4. Slater–Koster geometry and orbital symmetry
Position in the course: Lesson 4 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
Orbital orientation controls hopping. Slater–Koster parameterization separates a bond-distance-dependent radial integral from angular factors determined by the bond direction. A sigma interaction is aligned along the bond; pi interactions involve components transverse to it. This reduces many matrix elements to a smaller set of interpretable functions, provided the two-center approximation and orbital definitions are appropriate.
For p orbitals, split a unit Cartesian orbital vector into a component parallel to the bond and a component perpendicular to it. The longitudinal projector is Rhat Rhat transpose; the transverse projector is I minus that matrix. Thus the p–p hopping tensor is Vπ I+(Vσ−Vπ)Rhat Rhat transpose. Reading its xx and xy components gives the displayed equations. This projector derivation explains the factors l² and lm without memorizing a table.
For a bond along x, l=1 and m=n=0: px couples through sigma, while py and pz couple through pi. For a bond along the diagonal between x and y, l=m=1/√2: xx hopping averages sigma and pi, while xy is half their difference. The s–p term is odd under reversal of bond direction because a p orbital changes sign. Hermiticity is preserved only if reversed bonds and orbital phase conventions are handled consistently.
Radial functions are usually fitted or derived from reference electronic structure. Their decay may be represented by exponentials, power laws, or splines within a specified interval. A smooth cutoff is essential if forces are needed. Distance-only two-center functions can fail when coordination changes alter screening or orbital character. Check rotational covariance: rotating the geometry and orbital basis together should preserve eigenvalues. A model that depends spuriously on the global Cartesian orientation contains an implementation or parameterization inconsistency.
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. Derivatives need radial and angular terms
Differentiating a hopping with respect to atomic position changes both R and the direction cosines. A force implementation that differentiates only the radial function misses the angular response for p and d orbitals. The derivative of Rhat contains the transverse projector divided by R. Bond reversal must be checked separately from a global rotation because odd-parity orbitals change signs. Smooth radial cutoffs should make both the hopping and the derivatives continuous at the chosen boundary. Verify a small displaced geometry with finite differences before trusting forces from a multi-orbital Slater–Koster model.
3. Worked example
Take Vσ=2 and Vπ=−1 eV for a diagonal xy bond. Then Hpxpx=(2−1)/2=0.5 eV and Hpxpy=(2+1)/2=1.5 eV. For an x bond they are 2 and 0 eV. The radial parameters did not change; only geometry changed.
4. Exercises with explained solutions
Exercise. Show that the trace of the p–p hopping tensor is rotation independent.
Explained solution. The longitudinal projector has trace l²+m²+n²=1. Therefore Tr H=3Vπ+(Vσ−Vπ)=Vσ+2Vπ. A rotation changes individual entries but preserves this trace and the tensor eigenvalues.
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
Two-center tables are symmetry constructions, not universal radial data. Orbital phases, basis order, units, and bond direction must be documented.
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.
Original analytic teaching diagram under the lesson assumptions; no simulation results.
7. Related theory and practice
Quantum mechanics · Density functional theory · ABACUS · DFTB+