High-symmetry Configurations Research Articles

Low-frequency Raman active modes of twisted bilayer MoS2

We study the low-frequency Raman active modes of twisted bilayer MoS2 for several twist angles using a force-field approach and a parametrized bond polarizability model. We show that twist angles near high-symmetry stacking configurations exhibit stacking frustration that leads to significant buckling of the moiré superlattice. We find that atomic relaxation due to the twist is of prime importance. The periodic displacement of the Mo atoms shows the realization of a soliton network, and in turn, leads to the emergence of a number of frequency modes not seen in the high-symmetry stacking systems. Some of the modes are only seen in the XZ Raman polarization setup while others are seen in the XY setup. The symmetry of the normal modes, and how this affects the Raman tensors is examined in detail.

Cascading symmetry constraint during machine learning-enabled structural search for sulfur-induced Cu(111)-(43×43) surface reconstruction.

In this work, we investigate how exploiting symmetry when creating and modifying structural models may speed up global atomistic structure optimization. We propose a search strategy in which models start from high symmetry configurations and then gradually evolve into lower symmetry models. The algorithm is named cascading symmetry search and is shown to be highly efficient for a number of known surface reconstructions. We use our method for the sulfur-induced Cu (111) (43×43) surface reconstruction for which we identify a new highly stable structure that conforms with the experimental evidence.

Precise structure and energy of group 6 transition metal dichalcogenide homo- and heterobilayers in high-symmetry configurations

Two-dimensional group 6 transition metal dichalcogenide (2D TMDC) bilayers show various high-symmetry stacking configurations, which have also been observed in extended domains formed in their twisted homo- and heterobilayers. The interlayer energy varies for these stacking configurations, and the energy differences determine the relative size of the stacking domains. Therefore, the precise prediction of the composition- and stacking-dependent interlayer energy is crucial to model the domain structure of 2D TMDCs in their twisted bilayer homo- and heterostructures. For the validation of approximate methods that are necessary to tackle these systems encompassing thousands of atoms precise reference data is still lacking. Here, we employ the random phase approximation (RPA) on previously validated SCAN-rVV10 geometries to obtain interaction energies of state-of-the-art accuracy on the six high-symmetry stacking configurations ( Hhh , HhM , HhX , Rhh , RhM , and RhX ) of MX2 (M = Mo, W; X = S, Se) bilayers and compare them with the dispersion-corrected density-functional theory (DFT) functionals Perdew–Burke–Ernzerhof (PBE)+D3(BJ), PBE-rVV10L, and SCAN-rVV10. We identify SCAN-rVV10 as most reliable DFT variant with an average deviation of 1.2 meV/atom in relative energies from the RPA reference, and a root mean squared error of less than 2 meV/atom for interlayer interaction energies. We find interlayer distances obtained by PBE+D3(BJ) as being too short, with an impact on the electronic structure, resulting in the incorrect prediction of the band gap character in some cases. A further result of this work is the significant lowering of the interlayer energy and increasing of the interlayer distance in the high-energy stacking configurations. These stackings can be accessible via shear strain and promote exfoliation.

The Pseudo Jahn-Teller effect and NBO analysis for untangling the symmetry breaking in the planar configurations of M2X4+ (M = Si, Ge and X = Cl, Br, I): effect on electronic structure and chemical properties.

The Pseudo Jahn- Teller effect is a significant tool for evaluating molecular distortion and symmetry breaking. The PJT effect associated with NBO analysis can be a powerful method for studying the structural properties variations arising from D2h → C2h distortions. The theoretical studies on Si2X4+ and Ge2X4+ radical cations have been rare. The calculations have shown that C2h non-planar structures are more stable than planar structures with D2h symmetry. The [Formula: see text] PJTE problem of M2X4+ compounds is a result of the coupling between the ground B3u state and the exited B1u state in the Qb2g direction causes. Also, the difference in M and X atoms can affect the PJT instability of compounds. The findings of this work show that the energy gap between the ground and excited states that have D2h symmetry decreases from M2Cl4+ to M2I4+ and increases from Si2X4+ to Ge2X4+. In fact, there is a significant relationship between instability of high-symmetry configurations, geometric parameters, electron delocalization, chemical hardness, electronegativity, electrophilicity index, and PJT stabilization energies. These results may serve to evaluate the distortion of similar systems. The structures of Si2X4+ and Ge2X4+ are optimized by LC-BLYP, M06-2X, and B3LYP methods with def2-TZVPP basis set in GAMESS software. The details of the excited states of compounds are studied by the TD-DFT method. NBO analysis for planar and non-planar structures is carried out at B3LYP/def2-TZVPP level by the NBO 5. G program that demonstrates HOMO, LUMO, ED, bonding and antibonding orbital occupancies, bond order, and E2.

Intrinsic electronic and optical properties of monolayer and Bilayer CuI under many-body effects

Using the combination of density functional theory and many-body perturbation theory, we comprehensively investigate the electronic and optical properties of monolayer and bilayer CuI. For the bilayer, we build three high-symmetry stacking configurations i.e. AA, AB, and AC. The results reveal that monolayer CuI is a semiconductor with a direct band gap of 4.03 eV while the band gap of bilayer CuI is 1.62, 1.69, and 1.62 eV for AA, AB, and AC stacking configurations, respectively. The optical responses achieved from the Bethe–Salpeter equation are equal for light polarized along the x- and y-directions. The exciton binding energy of monolayer CuI is calculated to be 0.99 eV while it is 0.57, 0.59, and 0.56 eV for AA, AB, and AC stacking configurations, respectively. The maximum extinction coefficient of monolayer CuI is located in the ultraviolet area. Although, for the bilayers, the maximum extinction coefficient appears in the middle of the visible area. Overall, it is understood that the optoelectronic performance of the CuI binary compound improves from monolayer to bilayer. Because the AA configuration is more stable than the other two configurations, it is predicted to be the most suitable candidate for optoelectronic applications such as solar cells, photodetectors, and light-emitting diodes.

Four modifications of the Jahn-Teller effects. The problem of observables: spin-orbit interaction, tunneling splitting, and orientational polarization of solids.

In a semi-review paper, we first show that Landau's fundamental idea of the origin of spontaneous symmetry breaking (SSB) in atomic matter due to electronic degeneracy, termed the Jahn-Teller effect (JTE) and further developed into the pseudo-JTE (PJTE), was appended recently with two more modifications, the hidden JTE (h-JTE) and hidden PJTE (h-PJTE). All four versions of JTEs are defined in the adiabatic approximation by their adiabatic potential energy surfaces (APES), which possess a common feature - the lack of a minimum in the high-symmetry configuration, thus confirming (and extending) the Landau idea of SSB. However, although serving as a qualitative indication of the SSB and consequent possible (virtual) properties of the system, the APES by themselves are not experimentally observable directly, and this important feature of JTEs is often ignored. Taking spin-orbit interaction as an example, we show that just perturbation of the APES does not reveal its observable reduction by the JTE, which emerges only after solving the Schrödinger equation with this APES. Following the multi-minimum nature of the latter, this leads to tunneling splitting of the vibrational states in the minima wells or over-the-barrier (hindered) rotations, resulting in novel properties, one of them being the reduction of the spin-orbit coupling. We demonstrate the methodology of solving such problems by using the example of electric field polarization of the BaTiO3 crystal, which leads to a novel effect: orientational polarization of solids.

Chirality in the Solid State: Chiral Crystal Structures in Chiral and Achiral Space Groups.

Chirality depends on particular symmetries. For crystal structures it describes the absence of mirror planes and inversion centers, and in addition to translations, only rotations are allowed as symmetry elements. However, chiral space groups have additional restrictions on the allowed screw rotations as a symmetry element, because they always appear in enantiomorphous pairs. This study classifies and distinguishes the chiral structures and space groups. Chirality is quantified using Hausdorff distances and continuous chirality measures and selected crystal structures are reported. Chirality is discussed for bulk solids and their surfaces. Moreover, the band structure, and thus, the density of states, is found to be affected by the same crystal parameters as chirality. However, it is independent of handedness. The Berry curvature, as a topological measure of the electronic structure, depends on the handedness but is not proof of chirality because it responds to the inversion of a structure. For molecules, optical circular dichroism is one of the most important measures for chirality. Thus, it is proposed in this study that the circular dichroism in the angular distribution of photoelectrons in high symmetry configurations can be used to distinguish the handedness of chiral solids and their surfaces.

Enhanced ferromagnetic ordering of a Mn trimer symmetrically and fully exposed on iridium-doped graphene

The structure and magnetism of a Mn trimer adsorbed on iridium-doped graphene are studied using density functional theory calculations. Our calculation results show that the Mn trimer prefers to locate on top of the Ir atom and forms a fully exposed high-symmetry configuration with large binding energy and hardness of rotation. The ferromagnetic ordering of the Mn trimer fully exposed the on iridium-doped graphene is enhanced five times compared to a free Mn trimer. Our study shows that the enhancement originates from the fixed long bond and the C 3v symmetry of the Mn trimer constrained by the iridium-doped graphene.

The structural and electronic richness of buckled honeycomb AsP bilayers.

The sixteen different high-symmetry stacking configurations in buckled honeycomb AsP bilayers were identified using block diagrams and studied through several high-level computations, including the adiabatic-connection fluctuation-dissipation theorem in the random phase approximation (ACFDT-RPA). The lowest-lying energy form is an AA-type stacking, which is an indirect bandgap semiconductor, according to the G0W0 approach. All bilayers are indirect wide bandgap semiconductors, except for two systems, a narrow bandgap semiconductor and one with metallic behavior. This study shows the richness of structural and electronic properties in AsP hetero-bilayers with configurations found over a broad spectrum of interlayer distances and bandgaps.

Pseudo Jahn−Teller Origin of the Proton Tunneling in Zundel Cation Containing Water Clusters

The pseudo Jahn–Teller (PJT) origin of the proton transfer barrier in the Zundel cation at different O–O distances and in an H5O2+(H2O)4 cluster is revealed by means of ab initio calculations of their electronic structures and the adiabatic potential energy curves. The vibronic constants in this approach were estimated by fitting the ab initio calculated adiabatic potential to its analytical expression. It is shown also that the high-symmetry nuclear configurations ofproton-centered water clusters of the type H+(H2O)n (n = 6, 4, 3) are unstable with respect to the low-symmetry nuclear distortions leading to forming the dihydronium cation H5O2+ and the appropriate number of water molecules: H2n + 1On+ → (n – 2)H2O + H5O2+. The reason for this instability and the subsequent decay is the PJT coupling between the ground and excited electronic states.

The Jahn–Teller and Pseudo-Jahn–Teller Effects: A Unique and Only Source of Spontaneous Symmetry Breaking in Atomic Matter

In a mostly review paper, we show that the important problem of symmetry, broken symmetry, and spontaneous broken symmetry of polyatomic systems is directly related to the Jahn–Teller (JT) and pseudo-Jahn–Teller (PJT) effects, including the hidden-JT and hidden-PJT effects, and these JT effects (JTEs) are the only source of spontaneous symmetry breaking in matter. They are directly related to the violation of the adiabatic approximation by the vibronic and other nonadiabatic couplings (jointly termed nonadiabaticity) in the interaction between the electrons and nuclei, which becomes significant in the presence of two or more degenerate or pseudodegenerate electronic states. In a generalization of this understanding of symmetry, we suggest an improved (quantum) definition of stereo-chemical polyatomic space configuration, in which, starting with their high-symmetry configuration, we separate all atomic systems into three distinguishable groups: (1) weak nonadiabaticity, stable high-symmetry configurations; (2) moderate-to-strong nonadiabaticity, unstable high-symmetry configurations, JTEs, spontaneous symmetry breaking (SSB); (3) very strong nonadiabaticity, stable distorted configurations. The JTEs, inherent to the second group of systems, produce a rich variety of novel properties, based on their multiminimum adiabatic potential energy surface (APES), leading to a short lifetime in the distorted configuration. We show the role of the Curie principle in the possibilities to observe the SSB in atomic matter, and mention briefly the revealed recently gamma of novel properties of matter in its interaction with external perturbation that occur due to the SSB, including ferroelectricity and orientational polarization, leading to enhanced permittivity and flexoelectricity.

The Connection between Pseudo-Jahn–Teller Effect and Electron Delocalization for Proving the Origin of Equilibrium Geometry of Nitrosyl Halides (Halides = Cl, Br, and I)

The theoretical calculations on triplet linear and non-linear structures of nitrosyl halides XNO (X = Cl, Br, and I) confirm that the origin of instability and symmetry breaking of XNO linear structures is a pseudo-Jahn–Teller effect (PJTE). However, the PJTE associated with electron delocalization may prove the stability of XNO non-linear structures best. The distortion of high-symmetry (C∞v) configurations in XNOs emerges from mixing the PJT with ground 3Σ– and the excited 3Π states in the Qπ direction, which consequently make the ground state unstable. The PJT problem is formulated in the form (3Σ– + 3Π) $$ \otimes $$ π. The energy difference between high-symmetry C∞v geometries and low-symmetry Cs geometries energy gap between the ground and excited states decreases from ClNO to INO. Also, ΔEPJT (i.e., the difference of energy between the highest occupied molecular orbital (HOMO; LPX), and the lowest unoccupied molecular orbital (LUMO; $$\pi _{{{\text{NO}}}}^{*}$$ )) plays an important role in the instability of structures. Electron delocalization decreases and PJT instability increases from ClNO to INO in high-symmetry C∞v geometries. According to the results of the canonical molecular orbital (CMO) analysis, the contribution of the LPX non-bonding orbitals in the vibronic coupling constant (F) increases from ClNO to INO. Also, the NBODEL results indicate that the instability of compounds increases from ClNO to INO. Because the primary force constant is positive, the curvature of ground electronic state configuration curves in adiabatic potential energy surfaces (APESs) decreases from ClNO to INO. The variations of the chemical hardness differences between the non-linear (Cs) and linear (C∞v) structures decrease from ClNO to INO and of chemical potential differences increase from ClNO to INO, justifying the variation trend of ΔEPJT. Furthermore, based on the obtained results, Δ[η(Cs) – η(C∞v)] obeys the maximum hardness principle.

Theoretical Study on Structural Stability, Growth Behavior and Photoelectron Spectroscopy of Copper-Doped Germanium Clusters CuGen−/0 (n = 4–13)

Studies on Cu-doped germanium clusters in the neutral and mono-anion states CuGen−/0 (n = 4–13) are carried out employing a double-hybrid density functional mPW2PLYP scheme. The global minimal structure, spectral property, HOMO–LUMO gap, and stability of CuGen−/0 (n = 4–13) were confirmed. The results showed that the global minimal structures of the anionic clusters are Cu-substituted for a Ge in the ground state of anionic Gen+1− with n ≤ 8, and Cu atom is encapsulated into germanium cages starting from n = 9. For neutral, it is also Cu-encapsulated into cage-like Ge framework with n ≥ 9. The spectra information including adiabatic electron affinity, vertical detachment energy and simulated photoelectron spectroscopy were presented. The HOMO–LUMO gap, atomization energy, and second energy difference for CuGen−/0 (n = 4–13) clusters along with NICS of CuGe12− were evaluated to examine the thermodynamic and chemical stability. The results revealed that anionic cluster CuGe12− with a high-symmetry endohedral Ih configuration possessed a perfect thermodynamic stability and chemical reactivity, making it possible as appropriate building block for new multi-functional semiconductor materials.

Effects of organic molecule adsorption and substrate on electronic structure of germanene

The development potential of germanene-based integrated electronics originates from its high carrier mobility and compatibility with the existing silicon-based and germanium-based semiconductor industry. However, the small band gap energy band (Dirac point) of germanene greatly impedes its application. Thus, it is necessary to open a sizeable band gap without reducing the carrier mobility for the application in logic circuits. In this study, the effects of organic molecule (benzene or hexafluorobenzene) adsorption and substrate on the atomic structures and electronic properties of germanene under an external electric field are investigated by using density functional theory calculations with van der Waals correction. For benzene/germanene and hexafluorobenzene/germanene systems, four different adsorption sites are considered, with the center of the organic molecules lying directly atop the upper or lower Ge atoms of germanene, in the Ge-Ge bridge center, and on the central hollow ring. Meanwhile, different molecular orientations at each adsorption site are also considered. Thus, there are eight high-symmetry adsorption configurations of the systems, respectively. According to the adsorption energy, we can determine the most stable atomic structures of the above systems. The results show that the organic molecule adsorption can induce the larger buckling height in germanene. Both the adsorption energy and interlayer distance indicate that there is no chemical bond between the organic molecules and germanene. Mulliken population analysis shows that a charge redistribution in the two sublattices in germanene exists since benzene is an electron donor molecule and hexafluorobenzene is an electron acceptor molecule. As a result, the benzene/germanene system exhibits a relatively large band gap (0.036 eV), while hexafluorobenzene/germanene system displays a small band gap (0.005 eV). Under external electric field, germanene with organic molecule adsorption can exhibit a wide range of linear tunable band gaps, which is merely determined by the strength of electric field regardless of its direction. The charge transfer among organic molecules and two sublattices in germanene gradually rises with the increasing the strength of electric field, resulting in the electron density around the sublattices in germanene unequally distributed. Thus, according to the tight-binding model, a larger band gap at the <i>K</i>-point is opened. When germanane (fully hydrogenated germanene HGeH) substrate is applied, the band gaps further widen, where the band gap of benzene/ germanene/germanane system can increase to 0.152 eV, and that of hexafluorobenzene/germanene/germanane system can reach 0.105 eV. The sizable band gap in germanene is created due to the symmetry of two sublattices in germanene destroyed by the dual effects of organic molecule adsorption and substrate. Note that both of organic molecules and substrate are found to non-covalently functionalize the germanene. As the strength of the negative electric field increases, the band gaps can be further modulated effectively. Surprisingly, the band gaps of the above systems can be closed, and reopened under a critical electric field. These features are attributed to the build-in electric field due to the interlayer charge transfer of the systems, which breaks the equivalence between the two sublattices of germanene. More importantly, the high carrier mobility in germanene is still retained to a large extent. These results provide effective and reversible routes to engineering the band gap of germanene for the applications of germanene to field-effect transistor and other nanoelectronic devices.

Manifestation of descent symmetry phenomena in tetrahedral structure of M42+ (M = P, As, Sb) analogues

Manifestation of descent symmetry phenomena induced by pseudo Jahn–Teller interactions is reported for the P4 dicationic tetrahedral structure and its As4 and Sb4 analogues. The symmetry descent phenomena in the dicationic tetrahedral structure for the series is caused by the pseudo Jahn–Teller effect (PJTE) where the unstable (high Td symmetry) configuration distorts to the equilibrium geometry with a lower, C2 symmetry. State averaging six low-lying electronic states via CASSCF(8,8)/cc-pVTZ–(PP) computations determined the adiabatic potential energy surfaces along the distorting normal coordinate. The (E(I) + A1 + E(II)) ⊗ e for the M42+ (M = P, As, Sb) series has been formulated accordingly. Subsequently, the coupling constants were estimated by fitting energies obtained from the PJTE equations. Moreover, to understand how removing or adding electrons affects the PJTE in the M42+ series, electronic configurations were analysed for M4(0,2+,4+) analogues in which the M4(0,4+) are stable in their tetrahedral structure.

Hg adatoms on graphene: A first-principles study

The interest in understanding the interaction between graphene and atoms that are adsorbed on its surface (adatoms) spans a wide range of research fields and applications, for example, to controllably change the properties of graphene in electronic devices or to detect those changes in graphene-based sensors. We present a density functional theory study of the interaction between graphene and Hg adatoms. Binding energy, electronic structure and electric field gradient (EFG) were calculated for various high-symmetry atomic configurations, from isolated adatoms to a continuous Hg monolayer. Hg as isolated adatom was found to be the most stable configuration, with a binding energy of 188 meV. Whereas isolated adatoms have a minor effect on the electronic structure of graphene (small acceptor effect), Hg monolayer configurations induce a metallic state, with the Fermi level moving well above the Dirac point (donor behavior). Based on the EFG calculated for the various configurations, we discuss how hyperfine techniques (perturbed angular correlation spectroscopy, in particular) can be used to experimentally study Hg adsorption on graphene.

Blue Phosphorene Bilayer Is a Two-Dimensional Metal and an Unambiguous Classification Scheme for Buckled Hexagonal Bilayers.

High-level first-principles computations predict blue phosphorene bilayer to be a two-dimensional metal. This structure has not been considered before and was identified by employing a block-diagram scheme that yields the complete set of five high-symmetry stacking configurations of buckled honeycomb layers, and allows their unambiguous classification. We show that all of these stacking configurations are stable or at least metastable both for blue phosphorene and gray arsenene bilayers. For blue phosphorene, the most stable stacking arrangement has not yet been reported, and surprisingly it is metallic, while the others are indirect band gap semiconductors. As it is impossible to interchange the stacking configurations by translations, all of them should be experimentally accessible via the transfer of monolayers. The metallic character of blue phosphorene bilayer is caused by its short interlayer distance of 3.01Å and offers the exceptional possibility to design single elemental all-phosphorus transistors.

1. Quantum chemical study of the Jahn – Teller effect on the distortions of XO2 (X = O, S, Se, Te) systems

Preliminary researches provided essential information about the optimized configuration of triatomic XO2 (X = O, S, Se, Te) systems, which were bent in the ground state and linear in their first excited state. The Jahn-Teller effects including the Jahn-Teller (JTE), the Renner-Teller effect (RTE), and the pseudo Jahn–Teller effect (PJTE) are parts of the most important reasons for structural distortion in the high symmetry configurations for each molecular system. This study purpose was to investigate the dependence between PJT parameters including the vibronic coupling constant values ​​(F), energy gap between reference states (Δ), and initial force constant (K0). In all above mentioned molecules, stability were increased with the reduction in the symmetry level. This increment was attributed to the PJTE. The vibronic coupling interaction between the ground (Σg), and the first excited states (Πu) through the PJTE problem (PJT (Σg+ Πu) × Πu) was because of the asymmetry and molecules bending phenomenon. The hardness difference parameter Δ[η (C2V) -η (D∞h)] decreases from O to Te (30.42, 22.66, 22.65, 22.58 Kcal/mol). These changes could explain the trend, which were observed for the D∞h → C2V conversion process.

Polymorphous nature of cubic halide perovskites

It has been long known that numerous halide and oxide perovskites can have non-ideal octahedra, showing tilting, rotation, and metal atom displacements. It has also been known that compounds that have at low temperatures a single structural motif ("monomorphous structures") could become disordered at higher temperatures, resulting in non-ideal octahedra as an entropy effect. What is shown here is that in many cubic halide perovskites and some oxides compounds a distribution of different low-symmetry octahedra ("polymorphous networks") emerge already from the minimization of the systems internal energy, i.e., they represent the intrinsic, preferred low temperature pattern of chemical bonding. Thermal disorder effects build up at elevated temperatures on top of such low temperature polymorphous networks. Compared with the monomorphous counterparts, the polymorphous networks have lower predicted total energies (enhanced stability), larger band gaps and dielectric constants now dominated by the ionic part, and agrees much more closely with the observed pair distribution functions. The nominal cubic perovskites (Pm-3m) structure deduced from X-Ray diffraction is actually a macroscopically averaged, high symmetry configuration, which should not be used to model electronic properties, given that the latter reflect a low symmetry local configuration.

The Role of Pseudo Jahn–Teller Effect in Geometry and Electronic Parameters of Tetrahalodigermene Ge2X4 (X= Cl, Br, I)

The symmetry breaking of molecules depends on its context and origin. It is well shown the pseudo Jahn – Teller effect (PJTE) describes the origin of symmetry breaking of planar configurations. Based on the PJTE theory, the configurations, electronic parameters and conformational changes in molecular systems are predicted. Optimization of the D2h and C2h configurations of tetrahalodigermene Ge2X4 (X = Cl, Br, I) by ab initio calculations showed that the C2h configurations are more stable than planar ones. The different energy between high-symmetry D2h geometries and low-symmetry C2h geometries decrease from Ge2Cl4 to Ge2I4. Instability of the high-symmetry D2h configurations is arising from the PJT coupling of the ground Ag state and the exited B2g state in the Qb2g direction. The PJTE problem $$({{A}_{g}} + {{B}_{{2g}}}) \otimes {{b}_{{2g}}}$$ creates stable geometries of Ge2X4 (X = Cl, Br, I) analogs with C2h configurations. It is expected that when the energy gap between the desired states with D2h configurations decreases, the PJT stabilization energy (EPJT) increases. Despite this expectation, with increasing the energy gap from Ge2Cl4 to Ge2I4, EPJT decreases. According to the results of the canonical molecular orbital (CMO) analysis, the contributions of the b3u and b1u molecular orbitals in the vibronic coupling constant (F) decrease from Ge2Cl4 to Ge2I4. Consequently, assuming the primary force constant is positive, the negative curvatures of ground state electronic configuration curves in adiabatic potential energy surfaces (APESs) decreases from Ge2Cl4 to Ge2I4. The trend observed for the variations of the chemical hardness (η) of Ge2X4 analogs obey the maximum hardness principle.