Relativistic Density Functional Calculations Research Articles

Reactivity of Ts and At oxides and oxyhydrides with a gold surface from periodic DFT calculations.

Adsorption energies, Eads, of oxides and oxyhydrides of the superheavy element (SHEs) Ts and of its lighter homologue At on the gold surface are predicted on the basis of relativistic periodic density functional theory calculations via AMS BAND software. The following compounds were considered: MO, MO2, MOO, and MO(OH) (where M = At and Ts). The aim of this study is to support "one-atom-at-a-time" gas-phase chromatography experiments on reactivity/volatility of SHEs. The results obtained indicate that all the molecules investigated should interact fairly strongly with the gold surface, with those of Ts being more reactive than At ones. The similarity in the Eads values of all the considered At compounds would make it challenging to differentiate between them while measuring their adsorption enthalpies, given experimental uncertainty. However, the difference in Eads among Ts compounds is more pronounced, so that one should be able to differentiate between the species.

Front Cover: Fully Tin‐Coated Coinage Metal Ions: A Pincer‐Type Bis‐stannylene Ligand for Exclusive Tetrahedral Complexation (Chem. Eur. J. 27/2023)

A novel bis-stannylene pincer-type ligand has been prepared for the exclusive tetrahedral coordination of coinage metals. All compounds feature photoluminescent properties in the solid state. Detailed investigations by all-electron relativistic density functional theory calculations were performed. More information can be found in the Research Article by P. W. Roesky and co-workers (DOI: 10.1002/chem.202203583). Cover artwork by Alicia Schmidt.

A theoretical study of the adsorption behavior of superheavy 7p-elements and their compounds on a surface of gold in comparison with their lighter homologs.

Adsorption energies, Eads, of the 7th row superheavy elements (SHEs) Lv through Og, as well as of the homologous species of the 6th row elements Po through Rn on a gold surface are predicted on the basis of relativistic periodic density functional theory calculations via SCM BAND software. Since some of the elements can also form compounds such as hydrides and oxyhydrides under experimental conditions, the Eads values of the MH (M = Bi/Mc, Po/Lv, At/Ts and Rn/Og) and MOH (M = At/Ts and Rn/Og) molecules on a gold surface were also calculated. The aim of this study is to support "one-atom-at-a-time" gas-phase chromatography experiments on the reactivity/volatility of SHEs. The obtained results show that, in agreement with earlier predictions using somewhat different approaches and with experimental results on Hg, Cn and Rn, the adsorption strength of the elements on the Au(111) surface should follow the sequence: Hg > Fl > Og > Cn ≫ Rn, with the Eads values of less than 100 kJ mol-1. The other elements and their compounds under consideration should adsorb much more strongly on the gold surface with Eads values above 160 kJ mol-1, which should make them indistinguishable with respect to Eads in the chromatography column kept at room temperature and lower. However, with the further detector development, investigations of the chemical properties of these short-lived and less volatile SHEs and their compounds at high temperatures should be possible.

Superlattices of Gadolinium and Bismuth Based Thallium Dichalcogenides as Potential Magnetic Topological Insulators.

Using relativistic spin-polarized density functional theory calculations we investigate magnetism, electronic structure and topology of the ternary thallium gadolinium dichalcogenides TlGdZ2 (Z= Se and Te) as well as superlattices on their basis. We find TlGdZ2 to have an antiferromagnetic exchange coupling both within and between the Gd layers, which leads to frustration and a complex magnetic structure. The electronic structure calculations reveal both TlGdSe2 and TlGdTe2 to be topologically trivial semiconductors. However, as we show further, a three-dimensional (3D) magnetic topological insulator (TI) state can potentially be achieved by constructing superlattices of the TlGdZ2/(TlBiZ2)n type, in which structural units of TlGdZ2 are alternated with those of the isomorphic TlBiZ2 compounds, known to be non-magnetic 3D TIs. Our results suggest a new approach for achieving 3D magnetic TI phases in such superlattices which is applicable to a large family of thallium rare-earth dichalcogenides and is expected to yield a fertile and tunable playground for exotic topological physics.

Relativistic density functional investigation of the mono-lanthanum silicide clusters LaSin (n=1-6): geometries, electronic properties and IR spectra

The mono-lanthanum silicide clusters LaSin (n=1-6) have been studied adopting the relativistic density functional calculation with generalized gradient approximation. Considering different spin configurations, we calculated and discussed the equilibrium geometries, charge populations, the HOMO-LUMO gaps, as well as infrared (IR) absorption spectra of LaSin (n=1-6) clusters. It is found that: the lowest-lying LaSin (n=1-6) clusters basically maintain a similar framework to the low-lying Sin+1 clusters, and the La atoms prefer the surface sites. The relative stabilities are investigated based on the calculation of fragmentation energies and, showing that LaSi2, LaSi4, and LaSi5 clusters have enhanced stabilities. Charge populations analysis shows that the charges transfer from La atom to Sin framework and the La atom acts as an electron donor. HOMO-LUMO gaps indicate that LaSi2 and LaSi5 clusters have higher chemical stabilities. IR absorption spectrum and vibrational mode analysis show that the highest frequency absorption peaks all correspond to the breathing mode of the silicon framework, and the characteristic infrared absorption peaks caused by La atom vibration, except for LaSi dimer, all appeared in the low-frequency region.

Current density functional framework for spin–orbit coupling

Relativistic two-component density functional calculations are carried out in a non-collinear formalism to describe spin-orbit interactions, where the exchange-correlation functional is constructed as a generalization of the non-relativistic density functional approximation. Contrary to non-relativistic density functional theory (DFT), spin-orbit coupling, however, leads to a non-vanishing paramagnetic current density. Density functionals depending on the kinetic energy density, such as meta-generalized gradient approximations, should therefore be constructed in the framework of current DFT (CDFT). The latter has previously exclusively been used in the regime of strong magnetic fields. Herein, we present a consistent CDFT approach for relativistic DFT, including spin-orbit coupling. Furthermore, we assess the importance of the current density terms for ground-state energies, excitation energies, nuclear magnetic resonance shielding, and spin-spin coupling constants, as well as hyperfine coupling constants, Δg-shifts, and the nuclear quadrupole interaction tensor in electron paramagnetic resonance (EPR) spectroscopy. The most notable changes are found for EPR properties. The impact of the current-dependent terms rises with the number of unpaired electrons, and consequently, the EPR properties are more sensitive toward CDFT. Considerable changes are observed for the strongly constrained and appropriately normed functionals, as well as the B97M family and TASK. The current density terms are less important when exact exchange is incorporated. At the same time, the current-dependent kernel ensures the stability of response calculations in all cases. We, therefore, strongly recommend to use the framework of CDFT for self-consistent spin-orbit calculations.

BinPo: An open-source code to compute the band structure of two-dimensional electron systems

We introduce BinPo, an open-source Python code to compute electronic properties of two-dimensional electron systems. A bulk tight binding Hamiltonian is constructed from relativistic density functional theory calculations represented in the basis of maximally localized Wannier functions. BinPo has a Schrödinger-Poisson solver, integrating an electric field-dependent relative permittivity to obtain self-consistently the confining electrostatic potential energy term in the derived tight binding slab system. The band structure, energy slices, and other properties, along with different projections and orientations can be computed. High resolution and publishable figures of the simulations can be generated. In BinPo, priority has been given to ease-of-use, efficiency, readability and modularity, therefore becoming suitable to produce reliable electronic structures simulations at low computational cost. Along with the code itself, we provide files from first-principles calculations for some materials, instructions of use, and detailed examples of its wide range of capabilities. The code was developed with a focus on the ABO3 perovskite structure-based systems, such as SrTiO3 and KTaO3, because of their increasing impact in the materials community. Some features, such as the projection onto orbital states, are restricted to calculations using the relevant t2g orbitals for this family of materials, yet it is possible to include more elements in the basis for the band structure determination of other systems. The use of a relativistic approach allows for the inspection of the role of spin-orbit coupling and the resulting Rashba effect on the systems. We detail the approaches used in the code, so that it can be further exploited and adapted to other problems, such as adding new materials and functionalities which can strength the initial code scopes. Program summaryProgram title: BinPoCPC Library link to program files:https://doi.org/10.17632/zk4xy7vfy4.1Developer's repository link:https://github.com/emanuelm33/BinPoCode Ocean capsule:https://codeocean.com/capsule/0987578Licensing provisions: GPLv3Programming language: PythonExternal routines/libraries: NumPy, SciPy, Matplotlib, ASE.Nature of problem: Compute the band structure and other electronic properties in two-dimensional electron systems.Solution method: Construction of a slab Hamiltonian from one built using a basis of maximally localized Wannier functions. Solve the Schrödinger-Poisson scheme to get the self-consistent potential energy along the slab. Use this potential to compute the band structure and additional properties by solving the Schrödinger equation.Additional comments including restrictions and unusual features: Other features included are the visualization of the self-consistent process and final solutions, band structure and energy slices with different projections. High resolution and customizable figures can be obtained in all cases.

Chemical Bonding in the Intermetallic Compounds LaBeGe and ThBeGe

AbstractThe bonding situation in the intermetallic compounds LaBeGe and ThBeGe were investigated using the wavefunctions generated from the all‐electron relativistic density functional calculations. Following the quantum theory of atoms in molecules (QTAIM) the atomic basins were determined from the electron density and the effective charges computed. These were also compared to corresponding charges for the procrystal density. The bonding was analyzed utilizing the delocalization indices (DIs) computed from the overlap integrals over the QTAIM basins. The analysis was extended by the evaluation of the electron localizability indicator (ELI‐D) and completed by the calculation of DIs for the ELI‐D basins.

Spin and orbital magnetism of exohedral fullerene doped with single transition metal atom (Sc-Ni): A relativistic density functional theory study

Relativistic density functional theory calculations have been implemented to study the structural configurations, electronic and magnetic properties of exohedral 3d transition metal atoms doping on C60 fullerene (TM-C60) by a full potential local orbital method. The calculated binding energies revealed that the TM-C60 molecules are energetically stable in all adsorption sites. We also found that the TM-doped exohedral C60 complexes are more reactive than pure C60 fullerene. The spin and orbital magnetic moments were also determined. We have indicated that in all of the five adsorption sites, exohedral TM-C60 molecules (except for Ni-C60) are magnetized. Our magnetic calculations show that the total spin magnetic moments of TM-C60 molecules mainly come from the TM atoms. We found that the spin magnetic moments of adsorbed TM atoms in exohedral doping C60 fullerene from Sc to Mn (HH adsorption site) and for Sc to Cr (HP and P adsorption sites) are greater than their free states. Our calculations suggest that exohedral doping 3d TM-C60 complexes can almost maintain spin magnetic moments. We have indicated that the magnetic properties of exohedral TM-C60 molecules change in the presence of spin–orbit coupling. Significant orbital contributions to magnetic moments are recognized in these configurations while the spin moments are unaffected. We found that Co in exohedral doping of C60 fullerene shows a robust orbital moment in all the five adsorption sites.

Theoretical Investigation of Catalytic Water Splitting by the Arene-Anchored Actinide Complexes.

Actinide complexes, which could enable the electrocatalytic H2O reduction, are not well documented because of the fact that actinide-containing catalysts are precluded by extremely stable actinyl species. Herein, by using relativistic density functional theory calculations, the arene-anchored trivalent actinide complexes (Me,MeArO)3ArAn (marked as [AnL]) with desirable electron transport between metal and ligand arene are investigated for H2 production. The metal center is changed from Ac to Pu. Electron-spin density calculations reveal a two-electron oxidative process (involving high-valent intermediates) for complexes [AnL] (An = P-Pu) along the catalytic pathway. The electrons are provided by both the actinide metal and the arene ring of ligand. This is comparable to the previously reported uranium catalyst (Ad,MeArO)3mesU (Ad = adamantine and mes = mesitylene). From the thermodynamic and kinetic perspectives, [PaL] offers appreciably lower reaction energies for the overall catalytic cycle than other actinide complexes. Thus, the protactinium complex tends to be the most reactive for H2O reduction to produce H2 and has the advantage of its experimental accessibility.

Design of High‐Temperature Syntheses on the Example of the Heavy‐Atom Cluster Compound Sn[PtBi6I12

AbstractInvestigations into potential topological materials yielded the new subiodide Sn[PtBi6I12]. The combination of thermal analyses with phase analyses of the products of isothermal ex situ syntheses allowed the establishment of a complex high‐temperature synthesis protocol for the crystal growth of the target phase despite the lack of knowledge of the quaternary phase diagram. A special challenge was to prevent the formation of competing compounds of the solid solution series (Bi2xSn1–3x)[PtBi6I12] with x≠0. Sn[PtBi6I12] crystallizes, isostructural to Pb[PtBi6I12], in the rhombohedral space group R with lattice parameters a=1583.2(2) pm and c=1089.70(10) pm. The compound consists of cuboctahedral [PtBi6I12]2− clusters and Sn2+ cations in an octahedral coordination between the trigonal faces of two cluster units, thereby concatenating them into infinite linear chains. The chains are connected via Bi I inter‐cluster bridges, creating a high‐entropy variant of the NaCl structure type. Sn[PtBi6I12] is a semiconductor with an experimental bandgap of 0.8(1) eV. Fully relativistic density functional theory calculations including an implementation of the bifunctional formalism for the exchange energy indicate a topologically trivial bandgap of 0.81 eV between bands that are dominated by contributions of bismuth and iodine.

Bonding situation in isolable silver(I) carbonyl complexes of the Scorpionates.

The bonding situation of Ag(I)CO complexes having a Scorpionate ligand directly attached to the transition metal has been analyzed in detail by means of relativistic density functional theory calculations. To this end, different experimentally characterized complexes together with other representative species have been considered to rationalize the observed shift of the corresponding ν(CO) stretching frequencies and the influence of the substituents in the Scorpionate ligand. With the help of the energy decomposition analysis method combined with the natural orbital for chemical valence it is found that the main contribution to the bonding comes from the electrostatic attractions between the LAg(I) and CO fragments. Despite that, the LAg→CO π-backdonation is also significant in these species as well as in related LCu(I)CO complexes.

Exploring the heavy transition metal trihalide family: Two-dimensional magnetic materials with tunable band gap, huge magnetic anisotropy, and high-temperature magnetic ordering

In light of the forecast of recent topics of magnetic van der Waals materials [Burch et al., Nature (London) 563, 47 (2018); Gong and Zhang, Science 363, 706 (2019)], it was encouraged to explore new magnetic two-dimensional (2D) materials that contain heavy atoms so as to elevate the magnetic transition temperature. Here, relativistic density functional theory calculations, combined with ab initio molecular dynamics simulations, lattice dynamical calculations, and classical Monte Carlo (MC) simulations, were performed on transition metal trihalide (${\mathrm{TMI}}_{3}$; TM = La, Hf, Ta, W, Re, Os, Ir, Pt) layers, for which heavy $5d$-block and $5p$-block elemental atoms were considered owing to their strong spin-orbit coupling. We identified that these newly designed layers are energetically, dynamically, thermally, and thermomagnetically stable at room temperature. Our investigations revealed that most of them possess the intrinsic semiconducting characteristic with tunable band gap, diverse magnetic properties (ferromagnetic, antiferromagnetic, and nonferromagnetic orders), and large magnetic anisotropy energy (generally larger than 1.9 meV/f.u. and up to 29 meV/f.u.). Based on MC simulations, we proposed that ${\mathrm{TaI}}_{3}$, ${\mathrm{OsI}}_{3}$, and ${\mathrm{PtI}}_{3}$ layers have high Curie temperatures of 575, 225, and 170 K, respectively, which are much larger than those of most magnetic 2D materials such as the notable ${\mathrm{CrI}}_{3}$ (45 K), ${\mathrm{CrGeTe}}_{3}$ (28 K), and ${\mathrm{Fe}}_{3}{\mathrm{GeTe}}_{2}$ layers (130 K). Importantly, our calculations evidenced the high stability of the ${\mathrm{TaI}}_{3}$ layer under ambient conditions, large magnetic moment of $2\phantom{\rule{0.28em}{0ex}}{\ensuremath{\mu}}_{B}$, large magnetic anisotropy energy of $\ensuremath{-}7.3$ meV/f.u., and high Curie temperature of 575 K, making it possible to experimentally study this layer by more ex situ measurements and even to utilize it in future electronic devices.

Do nitrate ions preferentially bind to Ln/An ion in nuclear waste treatment? – Answers from DFT calculations

Stripping of radioactive minor actinides from lanthanides is an important process in the treatment of spent nuclear fuel (SNF). Almost in all the partitioning processes, the SNF is first introduced into the nitric acid medium before suitable reagents are added to selectively remove actinides. Therefore, in the present investigation an attempt has been made to understand the nature of interactions between the M3+ (M3+ = La, Nd, Pm, Sm, Eu, Gd, Ac, U, Np, Pu, Am, and Cm) and nitrate ions [(NO3)3]−3, metal–oxygen bond and stabilizing forces present in M(NO3)3 complexes using relativistic density functional theory calculations. Also, complexes with higher oxidation states [U(VI), Np(V), Pu(IV, VI)] along with nitrate ions are investigated. Optimized geometries of M(NO3)3 reveal that these metal ions readily form complexes with nitrate ions. Chemical reactivity is addressed from frontier molecular orbital analysis while the nature of metal–oxygen bonds is examined using Mayer and Nalewajski – Mrozek bond order analyses. Mulliken population analysis portrays the donor–acceptor interactions in these M(NO3)3 complexes. Results show that the ionic interactions persist in these complexes and QTAIM analysis endorses the ionic character of these complexes. Energy decomposition analysis gives an idea about the various energy components that stabilize the complexes. The water solvent environment does vary the various parameters involved in the complexation process. Overall, a comprehensive quantum chemical study is taken up to understand the nature of interactions present between M3+ and [(NO3)3]−3 ions to improve our understanding of the partitioning processes.

CO2 Cleavage Reaction Driven by Alkylidyne Complexes of Group 6 Metals and Uranium: A Density Functional Theory Study on Energetics, Reaction Mechanism, and Structural/Bonding Properties.

Designing novel catalysts is essential for the efficient conversion of metal alkylidyne into metal oxo ketene complexes in the presence of CO2, which to some extent resolves the environmental concerns of the ever-increasing carbon emission. In this regard, a series of metal alkylidyne complexes, [b-ONO]M≡CCH3(THF)2 ([b-ONO] = {(C6H4[C(CF3)2O])2N}3-; M = Cr, Mo, W, and U), have been comprehensively studied by relativistic density functional theory calculations. The calculated thermodynamics and kinetics unravel that the tungsten complex is capable of catalyzing the CO2 cleavage reaction, agreeing with the experimental findings for its analogue. Interestingly, the uranium complex shows superior catalytic performance because of the associated considerably lower energy barrier and larger reaction rate constant. The M≡C moiety in the complexes turns out to be the active site for the [2 + 2] cyclic addition. In contrast, complexes of Cr and Mo could not offer good catalytic performance. Along the reaction coordinate, the M-C (M = Cr, Mo, W, and U) bond regularly transforms from triple to double to single bonds; concomitantly, the newly formed M-O in the product is identified to have a triple-bond character. The catalytic reactions have been extensively explained and addressed by geometric/electronic structures and bonding analyses.

Relativity as a Synthesis Design Principle: A Comparative Study of [3 + 2] Cycloaddition of Technetium(VII) and Rhenium(VII) Trioxo Complexes with Olefins.

The difference in [3 + 2] cycloaddition reactivity between fac-[MO3(tacn)]+ (M = Re, 99Tc; tacn = 1,4,7-triazacyclononane) complexes has been reexamined with a selection of unsaturated substrates including sodium 4-vinylbenzenesulfonate, norbornene, 2-butyne, and 2-methyl-3-butyn-2-ol (2MByOH). None of the substrates was found to react with the Re cation in water at room temperature, whereas the 99Tc reagent cleanly yielded the [3 + 2] cycloadducts. Interestingly, a bis-adduct was obtained as the sole product for 2MByOH, reflecting the high reactivity of a 99TcO-enediolato monoadduct. On the basis of scalar relativistic and nonrelativistic density functional theory calculations of the reaction pathways, the dramatic difference in reactivity between the two metals has now been substantially attributed to differences in relativistic effects, which are much larger for the 5d metal. Furthermore, scalar-relativistic ΔG values were found to decrease along the series propene > norbornene > 2-butyne > dimethylketene, indicating major variations in the thermodynamic driving force as a function of the unsaturated substrate. The suggestion is made that scalar-relativistic effects, consisting of greater destabilization of the valence electrons of the 5d elements compared with those of the 4d elements, be viewed as a new design principle for novel 99mTc/Re radiopharmaceuticals, as well as more generally in heavy-element coordination chemistry.

Reactivity of the Superheavy Element 115, Mc, and Its Lighter Homologue, Bi, with Respect to Gold and Hydroxylated Quartz Surfaces from Periodic Relativistic DFT Calculations: A Comparison with Element 113, Nh.

Adsorption energies (Eads) of the superheavy element (SHE) Mc, its lighter homologue (Bi), as well as of another superheavy element Nh and some lighter homologues of SHEs on gold and hydroxylated quartz surfaces are predicted via periodic relativistic density functional theory calculations. The aim of this study is to support "one-atom-at-a-time" gas-phase chromatography experiments that are examining the reactivity and volatility of Mc. The obtained Eads values of the Bi and Mc atoms on the Au(111) surface are >200 kJ/mol. On the hydroxylated quartz surface, Mc should adsorb with a minimal energy of 58 kJ/mol. On both types of surfaces, Eads(Mc) should be ∼100 kJ/mol smaller than Eads(Bi) due to strong relativistic effects on its valence 7p electrons. A comparison with other SHEs under investigation shows that Mc should adsorb on gold more strongly than Cn, Nh, and Fl, while on quartz, Mc should adsorb like Nh, with both of them absorbing more strongly than volatile Cn and Fl. The highest reactivity of Mc in the row of the 7p elements is caused by the largest orbital and relativistic destabilization and expansion of the 7p3/2 atomic orbital. Using the calculated Eads, the distribution of the Nh and Mc events in the gas-phase chromatography column with quartz and gold-plated detectors is predicted via Monte Carlo simulations. As a result, Mc atoms should be almost 100% adsorbed in the first section of the chromatography column on quartz, while a few atoms of Nh can reach the second part of the column with gold-plated detectors.

Photoelectron angular distribution studies for two spin–orbit-split components of Xe 3d subshell: a critical comparison between theory and experiment

The photoelectron angular distribution asymmetry parameters β of the Xe 3d subshell were investigated using an x-ray free-electron laser (XFEL) at photon energies of 750 and 800 eV. Owing to the perfect polarization of the XFEL and two-dimensional momentum imaging capability of our velocity map imaging spectrometer, we determined the β values with high accuracy. The β values were also investigated based on relativistic time-dependent density functional theory calculations of up to 900 eV of photon energies. By comparing all the available experimental results including our data with the most reliable theories on the photon energy dependence of the β parameters, serious differences are noted between the experiments and theories. Further studies on resolving this difference will provide new insight into the photoionization processes of the deep inner shells.

Complete mapping of magnetic anisotropy for prototype Ising van der Waals FePS3

Several Ising-type magnetic van der Waals (vdW) materials exhibit stable magnetic ground states. Despite these clear experimental demonstrations, a complete theoretical and microscopic understanding of their magnetic anisotropy is still lacking. In particular, the validity limit of identifying their one-dimensional Ising nature has remained uninvestigated in a quantitative way. Here we performed the complete mapping of magnetic anisotropy for a prototypical Ising vdW magnet FePS3 for the first time. Combining torque magnetometry measurements with their magnetostatic model analysis and the relativistic density functional total energy calculations, we successfully constructed the three-dimensional mappings of the magnetic anisotropy in terms of magnetic torque and energy. The results not only quantitatively confirm that the easy axis is perpendicular to the ab plane, but also reveal the anisotropies within the ab, ac, and bc planes. Our approach can be applied to the detailed quantitative study of magnetism in vdW materials.

Momentum-resolved spin splitting in Mn-doped trivial CdTe and topological HgTe semiconductors

Exchange coupling between localized spins and band or topological states accounts for giant magnetotransport and magnetooptical effects as well as determines spin-spin interactions in magnetic insulators and semiconductors. However, even in archetypical dilute magnetic semiconductors such as Cd$_{1-x}$Mn$_x$Te and Hg$_{1-x}$Mn$_x$Te the evolution of this coupling with the wave vector is not understood. A series of experiments have demonstrated that exchange-induced splitting of magnetooptical spectra of Cd$_{1-x}$Mn$_x$Te and Zn$_{1-x}$Mn$_x$Te at the L points of the Brillouin zone is, in contradiction to the existing theories, more than one order of magnitude smaller compared to its value at the zone center and can show an unexpected sign of the effective Land\'e factors. The origin of these findings we elucidate quantitatively by combining: (i) relativistic first-principles density functional calculations; (ii) a tight-binding approach that takes carefully into account k-dependence of the potential and kinetic sp-d exchange interactions; (iii) a theory of magnetic circular dichroism (MCD) for $E_1$ and $E_1$ + $\Delta_1$ optical transitions, developed here within the envelope function $kp$ formalism for the L point of the Brillouin zone in zinc-blende crystals. This combination of methods leads to the conclusion that the physics of MCD at the boundary of the Brillouin zone is strongly affected by the strength of two relativistic effects in particular compounds: (i) the mass-velocity term that controls the distance of the conduction band at the L point to the upper Hubbard band of Mn ions and, thus, a relative magnitude and sign of the exchange splittings in the conduction and valence bands; (ii) the spin-momentum locking by spin-orbit coupling that reduces exchange splitting depending on the orientation of particular L valleys with respect to the magnetization direction.