docs/api/harrison_tb.md

Harrison Tight-Binding (universal parameters)

Dependency-free (numpy only) sp3 tight-binding Hamiltonian builder for real atoms and crystals -- no PySCF/OpenFermion, no SCF/DFT. See vhd_tb for the material-specific alternative when this module's accuracy isn't enough.

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. Atomic term values (ELEMENTS) and the universal eta coefficients (ETA) are transcribed from that book's Solid State Table, cross-checked against jarvist/HarrisonSolidStateTable.jl, an independent Julia implementation of the same table.

The method

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 between neighboring atoms' orbitals:

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

   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. This is what makes the method "universal": no fitting per material.

$\hbar^2/m_e = 7.62\ \text{eV·Å}^2$.

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

Off-diagonal sp3 matrix elements follow the standard Slater-Koster (1954) table for an (s, px, py, pz) basis and a bond of direction cosines $(l, m, n)$:

$$E(s,s)=V_{ss\sigma },E(s,x)=l{V}_{sp\sigma },E(x,s)=-l{V}_{sp\sigma }E(s,s)\; =\; V\_\{ss\backslash sigma\},\; \backslash quad\; E(s,x)\; =\; l\backslash ,V\_\{sp\backslash sigma\},\; \backslash quad\; E(x,s)\; =\; -l\backslash ,V\_\{sp\backslash sigma\}$$ $$E(x,x)=l^2{V}_{pp\sigma }+(1-l^2)V_{pp\pi },E(x,y)=lm(V_{pp\sigma }-V_{pp\pi })E(x,x)\; =\; l^2\; V\_\{pp\backslash sigma\}\; +\; (1-l^2)V\_\{pp\backslash pi\},\; \backslash quad\; E(x,y)\; =\; lm\backslash ,(V\_\{pp\backslash sigma\}-V\_\{pp\backslash pi\})$$

(and cyclic permutations for y, z). sp3_bond_block implements this; sp3_dimer_hamiltonian builds a 2-atom cluster from it, and zincblende_hamiltonian sums it with Bloch phases over the 4 nearest-neighbor bonds to build the full periodic crystal Hamiltonian.

Accuracy

This is a universal model -- one parameter table for every material, no per-material fitting, no d-orbitals. That buys zero setup cost per new material at the price of accuracy: gaps typically come out ~2-3x off from experiment, and for indirect-gap materials it can misplace the conduction-band minimum entirely (see vhd_tb for why and the fix). Validation numbers against real experimental gaps (GaAs, Si, Ge) are tracked in Dense-Evolution-Discovery.

::: dense_evolution.solvers.harrison_tb

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See also: vhd_tb for material-specific fitted parameters when the universal table's ~2-3x gap error isn't good enough.