Hartree-Fock binding of a metal, and what correlation adds

Units are SI, with energies per electron in rydbergs, 1 Ry = ħ²/(2ma₀²) = 13.606 eV, and lengths in Bohr radii, a₀ = 52.9 pm; nothing here is relativistic, so the course's metric (+, −, −, −) never enters. From topics 15.02 and 15.03, used and not re-derived, one Slater determinant of plane waves gives E(rₛ) = 2.21/rₛ² − 0.916/rₛ Ry, the Hartree term having cancelled against the background. Set the slope to zero: −4.42/rₛ³ + 0.916/rₛ² = 0, so rₛ* = 4.42/0.916 = 4.83 and E* = −0.916²/(4 × 2.21) = −0.0949 Ry = −1.29 eV per electron. At the minimum the kinetic energy is 0.0949 Ry and the exchange −0.1898 Ry, exactly −2 times it: when one term scales as rₛ⁻² and the other as rₛ⁻¹, stationarity forces 2K + Eₓ = 0, the virial theorem of a Coulomb system. Two limits check the shape. Switch the charge off and exchange vanishes, leaving only a kinetic energy that falls monotonically as the gas expands: a free Fermi gas has no minimum and must be held in by walls. As rₛ → ∞ both terms vanish, so −0.095 Ry is counted from electrons and background spread to zero density.

The minimum, rₛ* = 4.83, is a density of 1.43 × 10²⁸ electrons m⁻³, and the measured conduction-electron densities of the alkali metals bracket it: rₛ = 3.25 for lithium, 3.93 for sodium, 4.86 for potassium and 5.62 for cesium (quoted). In that sense exchange alone binds the gas near a real metal's density. The agreement is partly luck, and one argument shows it. Jellium has a single parameter, so it has a single equilibrium density, yet the simple metals run from aluminium at rₛ = 2.07 to cesium at 5.62: their ions, not their electron gas, choose the density. Nor is −1.29 eV a cohesive energy. Sodium's measured 1.11 eV per atom is counted from free neutral atoms, the jellium energy from electrons and background spread to zero density, and the two share no zero. Restoring the ions as point charges adds −1.8/rₛ Ry per electron, about twice the exchange term, and only the repulsion of the filled ion cores keeps the metal from shrinking to many times its measured density.

Now put all N electrons in one spin state. Each k-state holds one electron instead of two, so the occupied sphere doubles its volume and k_F grows by 2¹⁄³. Kinetic energy per electron, three fifths of the Fermi energy and so proportional to k_F², rises by 2²⁄³ = 1.587. Exchange acts only between parallel spins, and the exchange energy per electron is proportional to the k_F of the electron's own spin species, so it rises by 2¹⁄³ = 1.260. Hence E_P = 3.508/rₛ² − 1.154/rₛ Ry, derived. Polarising costs 1.298/rₛ² of kinetic energy and earns 0.238/rₛ of exchange. The cost has over five times the coefficient but falls off faster, so the two cross at rₛ = 1.298/0.238 = 5.45, which can be written exactly as (1 + 2¹⁄³)(2.21/0.916), because (2²⁄³ − 1)/(2¹⁄³ − 1) = 1 + 2¹⁄³. Beyond it Hartree-Fock prefers every spin aligned: the ferromagnetism of the free electron gas that Bloch found in 1929. Two checks: the threshold is a ratio of coefficients, dimensionless like rₛ itself, and at rₛ* = 4.83 the polarised gas still lies 6.4 mRy higher, so the bound minimum is unpolarised.

Partial polarisation ζ = (N↑ − N↓)/N interpolates. Each spin species fills its own Fermi sphere, so the kinetic and exchange energies acquire the factors [(1 + ζ)⁵⁄³ + (1 − ζ)⁵⁄³]/2 and [(1 + ζ)⁴⁄³ + (1 − ζ)⁴⁄³]/2, which grow as 1 + 5ζ²/9 and 1 + 2ζ²/9. Near ζ = 0 the energy is E(0) + (ζ²/9)(11.05/rₛ² − 1.832/rₛ) Ry, and the unpolarised gas stays a local minimum until rₛ = 6.03. Between 5.45 and 6.03 it is metastable while the fully polarised gas is lower, and no intermediate ζ is ever lowest, so Hartree-Fock predicts a first-order jump from ζ = 0 to ζ = 1, not a gradual onset. Cesium is the test. Its measured density gives rₛ = 5.62, past the threshold, where Hartree-Fock wants a saturated ferromagnet with μ₀M = μ₀nμ_B ≈ 0.11 T. Cesium is measured to be a weakly paramagnetic metal with no spontaneous moment (quoted). One determinant has answered a qualitative question wrongly.

Unit 10 defined the correlation energy, per electron here, as ε_c = E_exact − E_HF. The plane-wave determinant is a trial state, not even the lowest single determinant (Overhauser showed in 1962 that spin-density-wave determinants lie lower, quoted), and the variational principle bounds every trial state from above, so ε_c ≤ 0 at every density. Its origin is the pair the determinant ignores. Parallel spins are kept apart by the exchange hole, but antiparallel spins are left uncorrelated, each blind to where the other is; the true ground state digs a Coulomb hole in the opposite-spin density too. At high density the result is Gell-Mann and Brueckner's ε_c = 0.0622 ln rₛ − 0.096 Ry, quoted. Plain second-order perturbation theory diverges, because the Coulomb transform e²/(ε₀q²) grows without limit at small q; only after the most divergent terms are summed, which amounts to the gas screening its own interaction, does a logarithm survive. The limit checks: as rₛ → 0 the correction grows only as ln rₛ while the kinetic energy grows as rₛ⁻², so Hartree-Fock becomes exact at high density. The formula turns positive at rₛ = e^{(0.096/0.0622)} = 4.68, which the bound forbids.

At metallic densities correlation has to be computed, and the standard numbers are Ceperley and Alder's quantum Monte Carlo energies of 1980, quoted through Perdew and Zunger's fit: ε_c = −0.2846/(1 + 1.0529√rₛ + 0.3334rₛ) Ry for the unpolarised gas. At rₛ = 4 that is −0.064 Ry, 70 per cent of the Hartree-Fock energy there and six and a half times the Gell-Mann–Brueckner value of −0.0098 Ry. Because it weakens as rₛ grows, it tilts the curve and pulls the minimum in from 4.83 to rₛ = 4.18, at −0.155 Ry. It also decides the magnetism. For the polarised gas the fit is −0.1686/(1 + 1.3981√rₛ + 0.2611rₛ) Ry, far smaller, because the exchange hole had already kept parallel spins apart: at rₛ = 5.45 the two are −0.054 and −0.030 Ry, and a 0.024 Ry preference for the unpolarised gas swamps Hartree-Fock's millirydberg preference for alignment. The gas stays unpolarised across the metallic range and far beyond it. Only near rₛ = 100 does the potential energy win outright and freeze the electrons into Wigner's crystal (quoted).