Density Functional Tight Binding Calculations Research Articles

Charged Black-Hole-Like Electronic Structure Driven by Geometric Potential of 2D Semiconductors.

One of the exotic expectations in the 2D curved spacetime is the geometric potential from the curvature of the 2D space, still possessing unsolved fundamental questions through Dirac quantization. The atomically thin 2D materials are promising for the realization of the geometric potential, but the geometric potential in 2D materials is not identified experimentally. Here, the curvature-induced ring-patterned bound states are observed in structurally deformed 2D semiconductors and formulated the modified geometric potential for the curvature effect, which demonstrates the ring-shape bound states with angular momentum. The formulated modified geometric potential is analogous to the effective potential of a rotating charged black hole. Density functional theory and tight-binding calculations are performed, which quantitatively agree well with the results of the modified geometric potential. The modified geometric potential is described by modified Gaussian and mean curvatures, corresponding to the curvature-induced changes in spin-orbit interaction and band gap, respectively. Even for complex structural deformation, the geometric potential solves the complexity, which aligns well with experimental results. The understanding of the modified geometric potential provides us with an intuitive clue for quantum transport and a key factor for new quantum applications such as valleytronics, spintronics, and straintronics in 2D semiconductors.

Enhanced Photocatalytic Activity of Surface‐Modified TiO2 with Bimetallic AuPd Nanoalloys for Hydrogen Generation

Herein, commercial titania (TiO2‐P25) is modified with mono‐ and bi‐metallic (Au, Pd, and AuPd) nanoparticles synthesized by chemical reduction method using NaBH4 as a strong reducing agent at room temperature. Bimetallic AuPd nanoalloys homogeneous in size and well dispersed on the TiO2 surface are obtained. The charge‐carrier dynamics, which is a key factor in photocatalysis, is studied by time‐resolved microwave conductivity. The results reveal that surface modification plays a crucial role in charge‐carrier separation, increasing the activity under UV–vis light irradiation. The bimetallic AuPd nanoalloys formation is confirmed by high‐angle annular dark field scanning transmission electron microscopy and corroborated by semiempirical molecular dynamics simulations (Gupta‐LAMMPS). The surface‐modified TiO2 with bimetallic AuPd nanoalloys exhibits higher photocatalytic activity compared to TiO2 modified with their monometallic counterparts. The experimental results are also supported by density functional theory and density functional tight binding calculations, which show that alloying AuPd with low Pd content presents significant synergetic effects for hydrogen generation under UV–vis light from aqueous triethanolamine solutions. Additionally, the AuPd/TiO2 photocatalysts are stable with cycling.

Artificial kagome lattices of Shockley surface states patterned by halogen hydrogen-bonded organic frameworks

Artificial electronic kagome lattices may emerge from electronic potential landscapes using customized structures with exotic supersymmetries, benefiting from the confinement of Shockley surface-state electrons on coinage metals, which offers a flexible approach to realizing intriguing quantum phases of matter that are highly desired but scarce in available kagome materials. Here, we devise a general strategy to construct varieties of electronic kagome lattices by utilizing the on-surface synthesis of halogen hydrogen-bonded organic frameworks (XHOFs). As a proof of concept, we demonstrate three XHOFs on Ag(111) and Au(111) surfaces, which correspondingly deliver regular, breathing, and chiral breathing diatomic-kagome lattices with patterned potential landscapes, showing evident topological edge states at the interfaces. The combination of scanning tunnelling microscopy and noncontact atomic force microscopy, complemented by density functional theory and tight-binding calculations, directly substantiates our method as a reliable and effective way to achieve electronic kagome lattices for engineering quantum states.

Assessment of Hydrazone Derivatives for Enhanced Steel Corrosion Resistance in 15 wt.% HCl Environments: A Dual Experimental and Theoretical Perspective.

This study evaluates the corrosion inhibition capabilities of two novel hydrazone derivatives, (E)-2-(5-methoxy-2-methyl-1H-indol-3-yl)-N'-(4-methylbenzylidene)acetohydrazide (MeHDZ) and (E)-N'-benzylidene-2-(5-methoxy-2-methyl-1H-indol-3-yl)acetohydrazide (HHDZ), on carbon steel in a 15 wt.% HCl solution. A comprehensive suite of analytical techniques, including gravimetric analysis, potentiodynamic polarization (PDP), electrochemical impedance spectroscopy (EIS), and scanning electron microscopy (SEM), demonstrates their significant inhibition efficiency. At an optimal concentration of 5 × 10-3 mol/L, MeHDZ and HHDZ achieve remarkable inhibition efficiencies of 98% and 94%, respectively. EIS measurements reveal a dramatic reduction in effective double-layer capacitance (from 236.2 to 52.8 and 75.3 µF/cm2), strongly suggesting inhibitor adsorption on the steel surface. This effect is further corroborated by an increase in polarization resistance and a significant decrease in corrosion current density at optimal concentrations. Moreover, these inhibitors demonstrate sustained corrosion mitigation over extended exposure durations and maintain effectiveness even under elevated temperatures, highlighting their potential for diverse operational conditions. The adsorption process of these inhibitors aligns well with the Langmuir adsorption isotherm, implying physicochemical interactions at the carbon steel surface. Density functional tight-binding (DFTB) calculations and molecular dynamics simulations provide insights into the inhibitor-surface interaction mechanism, further elucidating the potential of these hydrazone derivatives as highly effective corrosion inhibitors in acidic environments.

Reinforcement learning for in silico determination of adsorbate-substrate structures.

Reinforcement learning (RL) methods have helped to define the state of the art in the field of modern artificial intelligence, mostly after the breakthrough involving AlphaGo and the discovery of novel algorithms. In this work, we present a RL method, based on Q-learning, for the structural determination of adsorbate@substrate models in silico, where the minimization of the energy landscape resulting from adsorbate interactions with a substrate is made by actions on states (translations and rotations) chosen from an agent's policy. The proposed RL method is implemented in an early version of the reinforcement learning software for materials design and discovery (RLMaterial), developed in Python3.x. RLMaterial interfaces with deMon2k, DFTB+, ORCA, and Quantum Espresso codes to compute the adsorbate@substrate energies. The RL method was applied for the structural determination of (i) the amino acid glycine and (ii) 2-amino-acetaldehyde, both interacting with a boron nitride (BN) monolayer, (iii) host-guest interactions between phenylboronic acid and β-cyclodextrin and (iv) ammonia on naphthalene. Density functional tight binding calculations were used to build the complex search surfaces with a reasonably low computational cost for systems (i)-(iii) and DFT for system (iv). Artificial neural network and gradient boosting regression techniques were employed to approximate the Q-matrix or Q-table for better decision making (policy) on next actions. Finally, we have developed a transfer-learning protocol within the RL framework that allows learning from one chemical system and transferring the experience to another, as well as from different DFT or DFTB levels.

Biginelli dihydropyrimidines carrying azole rings: Synthesis, computational studies, and evaluation of alpha‐glucosidase inhibitory and antimicrobial activities

In the current study, we designed three novel compounds by merging two valuable nitrogen-containing pharmacophores, namely dihydropyrimidine (DHPM) and azole, within a single molecule. To obtain the target molecules, azole-linked benzaldehydes were initially synthesized via nucleophilic aromatic substitution reaction between 4-fluorobenzaldehyde and pyrazole, imidazole, or 1,2,4-triazole. Afterward, Biginelli reaction was applied to yield azole-carrying DHPMs. Following structural confirmation, the obtained compounds were tested for their alpha (α)-glucosidase inhibitory and antimicrobial activities. According to the biological data, DHPM-P and DHPM-T showed strong inhibition values whereas DHPM-I failed to effectively inhibit the α-glucosidase enzyme. This situation was explained by molecular docking studies of all compounds into the binding site of alpha-glucosidase. The title compounds also presented moderate antibacterial and antifungal activities. Computational methods based on density functional tight binding (DFTB) and density functional theory (DFT) calculations, along with molecular dynamics (MD) simulations were used to analyze the reactive properties of the compounds. A combination of DFTB and DFT calculations was applied to obtain the lowest energy conformations of the molecules. Further computational analysis included calculations of local reactivity descriptors, such as molecular electrostatic potential and average local ionization energy, by applying DFT calculations. DFT calculations were also used to study intramolecular non-covalent interactions. MD simulations were employed to understand how the title molecules behave in water and to obtain their solubility parameters.

Self-Cleaning and Charge Transport Properties of Foils Coated with Acrylic Paint Containing TiO2 Nanoparticles

The study comprehensively investigates the design and performance of self-cleaning surfaces fabricated by coating aluminum foil with an acrylic paint matrix enriched with different content of titanium dioxide (TiO2) nanoparticles. The main goal was to assess the self-cleaning characteristics of the surfaces obtained. This study employs scanning electron microscopy (SEM) to analyze the morphology of TiO2-modified acrylic surfaces, revealing spherical particles. Raman spectroscopy elucidates signatures characterizing TiO2 incorporation within the acrylic matrix, providing comprehensive insights into structural and compositional changes for advanced surface engineering. Alternating current (AC) impedance spectroscopy was used to assess selected charge transport properties of produced self-cleaning surfaces, allowing us to gain valuable insights into the material’s conductivity and its potential impact on photocatalytic performance. The self-cleaning properties of these tiles were tested against three frequently used textile dyes, which are considered to pose a serious environmental threat. Subsequently, improving self-cleaning properties was achieved by plasma treatment, utilizing a continuous plasma arc. The plasma treatment led to enhanced charge separation and surface reactivity, crucial factors in the self-cleaning mechanism. To deepen our comprehension of the reactive properties of dye molecules and their degradation dynamics, we employed a combination of density functional tight binding (DFTB) and density functional theory (DFT) calculations. This investigation lays the foundation for advancing self-cleaning materials with extensive applications, from architectural coatings to environmental remediation technologies.

Towards graphene-based asymmetric diodes: a density functional tight-binding study.

Self-consistent charge density functional tight-binding (DFTB) calculations have been performed to investigate the electrical properties and transport behavior of asymmetric graphene devices (AGDs). Three different nanodevices constructed of different necks of 8 nm, 6 nm and 4 nm, named Graphene-N8, Graphene-N6 and Graphene-N4, respectively, have been proposed. All devices have been tested under two conditions of zero gate voltage and an applied gate voltage of +20 V using a dielectric medium of 3.9 epsilon interposed between the graphene and the metallic gate. As expected, the results of AGD diodes exhibited strong asymmetric I(V) characteristic curves in good agreement with the available experimental data. Our predictions implied that Graphene-N4 would achieve great asymmetry (A) of 1.40 at |VDS| = 0.2 V with maximum transmittance (T) of 6.72 in the energy range 1.30 eV. More importantly, while the A of Graphene-N4 was slightly changed by applying the gate voltage, Graphene-N6/Graphene-N8 showed a significant effect with their A increased from 1.20/1.03 under no gate voltage (NGV) to 1.30/1.16 under gate voltage (WGV) conditions. Our results open up unprecedented numerical prospects for designing tailored geometric diodes.

Polymorphism in S(+)Clopidogrel-Picrate: Insights from X-ray Diffraction, Vibrational Spectroscopy, Thermal Analysis, and Quantum Chemistry

The crystal structures of two pseudopolymorphic forms of S(+)clopidogrel–picrate are reported. Form 1 crystallizes in the monoclinic space group P21 with an ionic couple S(+)ClopH+·Pic− and a molecule of solvent ethanol in the asymmetric unit, while Form 2 crystallizes in the monoclinic space group C2 with two ionic couples in the asymmetric unit. The configurations and conformations of the ionic couples, held together by ionized +N-H···O hydrogen bonds, are nearly identical in the structures. The self-assembly properties are compared with reported clopidogrel salts, including those used in pharmaceutical formulations. The hydrogen bonds are discussed in reference to the general corresponding behavior of the N-bases picrates and the properties of the acid-base coformers. The preparations of the pseudopolymorphs were optimized toward two different methods: solvent evaporation and mechanochemical treatment. Reproducibility to generate the single crystalline phases was confirmed by thermal and vibrational spectroscopic properties. Periodic third-order density-functional tight binding (DFTB3) calculations predict rather small energy difference between the two pure phases of polymorphs 1 and 2. However, the included solvent molecules in Form 1 decrease the lattice energy for ~10.5 kcal mol−1, which leads to a lower ΔElatt. lattice energy in comparison to Form 2 (by ~7.3 kcal mol−1). All predicted trends are in line with the experimentally observed formation of Form 1 instead of its simulated non-solvated Form 1.

Exploring topological phase transition in Pt2Hg1−xTlxSe3

The transition from trivial to non-trivial phase in two-dimensional materials are called a topological phase transition (TPT). The Berry phase, non-local string order parameter, and edge states define the topological nature of the system. A newly discovered jacutingaite ma- terial Pt2HgSe3 is a layered material which occurs naturally in the form of minerals. The material can be exfoliated and was predicted as a quantum spin Hall insulator. Here, on the basis of density functional theory and tight-binding calculations, we explore Pt2Hg1−xTlxSe3 (x = 0.25, 0.50, 0.75, 1) to understand the electronic and topological properties. We start with the parent material Pt2HgSe3 wherein Hg is replaced partially with x amount of Tl, to tune the topological phases. From the electronic structure calculations, Pt2HgSe3 is found to be a non-trivial semimetal in it’s bulk. Upon electron doping, the material transforms to strong topological metallic phase. The topological Z2 invariant calculation shows TPT in Pt2Hg1−xTlxSe3 with weak topological insulating state (0;001) for x=0, to strong topological metal (1;000) for x=1, respectively.

Quantum Confinement in Epitaxial Armchair Graphene Nanoribbons on SiC Sidewalls.

The integration of graphene into devices necessitates large-scale growth and precise nanostructuring. Epitaxial growth of graphene on SiC surfaces offers a solution by enabling both simultaneous and targeted realization of quantum structures. We investigated the impact of local variations in the width and edge termination of armchair graphene nanoribbons (AGNRs) on quantum confinement effects using scanning tunneling microscopy and spectroscopy (STM, STS), along with density-functional tight-binding (DFTB) calculations. AGNRs were grown as an ensemble on refaceted sidewalls of SiC mesas with adjacent AGNRs separated by SiC(0001) terraces hosting a buffer layer seamlessly connected to the AGNRs. Energy band gaps measured by STS at the centers of ribbons of different widths align with theoretical expectations, indicating that hybridization of π-electrons with the SiC substrate mimics sharp electronic edges. However, regardless of the ribbon width, band gaps near the edges of AGNRs are significantly reduced. DFTB calculations successfully replicate this effect by considering the role of edge passivation, while strain or electric fields do not account for the observed effect. Unlike idealized nanoribbons with uniform hydrogen passivation, AGNRs on SiC sidewalls generate additional energy bands with non-pz character and nonuniform distribution across the nanoribbon. In AGNRs terminated with Si, these additional states occur at the conduction band edge and rapidly decay into the bulk of the ribbon. This agrees with our experimental findings, demonstrating that edge passivation is crucial in determining the local electronic properties of epitaxial nanoribbons.

Reticular Synthesis of Metal‐Organic Frameworks by 8‐Connected Quadrangular Prism Ligands for Water Harvesting

AbstractUtilization of rigid, highly connected organic linkers is critical for the reticular synthesis of functional metal–organic frameworks (MOFs). However, highly‐stable MOFs (e.g. Al/Cr/Zr‐based MOFs) based on rigid ligands with more than 6 coordinating functions have been rarely achieved thus far. Herein, we describe the construction of two bcu Zr‐based MOFs (named ZrMOF‐1 and ZrMOF‐2) from peripherally extended pentiptycene ligands (H8PEP‐1 and H8PEP‐2) with rigid quadrangular prism shape possessing 8 carboxylic groups at the prism vertices. Particularly, ZrMOF‐1 exhibits microporous structure with large Bruno‐Emmett‐Teller surface area and high water stability, endowing it a promising water harvesting material with a high water uptake capacity of 0.83 gH2O gMOF−1 at P/P0=0.90 and 25 °C, a steep uptake at a low P/P0 of 0.30, and excellent durability over 500 water adsorption‐desorption cycles. Moreover, self‐consistent charge density functional tight‐binding calculations were carried out, rationalizing the water adsorbing process and amount in ZrMOF‐1.

Reticular Synthesis of Metal-Organic Frameworks by 8-Connected Quadrangular Prism Ligands for Water Harvesting.

Utilization of rigid, highly connected organic linkers is critical for the reticular synthesis of functional metal-organic frameworks (MOFs). However, highly-stable MOFs (e.g. Al/Cr/Zr-based MOFs) based on rigid ligands with more than 6 coordinating functions have been rarely achieved thus far. Herein, we describe the construction of two bcu Zr-based MOFs (named ZrMOF-1 and ZrMOF-2) from peripherally extended pentiptycene ligands (H8 PEP-1 and H8 PEP-2) with rigid quadrangular prism shape possessing 8 carboxylic groups at the prism vertices. Particularly, ZrMOF-1 exhibits microporous structure with large Bruno-Emmett-Teller surface area and high water stability, endowing it a promising water harvesting material with a high water uptake capacity of 0.83 gH2O gMOF -1 at P/P0 =0.90 and 25 °C, a steep uptake at a low P/P0 of 0.30, and excellent durability over 500 water adsorption-desorption cycles. Moreover, self-consistent charge density functional tight-binding calculations were carried out, rationalizing the water adsorbing process and amount in ZrMOF-1.

Non-equivalent nature of acetylenic bonds in typical square graphynes and intricate negative differential resistance characteristics

The role of acetylenic linkage in determining the exotic band structures of 4, 12, 2- and 4, 12, 4- graphynes is reported. The Dirac bands, as confirmed by both density functional theory and tight-binding calculations, are robust and stable over a wide range of hopping parameters between -hybridized carbon atoms. The shifting of the crossing points of the Dirac bands along the k-path of these two square graphynes is found to be in opposite direction with the hopping along with the acetylenic bond. A real space decimation scheme has also been adopted for understanding this interesting behavior of the band structure of these two graphynes. The condition for the appearance of a nodal ring in the band structure has been explored and critically tested by appropriate Boron-Nitrogen doping. Moreover, both the graphynes exhibit negative differential resistance in their current–voltage characteristics, with 4, 12, 2- graphynes showing superiority.

Electronic transition dipole moments from time-independent excited-state density-functional tight-binding.

Computationally inspired design of organic electronic materials requires robust models of not only the ground and excited electronic states but also of transitions between these states. In this work, we introduce a strategy for obtaining electronic transition dipole moments for the lowest-lying singlet-singlet transition in organic chromophores from time-independent excited-state density-functional tight-binding (ΔDFTB) calculations. Through small-molecule benchmarks and applications to larger chromophores, we explore the accuracy, potential, and limitations of this semiempirical strategy. While more accurate methods are recommended for small systems, we find some evidence for the method's potential in high-throughput molecular screening applications and in the analysis of molecular dynamics simulations.

Efficient predictions of formation energies and convex hulls from density functional tight binding calculations

• DFTB was interfaced with CASM to enable efficient calculations of formation energies • Efficiency and accuracy of DFTB vs. DFT was carried out for SiC and ZnO materials • DFTB can be an order of magnitude faster than DFT for predicting formation energies Defects in materials significantly alter their electronic and structural properties, which affect the performance of electronic devices, structural alloys, and functional materials. However, calculating all the possible defects in complex materials with conventional Density Functional Theory (DFT) can be computationally prohibitive. To enhance the efficiency of these calculations, we interfaced Density Functional Tight Binding (DFTB) with the Clusters Approach to Statistical Mechanics (CASM) software package for the first time to calculate formation energies and convex hulls. Using SiC and ZnO as representative examples, we show that DFTB gives accurate results and can be used as an efficient computational approach for calculating and pre-screening formation energies/convex hulls. Our DFTB+CASM implementation allows for an efficient exploration (up to an order of magnitude faster than DFT) of formation energies and convex hulls, which researchers can use to probe other complex systems.

GPU-Enhanced DFTB Metadynamics for Efficiently Predicting Free Energies of Biochemical Systems.

Metadynamics calculations of large chemical systems with ab initio methods are computationally prohibitive due to the extensive sampling required to simulate the large degrees of freedom in these systems. To address this computational bottleneck, we utilized a GPU-enhanced density functional tight binding (DFTB) approach on a massively parallelized cloud computing platform to efficiently calculate the thermodynamics and metadynamics of biochemical systems. To first validate our approach, we calculated the free-energy surfaces of alanine dipeptide and showed that our GPU-enhanced DFTB calculations qualitatively agree with computationally-intensive hybrid DFT benchmarks, whereas classical force fields give significant errors. Most importantly, we show that our GPU-accelerated DFTB calculations are significantly faster than previous approaches by up to two orders of magnitude. To further extend our GPU-enhanced DFTB approach, we also carried out a 10 ns metadynamics simulation of remdesivir, which is prohibitively out of reach for routine DFT-based metadynamics calculations. We find that the free-energy surfaces of remdesivir obtained from DFTB and classical force fields differ significantly, where the latter overestimates the internal energy contribution of high free-energy states. Taken together, our benchmark tests, analyses, and extensions to large biochemical systems highlight the use of GPU-enhanced DFTB simulations for efficiently predicting the free-energy surfaces/thermodynamics of large biochemical systems.

Nucleation of zeolitic imidazolate frameworks: from molecules to nanoparticles.

We have studied the clusters involved in the initial stages of nucleation of Zeolitic Imidazolate Frameworks, employing a wide range of computational techniques. In the pre-nucleating solution, the prevalent cluster is the ZnIm4 cluster (formed by a zinc cation, Zn2+, and four imidazolate anions, Im-), although clusters such as ZnIm3, Zn2Im7, Zn2Im7, Zn3Im9, Zn3Im10, or Zn4Im12 have energies that are not much higher, so they would also be present in solution at appreciable quantities. All these species, except ZnIm3, have a tetrahedrally coordinated Zn2+ cation. Small ZnxImy clusters are less stable than the ZnIm4 cluster. The first cluster that is found to be more stable than ZnIm4 is the Zn41Im88 cluster, which is a disordered cluster with glassy structure. Bulk-like clusters do not begin to be more stable than glassy clusters until much larger sizes, since the larger cluster we have studied (Zn144Im288) is still less stable than the glassy Zn41Im88 cluster, suggesting that Ostwald's rule (the less stable polymorph crystallizes first) could be fulfilled, not for kinetic, but for thermodynamic reasons. Our results suggest that the first clusters formed in the nucleation process would be glassy clusters, which then undergo transformation to any of the various crystal structures possible, depending on the kinetic routes provided by the synthesis conditions. Our study helps elucidate the way in which the various species present in solution interact, leading to nucleation and crystal growth.

The Importance of Mg–Sb Interactions in Achieving High Conduction Band Degeneracy in Mg3Sb2 for High n-Type Thermoelectric Performance

In recent years, Mg3Sb2 has attracted much attention due to its excellent performance as an n-type thermoelectric material. The high performance of n-type Mg3Sb2 is largely due to the high valley degeneracy (NV) of the electron transport edge, which is the result of the conduction band minimum (CBM) being located at a low-symmetry, six-fold degenerate point (inside the ΓALM plane) in the first Brillouin zone. Furthermore, Bi alloying on the Sb site is well-known to improve electron mobility, and hence, performance. In previous works, this low-symmetry CBM has been attributed to interactions between the tetrahderally and octahedrally coordinated Mg-sites. However, these Mg–Mg interactions cannot explain changes in the shape of the CBM upon Bi alloying, while Mg–Sb (and Mg–Bi) interactions could explain them. Prior studies have neglected to fully consider the importance of the Sb orbitals in the existence of the low-symmetry CBM because at the CBM, there is very little contribution from the Sb orbitals. In this work, we use first-principles density functional theory and tight-binding calculations to show that the low-symmetry CBM is due to Mg-s–Sb-p interactions that are weakest at or near the CBM. Controlling the cation-s–anion-p interactions through anion- or cation-site alloying can be used as an engineering strategy to optimize thermoelectric performance.

Theory of triangulene two-dimensional crystals

Abstract Equilateral triangle-shaped graphene nanoislands with a lateral dimension of n benzene rings are known as [n] triangulenes. Individual [n] triangulenes are open-shell molecules, with single-particle electronic spectra that host n − 1 half-filled zero modes and a many-body ground state with spin S = ( n − 1 ) / 2 . The on-surface synthesis of triangulenes has been demonstrated for n = 3 , 4 , 5 , 7 and the observation of a Haldane symmetry-protected topological phase has been reported in chains of [3]triangulenes. Here, we provide a unified theory for the electronic properties of a family of two-dimensional honeycomb lattices whose unit cell contains a pair of triangulenes with dimensions n a , n b . Combining density functional theory and tight-binding calculations, we find a wealth of half-filled narrow bands, including a graphene-like spectrum (for n a = n b = 2 ), spin-1 Dirac electrons (for n a = 2 , n b = 3 ), p x , y -orbital physics (for n a = n b = 3 ), as well as a gapped system with flat valence and conduction bands (for n a = n b = 4 ). All these results are rationalized with a class of effective Hamiltonians acting on the subspace of the zero-energy states that generalize the graphene honeycomb model to the case of fermions with an internal pseudospin degree of freedom with C 3 symmetry.