7. Wannier orbitals and downfolding
Position in the course: Lesson 7 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
Wannier functions connect Bloch bands to localized orbitals. A k-dependent unitary transformation changes the basis within a chosen band subspace before Fourier transformation. Different gauges give different orbital shapes and hopping ranges while preserving the represented bands. Localization is therefore a carefully chosen representation, not an additional experimental measurement of unique atomic orbitals.
For an isolated group of bands, the dimension of the subspace is fixed at each k. Entangled bands require selecting a smooth subspace from a larger window; the outer and frozen energy windows influence the result. A good interpolation should reproduce target bands and relevant orbital character on independent k points. Smooth gauges aid localization, but symmetry or topological obstructions can complicate a fully localized representation under imposed constraints.
Downfolding eliminates unwanted degrees of freedom algebraically. Partition coefficients into retained A and eliminated B sectors. Solve the B block equation for cB, then substitute into the A equation. This yields an energy-dependent effective Hamiltonian. Far from the eliminated levels, expansion around a reference energy can give a simpler approximately energy-independent model. Near a discarded resonance, the inverse becomes large and the low-energy approximation can fail.
Truncating real-space hopping is a second approximation after choosing the subspace. Check interpolation versus range and ensure Hermitian pairs H(R)=H†(−R). Matching eigenvalues is necessary but not sufficient for all observables; matrix elements, Berry phases, and position operators can require consistent transformations. A model that accurately reproduces a narrow energy window should not be extrapolated to high excitations or drastically altered structures without 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.
2.1. Fourier interpolation has a convention
With the selected real-space convention, reconstruct H(k) by summing H(R)exp(ik·R). Intra-cell orbital positions may appear in additional phases depending on the basis convention. Use the same convention in hopping files, velocities, and optical matrix elements. Real-space truncation can spoil a symmetry if symmetry-related terms are dropped unevenly. Check reconstructed energies on a dense independent mesh and quantify errors near the target Fermi level. A localized orbital plot is useful intuition but does not establish that every matrix element needed for a response calculation has been transformed consistently.
3. Worked example
A retained level εA=0 couples by v=0.2 eV to a discarded level Δ=2 eV. Near E=0, the effective correction is v²/(E−Δ)≈−0.02 eV. The exact lower eigenvalue is [2−sqrt(4+0.16)]/2≈−0.0198 eV. Agreement follows from small v/Δ, not from downfolding being universally energy independent.
4. Exercises with explained solutions
Exercise. Explain why the correction diverges near the eliminated level and what to do.
Explained solution. The denominator E−Δ approaches zero, so perturbative elimination loses scale separation. Include that level in the retained subspace or solve the energy-dependent problem; do not hide the resonance with a fitted constant.
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
Wannier gauges are nonunique. Report windows, projections, gauge constraints, and hopping truncation when distributing a model.
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
- Slater–Koster, original orbital-geometry construction (1954)
- Su–Schrieffer–Heeger, solitons in polyacetylene (1979)
- Marzari et al., maximally localized Wannier functions review (2012)
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+