In an era of ab initio methods and many-body methods like the $GW$, there is not too much room for methods like the extended Hückel model to be the main method in any particular field of materials modeling. However, the method is still very much appreciated by the solid-state community particularly for it's accessibility. It is especially popular with those fond of Tight-Binding variants.
The extended Hückel model is an effective Hamiltonian that is used to approximate the Schrödinger eq. by making use of the variational theorem and the linear combination of atomic orbitals (LCAO). The key is to solve a set of linear Hückel equations of form
\begin{align}
\sum_{i,j}\left({H}_{ij} - \epsilon_\alpha S_{ij}\right) C_{ij} &= 0, &
\text{with } i,j &= 1, 2, 3, \dots \tag{1}\label{eq:huckel}
\end{align}
The diagonal elements of $\hat{H}$ are taken to be equal to the ionization energy of an electron in the $i$th valence $\phi$ of the isolated atom in the appropriate state, i.e. Valence State Ionization Potential (VSIP), expressed as $H_{ii} = - VSIP(\phi_{j})=\epsilon_\mathrm{onsite}$.
The off-diagonal elements of $\hat{H}$ are evaluated according to a modified Wolfsberg-Helmholtz relation, as mentioned by Felipe:
$$
H_{ij} = \mathcal{K} S_{ij} \left(\frac{H_{ii} + H_{jj}}{2}\right),
\tag{2}\label{eq:Hij}
$$
where $S_{ij}$ is the matrix of overlap integrals, $S_{ij} = \langle \phi_{i}\vert \phi_{j} \rangle$.
So... who is using it and how?
As mentioned by Felipe, the eH-TB method is fairly easy to work with and Tyberius highlighted that it's speed is primarily what gives it value. Hence, many researchers often write in-house codes for their research projects,[1] or build on community codes. Actually, my first formal encounter with materials modeling was learning the extended Hückel Tight-Binding method![2] I carried out a project under Prof. Donald H. Galván, who worked directly with Roald Hoffmann a couple years ago. When I began the project, I had no experience or knowledge of Quantum Mechanics or Solid State Physics, and worked with a modified version of the YAeHMOP code written by Greg Landrum, but that experience prepared me to really make sense of band theory and materials form an orbital perspective.
The method is a very good starting point for students interested in electronic structure theory, whether it be modeling of development. Therefore, it is a useful tool in the classroom. Recently, YaEHMOP was merged with the Avogadro molecular editor and visualizer to serve as a simple way for undergraduate quantum theory students to model band structures, Density of States (DOS) and Crystal Orbital Hamilton Population (COHP) using their personal computer. The authors published this in "Journal of Chemical Education".[3]
It is also implemented in the Quantum ATK platform for both academic research and materials development. In research, the method is often used to model the ground state and transport properties of systems of sized prohibitive by other methods (e.g. Carbon Nanotubes),[4] as a first-approximations to study a system's properties or model a material with an effective $\hat{H}$ to study physics that may not be accessible by traditional DFT. An example of this, is the work of A.S. Martins on the Hyper-Honeycomb lattice in J.Phys. Chem. C.[5] and on 2D materials with defects.[6] The method is still trusted enough to model the electronic structure in support of experimentally achieved materials.[7]
Recently, the Extended Hückel Hamiltonian has also found a home in quantum dynamics and charge transfer simulations.[8, 9, 10, 11]
It is still inspiring new TB models,[12] and the derivation of paper-and-pencil LCAO models help to understand complex materials. Last year, a LCAO model was proposed for double halide perovskites.[13] Additionally, due to its qualitative predictive power and speed, there are efforts to improve through Machine Learning.[14] Lastly, the eH is key ingredient in the novel GFNx-TB method.[15]
As seen, the extended Hückel method continues to play a strong role in quantum chemistry and materials modeling, though not in the same way it did in the twentieth century. Currently, I am not sure if it is the "standard" method of anything, but the model itself can teach us a lot about band theory at a qualitative level and that is useful for many things. If I were to give it three solid applications today, they would be:
- Education in band theory
- Method development in Quantum Chemistry
- Transport simulations
References:
El Khatib, M.; Evangelisti, S.; Leininger, T.; Bendazzoli, G. L. A theoretical study of closed polyacene structures. Phys. Chem. Chem. Phys. 2012, 14 (45), 15666. DOI: 10.1039/C2CP42144E.
Palos, E. I.; Paez, J. I.; Reyes-Serrato, A.; Galván, D. H. Electronic structure calculations for rhenium carbonitride: an extended Hückel tight-binding study. Phys. Scr. 2018, 93 (11), 115801. DOI: 10.1088/1402-4896/aae14c.
Avery, P.; Ludowieg, H.; Autschbach, J.; Zurek, E. Extended Hückel Calculations on Solids Using the Avogadro Molecular Editor and Visualizer. J. Chem. Educ. 2018, 95 (2), 331–337. DOI: 10.1021/acs.jchemed.7b00698.
Zienert, A.; Schuster, J.; Gessner, T. Extended Hückel Theory for Carbon Nanotubes: Band Structure and Transport Properties. J. Phys. Chem. A 2013, 117 (17), 3650–3654. DOI: 10.1021/jp312586j.
Veríssimo-Alves, M.; Amorim, R. G.; Martins, A. S. Anisotropic Electronic Structure and Transport Properties of the H-0 Hyperhoneycomb Lattice. J. Phys. Chem. C 2017, 121 (3), 1928–1933. DOI: 10.1021/acs.jpcc.6b10336.
Martins, A. d. S.; Veríssimo-Alves, M. Group-IV nanosheets with vacancies: a tight-binding extended Hückel study. J. Phys.: Condens. Matter 2014, 26 (36), 365501. DOI: 10.1088/0953-8984/26/36/365501.
Zhak, O.; Zdorov, T.; Levytskyy, V.; Babizhetskyy, V.; Zheng, C.; Isnard, O. Ternary antimonides Ln2Pd9Sb3 (Ln = La, Ce, Nd, Pr, and Sm): Crystal, electronic structure, and magnetic properties. J. Alloys Compd. 2020, 815, 152428. DOI: 10.1016/j.jallcom.2019.152428.
Tsuji, Y.; Estrada, E. Influence of long-range interactions on quantum interference in molecular conduction. A tight-binding (Hückel) approach. J. Chem. Phys. 2019, 150 (20), 204123. DOI: 10.1063/1.5097330.
Sato, K.; Pradhan, E.; Asahi, R.; Akimov, A. V. Charge transfer dynamics at the boron subphthalocyanine chloride/C60 interface: non-adiabatic dynamics study with Libra-X. Phys. Chem. Chem. Phys. 2018, 20 (39), 25275–25294. DOI: 10.1039/C8CP03841D.
Li, W.; Ren, W.; Chen, Z.; Lu, T.; Deng, L.; Tang, J.; Zhang, X.; Wang, L.; Bai, F. Theoretical design of porphyrin dyes with electron-deficit heterocycles towards near-IR light sensitization in dye-sensitized solar cells. Sol. Energy 2019, 188, 742–749. DOI: 10.1016/j.solener.2019.06.062.
Vohra, R.; Sawhney, R. S.; Singh, K. P. Contemplating charge transport by modeling of DNA nucleobases based nano structures. Curr. Appl Phys. 2020, 20 (5), 653–659. DOI: 10.1016/j.cap.2020.02.016.
Fujiwara, T.; Nishino, S.; Yamamoto, S.; Suzuki, T.; Ikeda, M.; Ohtani, Y. Total-energy Assisted Tight-binding Method Based on Local Density Approximation of Density Functional Theory. J. Phys. Soc. Jpn. 2018, 87 (6), 064802. DOI: 10.7566/JPSJ.87.064802.
Slavney, A. H.; Connor, B. A.; Leppert, L.; Karunadasa, H. I. A pencil-and-paper method for elucidating halide double perovskite band structures. Chem. Sci. 2019, 10 (48), 11041–11053. DOI: 10.1039/C9SC03219C.
Tetiana Zubatyuk, Ben Nebgen, Nicholas Lubbers, Justin S. Smith, Roman Zubatyuk, Guoqing Zhou, Christopher Koh, Kipton Barros, Olexandr Isayev, Sergei Tretiak. Machine Learned Hückel Theory: Interfacing Physics and Deep Neural Networks. arXiv:1909.12963 [cond-mat.dis-nn]
Bannwarth, C.; Ehlert, S.; Grimme, S. GFN2-xTB—An Accurate and Broadly Parametrized Self-Consistent Tight-Binding Quantum Chemical Method with Multipole Electrostatics and Density-Dependent Dispersion Contributions. J. Chem. Theory Comput. 2019, 15 (3), 1652–1671. DOI: 10.1021/acs.jctc.8b01176.
