TightBinding Models
The tightbinding (TB) method (and the closely related linear combination of atomic orbitals (LCAOs) method in quantum chemistry) is a computationally fast and robust approach to handle largescale molecular and condensed matter systems. In the TB approximation, electrons are assumed to be strongly bound to the nuclei. The oneparticle wavefunctions \(\{\psi_{\alpha}(\mathbf{x})\}\) are approximated by linear combinations of SlaterKostertype states (i.e., LCAOs) for isolated atoms, i.e.,
where \(i=\{1, \ldots, N\}\) runs over all sites and orbitals. The oneparticle states \( \psi_\alpha \rangle\) are eigenvectors of the parametrized Hamiltonian matrix (the TB Hamiltonian), \(\hat{H} = \sum_{i,j} t_{i,j} i\rangle \langle j \). The TB matrix elements — encoding onsite energies \((i = j)\) and hopping integrals between different atomic orbitals \((i \neq j)\) — can be estimated by means of other methods (e.g., SlaterKoster approach) or by matching the spectrum to that obtained by firstprinciples calculations in a suitable reference system^{1}^{2}^{3}^{4}.
Parameterized TB models provide an accurate description of molecular orbitals in molecules and Bloch wavefunctions in many solids. The complexity of TB models grows only linearly with the number of atomic orbitals, providing a basis for largescale calculations of a plethora of equilibrium and nonequilibrium physical properties, including optical absorption spectra, simulations of amorphous solids, and wavepacket propagation. Disorder, interfaces, and defects can be conveniently added to a TB model by modifying onsite energies and hopping integrals, and adding auxiliary sites. Such a multiscale approach has proven very successful in describing impurity scattering^{5}^{6}, moiré patterns^{7}, complex interactions induced by adatoms^{8}, optical conductivity of disordered 2D materials with up to tens of millions of atoms^{9}, and geometrical properties, vibrational frequencies and interactions of large molecular systems^{10}.

Simplified LCAO Method for the Periodic Potential Problem, J. C. Slater and G. F. Koster, Phys. Rev. 94, 1498 (1954). ↩

Elementary prediction of linear combination of atomic orbitals matrix elements, S. Froyen and W.A. Harrison, Phys. Rev. B 20, 2420 (1979). ↩

Tightbinding modelling of materials, C. M. Goringe, D. R. Bowler, and E. Hernández, Rep. Prog. Phys. 60, 1447 (1997). ↩

The Slater–Koster tightbinding method: a computationally efficient and accurate approach, D. A. Papaconstantopoulos and M. J. Mehl, Journal of Physics: Condensed Matter 15, R413 (2003). ↩

Resonant scattering by realistic impurities in graphene, T. O. Wehling et al. Phys. Rev. Lett. 105, 056802 (2010). ↩

Unified description of the dc conductivity of monolayer and bilayer graphene at finite densities based on resonant scatterers, A. Ferreira et al., Phys. Rev. B 83, 165402 (2011). ↩

Ab initio theory of moiré superlattice bands in layered twodimensional materials, J. Jung, A. Raoux, Z. Qiao, and A. H. MacDonald, Phys. Rev. B 89, 205414 (2014). ↩

Impact of complex adatominduced interactions on quantum spin Hall phases. F. J. dos Santos et al., preprint: arXiv:1712.07827 (2017). ↩

Numerical calculation of the CasimirPolder interaction between a graphene sheet with vacancies and an atom. T. Cysne et al., Phys. Rev. B 94, 235405 (2016). ↩

A Robust and Accurate TightBinding Quantum Chemical Method for Structures, Vibrational Frequencies, and Noncovalent Interactions of Large Molecular Systems Parametrized for All spdBlock Elements (Z = 1−86), S. Grimme , C. Bannwarth, and P. Shushkov, J. Chem. Theory Comput., 13 , 1989 (2017). ↩