Hubbard dimer: superexchange and one determinant's failure

Units are SI with energies in eV and ħ explicit in spin; nothing here is relativistic, so no metric signature enters (the course uses (+, −, −, −) when four-vectors appear). The model is quoted, not derived: one orbital per site, hopping −t between them, and of all the Coulomb integrals of topic 10.04 only the on-site one, U. Order the spin-orbitals 1↑, 1↓, 2↑, 2↓; two electrons in four give 6 states. H conserves N, S_z and S², and commutes with the swap P of the sites, so most of the work is done before any algebra. In c†₁↑c†₂↑|0⟩ a hop would put two parallel spins in one orbital, which the Pauli principle forbids, so it is an eigenstate at exactly 0; so is c†₁↓c†₂↓|0⟩, and by S² symmetry the third triplet state. Of the remaining S_z = 0 states, the odd ionic combination (|d₁⟩ − |d₂⟩)/√2, with |dᵢ⟩ = c†ᵢ↑c†ᵢ↓|0⟩, is the only singlet odd under P, so it sits alone at U. A 2 × 2 block is all that is left.

Write |a⟩ = c†₁↑c†₂↓|0⟩ and |b⟩ = c†₁↓c†₂↑|0⟩. The hop c†₁↓c₂↓ acting on |a⟩ passes c₂↓ through c†₁↑ for one sign, giving −c†₁↓c†₁↑|0⟩ = +|d₁⟩ after reordering; c†₂↑c₁↑ gives |d₂⟩ directly. So T|a⟩ = −t(|d₁⟩ + |d₂⟩), while on |b⟩ each matching hop differs by one swap, so T|b⟩ = +t(|d₁⟩ + |d₂⟩). The triplet (|a⟩ + |b⟩)/√2 is therefore annihilated by T, confirming its 0, and the covalent singlet |cov⟩ = (|a⟩ − |b⟩)/√2 reaches the even ionic state |ion⟩ = (|d₁⟩ + |d₂⟩)/√2 with ⟨ion|H|cov⟩ = −2t, the 2 counting either electron hopping. The block [[0, −2t], [−2t, U]] has eigenvalues (U ∓ √(U² + 16t²))/2, derived. The trace checks it, 3 × 0 + U + U = 2U, and so does U = 0, where −2t, 0 (four states) and +2t are two free electrons in bonding (−t) and antibonding (+t) orbitals. At U = 6.0 eV and t = 1.5 eV, E₀ = −1.24 eV and the ground state's ionic weight, (1 − U/√(U² + 16t²))/2, is 0.146.

For U ≫ t the root expands to E₀ = −4t²/U + 16t⁴/U³ − …, derived. Second-order perturbation theory, used from Quantum Mechanics II, says why. At t = 0 the singlet and triplet with one electron per site are degenerate at 0; the hop takes the singlet to the ionic state at U with element −2t, giving (2t)²/(0 − U) = −4t²/U, and takes the triplet nowhere: a doubly occupied orbital must be a spin singlet, and hopping conserves spin. So J = 4t²/U, and H_eff = J(S₁⋅S₂/ħ² − 1/4) reproduces both levels: S₁⋅S₂ = −3ħ²/4 gives −J, +ħ²/4 gives 0. The sign is the point. Helium's exchange integral K favours parallel spins; this kinetic exchange favours antiparallel ones, because Pauli blocks the parallel pair's virtual hop. The ionic admixture is small, 3.6% at U = 10t, yet it carries all of J. Measured: in copper(II) acetate monohydrate two Cu²⁺ spins form a dimer whose singlet lies about 290 cm⁻¹, 0.036 eV, below its triplet (quoted). With U ≈ 5 eV that needs t = √(JU)/2 ≈ 0.21 eV through the acetate bridges, an estimate rather than a fit.

Put both electrons in b†σ = (c†₁σ + c†₂σ)/√2. Each site then carries ½ of each spin, so the Fock operator of topic 10.06 adds the same U⟨nᵢ↓⟩ = U/2 to every site for a ↑ electron, and its eigenvectors stay bonding and antibonding: self-consistent at every U, with ε_b = −t + U/2. Contractions give the energy: −t per electron from hopping, U⟨nᵢ↑⟩⟨nᵢ↓⟩ = U/4 on each site, and no exchange term, because U acts only between opposite spins, whose exchange contraction vanishes. So E_RHF = −2t + U/2, derived, which is 2ε_b less the U/2 that adding orbital energies counts twice. Expanded on sites the determinant is (|cov⟩ + |ion⟩)/√2: ionic weight ½ whatever U is. At U = 0 that is exact. Pull the bond apart and t falls towards 0: the exact energy rises to 0, two neutral atoms, but E_RHF rises to U/2, as half the state is still ionic. For hydrogen U is I − A = 13.598 − 0.754 = 12.84 eV (measured values, quoted), the cost of H⁺ + H⁻ over H + H, so the model leaves RHF 6.42 eV too high at dissociation. Real H₂ fails the same way: its RHF curve ends several eV above two separated atoms (quoted).

Give the spins different orbitals: ↑ in cos θ φ₁ + sin θ φ₂ and ↓ in the mirror image sin θ φ₁ + cos θ φ₂. Each hop gives −t sin 2θ and the double occupancy costs 2U cos²θ sin²θ, so with s = sin 2θ, E(s) = −2ts + (U/2)s². RHF is θ = 45°, stationary at every U as Brillouin's theorem requires, but once U exceeds 2t its curvature in 2θ, −(U − 2t), turns negative and the minimum moves to s = 2t/U, where E_UHF = −2t²/U, derived. The Fock operator drives it: the ↑ electron sees U⟨n↓⟩, larger on site 2, so it leans to site 1 and pushes the ↓ electron further onto site 2. The price is spin. ⟨S²⟩ = ħ²(1 − 4t²/U²), the triplet weight is (1 − 4t²/U²)/2, and site 1 carries a moment ⟨n₁↑ − n₁↓⟩ = √(1 − 4t²/U²) where the exact singlet has none. On H_eff this is the Néel product state, ⟨S₁⋅S₂⟩ = −ħ²/4, with energy −J/2 = −2t²/U: correct dissociation, half the superexchange, because the spin flip that makes ↑↓ a singlet is missing. In H₂ this instability, the Coulson-Fischer point, lies near 1.2 Å, 1.6 times the 0.741 Å bond (quoted).

In molecular orbitals, with a†σ = (c†₁σ − c†₂σ)/√2, the even singlet space is spanned by |RHF⟩ = (|cov⟩ + |ion⟩)/√2 and the double excitation |D⟩ = a†↑a†↓|0⟩ = (|ion⟩ − |cov⟩)/√2. There H is [[−2t + U/2, U/2], [U/2, 2t + U/2]], derived: the only coupling is U/2, and the single excitations b†↑a†↓|0⟩ and a†↑b†↓|0⟩ are odd under the swap of sites, so they cannot mix, Brillouin's theorem made exact by symmetry. Its lower eigenvalue is E₀ again, so two-determinant configuration interaction, here the full CI, is exact; CI in general is only named in this course. The weight of |D⟩, (1 − 4t/√(U² + 16t²))/2, is 0.15 at U = 4t, 0.31 at U = 10t and tends to ½ as t → 0, when the exact state becomes the purely covalent Heitler-London state (|RHF⟩ − |D⟩)/√2 and no single determinant of the right spin can follow it. That is the line this topic draws: correlation that reweights determinants at order one, as a stretched bond does, defeats any single determinant, while closed-shell helium's, topic 10.08, is a 1.4% correction to one.