8. Parameter fitting and transferability
Position in the course: Lesson 8 of 10. Complete the preceding derivation and use the explained exercises to check understanding.
1. Purpose and assumptions
A tight-binding parameter set encodes a training domain. Specify which structures, strains, compositions, magnetic states, and energy windows supplied the reference. A model fitted only to one equilibrium band structure may fail under compression or bond breaking. Transferability is demonstrated by independent tests across the intended domain, not by a small training residual.
Construct an objective from observables relevant to the goal. Band energies can be weighted near the Fermi level or across a chosen window; forces and total energies require a consistent energy model beyond a one-electron spectrum. Symmetry-related parameters should be constrained together. Regularization can discourage unstable or excessively large parameters but introduces a strength that must be chosen and documented. Units determine the relative effect of mixed energy and force terms.
Energy-only fitting can be underdetermined. A common onsite shift changes every band equally; orbital gauge transformations can change individual matrix elements without changing eigenvalues. Band crossings also complicate naive band-index matching: matching the nth energy blindly can exchange orbital character. Use symmetry labels, subspace comparisons, or overlaps where appropriate. Identifiability analysis distinguishes uniquely constrained combinations from arbitrary parameter choices.
Split validation by structures rather than randomly by neighboring k points from the same structure. Closely related samples share information and make apparent test error overly optimistic. Hold out strains or chemical environments, evaluate worst cases as well as average error, and inspect physical constraints. Extrapolation should be flagged when bond lengths or coordination leave the training region. Smooth radial functions and cutoffs matter for forces even when discrete energy data fit well.
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. Total energy fitting needs more than eigenvalues
A model can reproduce occupied band energies and still predict the wrong equilibrium bond length if its remaining energy terms are absent or inconsistent. Separate electronic, repulsive, charge, and interaction contributions when constructing a practical force model. If reference data combine different spin states or functionals, decide whether those differences are intended targets or incompatible labels. Cross-validation should hold out complete environments and report the domain of applicability. Examine parameter sensitivity: if widely different parameter vectors yield the same training loss, a unique physical interpretation of individual parameters is not justified. Predictions may still be stable within the fitted domain, but extrapolation becomes especially uncertain.
3. Worked example
A two-site model with known ε and no overlap has splitting ΔE=2|t|, so a 4 eV splitting fixes |t|=2 eV. It does not fix the sign of t without a phase convention. If ε is unknown but the two energies are known, their average fixes ε and half their difference fixes |t|.
4. Exercises with explained solutions
Exercise. If t(R)=t0 exp[−α(R−R0)], show the small-strain response.
Explained solution. For δR small, exp(−αδR)≈1−αδR, hence δt≈−αt0δR. The parameter α has inverse-length units. A fit that ignores derivative behavior may reproduce energies yet predict poor electron–lattice coupling.
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
A good RMSE does not prove a correct ordering of near-degenerate phases. Compare error scales with the specific energy differences being interpreted.
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+