LESSON NOTES · 09

9. Hubbard interactions and self-consistent-charge DFTB

Position in the course: Lesson 9 of 10. Complete the preceding derivation and use the explained exercises to check understanding.

1. Purpose and assumptions

A one-electron tight-binding Hamiltonian does not by itself solve electronic correlation. The Hubbard model adds an onsite repulsion for simultaneous opposite-spin occupation. Its operators act in a many-particle Fock space, whose size grows exponentially with sites. Diagonalizing the original one-particle hopping matrix is therefore not diagonalizing the interacting Hubbard model.

Mean-field decoupling replaces an occupation product by products involving expectation values and subtracts the duplicated constant. This creates spin-dependent onsite potentials U〈n opposite〉 and a self-consistency loop. It can reveal broken-symmetry patterns but neglects fluctuations. A symmetry-broken mean-field insulator is not a general proof of a Mott state. At half filling and large U/t, virtual hopping produces an antiferromagnetic exchange scale J≈4t²/U under the appropriate low-energy assumptions.

DFTB is a different construction: it expands a density-functional energy around a reference density and uses a small localized basis plus parameterized integrals. Self-consistent-charge DFTB adds a quadratic charge-fluctuation term. For a symmetric γ matrix, differentiating its half-weighted double sum gives the charge potential displayed. Atomic charge definitions are part of the model, often tied to a nonorthogonal basis population analysis; the sign convention for Δq must be stated.

The repulsive term is essential to practical total energies and forces. Band energies alone omit interactions and reference contributions that the fitted repulsive component approximates. Parameter sets have element-pair coverage and validated chemical environments; combining incompatible sets can be physically inconsistent. DFTB is neither identical to an arbitrary empirical TB Hamiltonian nor a full self-consistent Kohn–Sham calculation. It can offer useful speed within its domain, but charge transfer, unusual coordination, magnetism, and excited states require method-specific validation.

2. Derivation step by step

Read each equality with its assumptions. Atomic units are used for DFT equations unless another unit is stated; TB parameters retain explicit energy and length units. The conjugate transpose is denoted by a dagger, and a prime on a coordinate denotes a separate integration variable.

\[ \begin{aligned} H_{Hub}&=-t\sum_{\langle ij\rangle\sigma}(c_{i\sigma}^\dagger c_{j\sigma}+\mathrm{h.c.})+U\sum_i n_{i\uparrow}n_{i\downarrow},\\ n_\uparrow n_\downarrow&\simeq n_\uparrow\langle n_\downarrow\rangle+\langle n_\uparrow\rangle n_\downarrow-\langle n_\uparrow\rangle\langle n_\downarrow\rangle,\\ E_{SCC}&=E_{\mathrm{band}}+\tfrac12\sum_{AB}\gamma_{AB}\Delta q_A\Delta q_B+E_{\mathrm{rep}},\\ \frac{\partial E_{\mathrm{charge}}}{\partial\Delta q_A}&=\sum_B\gamma_{AB}\Delta q_B. \end{aligned} \]

2.1. Avoid double counting in self-consistent models

An effective one-particle potential can count an interaction contribution once for each occupied particle, whereas the total interaction energy counts each pair once. Therefore the sum of self-consistent eigenvalues generally requires a correction. The mean-field constant subtracted in the Hubbard decomposition illustrates this bookkeeping. SCC-DFTB similarly has a specified total-energy expression and cannot be reduced to its final eigenvalue sum. Charges should conserve the chosen total charge; a residual charge imbalance signals an occupation, overlap, or convergence problem. Different parameterizations can define populations differently, so numerical charges must be compared under matched definitions.

3. Worked example

For t=0.1 eV and U=4 eV, the large-U exchange estimate is J=4×0.01/4=0.01 eV. This is an effective-model estimate, not a temperature or a full magnetic phase diagram. If U becomes comparable to t, the strong-coupling expansion loses its justification.

4. Exercises with explained solutions

Exercise. For two charges +q and −q with γAA=γBB=γ0 and γAB=γ1, compute the quadratic charge energy.

Explained solution. The double sum gives [γ0q²+γ0q²−2γ1q²]/2=(γ0−γ1)q². Both ordered cross terms are required. Whether this is stabilizing depends on the consistent parameterized kernel and convention.

Further check. State the units and the allowed regime for every parameter in the worked example. Change one assumption and identify which derivation step must be revisited. A correct explanation names the affected constraint, operator, or boundary condition rather than merely saying that the answer changes.

5. Misconceptions and limitations

Self consistency solves the chosen mean-field or charge model, not the full many-body problem. A converged charge does not guarantee parameter transferability.

The illustration is an original teaching schematic. It is not output from a numerical materials simulation.

6. Connections and sources

Related: density-functional foundations · Interacting Monte Carlo methods

DFTB+

• Slater–Koster, original orbital-geometry construction (1954)
• Su–Schrieffer–Heeger, solitons in polyacetylene (1979)

• Marzari et al., maximally localized Wannier functions review (2012)

• Elstner et al., self-consistent-charge DFTB (1998)

The explanations, algebra, and invented worked examples are original teaching synthesis. The cited papers establish the underlying theories, not the numerical toy values.

7. Related theory and practice

Quantum mechanics · Density functional theory · ABACUS · DFTB+

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