Electronic band structure and chemical bonding in trigonal Se and Te

Se and Te are chalcogens, elements of the 16th group of the Periodic Table, with valence electron configurations of 4s²4p⁴ and 5s²5p⁴, respectively. Se and Te, as well as their semiconducting compounds, chalcogenides, are in demand in electronics, photovoltaics, optics, sensors, xerography, detection, data recording, processing and storage, and various other fields of practical applications³ due to their experimental semiconducting bandgaps of about 1.6¹ and 0.3² eV, respectively, and the large values of absorption coefficient and optical conductivity.

Recently, significant attention has been paid to the unusual properties of the electronic band structure of trigonal Se and Te caused by the chirality (absence of space inversion and mirror-rotation axes) of their crystal lattices (space group P3₁21 or P3₂21),⁴ wherein atoms are located in parallel helical chains that form a hexagonal structure [Fig. 1(a)].

Previous studies have particularly focused on semiconductor–three-dimensional Weyl semimetal–metal phase transitions in Se and Te under increasing pressure,⁶ unconventional electronic structures and spin texture of trigonal Te,^7–11 current-induced optical activity¹² and magnetization¹³ in bulk trigonal Te, and electrical magnetochiral anisotropy in Te.¹⁴ The discovery of kinetic magnetoelectric effect¹⁵ in bulk nonmagnetic Te showed that nonmagnetic chiral crystals can be used in spintronics and devices for the generation of magnetization without electric current coils.

Unusual linear and nonlinear optical properties of quasi-one- and two-dimensional structures of Se and Te helical chains have also been studied.^16–18 

Both topological peculiarities of the electronic band structure of bulk crystals and thin film properties are closely related to the types of chemical bondings existing in trigonal Se and Te. Aiming to understand the nature of the chemical bondings in semiconducting chalcogenides, previous studies^19–21 primarily discussed the electronic band structure of chalcogens. Pauling’s ideas²² on valence bonds, resonance bonding, and lone pairs of valence electrons were used in these studies. Various assumptions about the contributions of valence s- and p-electrons to bonding and the types of chemical bondings in trigonal Se and Te have been made.^19–21 For example, a previous study²⁰ claimed that (1) valence s-electrons can be ignored considering the nature of bonding in trigonal Se and Te since their binding energies are significantly larger than those of valence p-electrons (in contrast to the sp³ hybridization of s- and p-electrons in Ge and Si), and (2) valence p-electrons can be divided into two groups: low-lying p-electrons providing covalent bonding between nearest atoms in helical chains and upper nonbonding (lone-pair) p-electrons. Based on the analysis of the compatibility between the number of valence electrons and coordination numbers of atoms and comparison of interatomic distances and doubled atomic and van der Waals radii, some studies^19,21 supposed that each atom of a chain is covalently bonded with two nearest atoms of the same chain and the bonding with the four next nearest atoms in the neighboring chains of trigonal Se and Te is of the resonant and van der Waals types. However, the latter conclusion was not accompanied by any estimation of the van der Waals forces in Se and Te.

A recently published study²³ presented a similar point of view: van der Waals bonding between covalently bonded helical chains and neutrality of valence s- and lone-pair p-electrons with respect to bondings in trigonal Se.

Empirical pseudopotential methods and tight-binding models have been used²⁴ to calculate the electronic band structure and charge density distribution in trigonal Se and Te. Based on the charge density distribution analysis in Joannopoulos et al.,²⁴ various conclusions have been made: (1) the covalent bonding between neighboring atoms within helical chains is due to the lower lying states of valence s- and bonding p-electrons and (2) the upper part of the bonding p-electron states is responsible for the interchain bonding, which is characterized as covalent-like but is significantly weaker than the bonding between atoms within the same helical chain. Joannopoulos et al.²⁴ did not consider the space charge distribution of lone-pair p-electrons, but it was presented in preceding papers.^25,26 However, these preceding papers do not clearly describe the role of lone-pair p-electrons in chemical bondings in trigonal Se and Te. Note that the electronic band structure was calculated in the previous papers^24–26 using nonrelativistic approximation and van der Waals forces were not considered.

Some studies^16–18 defined interchain bonding in trigonal Se and Te as covalent-like quasi-bonding by analogy with bonding between layers of atoms in black phosphorous²⁷ and dichalcogenides of transition metals.²⁸ 

The nature of chemical bonding in bulk trigonal Te has been theoretically studied²⁹ using the Vienna ab initio simulation package code. Based on the charge density distribution analysis and results of electron localization function (ELF)³⁰ calculations, Ye et al.²⁹ concluded that in trigonal Te the ordinary covalent bonding between neighboring atoms in helical chains coexists with the much weaker coordinate covalent bonding with atoms of neighboring chains.

Our previous study³¹ showed that valence s-electrons of trigonal Te with binding energies of 8–14 eV significantly contribute to the charge density of bond critical points (BCPs), affording the chemical bonding between neighboring atoms of helical chains. Calculations were performed using the full potential linearized augmented plane wave method implemented in the program package WIEN2k.³² CRITIC2³³ program was used to analyze the charge density distribution. Furthermore, in agreement with the results of previous studies,^24–26 we³¹ showed that valence p-electrons are responsible for both intra- and interchain bonding. However, note that due to the overlapping of bands of valence p-electrons in trigonal Te, their separation into bonding and lone-pair parts is possible only with large errors.³¹ Despite the fact that trigonal Te is a semiconductor and rhombohedral Sb is a semimetal, the parameters of their BCPs are similar.³¹ 

One study analyzed the possibility of the existence of lone-pair electrons in several substances, particularly in trigonal Se and Te and in semimetal Sb.³⁴ 

Considering the large range of views on the electronic structure and the nature of chemical bonding in trigonal Se and Te and the importance of this problem for predicting the practical usage of bulk and low-dimensional samples of these materials, the quantum theory of atoms in molecules (QTAIM) method^35–37 is applied herein to specify the types of chemical bonding existing in bulk trigonal Se and Te based on the results of their electronic band structure calculations obtained using the WIEN2k program package.³² 

The WIEN2k program,^32,38 which is based on the augmented plane wave plus local orbital method of density functional theory (DFT), is used herein to determine the electronic band structure of bulk trigonal Se and Te, optimize their crystal structure, and find charge density distribution in the crystal lattice. The most difficult problem of DFT application while studying the physical properties is the proper choice of the exchange–correlational functional. Herein, the exchange potential of Becke and Johnson that is modified by Tran and Blaha (mBJ)³⁹ is used. This allows the determination of bandgaps in semiconductors and oxides closest to the experimental results.⁴⁰ To perform the crystal structure optimization, the Perdew–Burke–Ernzerhof (PBE)⁴¹ exchange–correlation functional was used. PBE is the second rung of the so-called “Jacob’s ladder”^42,43 of exchange–correlation functionals, essentially improving on the local density approximation (LDA), the first rung of the ladder.

To determine the contribution of van der Waals forces to bondings in trigonal Se and Te, nonlocal van der Waals exchange–correlation functional rev-van der Waals (vdW)-DF2, suggested by Hamada,⁴⁴ was used. This functional has been tested on various substances and is known⁴⁵ to afford the smallest error in binding energy calculations and crystal structure optimization.

For valence states, the relativistic effects were considered with the second variational method, including spin–orbital coupling, and core states were fully relativistically treated.³² To ensure the required accuracy, the following parameters were used: the expansion in the l-orbital moment was performed within atomic spheres to l_max = 10, and the maximum wave vector k_max used in the expansion in plane waves was defined by-product R_mt · k_max = 8.5, where R_mt is the atomic sphere radius. The total number of points in the Brillouin zone was taken as 5000. The convergence criterion for the total energy was set as 10⁻⁵ Ry and that for the charge distance (difference between the charge density integrated over the unit cell for two consecutive iterations of the self-consistent field cycle) was set as 10⁻⁵ e (e is the positive value of the elementary charge).

The charge density distribution ρ(r) in the crystal lattices of trigonal Se and Te was taken from the band structure calculation results. The program CRITIC2,³³ which is designed for the topological analysis of scalar fields in periodic structures based on the QTAIM method,^35–37 was used to analyze ρ(r). According to Bader’s QTAIM, points r_c, wherein the charge density gradient equals zero, i.e., ∇ρ(r_c) = 0, are called critical points. The critical points in the charge density distribution are characterized by the rank ω [the number of nonzero eigenvalues λ_i, main values of curvature of the Hesse matrix of the second derivatives of the charge density over coordinates (∂²ρ/∂x_i∂x_j)] and signature ϭ, the algebraic sum of λ_i signs: (ω, ϭ). Four types of stable critical points exist:^35–37 (3, −3)—nucleus (local maximum), (3, +3)—cage (local minimum), (3, +1)—ring (the first saddle critical point), and (3, −1)—bond (the second saddle critical point). BCPs play an important role in the classification of the chemical bonding types. Covalent bonding is specified by the following Bader’s parameters of BCPs:^35–37 negative sign of Laplacian ∇²ρ_b < 0; negative curvatures λ_1,2 < 0, which are large in absolute values $λ1,2>λ3$⁠; and high charge density ρ_b. In contrast, for ionic bonding ∇²ρ_b > 0, $λ1,2≪λ3$⁠, ρ_b is small, and the charge density is mainly concentrated on atoms.

In the following, Bader’s parameters of BCPs (eigenvalues λ_i—main values of the curvature of Hesse matrix, the sign and value of charge density Laplacian ∇²ρ_b, and the charge density ρ_b) are used for the classification of chemical bonding types existing in crystals of trigonal Se and Te. Additionally, two dimensionless parameters are employed:⁴⁶ flatness f (ratio of the minimum charge density in cage critical points $ρcmin$ to the maximum charge density in BCPs $ρbmax$⁠), which characterizes the uniformity of the charge density distribution in crystals, and molecularity

The third parameter⁴⁶ charge transfer c = 0 for crystals of Se and Te as they comprise only one type of atom and due to the absence of non-nuclear maxima in their charge density distribution.

Calculations of the electronic band structures of trigonal Se and Te were performed via the WIEN2k program using the mBJ exchange potential, LDA short-range correlations, and experimental lattice parameters (Table I). Particularly, Table I lists the title of the chalcogen crystal, space group, lattice parameters, types of Wyckoff positions occupied by atoms in the crystal lattice, coordinates of nonequivalent atoms in the unit cell, and references to studies where the experimental data on lattice parameters are published.

Se and Te atoms have similar configurations of valence electrons 4s²4p⁴ and 5s²5p⁴. Partial densities of energy states (DOS) for valence s- and p-electrons in trigonal Se and Te crystals are shown in Fig. 2.

Note the most significant features of DOS in Fig. 2. In trigonal Se, the upper states of valence p-electrons (from 0 to −3 eV), which were formerly called lone-pair p-electrons,²⁰ do not overlap with the lower states (from −3 to −6 eV), previously called bonding p-electrons.²⁰ In contrast, in trigonal Te, the lower and upper states of valence p-electrons overlap. In both Se and Te, the binding energies of valence s-electrons are considerably larger than those of valence p-electrons. The mixing of s- and p-states in Se and Te is low (at the level of a few percent), which is completely different from that for semiconducting Ge (Fig. S1 of the supplementary material), for which the hybridization of s- and p-states is nearly 100%.

The positions and intensities of maxima for valence s- and p-electrons DOS in Fig. 2 qualitatively agree with the peculiarities of DOS calculated in Joannopoulos et al.²⁴ as well as with features identified in the experimental x-ray and ultraviolet photoelectron spectra for trigonal Te^49,50 and Se.^1,49,51,52 The calculated energy bandgaps presented in Table II agree well with the experimental bandgaps for trigonal Se and Te.

The electronic band structures of trigonal Te and Se, including three doubly degenerated bands of valence s-electrons, six lower and six upper bands of valence p-electrons, and unoccupied states of conductivity bands, are shown in Figs. 3 and 4, respectively. Band paths within the first Brillouin zone of trigonal Se and Te [Fig. 1(b)] are chosen in accordance with recommendations⁵ based on crystallographic convention. Results of the band structure calculations for trigonal Te and Se well agree with the theoretical and experimental data of previous studies.^6–11 

Note that the mBJ exchange potential and short-range LDA correlations not only provide bandgap values that are close to the experimental values but also correctly describe the peculiarities of Se and Te electronic bands related to unusual topological properties of these substances actively studied in recent years.^6–11 Particular, doubly degenerate Weyl nodes at the H point close to the Fermi energy (H₆) and on the H–K line in the electronic band structures of Se and Te, which are discussed in the literature,^6,7 and the Weyl nodes in the Te band structure on the Г–A line for lower p-electrons⁸ and upper p-electrons⁹ are well reproduced in Figs. 3 and 4. Spin–orbit interaction and chiral crystal structure splittings of valence p-electrons energy bands in trigonal Te, which are visible in Fig. 3, are in good agreement with the experimental and theoretical results obtained in Ref. 9 via angle-resolved photoemission spectroscopy (ARPES) and DFT calculations. Several crossings (Weyl nodes) of valence p-electrons bands of trigonal Te that were revealed by ARPES in Ref. 11 and classified as composite Weyl fermions, Kramers–Weyl fermions, and accordion Weyl points are also presented in Fig. 3. Thus, the calculated electronic band structures of trigonal Te and Se that are shown in Figs. 3 and 4, respectively, can be considered reliable.

Table III presents parameters of two types of BCPs in the charge density distribution for crystals of trigonal Se and Te. Particularly, Table III presents the space groups describing the crystal symmetry; Wyckoff positions occupied by BCPs in the crystal lattice; number of BCPs of each type for nonequivalent atoms in the unit cell (N BCPs); interatomic distances d between bonding atoms; calculated Bader’s characteristics of BCPs (ratio of main eigenvalues of Hesse matrix |λ_1,2|/λ₃, Laplacian value ∇²ρ_b, and density of charge in BCPs ρ_b); ELF values in BCPs η(r_b); contributions of s-electrons, lower p_l-electrons, and upper p_u-electrons to ρ_b; flatness f; and molecularity μ. For trigonal Te, due to the overlapping of lower and upper energy bands of valence p-electrons (Figs. 2 and 3), only the summarized contribution of p-electrons to ρ_b is presented in Table III. For trigonal Se, the lower and upper valence p-electrons approximately range from −6 to −3 and −3 to 0 eV, respectively (Figs. 2 and 4).

In the unit cell of trigonal Te and Se, only one type of nonequivalent atoms is present (Table I). Therefore, the number of BCPs per atom in the formula unit N_at for trigonal Te and Se is equal to 6. This number coincides with the number of valence s- and p-electrons in Te and Se atoms.

Table III shows that trigonal Te and Se have two types of BCPs with significantly different Bader’s parameters. Valence s- and lower p-electrons contribute nearly equally to ρ_b of BCPs of the first type in trigonal Se, while lower and upper p-electrons mainly contribute to the ρ_b of BCPs of the second type. This conclusion is illustrated in Fig. 5, wherein the charge density distribution in areas of BCPs in trigonal Se is shown.

The picture plane in Fig. 5 is parallel to the plane (a, b) of the coordinate system of trigonal Se and crosses the c axes at half of the height. One of the three BCPs of the first type with coordinates (0.11, 0.11, 0.5) is located in this plane at half of the distance between the two neighboring Se atoms with coordinates (0.2254, 0.0, 1/3) and (0.0, 0.2254, 2/3) (Table I), belonging to the same turn of the helical chain. Two of the six BCPs of the second type with coordinates (0.598, 0.102, 0.491) and (0.102, 0.598, 0.509) are present in the nearest vicinity to the picture plane, providing bonding between neighboring Se helical chains. This plane also comprises one of the three cage critical points with coordinates (0.606, 0.606, 0.5) and minimal charge density ρ_c = 0.01 e/ų. The charge density distribution in areas of BCPs of trigonal Se obtained via the charge density difference (CDD) method is demonstrated in Figs. 6(a)–6(c) in the same picture plane as in Fig. 5. Figure 6(d) shows the total charge density distribution.

The CDD method, available in the WIEN2k software package, allows the determination of the difference between the charge density distribution in crystals and atoms. CDD is an analog of the so-called deformation density distribution method used in quantum chemistry to distinguish between covalent and ionic types of bonding in molecules.⁵³ Note that a relatively large positive deformation density exists between atoms in covalently bonded molecules, such as H₂.⁵³ However, sometimes the deformation density is close to zero (F₂) or even negative.⁵⁴ To overcome this difficulty, a new type of chemical bonding, charge-shift bonding, has been suggested,^55,56 which considers the possibility of charge fluctuations.

To determine the influence of van der Waals forces on the charge density distribution in trigonal Se and Te, electronic band structure calculations were performed using the WIEN2k program package and the exchange–correlation functional rev-vdW-DF2, as recommended in Tran et al.⁴⁵ For comparison, the PBE⁴¹ exchange–correlation functional was also used in the calculations. For completeness, the band structure calculations were conducted twice, with experimental lattice parameters of trigonal Se and Te listed in Table I and with optimized lattice parameters obtained via the procedure implemented in the WIEN2k program package.

Based on Hamada,⁴⁴ the revised Becke B86R exchange functional (original Becke exchange functional, B86⁵⁷) was taken as the exchange energy E_x and the short-range LDA was treated as the correlation energy. Additionally, Hamada⁴⁴ described van der Waals forces using the nonlocal long-range correlation functional vdW-DF2.⁵⁸ To assess the effectiveness of the rev-vdW-DF2 functional, all three components of the functional need to be simultaneously considered. However, aiming to estimate the influence of vdW forces on BCP parameters in trigonal Se and Te, the first two parts of rev-vdW-DF2 functional (B86R exchange functional and LDA short-range correlations) were also used for the electronic band structure calculations independent from the vdW-DF2 functional. The charge density distribution in trigonal Se and Te obtained after the electronic band structure calculations was analyzed using the CRITIC2 program.³³ The characteristics of BCPs in trigonal Se and Te are presented in Tables IV and V, respectively. Particularly, Tables IV and V list the type of exchange–correlation functional used in calculations, lattice parameters (experimental or optimized), Wyckoff positions occupied by BCPs in the crystal lattice, interatomic distances d between bonding atoms, calculated Bader’s characteristics of BCPs (ratio of main eigenvalues of Hesse matrix |λ_1,2|/λ₃, Laplacian value ∇²ρ_b, density of charge in BCPs ρ_b), flatness f, and molecularity μ.

Tables IV and V show that when the charge density distribution is calculated with fixed experimental lattice parameters, the B86R exchange functional plus LDA short-range correlations afford nearly the same BCP characteristics as the complete rev-vdW-DF2 exchange–correlation functional. Despite the fact that the optimized lattice parameters markedly differ for the two sets of exchange–correlation functionals, the values of Bader’s characteristics of BCPs for trigonal Se and Te only slightly changed after the inclusion of the vdW-DF2 functional. This conclusion agrees with the statement:³² “The non-local vdW potential affects only very little the density and electronic structure and is therefore essential only for the forces.”

Semiconducting Ge and semimetal Sb can be considered prototypical substances with covalent and semimetallic types of bonding. Previous studies^31,59 presented Bader’s parameters of BCPs for Ge as typical parameters for BCPs of covalent type (see Table S1 of the supplementary material). Each atom has four nearest neighbors in the crystal lattice of Ge and only one type of BCPs is present in Ge, located in 16c Wyckoff position (space group N 227$Fd3̄m$⁠).⁴ The charge density distribution in the basal crystal plane (a, b) of a Ge unit cell obtained by the CDD method is shown in Fig. S2 of the supplementary material. This type of BCPs is characterized by a relatively high value of the charge density $ρbCDD$ = 0.17 e/ų and limited area of the charge density location around BCPs in the CDD image. Bader’s parameters of BCPs (Table S1 of the supplementary material) and the charge density distribution in areas of BCPs of Ge (Fig. S2 of the supplementary material) can be considered typical for the covalent type of chemical bonding. According to Fig. S2 of the supplementary material, the charge density in semiconducting Ge is nonuniformly distributed and flatness f = 2.5%.⁵⁹ Partial contributions of valence s- and p-electrons to bondings in Ge cannot be separated due to the strong overlapping (hybridization) of s- and p-valence bands (Fig. S1 of the supplementary material).

In semimetal Sb, two types of BCPs^31,59 are present, located at the 3d and 3e Wyckoff positions in the rhombohedral crystal lattice (space group N 166 $R3̄m$⁠)⁴ (Table S1 of the supplementary material). Charge density distributions in the areas of BCPs of the first and the second types for Sb crystal are shown in Figs. S3 and S4 of the supplementary material, respectively. Both Bader’s parameters for BCPs of the first type (Table S1 of the supplementary material) and the character of the charge density distribution in their areas (Fig. S3 of the supplementary material) denote a large degree of covalence of the first type of BCPs in Sb.³¹ 

The charge density distribution for BCPs of the second type in the crystal lattice of Sb is shown in Fig. S4 of the supplementary material. In addition to the lower charge density values in Fig. S4 than those in Fig. S3, the difference between the charge density distribution patterns in Figs. S3 and S4 is especially evident in the lower left parts of these figures. For BCPs of the second type, the CDD image of the combined contributions of the s- and p-electrons looks like a ribbon, which can be regarded as specific for semimetal Sb. Similar to Ge, the charge density distribution in Sb is not uniform and flatness f = 4.8%.⁵⁹ For comparison, in pure alkaline metals, f is close to 100%.⁴⁶ Notably, the number five of the valence electrons in Sb atoms with configuration s²p³ is less than the number six of BCPs of both types per atom in Sb crystals, which can promote the delocalization of valence electrons and close the bandgap.

BCPs of the first type located in 3b Wyckoff position in trigonal Se and Te can be classified as covalent, although the degree of covalence in Se is greater than that in Te. This conclusion is based on BCP characteristics presented in Table III, which are specific for covalent-like BCPs. Features of the charge density distribution in areas of BCPs for trigonal Se and Te shown in Figs. 5 and 6 and Fig. S5 also support this conclusion.

The second type of BCPs in trigonal Se and Te, providing bonds between neighboring helical chains in the crystal structure, has small charge densities ρ_b, $ρbCDD$⁠, and ELF values η(r_b) (Table III). Large positive values of the charge density Laplacian for the second type of BCPs denote the expulsion of the charge density from the area of this BCP.^35–37 The characteristics of the charge density distribution in areas of the second BCPs in trigonal Se and Te resemble that of the semimetal Sb (Fig. S4 of the supplementary material). Similarity with Sb is especially large for Te, which is often classified as a metalloid (Fig. S5 of the supplementary material). Therefore, we can conclude that two types of chemical bonding coexist in trigonal Se and Te: covalent and semimetallic.

The electronic band structures of trigonal Se and Te calculated via the WIEN2k program package using the PBE and rev-vdW-DF2 exchange–correlation functionals are presented in Figs. S6–S9 of the supplementary material. Calculations were performed using the optimized lattice parameters listed in Tables IV and V. The electronic bands for trigonal Se and Te, shown in Figs. S6 and S8 of the supplementary material, were calculated using the PBE exchange–correlation functional. They are similar to those shown in Figs. 3 and 4, except for the energy bandgap values. The bandgap for trigonal Se in Fig. S6 of the supplementary material is about 0.85 eV, which is noticeably less than the bandgap in Fig. 4, and the bandgap is absent for trigonal Te in Fig. S8 of the supplementary material. The bandgap for trigonal Te is also absent in Fig. S9 of the supplementary material, which was obtained using the rev-vdW-DF2 functional. In Fig. S8 of the supplementary material, in addition to the much lower bandgap (0.4 eV) for trigonal Se, various important features of the electronic bands related to the chiral symmetry of the Se crystal structure differ from those of Fig. 4.

Bader’s parameters of BCPs (ratio of main eigenvalues of Hesse matrix |λ_1,2|/λ₃, Laplacian value ∇²ρ_b, and density of charge in BCPs ρ_b) in trigonal Se and Te that were calculated using the mBJ exchange potential plus LDA correlations (Table III) numerically differ from those obtained using the PBE and rev-vdW-DF2 exchange–correlation functionals (see Tables IV and V). However, the found distinctions do not change the relation of Bader’s parameters for the first and second types of BCPs in trigonal Se and Te. Thus, the conclusion of the coexistence of covalent and semimetallic types of chemical bonding in trigonal Se and Te remains valid for all variants of exchange–correlation potentials and functionals used herein for the band structure calculations.

Analysis of the charge density distribution in trigonal Se reveals that all types of valence electrons, s-electrons with binding energies below 8 eV and lower (from −6 to −3 eV) and upper (from −3 to 0 eV) p-electrons, participate in bondings. Valence s-electrons and lower p-electrons nearly equally contribute to the charge density of BCPs located between nearest neighboring atoms within helical chains, while the lower and upper valence p-electrons are decisive for BCPs, providing cohesion between neighboring helical chains in trigonal Se and Te. A comparison of the charge density distribution features in the crystals of trigonal Se and Te with those of semiconducting Ge and semimetallic Sb as well as matching of Bader’s parameters of BCPs shows the coexistence of two types of chemical bondings in trigonal Se and Te: covalent and semimetallic. Inclusion of the nonlocal long-range vdW-DF2 correlation functional in the calculation scheme insignificantly changes the charge density distribution parameters in trigonal Se and Te, characterizing intra- and interchain bonding. The mBJ exchange potential plus LDA short-range correlations correctly describe the energy bandgaps as well as peculiarities of the electronic band structure of trigonal Se and Te related to their chiral crystal symmetry.

See the supplementary material for contributions of valence s- and p-electrons to DOS in Ge; characteristics of BCPs in Ge and Sb crystals; charge density distributions in the areas of BCPs in Ge, Sb, and Te crystals; and electronic band structures of trigonal Te and Se calculated with the use of PBE and rev-vdW-DF2 exchange–correlation functionals.

This work was conducted using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Mega-science Facilities at NRC “Kurchatov Institute,” http://ckp.nrcki.ru/. The authors are grateful to Eugene Maryin for help in preparing the illustrations.

The authors have no conflicts to disclose.

The data that support the findings of this study are available within the article and its supplementary material.