The Electronic Structure and Stability of Germanium Tubes Ge30H12 and Ge33H12

The geometries of non-tetrahedral and ultrastable silicon and germanium nanocrystals X18H12 and X19H12 (X = Si, Ge) have recently been predicted for the development of cluster-based nanomaterials for energy and microengineering purposes. To further explore the possibility of larger Ge clusters, we investigated in this work the molecular and electronic structure of the germanium tube Ge30H12, composed of six parallel, planar hexagons using DFT calculations. Insertion of Ge atoms at the center of three inner hexagons of Ge30H12 leads to a Ge33H12 tube, which is also an energy minimum structure. The electronic structure and molecular orbital shapes of these tubes can be predicted by the wavefunctions of a particle on a hollow cylinder model and a cylinder model. Different aromaticity indices including PDI, Iring, ING, MCI, and INB, as well as the electron localization function (ELF) were calculated to evaluate the electron delocalization and the aromaticity of the Ge tubes considered.


Abstract
Geometries of non-tetrahedral and ultrastable silicon and germanium nanocrystals X18H12 and X19H12 (X = Si, Ge) have recently been predicted for the development of cluster-based nanomaterials for energy and microengineering purposes. To further explore the possibility of larger Ge clusters, we investigated in this work the molecular and electronic structure of the germanium tube Ge30H12, composed of six parallel, planar hexagons using DFT calculations.
Insertion of Ge atoms at the center of three inner hexagons of Ge30H12 leads to a Ge33H12 tube, which is also an energy minimum structure. The electronic structure and molecular orbital shapes of these tubes can be predicted by the wavefunctions of a particle on a hollow cylinder model and a cylinder model. Different aromaticity indices including PDI, Iring, ING, MCI, and INB, as well as the electron localization function (ELF) were calculated to evaluate the electron delocalization and the aromaticity of the Ge tubes considered.

Introduction
Since the discovery of benzene in 1825 by Faraday, aromaticity was introduced as the key concept in the realm of organic chemistry to describe the stability, molecular structure and reactivity of many organic molecules. [1][2][3][4] Due to the persistent ambiguity in the definition of aromaticity and its non-observable nature, it has been a subject of continuing debate among chemists. Despite the controversy in the definition of aromaticity, aromaticity still remains as a useful, even fundamental, concept for the interpretation of structural and chemical properties of a variety of classes of organic molecules. 5 Moreover, the discovery of aromatic inorganic/organometallic clusters promises to expand its scope of application. [6][7][8][9][10][11] Many indices have been proposed in the literature for direct and quantitative measurement of aromaticity, which can be categorized into four main groups, including energy, structure, electron delocalization, and magnetic based indices. 12,13 Accordingly, it has been recommended that a set of indices, rather than a sole index, should employed for characterization of aromatic compounds. 14 The final results would be more reliable if different aromaticity indices provided consistent results for a set of compounds. The necessity of different aromaticity indices has been discussed in several recent reviews. 1,5,[15][16][17][18] Triggered by benzene discovery, chemists have long been seeking to expand the concept of aromaticity to inorganic molecules, especially to analogous silicon compounds. 19,20 Regarding this interest, many attempts were made to synthesize and predict stable aromatic silicones. For example, Scheschkewitz et al. synthesized a silicon counterpart to benzene, with bulky organic substituents instead of hydrogen, showing high stability based on electron delocalization and aromaticity. 21 Silicon and germanium nanocrystals and nanostructures have drawn the attention of researchers for their potential application in energy conversion, energy storage, light-emitting diode, and memory devices. Nowadays solar cells based on silicon nanocrystals find their way into the energy production market and germanium nanostructures are being studied as high-capacity anode materials for Li-ion batteries. [22][23][24][25][26] Recently, Vach 27 predicted the aromaticity of the electron-deficient tubular Si19H12. The predicted Si19H12 contains three parallel and planar hexagons and one additional Si atom located in the middle hexagon. The central Si atom is multicoordinated and Si-Si bonds are characterized by electron deficient properties. Electron deficiency keeps electrons delocalized through the whole structure, and ultimately provides aromatic character to the system. Vach's calculations showed that Si19H12 is thermodynamically more stable than Si18H12 and such a stability is related to a strong electron delocalization. 27 Using electronic and magnetic criteria, it was shown that the overcoordinated Si19H12 has electron deficient bonds and is more aromatic than benzene. Due to electron-deficient aromaticity, electronic and optical properties of Si19H12 are totally different from the Si18H12 nanotube, 28 which may result in light-harvesting applications such as solar cells and optoelectronic devices. 29 It was also shown that Ge18H12 nanocrystals, similar to Si18H12, could be further stabilized by insertion of one germanium atom into the center of the middle hexagon, and exhibit some electron-deficient aromaticity. Our previous results showed that the electrondeficient aromaticity concept could be extended to Ge based materials. 30 The presence of such aromaticity is intriguing and has stimulated us to further explore the possibility of larger Ge clusters being stabilized by this bonding phenomenon. 31 It is likely that the feature observed in tubular silicon and germanium species arises from a type of aromaticity that is called tubular or cylindrical aromaticity, as recently reported by some of authors of the present study. 32,33 This aromaticity is explained by the hollow cylinder model (HCM) in which the Schrödinger equation can be solved for a particle moving in a hollow cylindrical box. 34 Accordingly, the shape of the eigenstates obtained from the HCM is quasiidentical with that of molecular orbitals calculated from quantum chemical methods.
In this context, we set out to examine the electronic structures of two extended tubular germanium tubes including Ge30H12 and Ge33H12 making use of the hollow cylinder model (HCM) and cylinder model (CM), respectively. Also, different electronic and magnetic indices are analyzed to further shed light on the aromaticity of these compounds. The proposed germanium tubes in this study and their silicon counterparts are expected to play a role in the following generation of energy storage and energy conversion devices.

Computational Methods
Standard electronic structure calculations and geometry optimizations are performed within the framework of density functional theory (DFT) using hybrid B3LYP functional and the 6-31G(d) basis set [35][36][37] with the aid of the Gaussian 09 program. 38 Harmonic vibrational frequency calculations at the same level reveal only real vibrational frequencies and thus reported structures are minima of the potential energy surface. As for aromaticity indices, several criteria based on electron delocalization measures are employed. 12,13,39 These indices measure the cyclic electron delocalization of mobile electrons in aromatic rings. First, we study the para-delocalization index (PDI) 17,35 which is obtained using the delocalization index (DI) 41,42 as defined in the framework of the QTAIM of Bader. [43][44][45] The PDI is an average of all DI of para-related atoms in a given six-membered ring. For mono-determinantal closed-shell wavefunctions, the DI between atoms A and B is given by: . , Hilbert space, the latter quantity is known as the Mayer bond order. 46 We also employ a set of four  15 for a closed-shell monodeterminantal wavefunction is defined as follows: Some of us proposed a normalized version of the Iring index, 47 ING, which is expected to be less dependent on the ring size than its unnormalized analogs, and for aromatic species is given by: where N is the total number of atoms in the ring and N the total number of  electrons. ING has the peculiarity of reproducing the so-called TREPE 48 values at the Hückel MO theory. 49 According to Bultinck and coworkers 50 , summing up all the Iring values resulting from the permutations of indices A1, A2, ... AN defines a new index of aromaticity, the multicenter index (MCI) whose formula reads: where P(A) stands for a permutation operator acting over string A interchanging the atomic labels that none of the multicenter indices aforedescribed can be easily computed for rings of more than twelve members. 51 Calculation of atomic overlap matrices (AOM) and computations of the DI, Iring, ING, MCI, and INB are performed with the AIMPAC 52 and ESI-3D 53-55 collection of programs. Calculations of the DIs with DFT cannot be performed exactly because the electron-pair density is not available at this level of theory. 56 As an approximation, we use the Kohn-Sham orbitals obtained from a DFT calculation to compute Hartree-Fock-like DIs through Eq. (1) and, therefore, we do not expect to recover electron correlation effects.
We also analyze the electron localization function (ELF) using the TopMod program. 57 As shown by Savin et al., 58 the ELF measures the excess of kinetic energy density due to the Pauli repulsion. In the region of space where the Pauli repulsion is strong the ELF is close to one, whereas where the probability of finding same-spin electrons close together is high, the ELF tends to zero. For an N-electron single determinantal closed-shell wavefunction built from Hartree-Fock (HF) or Kohn-Sham orbitals, the ELF is given by 59, 60 : , Where , , (9) and where N is the number of electrons and Γ (2) ( ⃑ 1 , ⃑ 2 ) is the same-spin contributions to the pair density. Since the ELF is a scalar function, an analysis of its gradient field can be carried out to locate its attractors (local maxima) and the corresponding basins. There are basically two chemical types of basins: the core basins (C) and the valence (V) ones, which are characterized by their synaptic order, i.e., the number of core basins with which they share a common boundary. 61 Graphical representations of the bonding are obtained by plotting isosurfaces of the ELF. These isosurfaces delimit volumes within which the Pauli repulsion is rather weak. The localization domains are called irreducible when they contain only one attractor, and reducible otherwise. The reduction of reducible domains is another criterion of discrimination between basins, and the reductions occur at a critical value of the bonding isosurface. The domains are ordered with respect to the ELF critical values, yielding bifurcations (tree diagrams). The ELF bifurcation values can be also taken as a measure of aromaticity. According to Santos and co-workers 62 , aromatic compounds are characterized by high bifurcation values and small differences in bifurcation values of different basins.
The electronic structure of Ge30H12 is also examined by the hollow cylinder model (HCM) 32,33 while the electronic structure of Ge33H12 is examined by the cylinder model (CM). 63 The latter can be considered as a special case of the former when the inner radius of the hollow cylinder is neglected. The Schrӧdinger equation for a particle moving in a hollow cylinder was solved by Gravesen and co-workers 34 and later on by Miliodoros for a cylinder with Möbius topology. 64 The key point, which causes a difference between the two models is related to the different boundary conditions, i.e., the CM is a special case of the HCM when the inner radius (R0) is neglected. A change in the boundary conditions leads to a change in the Schrӧdinger equation solution for P(ρ) function among the separable solution ψ(ρ, Θ, z) = P(ρ)Θ(θ)Z(z). This leads to a change in the ordering of eigenstates (wavefunctions), such that the cylinder model becomes more suitable for a narrow nanotube-like molecular structure. 63 In general, the energy spectrum in both HCM and CM is quantized with three quantum numbers, namely, the rational k (k = 1, 2, 3, …), the rotational l (l = 0, ±1, ±2, ±3, …), and the radial n (n = 1, 2, 3, …), and it is defined by the expressions: for HCM (eq.10) and CM (eq.11), respectively. In Eqs. (10) and (11), m is the mass of the particle moving in the cylinder, R is the radius of the cylinder, L f = L/R with L being the length of the cylinder, and klnR is obtained from boundary conditions.

Results and Discussion
The optimized structures of Ge30H12 and Ge33H12 and the nomenclature used for Ge atoms are shown in Figure 1. The tube Ge30H12 consists of five Ge6 rings containing 12 GeA, 12 GeB, and 6 GeC atoms, piled on top of each other. The tube Ge33H12 is also comprised of two Ge6 rings (12 GeA) at the outermost and three innermost Ge6 units (12 GeB, 6 GeC, 2 GeE, and one GeD atoms).
In hosting three additional Ge atoms along the symmetry axis, the tubular Ge30H12 maintains a Our previous results pointed out that after introduction of one Ge atom into Ge18H12, the resulting Ge19H12 structure stabilizes 4.5 eV with respect to the isolated units. 30 We attribute this stabilization to the increased aromatic character, as proved by different analysis. In this work, upon insertion of three Ge atoms into the Ge30H12 tube, Ge33H12 stabilizes 21.0 eV (see Figure 1). In the two following subsections, we analyze in detail the electronic structure of both tubular forms.

Ge30H12
The tubular Ge30H12 includes 54 Ge-Ge and 12 Ge-H bonds and it is also contains 132 valence electrons. According to the Lewis theory, this molecule fits perfectly with a structure of 132 valence electrons organized in 66 2c-2e bonds. However, the natural geometry of the tubular moiety tends to create some delocalized electrons within the structure and decrease the number of localized electrons in the structure. Ge-H BCPs is positive, suggesting that the 54 Ge-Ge bonds behave as classical covalent bonds, and the 12 Ge-H are closed-shell bonds. 65,66 The electron densities at Ge-H BCPs are also remarkable larger than those at Ge-Ge BCPs, suggesting a great "donor-acceptor" character of the Ge-H bonds.
The ∇ 2 Laplacian contour maps in different planes of Ge30H12 are shown in Figure 3.a to 3.e. Figure 3.a displays great charge concentration regions belonging to the hydrogen atoms. tangential (t-MOs). Recently, the HCM has been applied to a prism structure, the skeleton Si12 of Cr@Si12, whose whole valence MOs fit in a unique HCM. 63 The fact that only one HCM is required for all valence MOs is due to the sp 3 hybridization of silicon atoms. However, the latter destabilizes the Si12 skeleton.
Germanium is isovalent with silicon, and the tubular skeleton Ge30 consists of five Ge6 rings.
Therefore, we also find the whole set of valence   Let us define the difference between MOs energies as: where EMO(Ge30H12) and EMO(Ge30) are the energies of the MOs in Ge30H12 and Ge30, respectively, which are assigned the same quantum number from the HCM. All ΔMO values are positive for the π-MOs interaction.
In both (k 0 2)-orbital and (k ±1 2)-orbital, the ΔMO values increase along with the rational quantum number k. Therefore, in Ge30, the occupied MOs with high k rapidly become the unoccupied MOs in Ge30H12. As a result, 10 π-MOs in Ge30 are formed while the π-MOs set of Ge30H12 has now 6 MOs. Hence, reduction of delocalized radial electrons is a way to stabilize the tubular structure.
In The results obtained for different indices of aromaticity are given in To further analyze the aromaticity of Ge30H12 we have also performed an ELF bifurcation analysis of this system (see Figure 7). We find that the separation of the six tangential Ge-Ge bond   Ge33 skeleton can move in a cylinder whose radius is larger than the radius of the structure by ΔR = 1.62 Å.

Ge33H12
Although the Ge33 skeleton is actually larger than the Ge30 counterpart, the height of the cylinder in which the electrons of Ge33 are moving becomes smaller than the height of the hollow cylinder in which electrons of Ge30 circulate. The extended height of the cylinder of Ge33 is also smaller than the extended height of the hollow cylinder of Ge30. The shape of the ground state valence MOs can partly be predicted considering this phenomenon.
According to the HCM, the electrons in the Ge30 can move in and out of the tubular structure to distances up to 1.88 Å whereas, in Ge33, the CM predicts that the electrons can move in and out of the tubular structure up to 1.62 Å.
In general, the presence of three central Ge atoms vertically placed inside the tubular structure reduces the available volume for moving electrons. As it has three more Ge atoms, more valence electrons are added to the system, and the tubular Ge30 also donates electrons to these central atoms. The net charges given in Table 3 show that large negative charges are located on GeE and GeD positions. Moreover, Table 1 displays that the largest Mayer bond orders correspond to the GeE-GeD bonds, while the Mayer bond orders of GeB-GeB, GeB-GeC, GeC-GeC, GeB-GeD, and GeC-GeE bonds are significantly smaller.  can also be considered as three Ge7 units (GeB and GeC atoms Figure 1b). Results in Table 4 show that the aromatic character of Ge6 rings is small for the outer rings, but substantial for the inner

Concluding Remarks
This paper investigates the geometries, chemical bonding, MOs analysis, and aromaticity of two hydrogenated germanium tubes Ge30H12 and Ge33H12. We have shown that the MOs of