Harrison / VHD Tight-Binding Validation

Note

The implementation lives in the main library: harrison_tb.py / vhd_tb.py. This page is the full experimental validation record.

Knowledge base and experiment log for Walter A. Harrison's empirical tight-binding method and its Vogl-Hjalmarson-Dow material-specific fix, run against real GaAs, Si, and Ge and checked directly against experimental band gaps.

Source

Walter A. Harrison, Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond. Originally published by W. H. Freeman, 1980; reprinted by Dover Publications (Dover Books on Physics), 1989, ISBN 0-486-66021-4.

This repo does not have access to the book's full text (copyrighted, non-open) and none was reproduced. What follows is the small set of facts from the book that are public/standard enough to appear identically across independent secondary sources and directly checkable by running the code -- transcribed and cross-checked against jarvist/HarrisonSolidStateTable.jl, an independent Julia implementation of the same table, fetched 2026-08-05.

The method, in brief

Harrison's tight-binding model builds a solid's electronic Hamiltonian from two ingredients only, both universal (materials-independent functional form):

1. Atomic term values -- the free-atom s and p orbital energies (on-site Hamiltonian diagonal), tabulated per element.
2. A universal bond-scaling law for the off-diagonal (hopping) matrix elements:

$\(V_{ll'm} = \eta_{ll'm} \cdot \frac{\hbar^2}{m_e d^2}\)$

where \(d\) is the bond length and the four dimensionless \(\eta\) coefficients are the same for every element pair -- only \(d\) and the atomic term values change between materials.

\(\hbar^2/m_e = 7.62\ \text{eV·Å}^2\).

coefficient	value
\(\eta_{ss\sigma}\)	-1.40
\(\eta_{sp\sigma}\)	+1.84
\(\eta_{pp\sigma}\)	+3.24
\(\eta_{pp\pi}\)	-0.81

Experiment 1: sp3 dimer sanity check (run 2026-08-05)

Two-atom sp3 dimer, sp3_dimer_hamiltonian(a, b, d):

• Si-Si at d=2.35 Å (diamond-Si nearest-neighbor bond length): eigenvalues [-16.626, -12.25, -9.847, -7.638, -7.638, -5.402, -5.402, -1.418] eV. Hermitian; bonding states pushed well below the atomic eps_p=-6.52 eV level, antibonding above, with the expected doubly-degenerate pi pair -- physically sane sp3 bonding/antibonding splitting.
• Ga-As at d=2.45 Å (zinc-blende bond length): eigenvalues [-18.386, -12.344, -9.239, -8.228, -8.228, -4.582, -4.582, -1.541] eV. Hermitian; asymmetric splitting relative to Si-Si, consistent with the heteropolar (partly ionic) Ga-As bond.

These confirm the Hamiltonian construction is correct -- not yet a periodic-crystal band gap.

Experiment 2: periodic GaAs band gap, Harrison universal (run 2026-08-05)

zincblende_hamiltonian(k, cation, anion, lattice_constant_angstrom) builds the real periodic Bloch Hamiltonian (two-atom zinc-blende basis, 4 nearest-neighbor bonds per atom, Bloch phase per bond). Diagonalized at \(\Gamma\) (\(k=0\)) for real GaAs (lattice constant a=5.6533 Å, the standard experimental zinc-blende value):

• \(\Gamma\) eigenvalues (eV): -22.069, -9.537 (x3), -6.631, -3.273 (x3)
• valence band max = -9.537 eV, conduction band min = -6.631 eV
• computed direct gap = 2.906 eV
• experimental GaAs direct gap = 1.42 eV -- 104.7% error

X and L point eigenvalues (eV), for context on band ordering away from \(\Gamma\):

• X: -19.351, -15.314, -13.435 (x2), -3.966, -2.880, 0.625 (x2)
• L: -20.202, -15.582, -11.442 (x2), -6.194, -1.368 (x2), 0.467

Hermitian and structurally sane -- the gap error is a known, documented limitation of Harrison's universal (same 4 eta coefficients for every element pair, no per-material fitting, no d-orbitals) parameter set on polar/ionic compound semiconductors like GaAs, not a code bug.

Experiment 3: GaAs with Vogl-Hjalmarson-Dow material-specific parameters (run 2026-08-05)

P. Vogl, H. P. Hjalmarson, J. D. Dow, "A Semi-empirical tight-binding theory of the electronic structure of semiconductors", J. Phys. Chem. Solids 44 (5), 365-378 (1983) -- sp3s* basis (5 orbitals/atom, including an extra fitted s* orbital with no literal physical meaning, added purely to get the lowest conduction band right). Parameter values transcribed from an independent open-source implementation, github.com/rpmuller/TightBinding (TB.py, fetched 2026-08-05), which cites the same paper.

Implemented as dense_evolution/vhd_tb.py (sp3s_star_hamiltonian, direct_gap_at_gamma, MATERIALS for 16 zinc-blende/diamond semiconductors: C, Si, Ge, Sn, SiC, AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP, InAs, InSb, ZnSe, ZnTe).

Result: GaAs direct gap at \(\Gamma\) = 1.55 eV vs. experimental 1.42 eV -- 9.2% error, vs. Harrison universal's 104.7% error for the same material. All 16 materials in the table verified Hermitian at a generic (non-\(\Gamma\)) k-point.

Bug found and fixed during porting: the reference implementation's own Hamiltonian assembly (Hac.conjugate() without transposing) does not in general produce a Hermitian matrix -- vhd_tb.py fixes this by assembling with the proper conjugate transpose (.conj().T), verified Hermitian for all 16 materials.

Experiment 4: silicon, indirect gap (run 2026-08-05)

Si ("silicio ibrido" in this repo's older band-comparison pipeline, scripts/next_gen_silicon.py / data/bande_silicio_ibrido.csv) is an indirect-gap material -- unlike GaAs, its conduction-band minimum is not at \(\Gamma\), it's off-axis along the \(\Gamma \to X\) (Delta) line. direct_gap_at_gamma alone would give the wrong (too-large) number. Used band_extrema_along_path(material, k_start, k_end) to scan \(\Gamma \to X\) and find the true valence-band max and conduction-band min:

• VHD sp3s* (material-specific): VBM=0 eV at \(\Gamma\) (correct), CBM=1.1713 eV at k=0.732 of the way from \(\Gamma\) to X (literature value is closer to k~0.85 -- same Delta-valley, not an exact match, but right regime). Computed indirect gap = 1.171 eV vs. experimental 1.12 eV (300 K) -- 4.6% error. \(\Gamma\)-\(\Gamma\) direct transition (context only, not the fundamental gap): 3.43 eV -- in the right ballpark for Si's known E0' direct transition (~3.4 eV).
• Harrison universal sp3 (no s*), for comparison: puts the conduction-band minimum at \(\Gamma\) too (misses the indirect character entirely) and gives gap = 3.66 eV vs. 1.12 eV -- 227% error, worse than its own GaAs result -- exactly the known failure mode Vogl-Hjalmarson-Dow's s* orbital was added to fix.

Experiment 5: germanium, indirect gap at L not X (run 2026-08-05)

Ge is also indirect-gap, but unlike Si its conduction-band minimum is at the L point (\(k=(0.5,0.5,0.5)\), \(\Gamma \to L\) direction), not along \(\Gamma \to X\). Scanned both directions to check the model gets the right valley:

• VHD sp3s*: \(\Gamma \to X\) scan: CBM=0.7895 eV at k=0.808 of the way to X. \(\Gamma \to L\) scan: CBM=0.7649 eV at k=(0.5,0.5,0.5), i.e. exactly at L. L is lower than X (0.765 < 0.790 eV) -- correctly identifies L as the true global conduction-band minimum, matching real Ge physics (unlike Si, where X/Delta wins). Computed indirect gap = 0.765 eV vs. experimental 0.66 eV (300 K) -- 15.9% error. \(\Gamma\)-\(\Gamma\) direct transition (context only): 0.90 eV, in the right ballpark for Ge's known E0 direct gap (~0.8 eV).
• Harrison universal sp3, for comparison: puts the conduction-band minimum at \(\Gamma\) on both paths (same failure mode as Si), giving gap = 1.831 eV vs. 0.66 eV -- 177.5% error.

Comparison plot

Produced by scripts/harrison_vhd_validation.py, which also writes the raw numbers to data/harrison_vhd_gap_comparison.csv:

Summary across all three materials

Material	Gap type	Harrison universal	VHD material-specific	Experimental
GaAs	direct, at \(\Gamma\)	2.906 eV (104.7% error)	1.55 eV (9.2% error)	1.42 eV
Si	indirect, \(\Gamma \to X\)	3.66 eV (227% error), CBM misplaced at \(\Gamma\)	1.171 eV (4.6% error)	1.12 eV
Ge	indirect, \(\Gamma \to L\)	1.831 eV (177.5% error), CBM misplaced at \(\Gamma\)	0.765 eV (15.9% error), correctly finds L below X	0.66 eV

Harrison's universal parameters are qualitatively sane (Hermitian, correct bonding/antibonding structure) but consistently ~2-3x off quantitatively, and for the indirect-gap materials (Si, Ge) it structurally cannot place the conduction-band minimum correctly -- it always lands at \(\Gamma\), missing the real off-axis valley entirely. Vogl-Hjalmarson-Dow (1983) material-specific sp3s* parameters get within 5-16% of experiment and correctly identify which valley (X for Si, L for Ge) is the true minimum.

What this does NOT replace

Neither model replaces the DFT/SCF path already in this repo for GaAs (see scripts/vqe_tmi_material_design.py's DFT-derived hopping, T1_GAAS_DFT_EV=7.917 eV) when first-principles accuracy is needed. Their value is as fast, dependency-free (no PySCF/OpenFermion, numpy only) estimates or starting points -- VHD close enough to experiment to be useful quantitatively, Harrison useful mainly for qualitative/order-of-magnitude checks or when no per-material fit exists yet.