau as in HF.
Reconstruction: this is the philosophical twist. The KS Slater determinant is NOT the true ground-state wavefunction. It is an auxiliary object whose density matches the true ground-state density. The reconstruction goes through the density, not the wavefunction.

The strange consequence: KS orbital energies $ε_{α}$ are mathematical artifacts of the auxiliary system. Strictly, only the HOMO energy has a direct physical interpretation (it equals minus the ionization potential, for the exact functional). Everything else is bookkeeping. But the bookkeeping is so useful that practitioners interpret KS orbital energies as if they were physical excitation energies — which is wrong in principle but works well enough that nobody can shake the habit.

The trick of using the same matrix machinery (single-particle eigenvalue problem, occupation by Aufbau, density built from the orbitals) means HF and KS-DFT look identical in code. Swap the Fock operator for the KS operator, swap the exchange integral for the XC kernel, and you have the same SCF loop.

Catalog: nuclear shell model

Same skeleton, different starting potential. For a nucleus with $A$ nucleons, write down a phenomenological one-body Hamiltonian that captures the mean field nucleons see:

\hat{h}_{eff} = - \frac{ℏ ^{2}}{2 m} \nabla^{2} + \frac{1}{2} m ω^{2} r^{2} - V_{L S} \hat{L} \cdot \hat{S} .

The harmonic oscillator is the simplest mean-field model that gets the bulk of the central potential right (real nuclei look more Woods-Saxon, but harmonic is analytically nice and qualitatively close). The crucial extra piece is the spin-orbit coupling $- V_{L S} \hat{L} \cdot \hat{S}$ , which splits levels of given $ℓ$ into $j = ℓ + 1/2$ and $j = ℓ - 1/2$ . Without spin-orbit you don't get the magic numbers right; with it you do.

Diagonalize, get a stack of single-nucleon levels (1s½, 1p³⁄₂, 1p½, 1d⁵⁄₂, …). Fill them with neutrons and protons independently (separate Aufbau for each isospin). The shell closures fall at $A = 2, 8, 20, 28, 50, 82, 126$ — the magic numbers, where adding one more nucleon would have to go into a shell with a much larger energy gap.

Slot-by-slot:

Effective Hamiltonian: phenomenological mean-field harmonic + spin-orbit. Prescribed, not self-consistent in the simplest version (refined self-consistent versions exist as Hartree-Fock with the nuclear force).
Eigenvalue problem: single-nucleon Schrödinger equation in the mean field.
Occupation rule: Aufbau, separately for protons and neutrons.
Reconstruction: Slater determinant of filled levels for the inert core; configuration mixing among valence-shell nucleons handles the residual interaction.

The shell model is the cleanest example of the single-particle approximation working in a system where the underlying "interaction" (the nuclear force) is wildly more complicated than Coulomb. The fact that it works tells you the mean-field idea is robust — the central potential captures the bulk of the physics whether the residual is electromagnetic, strong, or anything else short-ranged. See the nuclear shell model page for more.

Catalog: BCS and HFB

Now we extend the skeleton. For superconductors, superfluids, and pairing-correlated nuclei, the relevant "particle" is not a bare electron or nucleon — it's a Bogoliubov quasiparticle, a coherent superposition of a particle at energy $+ E$ and a hole at energy $- E$ . The single-particle (now: single-quasiparticle) Hamiltonian acts on a two-component Nambu spinor:

(h_{HF} - μ Δ^{*} Δ - (h_{HF} - μ)^{*}) (u_{α} v_{α}) = E_{α} (u_{α} v_{α}) .

The diagonal blocks are the HF Hamiltonian shifted by the chemical potential. The off-diagonal $Δ$ is the pair field — what mixes particle and hole and opens the energy gap. Solve the eigenvalue problem to get quasiparticle energies $E_{α}$ and Nambu amplitudes $(u_{α}, v_{α})$ . The many-body ground state is a coherent superposition of pairs — a vacuum for the quasiparticles. When $Δ \to 0$ the equation block-diagonalizes and you recover HF.

Slot-by-slot:

Effective Hamiltonian: the HFB matrix above, self-consistent in both the density (drives $h_{HF}$ ) and the pair density (drives $Δ$ ).
Eigenvalue problem: a doubled-dimension eigenproblem on Nambu spinors. The doubling encodes the particle-hole mixing.
Occupation rule: the quasiparticle vacuum is annihilated by all quasiparticle operators. In terms of original particles, this corresponds to a probability $∣ v_{α} ∣^{2}$ of finding the particle in level $α$ .
Reconstruction: a quasi-Slater state (BCS vacuum) that is a product over pair operators — it does not have a fixed particle number, but it has a fixed average particle number.

HFB is the canonical example of how the skeleton stretches: keep the four ingredients, generalize what "particle" means, and you can handle pairing physics that the bare single-particle picture cannot. See the HFB page for the full derivation in the nuclear context.

Catalog: tight-binding and band theory

For crystals, lattice translation symmetry hands you the single-particle picture almost for free. The effective Hamiltonian commutes with lattice translations, so it's block-diagonal by crystal momentum $k$ . Within each $k$ -block you diagonalize a finite matrix (a sum of intra-cell hoppings) and get a discrete set of energies — the bands $ε_{n} (k)$ .

\hat{h}_{eff} (k) ψ_{n k} = ε_{n} (k) ψ_{n k} .

In tight-binding, $\hat{h}_{eff}$ is a hopping matrix between lattice sites built by hand or fit to first-principles data. In ab-initio band theory (DFT in a crystal) it's the KS Hamiltonian. Either way: diagonalize per $k$ , fill up to the Fermi level, you have the electronic ground state.

Slot-by-slot:

Effective Hamiltonian: the lattice-periodic one-body Hamiltonian, block-diagonal by $k$ .
Eigenvalue problem: separate small eigenproblem per $k$ -point.
Occupation rule: Fermi-Dirac at the system temperature, sampled over the Brillouin zone. At zero temperature: fill bands up to $ε_{F}$ .
Reconstruction: Slater determinant of all filled Bloch states. The Fermi surface — the set of $k$ points with $ε_{n} (k) = ε_{F}$ — is the geometric object that determines whether the material is a metal, insulator, or semimetal.

Band gaps, Fermi surfaces, density of states at the Fermi level, electrical conductivity, magnetic order — all are derived from the single-particle level diagram $ε_{n} (k)$ . Solid-state physics is, to a first approximation, the study of how this diagram looks for different materials.

Side by side

Same four ingredients, different instantiations:

Method	Effective Hamiltonian	Built how	Many-body state	What's missing
Hartree-Fock	Fock operator $\hat{F} = \hat{h}_{0} + \hat{J} - \hat{K}$	Self-consistent variational	Single Slater determinant	Dynamic correlation
Kohn-Sham DFT	KS Hamiltonian with $v_{xc} [ρ]$	Self-consistent density	Auxiliary determinant; physical observable is density	Exact $v_{xc}$ unknown
Nuclear shell model	Harmonic + spin-orbit	Prescribed phenomenologically	Core determinant + valence configuration mixing	Residual nucleon-nucleon force inside valence space
BCS / HFB	HFB matrix with pair field $Δ$	Self-consistent in density AND pair density	Quasiparticle vacuum (BCS state)	Number violation; correlations beyond mean-field pairing
Tight-binding	Lattice hopping matrix	Phenomenological or fit to DFT	Filled Bloch states up to $ε_{F}$	Electron-electron interactions beyond the parameters

The structural sameness is more important than the differences. Once you see a single-particle level diagram and an occupation rule, you know what you're looking at, regardless of subfield.

Why this works (when naively it shouldn't)

Here's the puzzle. The true ground state of an $N$ -body interacting system is, in general, a superposition of $(N M)$ Slater determinants. Restricting to a single determinant is an exponentially small fraction of the available configuration space. How does it ever land on the right answer?

Four reasons, layered:

(1) The variational principle bounds the error. For methods like HF that minimize $⟨ Φ∣ \hat{H} ∣Φ ⟩$ over Slater determinants, the best determinant is closer to the truth than any other determinant. The error is the correlation energy, and for most molecules in equilibrium it's a few percent of the total energy. Small in relative terms, even though absolute energies are enormous.

(2) Pauli exclusion enforces a lot of correlation for free. Even at the single-determinant level, the antisymmetry of the wavefunction means electrons of the same spin keep apart — the exchange (or Pauli) hole. This is half of the electron-electron avoidance you'd want, baked into the formalism without you having to ask. The piece you miss (the Coulomb hole, for opposite-spin pairs) is smaller.

(3) Mean-field captures the average, fluctuations are small. Each electron sees the time-averaged field of the others. The instantaneous deviations from this average are the fluctuations, and for most condensed-matter and chemical systems the fluctuations are weak relative to the mean field. The empirical sign of this in nuclear physics is the long mean free path of nucleons inside nuclei — they rarely scatter because Pauli blocks most final states.

(4) Symmetry does a lot of the work. Spin symmetry, translation symmetry in solids, rotational symmetry in atoms — these dictate the structure of the single-particle levels even before you do any dynamical calculation. The shell structure of atoms, the band structure of crystals, the magic numbers of nuclei — these are largely consequences of the symmetry of $\hat{h}_{eff}$ , not of intricate many-body physics.

Where it breaks

Single-particle theory is not universal. Several regimes break it, and recognizing them is part of the job.

Strong correlation. When the interaction energy scale exceeds the single-particle bandwidth, the picture of independent quasiparticles falls apart. The textbook example is the Hubbard model at half-filling: small $U / t$ gives a metal (well described by band theory); large $U / t$ gives a Mott insulator (band theory predicts a metal, the system insists on being insulating). The crossover is non-perturbative; no single-particle theory captures it without significant extensions (DMFT, DFT+U, etc.).

Bond breaking and multireference. Stretch H2 to large bond length. The exact ground state becomes a superposition of two equally important Slater determinants (one electron on each atom, in either spin orientation). RHF — which forces both electrons into the bonding orbital — gives a qualitatively wrong dissociation, predicting an ionic state that's far too high in energy. UHF rescues the energy by breaking spin symmetry but at the cost of producing a wavefunction that isn't a spin eigenstate. The right answer needs multireference methods: CASSCF, MRCI, CCSD with multireference references.

Open-shell singlets, diradicals. Molecules with two unpaired electrons coupled into a singlet (carbenes, ozone, the para-benzyne diradical) require at least two determinants to describe properly. Single-determinant theories either lie about the spin or get the energy wrong.

Quantum magnetism, spin liquids, frustrated systems. When the many-body ground state has long-range entanglement that no Slater determinant can mimic — frustrated antiferromagnets, gapped spin liquids, fractional quantum Hall states — single-particle theory tells you essentially nothing about the low-energy physics. You need new tools (tensor networks, DMRG, anyons).

The recurring theme: single-particle theory works when one Slater determinant dominates and the rest are small corrections. It fails when there's no dominant determinant.

The constructed-object insight

Here is the point that grinds my gears and is the actual reason this page exists. The single-particle levels ${ε_{α}, ψ_{α}}$ are not eigenstates of the true many-body Hamiltonian. They are eigenstates of an effective one-body operator that the method constructed. The Slater determinant of the lowest $N$ of them is not the true ground state. The "virtual orbitals" sitting above the Fermi level are not predictions of where excited electrons live — they are wherever the diagonalization of $\hat{h}_{eff}$ happened to put extra eigenvectors.

These objects are bookkeeping. They are auxiliary constructions whose properties happen to encode useful pieces of the true physics. And yet:

Atomic shell structure (1s, 2s, 2p, 3s, 3p, 3d, 4s, …) is explained by them.
The periodic table is organized by them.
Band gaps and metallicity follow from them.
Nuclear magic numbers are predicted by them.
Optical absorption spectra are computed from transitions between them.
Magnetic order is described in their language.

The level diagram became the conceptual vocabulary of three different subfields. Chemists draw molecular orbital diagrams; condensed-matter physicists draw band diagrams; nuclear physicists draw shell diagrams. They're all drawing the same kind of picture: a set of single-particle energies, populated by some rule. The picture is not the truth, but it is so close to the truth that no one ever found a better one.

Koopmans' theorem (for HF) gives the orbital energies physical meaning: the energy $ε_{i}$ of an occupied orbital is, to first approximation, minus the ionization potential from that orbital; the energy $ε_{a}$ of a virtual orbital is minus the electron affinity for adding an electron to that orbital. These are frozen-orbital approximations — the other orbitals are assumed not to relax — and they're surprisingly accurate. In DFT the same heuristic works with caveats (only the HOMO is rigorous; everything else is an approximation, but a useful one).

The linear-response pattern

Once you have a single-particle level diagram, you can do linear response on top of it. Perturb the system slightly, ask how the density changes, and the answer is encoded in matrices that couple occupied levels to virtual levels:

(A B^{*} B A^{*}) (X Y) = ω (10 0 - 1) (X Y) .

This is the Casida block equation for TDDFT/TDHF. The same structure shows up in:

RPA (random-phase approximation) — linear response of HF.
TDHF / TDDFT — linear response of HF / KS-DFT.
QRPA — linear response of HFB; what nuclear physicists call collective excitations on top of pairing.
BdG-RPA — linear response of BdG (the condensed-matter cousin of HFB).

The matrix elements differ in detail, but the block structure — two coupled amplitudes $X$ (excitation) and $Y$ (de-excitation), indefinite metric on the right-hand side, ordinary eigenvalue problem after the $(A - B)^{1/2} (A + B) (A - B)^{1/2}$ fold — is universal. Whenever you see this block structure in a paper, you know it's a linear-response calculation built on a single-particle theory. Conversely, if you understand it for one method, you understand it for all of them.

Worked example: filling levels in a 1D box

The smallest concrete example of the move: non-interacting fermions in a 1D infinite well. The single-particle problem is the textbook particle-in-a-box, with levels $ε_{n} = (nπ / L)^{2} /2$ for $n = 1, 2, 3, \dots$ . The "effective Hamiltonian" is just the kinetic energy operator with the box boundary conditions; there's no interaction so no self-consistency needed. The whole single-particle theory is solvable in closed form, and the only nontrivial step is the level-filling.

For different particle numbers $N$ , fill the lowest $N /2$ spatial orbitals with two electrons each (spin up and spin down) and sum the energies:

"""
Single-particle picture for non-interacting fermions in a 1D infinite well.

  Length L, m = hbar = 1 (atomic units).
  Single-particle energies:  eps_n = (n pi / L)^2 / 2,   n = 1, 2, 3, ...

  Fill the lowest levels by Aufbau (2 fermions per spatial orbital).
  Total energy = sum of filled orbital energies, times 2 for spin.
"""
import numpy as np

L = 10.0
levels = [(n, (n * np.pi / L)**2 / 2) for n in range(1, 11)]

print(f"L = {L},  single-particle energies eps_n = (n*pi/L)^2 / 2:")
for n, eps in levels[:8]:
    print(f"  n = {n}:  eps = {eps:.4f} Ha")

print()
print("Filling N fermions by Aufbau (2 per spatial level):")
print("-" * 60)
for N in [2, 4, 6, 8, 10]:
    n_filled = N // 2
    filled = levels[:n_filled]
    E = 2 * sum(eps for _, eps in filled)
    eps_F = filled[-1][1]
    config = ', '.join(f"n={n} (up,dn)" for n, _ in filled)
    print(f"N = {N:2d}:  E_total = {E:.4f} Ha   eps_F = {eps_F:.4f} Ha")
    print(f"          configuration: {config}")

Output:

L = 10.0,  single-particle energies eps_n = (n*pi/L)^2 / 2:
  n = 1:  eps = 0.0493 Ha
  n = 2:  eps = 0.1974 Ha
  n = 3:  eps = 0.4441 Ha
  n = 4:  eps = 0.7896 Ha
  n = 5:  eps = 1.2337 Ha
  n = 6:  eps = 1.7765 Ha
  n = 7:  eps = 2.4181 Ha
  n = 8:  eps = 3.1583 Ha

Filling N fermions by Aufbau (2 per spatial level):
------------------------------------------------------------
N =  2:  E_total = 0.0987 Ha   eps_F = 0.0493 Ha
          configuration: n=1 (up,dn)
N =  4:  E_total = 0.4935 Ha   eps_F = 0.1974 Ha
          configuration: n=1 (up,dn), n=2 (up,dn)
N =  6:  E_total = 1.3817 Ha   eps_F = 0.4441 Ha
          configuration: n=1 (up,dn), n=2 (up,dn), n=3 (up,dn)
N =  8:  E_total = 2.9609 Ha   eps_F = 0.7896 Ha
          configuration: n=1 (up,dn), n=2 (up,dn), n=3 (up,dn), n=4 (up,dn)
N = 10:  E_total = 5.4283 Ha   eps_F = 1.2337 Ha
          configuration: n=1 (up,dn), n=2 (up,dn), n=3 (up,dn), n=4 (up,dn), n=5 (up,dn)

The total energy grows like $N^{3}$ for large $N$ — which is the 1D Fermi-gas scaling — because filling $N /2$ levels means the last filled level has energy $\propto (N /2)^{2}$ , and you sum $N /2$ of them. The Fermi energy $ε_{F}$ is the highest filled level. Adding two more fermions moves them into the next empty level, raising $ε_{F}$ discretely. This is the whole Fermi-gas physics in five lines of Python.

Notice what just happened. We took an $N$ -body problem, identified it as non-interacting, reduced it to a single-particle eigenvalue problem (which we knew by hand), and reconstructed the many-body energy by summing single-particle energies with an occupation rule. That's the move. In HF, DFT, the shell model, and band theory, the only difference is that $\hat{h}_{eff}$ has to be built self-consistently or phenomenologically — but the structural moves are the same.

The take-away

If you remember one thing from this page: every successful many-body theory in the catalog above is, structurally, the same move. Choose a one-body operator, diagonalize, fill, declare that an approximation. The success of the move across chemistry, condensed matter, and nuclear physics is one of the most underappreciated coincidences in physics. Or maybe it's not a coincidence — maybe the physical world is just kind enough to interacting fermions that mean-field theory mostly works, and the cases where it doesn't are interesting precisely because they're exceptions.

Either way: the next time you see a single-particle level diagram, you'll recognize it as the same picture you've seen a hundred times before, just relabeled. That recognition is what these notes were trying to surface.

Related on this site

Hartree-Fock and the from-scratch H2 series implement the HF case end to end. Casida's equation is the canonical linear-response application of the single-particle picture. The nuclear shell model and HFB are the nuclear-physics instances. Slater determinants are the wavefunction object every single-particle theory ultimately produces.