domain: chim phys spi

text:
C Siouani

S Mahtout

S Safer

F Rabilloud
email: franck.rabilloud@univ-lyon1.fr

Structure, Stability and Electronic and Magnetic Properties of VGe n (n=1-19) Clusters

We systematically study the equilibrium geometries, electronic and magnetic properties of Gen+1 and VGen (n = 1-19) clusters using the density functional theory approach (DFT) within the generalized gradient approximation (GGA). Endohedral structures in which the vanadium atom is encapsulated inside a Gen cage are predicted to be favored for n ≥ 10. The dopant V atom in the Gen clusters has not an immediate effect on the stability of small germanium clusters (n<6) but it largely contributes to strengthen the stability for n ≥ 7. Our study enhances the large stability of the VGe14 cluster which presents an Oh symmetry cagelike geometry and a peculiar electronic structure in which the valence electrons of V and Ge atoms are delocalized and exhibit a shell structure associated to the quasi-spherical geometry. Consequently, this cluster is proposed to be a good candidate to be used as the building blocks for developing new materials. The cluster size dependence of the stability, the vertical ionization potentials and electron affinities of Gen+1 and VGen are presented. Magnetic properties and the partial density of states of the most stable VGen clusters are also discussed. VGe-a C∞v VGe3-a C2v VGe2-b C2v VGe2-a D∞h VGe3-b C3v VGe4-a Cs VGe5-c C2v VGe4-b C2v VGe4-c C2v VGe4-d C3v VGe5-a Cs VGe5-b Cs VGe12-a C2v VGe12-b C1 VGe12-c C1 VGe12-d Cs

Introduction

For several decades, the properties of small, intermediate and large sized clusters have been intensively investigated because of their potential applications in nanotechnologies. It is well know that most of physical and chemical properties of small and intermediate sized clusters are greatly determined by the size, composition and shape. In particular, the properties may change dramatically with the addition or substitution of one or few atoms in the cluster.

Semiconductor clusters doped with transition metal (TM) atoms are expected to yield interesting magnetic, electronic and optical properties for eventual applications in materials science, microelectronics, biological and medical area. The doping is used to control electronic and magnetic properties, or to stabilize peculiar structures. In particular, the experimental evidences of the formation of stable Si cages in which Si atoms encage a metal atom, as suggested earlier by theoretical calculations, has opened new perspectives to build cluster-assembled materials [1][2][3][4][5] . Germanium is also an important semiconductor material.

Extensive experimental and theoretical investigations on the pure and metal-doped Ge clusters were reported in the last decade . An experimental atomic scale observation of the dynamical behaviour of small germanium clusters combined with ab initio calculations have suggested that seven-membered rings, trigonal prisms and some smaller subunits are possible building blocks that stabilize the structure germanium cage clusters [START_REF] Bals | Atomic Scale Dynamics of Ultrasmall Germanium Clusters[END_REF] . From the theoretical point of view, several calculations have suggested that a Gen cage can be stabilized by encapsulating a guest metal atom inside the cage, as it was previously shown in the case of Sin cages. However the growth patterns of the TM-doped germanium clusters are dependent on the metal atom. For examples, density functional theory (DFT) calculations have shown that the doping Cu atom enhances the stability of the Gen cluster from n=10 and the electron charge transfers from the Cu atom to the neighbooring Ge atoms [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF][START_REF] Bandyopadhyay | Architectures, Electronic Structures, and Stabilities of Cu-doped Gen Clusters: Density Functional Modeling[END_REF] . Using First-principles calculations, Jing et al. [START_REF] Jing | No Quenching of Magnetic Moment for the GenCo (n=1 -13) Clusters: First Principles Calculations[END_REF] observed that the doping by Co atom enhances the stability of germanium cluster host without quenching the total magnetic moment. More recently, in a photoelectron spectroscopy study combined with DFT calculations, Deng et al. [START_REF] Deng | Structural and Magnetic Properties of CoGen -(n=2-11) Clusters: Photoelectron Spectroscopy and Density Functional Calculations[END_REF] found that the magnetic moment of the anionic and neutral CoGen (-) clusters decreases to the lowest values for sizes n = 10 and 11. For Ni doped germanium clusters, Ni atom was found to be encapsulated inside a germanium cage for clusters containing more than 8 atoms, while the large and peculiar stability of NiGe10 were explained by the electron counting rule as the cluster size corresponds to 20 valence electrons [START_REF] Wang | A Theoretical Study on Growth Patterns of Ni-Doped Germanium Clusters[END_REF][START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF] . Wang and Han [START_REF] Wang | Geometries and Electronic Properties of the Tungsten-Doped Germanium Clusters: WGen (n = 1-17)[END_REF] also investigated the doping with tungsten and demonstrated that WGe12, WGe14, WGe16 structures enhance stabilities over their neighboring clusters while the electron charge transfer always is done from the germanium framework to the W atom. In their ab initio investigation on the properties of thorium encapsulated germanium cages clusters with the size of 16, 18 and 20 atoms, Singh et al. [START_REF] Singh | Thorium Encapsulated Caged Clusters of Germanium: Th@Gen, n= 16, 18, and 20[END_REF] have reported that the doping atom is located at the center of the cage and the binding energies of doped clusters are enhanced in comparison to the pure germanium clusters with the same sizes. According to the DFT study by Kapila et al. [START_REF] Kapila | First Principle Investigation into Structural Growth and Magnetic Properties in GenCr Clusters for n=1-13[END_REF] the growth behavior of CrGen for n≤13 shows preference of Cr atom to stabilize at the exohedral position but the doping with Cr atom does not enhance the cluster stability, while Hou et al. [START_REF] Hou | Chromium-Doped Germanium Clusters CrGen (n = 1-5): Geometry, Electronic Structure, and Topology of Chemical Bonding[END_REF] have found that small CrGen clusters prefer structures with high-spin ground state and large magnetic moments. More recently, Mahtout et al. [START_REF] Mahtout | Electronic and Magnetic Properties of CrGen (15 ≤ n ≤ 29) Clusters: A DFT Study[END_REF] show that the encapsulation of Cr atoms within Gen clusters leads to stable CrGen (n=15-29) clusters and the electronic and magnetic properties depend on the structure and the position of Cr atom in the clusters. The FeGen clusters were investigated by Zhao and Wang [START_REF] Zhao | Stabilities, and Electronic Properties of FeGen (n = 9-16) Clusters: Density-Functional Theory Investigations[END_REF] , while anionic AuGen -species were considered by Li et al. [START_REF] Li | Size-Selective Effects in the Geometry and Electronic Property of Bimetallic Au-Ge Nanoclusters[END_REF] . The properties of bimetallic Mo2 doped germanium clusters in the size range of 9 to 15 atoms were also investigated at DFT level [START_REF] Wang | Stabilities, and Vibrational Properties of Bimetallic Mo2-Doped Gen (n = 9-15) Clusters: A Density Functional Investigation[END_REF] . It was shown that the most stable structures are obtained when one Mo atom is inside of the Gen cage and the second is located at the surface for n ≤ 13 atoms. But Mo2 dimer was completely encapsulated into the germanium cage when the cluster size is larger. Considering Mn doped germanium clusters, Zhao and Wang [START_REF] Zhao | Stabilities, and Magnetic Properties of MnGen (n = 2-16) Clusters: Density-Functional Theory Investigations[END_REF] found that the doping by one Mn atom contributes to enhance the stability of the germanium cage clusters while the charge transfers were done from Mn to the neighboring Ge atoms. However they found that the magnetic moment of the Mn atom does not quench in all Ge cage clusters. Kapila et al. [START_REF] Li | Size-Selective Effects in the Geometry and Electronic Property of Bimetallic Au-Ge Nanoclusters[END_REF] investigated the effects of doping with several transition metal atom (Mn, Co and Ni) on the properties of Ge cage clusters. They found that the total magnetic moment is localized at the doping atom and neighboring Ge atoms and the Ni doped Gen clusters are the most stable compared to other Co and Mn doped Gen clusters. Very recently, an inherent tendency of formation of an endohedral cage were reported for multicharged ruthenium-doped Ge clusters 9 .

The reading of the previous works shows that geometric, electronic and magnetic properties of transition metal doped germanium clusters are significantly different from those of pure Gen ones, and change dramatically with their size and the nature of doping metal atoms. To our knowledge, the doping of germanium clusters with vanadium atoms has been considered in two previous works 7,8 . First, studying the relative stability of Sc, Ti, and V encapsulating Gen (n=14-20) clusters in both neutral and charged states, it was explored how far any enhanced stability can be explained by the formation of a closed shell free-electron gas inside the Ge cages, and if an electron-rich region spread uniformly around the metal atom can be unequivocally identified and visualized using the topological features of the molecular electrostatic potential 7 . The analysis was based on a naive electron counting, where each Ge atom is supposed to donate one electron to the free-electron gas while V atom gives 5 electrons. Therefore VGe15 and VGe16 + are possible 20-electron clusters, and were predicted to have an enhanced stability. Recently, in a joint experimental and theoretical investigation, the photoelectron spectra of VGen -(n ≤ 12) were measured and compared to DFT calculations on putative structures for both anionic and neutral species 8 . However, both studies focused on a restricted cluster size range, and there is a clear need to rationalize these studies to a wider size range in order to highlight a growth pattern, while performing a more extended search for the lowest isomer with new putative structures.

The present work aims to understand the influence of one vanadium atom on germanium clusters and to study the different properties of the VGen clusters that will be compared to other metal doped germanium clusters. So, the geometric structures, relative stabilities, electronic and magnetic properties of small VGen (n=1-19) clusters have been systematically investigated by using first principles calculations. The structures of low-lying energy isomers of each cluster size have been optimized and analysed, as well as the binding energy, HOMO-LUMO gaps, vertical ionization potential and vertical electron affinity. This paper is organized as followed: we describe in section 2 the computational method used in this work, while results and discussion on the clusters geometries, stabilities, electronic and magnetic properties of VGen clusters, and also the peculiar case of VGe14, are presented in Section 3. Finally, our main conclusions are given in Section 4.

Computational methodology

All calculations were performed using the spin polarized density functional theory implemented in the SIESTA package [START_REF] Soler | The SIESTA Method for Ab Initio Order-N Materials Simulation[END_REF] . They were carried out using the generalized gradient approximation formulated by Perdew, Burke, and Ernzerhof 46 (PBE) to describe the exchange-correlation energy functional. The norm-conserving Troullier-Martins nonlocal pseudopotentials [START_REF] Troullier | Efficient Pseudopotentials for Plane-Wave Calculations[END_REF] were used together with a flexible basis set of localized Gaussian-type atomic orbitals. Then, core electrons were replaced by non-local, norm-conserving pseudopotentials factorized in the Kleinman-Bylander form [START_REF] Kleinman | Efficacious Form for Model Pseudopotentials[END_REF] . We have used 3d 3 4s 2 configuration for V and 4s 2 4p 2 for Ge. The geometries have been optimized without any symmetry constraints and the optimization of electronic structure was obtained by solving the Kohn-Sham equations [START_REF] Kohn | Self-Consistent Equations Including Exchange and Correlation Effects[END_REF] self-consistently with a convergence criterion of 10 -4 a.u. on the energy and electron density. We have used the k=0 (Γ) point approximation for Brillouin zone sampling. We have used the double ζ (DZ) basis for Ge atoms and double ζ (DZP) bases with polarization function for V atom. In the optimization process, the volume of the system was kept constant and with a big supercell of 40 Å was used in order to avoid interaction between the neighboring clusters. Structural optimizations are performed using conjugate gradient algorithm and the convergence criterion on the Hellmann Feynman forces imposed that the residual forces were smaller than 0.04 eV/Å. Several spin multiplicity states were tested. The Mulliken population analyses were done in order to obtain the atomic charge and the unpaired spin population. The validity of current computational method has been tested on germanium dimer properties. The obtained results are summarized in Table 1 and compared to the available theoretical and experimental data. Our calculated bond length of 2.450 Å and dissociation energy of 1.446 eV are consistent with the literature. They show the reliability of current computational scheme to describe small germanium clusters. For the dimer VGe, the electronic ground state is found to be a sextet with a bond distance of 2.526 Å and a binding energy of 1.198 eV. To the best of our knowledge, there is no available experimental or calculated value.

Results and discussion

Structural properties

It is well known that the number of isomers increases exponentially with increasing cluster size, making it challenging job to search for the most lowest-energy structure. In order to obtain the global minimum structures of Gen+1 and VGen clusters, we have optimized a large number of possible isomeric structures. Lots of possible initial structures including one, two or three dimensional (3D) configurations have been considered for each size, amongst others some initial structures of Gen+1 and metal-doped Gen clusters were taken from literature. Also, the putative structures of VGen have been obtained by local relaxation after the substitution of one Ge atom by V atom in several isomers of the original pure Gen+1 cluster. The different initial positions of the V atom in the Gen+1 clusters lead to different VGen isomers. Of course, the explicit treatment of the electronic structure constitutes a demanding computational task, and the search for the lowest isomer cannot include a global optimization procedure of the potential energy surface. So we cannot be sure that a more stable cluster than those found in our calculations does not exist. In this work we only show the best calculated structures for each cluster size. The stability of the structures was checked by calculating their harmonic vibrational frequencies. If any imaginary frequency was found, further relaxation was carried out until the true local minimum was obtained.

In the case of pure Gen+1 clusters, our calculations show a growth pattern in which the planar structures only appears in the very small clusters while the tridimensional structures dominate from n+1=5. Up to n+1=20, prolate-type structures compete with nearly spherical structures, and almost all atoms are located in surface. Many of the obtained best structures are in agreement with the previous theoretical studies of the literature. Our lowest-energy isomers are shown in Figure 1, while the energetic ordering of isomers are given Table 2. For each size, data for most stable isomer are reported in bold character. For clusters with n=2 and 3, the ground state structures adopt a planar form in agreement with previous works [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF][START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF][START_REF] Shi | A Computational Investigation of Aluminum-Doped Germanium Clusters by Density Functional Theory Study[END_REF] by using different density functional calculations. Both linear and triangular structures are considered for the Ge3. The triangular geometry with D3h symmetry and average bond length of 2.342 Å is found to be the most stable structure. The most stable isomer of the tetramer Ge4 has D2h symmetry and an average bond length of 2.577 Å which are in good agreement with previous works [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF][START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF][START_REF] Deutsch | Binding Energies of Germanium Clusters, Gen (n = 2-5)[END_REF] . The three dimensional geometries appears for n=4. The Ge5 pentamer ground state is a triangular bipyramid structure with D3h symmetry and an average bond length of 2.548 Å. This result is consistent with the previous theoretical results [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF][START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF][START_REF] Kapila | First Principle Investigation into Structural Growth and Magnetic Properties in GenCr Clusters for n=1-13[END_REF] . For Ge6, two isomers are found to be quasi-degenerate, the first one is a bicapped rectangular structure with Cs symmetry and an average bond length of 2.744 Å, while the second one is a bicapped quadrilateral structure with D4h symmetry. The Cs-symmetry structure has been predicted to be the lowest-energy isomer in some previous works [START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF] , while the D4h symmetry is preferred in other works [START_REF] Kapila | First Principle Investigation into Structural Growth and Magnetic Properties in GenCr Clusters for n=1-13[END_REF] . In our calculation, the high-symmetry structure is found to lie 0.006 eV above the Cs one, but it seems clear that the ordering may change with the functional or the basis set used. The lowest-energy Ge7 cluster displays a pentagonal bipyramid structure of D5h symmetry and Ge-Ge bond at 2.747 Å which are in good agreement with the results of Wang and Han [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF] and Kapila et al. [START_REF] Kapila | First Principle Investigation into Structural Growth and Magnetic Properties in GenCr Clusters for n=1-13[END_REF] . The most stable isomer for Ge8 cluster presents a capped pentagonal bipyramid structure with Cs symmetry, in agreement with previous works [START_REF] Wang | A Computational Investigation of Copper-Doped Germanium and Germanium Clusters by the Density-Functional Theory[END_REF][START_REF] Kapila | First Principle Investigation into Structural Growth and Magnetic Properties in GenCr Clusters for n=1-13[END_REF] . For Ge9 cluster, a capped cube-like structure with C4v symmetry is found to be the most stable isomer. Its average bond length is calculated to be 2.663 Å. A similar result of Ge9 is obtained by Bandyopadhyay and Sen [START_REF] Bandyopadhyay | Density Functional Investigation of Structure and Stability of Gen and GenNi (n = 1-20) Clusters: Validity of the Electron Counting Rule[END_REF] . In the case of Ge10, the most stable structure is calculated to be a capped irregular pentagonal prism structure with C1 symmetry and an average bond length of 2.748 Å (labeled Ge10-c in Figure 1). A more regular structure with D2h symmetry (labeled Ge10-a in Figure 1) is found to lie 0.06 eV above the previous one. The shape of Ge11 is a caped pentagonal basis structure and with a C2v symmetry and an average bond length of 2.839 Å. For Ge12 clusters, the most stable structure (Ge12-f) is a compact near spherical structure with C2v symmetry, but a more prolate-like structure (Ge12-a) is found to lie 0.04 eV above. From Ge13 to Ge16 clusters, a prolate structure with C1 symmetry is always favored. The average bond lengths are 2.811, 2.725, 2.791 and 2.759 Å for n=13, 14, 15, 16 respectively. The best isomers of Ge17 (Ge17-a) has a near spherical shape with Cs symmetry and an average bond length of 2.805 Å. In the case of the Ge18 cluster, our calculations suggest that the prolate-like isomer Ge18-b with C2 symmetry and an average bond length of 2.732 Å is the most stable one. The lowest-energy isomer for Ge19 has a prolate shape (Ge19a) and an average bond length of 2.769 Å whereas the most stable structure of Ge20 (Ge20-b) adopts a C1 symmetry with a larger average bond length of 2.812 Å.

We now present our results concerning VGen clusters. Optimized structures are showed in Figure 2, and energies are given in Table 3. Up to n=9, VGen clusters adopt a somewhat similar structure than their corresponding Gen+1, with two exceptions, VGe3 which prefers a compact tridimensional structure while Ge4 is a rhombus and VGe8 which significantly differs from the C4v-symmetry structure of Ge9. Interestingly, for n=10-16, the vanadium atom forms a core surrounded by all Ge atoms which in turn form an opened cage for n=10 and 11 and a fully closed cage for n=12-16 in which V is encapsulated. Then, for larger size, we see a competition between some prolate-and cage-like structures. The most stable structure of VGe19 cluster (labelled VGe19-c in figure 3) can be described as a cage-like structure V@Ge11 anchored by three Ge atoms to the remainder of the cluster. This could mean that the cage V@Gen may be used as the building blocks to make cluster-assembled materials. Clearly, the substitution of one Ge atom by a V atom leads to significant deformation for n>9. Similar conclusions can be done if we look at the deformation induced by introducing a V atom in a host Gen cluster.

We now describe the structures of VGen clusters in more details. The lowest-energy VGe2 cluster displays a triangular structure of C2v symmetry. This ground-state structure has two equivalent V-Ge bonds 2.536 Å and one Ge-Ge bond 2.405 Å. The Ge-Ge distance is 0.063 Å larger than in the case of Ge3 trimer and the V-Ge bond distance is 0.01 Å larger than in VGe dimer. For the VGe3 tetramer, two low-lying structures are found as local minima. The first one presents a planar C2v symmetry and the second one is a triangular pyramid with C3v symmetry. This later is the lowest-energy isomer. Its calculated binding energy equals 2.350 eV/atom and is much smaller than that of the ground state of pure germanium tetramer Ge4 (2.556 eV/atom). The most stable structure of VGe4 cluster is a distorted rectangular pyramid with Cs symmetry. The V-Ge and Ge-Ge bond lengths are 2.600 and 2.713 Å respectively. Its binding energy is 0.147 eV/atom and is smaller than that Ge5. The lowestenergy structure of VGe5 is a quasi-rectangular bipyramid with C2v symmetry and the V atom located at the convex site. The different V-Ge bond distances vary from 2.643 Å to 2.901 Å and the average Ge-Ge bond distance is found to be 2.704 Å. For VGe6, two structures, with C2v and C3v point group symmetry respectively, compete for the lowest-energy isomer. In the latter, the aV-Ge bond lengths are much shorter (2.494 Å versus 2.746 Å). The lowest energy isomer for VGe7 cluster (VGe7-b) can be described as a germanium atom capping a pentagonal bipyramid with the V atom at the vertex. This stable structure with Cs symmetry is obtained by the substitution of a Ge atom by V in the ground state geometry Ge8-b of pure cluster Ge8. Its binding energy is only 0.002 eV/atom larger than the case of corresponding pure cluster. The global minimum structure of the VGe8 cluster is the tricapped trigonal prism geometry (VGe8-d) with the V atom at the surface (structure similar to that obtained in ref 8). The total spin magnetic moment is only 1 µB. Its binding energy is 2.966 eV/atom and so 0.12 eV/atom higher than that of its corresponding pure isomer Ge9-d. In VGe9 cluster, the ground state geometry (VGe9-e) can be described as a combination of two distorted hexagonal prism with V atom at the top of one of them. Let us note that the second isomer (VGe9-f) found to lie 0.04 eV/atom above VGe9-e will be used to build cage structures for n ≥ 10. From n=10 our obtained clusters show an increasing tendency to encapsulate the vanadium at the center of the structure. In other words, the V-encapsulated germanium clusters become the dominant geometries. For VGe10 cluster, the named VGe10-c structure with V atom partially encapsulated in an opened Ge10 cage with C2v symmetry is the ground state isomer. Its binding energy is 0.099 eV/atom larger than that of the corresponding pure cluster Ge11-c. For VGe11, the V atom tends to stabilize at the surface rather than at the central encapsulated position. Our lowest energy structure (VGe11-a) is a distorted superposition of two rhombic structure with C2v symmetry and bond-lengths of 2.689 Å and 2.920 Å for Ge-Ge and V-Ge respectively. The lowest-energy structure (VGe12-f) of the VGe12 cluster is a perfect hexagonal prism geometry with V atom encapsulated at the center of the germanium cage with D3d symmetry. The second (VGe12-d) and third (VGe12-e) lowest isomers are also closed cage with Cs and D4h symmetry. Endohedral structures have been already found for Mn-, Co-, Ni-Ti-doped Ge12 [START_REF] Jing | No Quenching of Magnetic Moment for the GenCo (n=1 -13) Clusters: First Principles Calculations[END_REF][START_REF] Singh | Thorium Encapsulated Caged Clusters of Germanium: Th@Gen, n= 16, 18, and 20[END_REF][START_REF] Kapila | Electronic and Magnetic Properties of Mn, Co, Ni in Gen for (n=1-13[END_REF][START_REF] Zhao | Stabilities, and Magnetic Properties of MnGen (n = 2-16) Clusters: Density-Functional Theory Investigations[END_REF] . For n=13, the ground-state (VGe13-a) is a capped hexagonal prism structure with the V atom totally encapsulated at the center of the cage. The high C6v symmetry is found for this doublet spin state. Four V-encapsulated geometries have been found for VGe14 cluster. The most stable one (VGe14-a) presents the high Oh symmetry with a large binding energy of 3.183 eV/atom. That structure has already been found as the lowestenergy isomer in the case of WGe14 [START_REF] Wang | Geometries and Electronic Properties of the Tungsten-Doped Germanium Clusters: WGen (n = 1-17)[END_REF] and one of the stable structures but not in competition for the most stable isomer in NiGe14 28 or CuGe14 [START_REF] Bandyopadhyay | Architectures, Electronic Structures, and Stabilities of Cu-doped Gen Clusters: Density Functional Modeling[END_REF] . For n=15, the optimized ground state structure is the endohedral structure VGe15-c. It can be viewed as addition of one Ge atom at vertices of the isomer VGe14-a. This geometry with Cs symmetry is 0.127 eV/atom more stable than the corresponding most stable germanium clusters with the same size (Ge17). It has the shortest Ge-Ge and largest V-Ge average bond lengths of 2.755 Å and 3.005 Å respectively. Kumar et al. [START_REF] Kumar | Architecture, Electronic Structure and Stability of TM@Gen (TM = Ti, Zr and Hf; n = 1-20) Clusters: a Density Functional Modeling[END_REF] have also reported that adding one Ge atom to the bottom of the rhombi in a symmetrical hexagonal bi-capped structure TM-Ge14 may leads to the ground state structure of TM-Ge15 clusters. All of the ground state geometries for the bigger clusters (n ≥ 16) in the present study show distorted structures. Particularly the endohedral structures become strongly deformed cages because a unique V atom cannot stabilize a perfect cage when the number of Ge atoms is too high. In the case of VGe16, the irregular cage-like structure VGe16-d is found to be the lowest isomer while the more regular D2h isomer (VGe-a) lie 0.68 eV above. For VGe17, a prolate structure were found to be the ground state (VGe17-a) while no cage-like isomer was found to be stable. The lowest-energy structure of VGe18 is VGe18-e, in which the V atom is highly coordinated and located at the center of the cage of germanium. However, the isomer VGe18-b is found to lie only 0.003 eV/atom above. For n=19, our six best putative isomers are given in Figure 2. As discussed before, the best isomer (VGe19-c) combines a prolate-like structure and a cagelike structure. It can be described as a cage-like structure V@Ge11 anchored by three Ge atoms to the remainder of the cluster. The binding energy of VGe19 is 0.062 eV/atom larger comparatively to the ground state structure of the pure Ge20 clusters.

Energetics and electronic properties

To compare the relative stability of the germanium and vanadium doped germanium clusters, the binding energy were calculated as followed:

Eb (Gen+1) = ((n+1) E(Ge) -E(Gen+1)) / (n+1), Eb (VGen) = (n E(Ge) + E(V) -E(VGen)) / (n+1),
where E represents the total energies of the relevant system. The obtained results are reported in tables 2 and 3. The evolution of binding energies of Gen+1 and VGen clusters as a function of the size n is shown in Figure 3 for the most stable isomers. As expected, the binding energy of Gen+1 and VGen clusters generally increases with increasing size, which means that these clusters can continuously gain energy during the growth process. For VGen, the binding energy increases rapidly from 1.19 eV for the dimer to 2.8 eV for cluster with n=6 then a slow and non-monotonic growth is observed up to n= 19, with two local maxima at n=10 and 14. Also, we observe that the binding energy of VGen clusters is larger than the corresponding Gen+1 clusters, except for the cases in range of n=3-6. This proves that the doping V atom generally enhances the stability of germanium clusters. This may appear surprising if we consider the binding energy of VGe and Ge2 dimers, 1.198 and 1.446 eV respectively, which seems to be unfavorable to form V-Ge bonds. The great stability is enhanced for the endohedral structures of n=10, 12-15, in which V is encapsulated in a quasi-perfect Gen cage. This behavior is due to the absorption of the dangling bonds of the germanium cage by the doping vanadium atom located in a central position. The high binding energies for VGe10, VGe12, VGe13, VGe14, and VGe15 are calculated to 3.058, 3.127, 3.131, 3.183, 3.179 eV/atom, they are ~0.15 eV/atom larger than those of the corresponding Gen+1 clusters. The highest value of the binding energy is obtained for VGe14. The high stability of the VGe14 cluster might stem from its high symmetry (Oh) and the fact that the enclosed V atom inside the cage of germanium interacts with all germanium atoms (high coordination number).

In order to get insight into the kinetic stability of the clusters, the HOMO-LUMO gap (energy difference between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO)) of VGen clusters was also calculated and analyzed. It is considered to be an important criterion in terms of the electronic stability of clusters and represents the ability for the cluster to participate in chemical reactions. In general as the HOMO-LUMO gap increases, the reactivity of the cluster decreases [START_REF] Yoshida | Validity of the Weighted HOMO-LUMO Energy Separation as an Index of Kinetic Stability for Fullerenes with up to 120 Carbon Atoms[END_REF] . The obtained results are shown in Figure 4 for the most stables isomers of each size. For comparison, the HOMO-LUMO gaps of the most stable pure Gen+1 clusters are also reported. Data can be found in Table 2 and3. The decreasing tendency with increasing size is strong for pure germanium clusters but less significant for VGen clusters. The gap of VGen is much lower than that of the pure Gen+1 cluster, except for sizes n= 11, 13 and 14, indicating that the vanadium dopant in the germanium clusters generally increases the chemical activity of clusters. The value for VGen oscillates roughly between 0.5 and 1 eV but the case n=14 is an exception with a value of 1.495 eV. This points the high stability of VGe14 and reflects that this cluster is chemically stable and is a good candidate to be used as a building block for developing new materials.

It is very know in cluster physics that the second order energy difference (Δ2E) is a sensitive quantity which can directly reflect the relative stability of the corresponding clusters [START_REF] Wang | Structure and Electronic Properties of Gen (n = 2-25) Clusters from Density-Functional Theory[END_REF] . It is generally compared to the relative concentration of species determined in mass spectroscopy measurements. Here, the calculated second order energy differences of Gen+1 and VGen clusters are obtained as followed ∆2E (Gen+1) = E(Gen+2) + E(Gen) -2E(Gen+1), ∆2E(VGen) = E(VGen+1) + E(VGen-1) -2 E(VGen), and are shown in Figure 05. The large positive value for n= 10 together with the negative values for n=9 and 11 reflect the relative high stability of the opened cage VGe10 compared to the exohedral structures of VGe9 and VGe11. Similarly, the large positive values for n=12-15 together with the large negative values for n=11 and 16 reflect the high stability of the perfect cages for VGe12-15 compared to the exohedral structure of VGe11 and the distorted cage of VGe16. It is interesting to note that the Δ2E values for Ge13 and Ge16 are also large, and so before the substitution of a Ge atom by a V atom (resulting in VGe12 and VGe15 respectively), the sizes n=12 and 15 are favored. This result indicates that the high stability of VGe12 and VGe15 might also be due to the high stability of their corresponding initial pure germanium cage Ge13 and Ge16.

Vertical ionization potential (VIP) and vertical electron affinity (VEA) are defined as:

VIP = E + -E VEA = E -E - where E is the total energy of the neutral cluster, E + and E -represent the total energies of the cationic and anionic clusters respectively with the same geometry as the neutral cluster. These data could be compared to experiment even if to our knowledge no experimental is available nowadays. Results for Gen+1 and VGen (n = 1-19) clusters are listed in Tables 2 and3. The VIP defines the quantity of energy needed to extract an electron from the neutral clusters and it is a useful for determining the stability of clusters. In general, the larger the values of VIP, the more stable is the cluster. Here, the obtained value of VIP for the most stable isomer of Gen+1 and VGen clusters are shown in Figure 6. The VIP of VGen clusters show an oscillating behavior until size n=14 and a decreasing tendency from n=15. All values are between 6.2 and 7.12 eV. Pronounced peaks are observed at n= 2, 4, 8 and more apparent peak at n= 14. Again, VGe14 cluster seems to be particularly stable. The VIP of pure germanium clusters show a decreasing tendency with the increasing of the cluster size, in line with the results obtained for HOMO-LUMO gaps showed in Figure 4. The cluster size dependent of VEA for Gen+1 and VGen clusters are plotted in Figure 7. We can see that the electron affinity is sensitive to the size of the cluster and it roughly shows an increasing tendency with the increasing cluster size and thus the larger clusters will capture electrons more easily. The relative minimum located at n=14 indicates a relative larger stability for VGe14. Our calculated VEA values are found to be 0.4-0.8 eV smaller than the calculated vertical electron detachment energies of the anion VGen -. 8

It has been established that chemical hardness (η) is an important quantity which may be used to characterize the relative stability of molecules and clusters through the principle of maximum hardness (PMH) of Pearson [START_REF] Parr | Density Functional Theory of Atoms and Molecules[END_REF][START_REF] Parr | Absolute Hardness: Companion Parameter to Absolute Electronegativity[END_REF] . On the basis of a Koopmans theorem, the chemical hardness η is defined as:

η =VIP-VEA
The obtained results are reported in tables 2 and 3 for Gen+1 and VGen clusters respectively. The relationships of η and the size n are shown in Figure 8 for both species. The chemical hardness has a roughly decreasing evolution as the size n increases for Gen+1 and VGen clusters. The relative high value for VGe12-14 indicates a relative high stability for the three clusters, in particular for VGe14 for which the value of η is as large as for a small cluster like VGe4.

The magnetic behavior of the VGen clusters has been investigated by calculating the total spin magnetic moment (TSMM) and the partial density of states (PDOS) 3d, 4s and 4p orbitals for all of the most stables clusters. The TSMM is defined as the difference between the occupation numbers of spin-up and spin-down states. The TSMM values of VGen clusters are shown in Table 3 and Figure 9 for the most stables structure in each size. The PDOS are shown in Figure 10 where the Fermi level is shifted to zero position, the spin-down density is plotted as negative and the spin-up density is plotted as positive. We observe that the magnitude of TSMM oscillates between 1 and 5 μB. A large TSMM value of 5 μB is observed in clusters with n=1 and 5. For many clusters (n=2,3,4,6,7,9, 11,12,14,17,18,19) the value of 3μB is found as for the isolated V atom. The PDOS indicate that the large differences between the up and down electronic states around the Fermi level mainly come from the d states while a very few of magnetic moments are found on the 4s and 4p states. So the magnetic moment of clusters is mainly due to 3d electrons of vanadium. However, the magnetic moment is quenched (1 μB) for clusters corresponding to n= 8, 10, 13, 15, 16.

The case VGe 14

Our results, detailed in previous sections, enhance the high stability of VGe14 cluster. The cluster presents the highest stability from both the binding energy (figure 3) and the second-order energy difference (figure 5). It also has a relatively high HOMO-LUMO gap (figure 4), a relatively high VIP (figure 6) and small VEA (figure 7), and a large chemical hardness (Figure 8). Also its geometrical structure adopts the Oh symmetry point group. As a matter of fact, we have checked that all harmonic frequencies are positive, since the lowest ones were calculated to be a T2u mode at 54 cm -1 and a T2g mode at 105 cm -1 . To further investigate that peculiar cluster, we consider the possibility that the high symmetry maximizes the stability by allowing the formation of a quasi-spherically confined free-electron gas inside the cage. The electronic structure of VGe14 is detailed in Figure 11. We show the density of states (DOS) and the Kohn-Sham orbitals calculated at PBE/cc-pvtz level with the software Gaussian09 [START_REF] Frisch | Gaussian09, Revision D.01[END_REF] . The 3s and 3p valence electrons of Ge and the 4s and 3d valence electrons of V are found to be delocalized and exhibit a shell structure associated to the quasi-spherical geometry. Although some little deviations from a perfect sphere due to explicit location of atoms and the Oh symmetry instead of Kh (the symmetry of the atom), we can easily distinguish the orbitals S, P, D, F, G. The 61 valence electrons of the cluster are organized with the following occupations: 1S 2 1P 6 1D 10 2S 2 1F 14 3S 2 2P 6 1G 15 2D 4 . Although the number of electrons does not exactly fit with shell closings numbers, such an organization contributes to the high stability of the cluster through the pooling of electrons. The shell model is well known in metal cluster and has been predicted by the jellium model [START_REF] De Heer | The Physics of Simple Metal Clusters: Experimental Aspects and Simple Models[END_REF][START_REF] Brack | The Physics of Simple Metal Clusters: Self-Consistent Jellium Model and Semiclassical Approaches[END_REF] which considers the valence electron confined in a spherically symmetric potential while the ionic core structure is replaced by a uniform positive background. To our knowledge, such an electronic organization in metal-doped germanium clusters has not been previously reported.

Our present results contrast with previous works 7 in which the enhanced stability of VGe15 was highlighted and was supposed to originate from a 20-electron filled shell electronic configuration. The analysis was based on the electron counting, where each Ge atom donates one electron to the free-electron gas while V atom gives 5 electrons. Therefore VGe15 was a possible 20-electron cluster, and was predicted to have enhanced stability. In our work, an enhanced stability is found for VGe14, while VGe15 has also a large stability. The stability of VGe14 cannot be explained by the 20-electrons rule. The molecular orbitals are clearly in favor of a reorganization of the 61 valence electrons in a super-shell occupations, and not of a 20-electrons rule (or 18-electron as suggested for other metals).

Conclusion

In this work we have systematically studied the properties of Gen+1 and VGen (n = 1-19) clusters using first principles DFT calculations. The lowest energy structures and some low lying isomers have been identified for both pure and vanadium doped germanium clusters. In optimized structures of pure germanium clusters, Ge atoms are generally localized at the surface of the cluster. Up to n=9, VGen clusters adopt a somewhat similar structure than their corresponding Gen+1, except VGe3 and VGe8. For n ≥ 10 the endohedrally vanadium doped germanium structures are predicted to be the most stable. For n=10-16, the vanadium atom forms a core surrounded by all Ge atoms which in turn form an opened cage for n=10 and 11 and a fully closed cage for n=12-16 in which V is encapsulated. Then, for larger size, we see a competition between some prolate-and cage-like structures. For VGe19, the best isomer combines a prolate-like structure and a cagelike structure. It can be described as a cage-like structure V@Ge11 anchored by three Ge atoms to the remainder of the cluster. Hence, we can expect similar features for larger size VGen clusters with a cage-like V@Gen, n~14, anchored to the other germanium atoms.

The dopant V atom in the Gen clusters has not an immediate effect on the stability of small germanium clusters (n<6) but it contributes largely to strengthen the stabilities of the germanium cages from n=10 to n=19. The HOMO-LUMO gaps of the VGen clusters are generally smaller than those of the corresponding Gen+1 clusters. The calculated binding energy, the second-order energy difference, the HOMO-LUMO gaps, vertical ionization potential, chemical hardness manifest the large stability of the VGe14 cluster with Oh symmetry. That cluster presents a peculiar electronic structure in which the valence electrons of V and Ge atoms are delocalized and exhibit a shell structure associated to the quasispherical geometry. Consequently, this cluster can be a good candidate to be used as the building blocks for developing new materials. The Mulliken electrons population analysis explored in the partial density of states (PDOS) indicates that the total spin magnetic moment of VGen clusters is mainly due to 3d electrons.

Figure captions

Fig. 01: Lowest energy structure and their corresponding isomers of Gen+1 (n=1-19) clusters.

For each size, the lowest-energy isomers are reported in bold character.

Fig02. Lowest energy structure and their corresponding isomers of VGe n (n=1-19) clusters. For each size, the lowest-energy isomers are reported in bold character. For each size, the lowest-energy isomers are reported in bold character.

Fig. 03 :Fig. 04 :

 0304 Fig. 03: Size dependence of the binding energy of Gen+1 and VGen (n=1-19) clusters.

Fig. 05 :Fig. 06 :

 0506 Fig. 05: Size dependence of the second order energy difference of Gen+1 and VGen (n=1-19) clusters.

Fig. 07 :Fig. 08 :

 0708 Fig. 07: Size dependence of the vertical electronic affinity of Gen+1 and VGen (n=1-19) clusters.

Fig. 09 :

 09 Fig. 09: Size dependence of the total spin magnetic moment of VGen (n=1-19) clusters.

Fig. 10 :Fig. 11 :Fig. 01 :

 101101 Fig. 10: The projected density of states (PDOS) for VGen (n=1-19) clusters.

Fig02.

  Fig02. Lowest energy structure and their corresponding isomers of VGen (n=1-19) clusters. For each size, the lowest-energy isomers are reported in bold character.

Fig. 03 :Fig. 04 :Fig. 05 :Fig. 06 :Fig. 07 :Fig. 08 :Fig. 09 :

 03040506070809 Fig. 03: Size dependence of the binding energy of Gen+1 and VGen (n=1-19) clusters.

Table 1 :

 1 The binding energy and bond length of Ge2 dimers: comparisons with other available experimental and theoretical data.

	Parameters	Our work	Theoretical value	Experimental value
	E b (eV/atom)	1.446	1.44 a , 1.772 e , 1.92 k , 1.34 l , ~1.45 i 1.35 m
	R (Å)	2.450	2.548 a , 2.399 b , 2.54 c , 2.421 e ,	2.57 n . 2.44 o
			2.528 d , 2.413 f , 2.36 g , 2.375 h ,
			2.48 j , 2.44 l , 2.422 p
	a Ref. [11].

b

Ref.

[14]

. c Ref.

[18]

. d Ref.

[23]

. e Ref.

[24]

. f Ref.

[30]

. g Ref.

[31]

. h Ref.

[33]

. i Ref.

[34]

. j Ref.

[35]

. k Ref.

[37]

. l Ref.

[40]

. m Ref.

[50]

. n Ref.

[29]

. o Ref.

[51]

. p Ref.

[52]

.

Table 2 :

 2 Symmetry group, binding energy Eb (eV/atom), HOMO-LUMO gap ΔE (eV), vertical ionization potential VIP (eV), vertical electron affinity VEA (eV) and chemical hardness η (eV) for Gen+1 (n=1-19) clusters.

	n	Symmetry	E b (eV/atom)	ΔE (eV)	VIP (eV)	VEA (eV)	η (eV)	a Ge-Ge
	1	a-D ∞h	1.446	1.816	7.844	1.900	5.944	2.450
	2	a-D ∞h	2.049	1.267	7.789	1.427	6.362	2.342
		b-D3h	2.059	2.426	7.804	2.200	5.604	2.546
	3	a-C 2v	2.212	0.207	7.197	2.046	5.151	2.502
		b-D2h	2.556	1.179	7.855	1.558	6.298	2.577
		c-D 2d	2.277	1.067	7.243	2.230	5.013	2.517
	4	a-C 4v	2.490	0.643	7.664	2.176	5.488	2.668
		b-D3h	2.707	2.068	8.062	1.213	6.849	2.548
	5	a-D4h	2.847	1.995	7.863	1.269	6.594	2.783
		b-C s	2.739	0.925	7.474	1.952	5.522	2.703
		c-Cs	2.848	1.957	7.828	1.251	6.577	2.744
	6	a-D5h	2.975	1.836	7.939	1.700	6.239	2.747
		b-C 2v	2.695	0.243	6.952	2.428	4.524	2.757
		c-C s	2.700	0.292	6.849	2.278	4.571	2.772
		d-C 2	2.844	1.171	7.169	1.657	5.512	2.755
	7	a-C 2v	2.867	0.980	7.300	2.157	5.143	2.777
		b-1-Cs	2.903	1.214	7.122	1.714	5.407	2.762
		c-2-C s	2.782	1.057	7.064	1.915	5.149	2.721
		d-C 2v	2.838	1.197	7.233	1.897	5.335	2.768
	8	a-C4v	2.874	1.360	7.258	1.915	5.343	2.663
		b-D 3h	2.716	0.397	6.817	2.600	4.217	2.690
		c-C 2v	2.675	0.508	6.783	2.401	4.382	2.787
		d-C 2v	2.837	0.898	6.972	2.087	4.885	2.826
	9	a-D 2h	2.947	1.293	7.139	1.948	5.191	2.766
		b-D 4h	2.735	0.001	6.551	2.678	3.873	2.740
		c-C1	2.953	1.297	7.013	1.851	5.162	2.748
		d-C 1	2.895	1.106	6.936	1.980	4.956	2.771
		e-C 2v	2.846	0.997	6.933	2.077	4.855	2.686
	10	a-C s	2.928	0.807	6.850	2.330	4.520	2.786
		b-C 2	2.954	1.229	6.784	1.837	4.946	2.758
		c-C2v	2.959	0.861	6.923	2.339	4.584	2.839
		d-C 2v	2.850	1.051	6.643	1.965	4.679	2.695
	11	a-C 2h	2.976	0.578	6.586	2.390	4.195	2.755
		b-D 2h	2.899	0.491	6.839	2.690	4.149	2.706
		c-D 4h	2.872	0.268	6.928	2.986	3.942	2.787
		d-C s	2.886	0.754	6.847	2.507	4.340	2.738
		e-C 1	2.935	0.913	6.641	2.123	4.517	2.795
		f-C2v	2.979	0.442	6.689	2.517	4.171	2.817
	12	a-C 2v	2.824	0.159	6.129	2.373	3.756	2.885
		b-C 4v	2.917	0.708	6.998	2.687	4.311	2.824
		c-C 1	2.984	1.095	6.902	2.268	4.634	2.771
		d-C1	2.994	1.127	6.978	2.318	4.660	2.721
		e-C 1	3.010	1.127	6.906	2.299	4.607	2.811
	13	a-O h	2.940	1.888	7.149	3.608	3.541	2.810
		b-C1	2.987	0.812	6.472	2.238	4.234	2.725
		c-C 1	2.970	1.138	6.675	2.220	4.455	2.752
		d-C 1	2.952	0.853	6.636	2.395	4.240	2.753
		e-C 1	2.955	0.791	6.528	2.295	4.233	2.832
		f-C 1	2.948	0.953	6.633	2.337	4.296	2.781
	14	a-O h	2.912	0.525	6.135	2.278	3.857	2.873
		b-C1	3.018	1.108	6.679	2.207	4.472	2.772
		c-C1	3.019	0.958	6.856	2.514	4.342	2.791
		e-C 1	2.955	0.848	6.394	2.256	4.138	2.737
		f-C s	2.964	1.136	6.689	2.177	4.512	2.726
	15	a-D2h	3.052	1.106	6.661	2.278	4.383	2.759
		b-C s	2.980	0.900	6.627	2.415	4.212	2.751
		c-C 2v	3.022	0.594	6.467	2.519	3.948	2.829
		d-C 1	2.982	1.172	6.824	2.347	4.477	2.794
		e-C 1	2.976	0.935	6.526	2.392	4.134	2.732
	16	a-Cs	3.010	0.719	6.584	2.682	3.902	2.805
		b-C 2	2.959	0.541	6.442	2.774	3.668	2.792
		c-C 1	2.990	1.130	6.597	2.302	4.295	2.797
	17	a-C 2v	2.948	0.671	6.420	2.554	3.866	2.821
		b-C2	3.046	0.777	6.531	2.599	3.932	2.732
		c-C 1	2.990	0.991	6.518	2.415	4.103	2.772
		d-C s	2.976	0.693	6.619	2.827	3.792	2.764
		e-C s	2.942	0.274	6.265	2.823	3.442	2.726
	18	a-C2v	3.054	1.074	6.811	2.617	4.194	2.769
		b-C 1	3.039	0.829	6.392	2.450	3.942	2.766
		c-C 1	3.017	1.065	6.652	2.543	4.109	2.728
		d-C 3v	3.024	1.255	6.504	2.168	4.336	2.748

Table 3 :

 3 Symmetry group, binding energy Eb (eV), HOMO-LUMO gap ΔE (eV), total spin magnetic moment µ (µb), vertical ionization potential VIP (eV) and electron affinity VEA (eV), chemical hardness η (eV) and bond length a(Ge-Ge. Ge-V) (Å) for VGen (n=1-19) clusters.

	n	Symmetry	Eb (eV)	ΔE	µ (µB)	VIP (eV)	VEA (eV)	η(eV)	aGe-Ge	aGe-V
				(eV/atom)
	1	a-C∞v	1.198	0.746	4.999	6.257	0.608	5.648	-	2.526
	2	a-D∞h	1.450	0.922	8.909	6.786	1.595	5.191	-	2.588
		b-C2v	2.159	0.942	3.000	6.938	0.845	6.093	2.405	2.536
	3	a-C2v	2.330	0.747	4.999	6.392	1.369	5.023	2.423	2.684
		b-C3v	2.350	0.533	2.999	6.352	0.927	5.425	2.676	2.599
	4	a-Cs	2.560	0.606	2.999	6.820	1.515	5.305	2.713	2.600
		b-C2v	2.409	0.186	4.988	6.747	2.275	4.472	2.496	2.696
		c-C2v	2.347	0.430	3.000	5.937	1.337	4.600	2.580	2.493
		d-C3v	2.494	0.306	0.999	6.625	1.736	4.889	2.746	2.588
	5	a-Cs	2.620	0.908	3.000	6.528	1.657	4.871	2.577	2.655
		b-Cs	2.602	0.499	4.953	6.361	1.930	4.431	2.726	2.679
		c-C2v	2.733	0.726	4.997	6.648	1.817	4.831	2.704	2.770
		d-C1	2.694	0.730	3.000	6.916	1.932	4.983	2.699	2.573
	6	a-C2v	2.838	0.363	3.000	6.392	1.745	4.647	2.746	2.750
		b-C2v	2.686	0.341	2.999	6.687	2.288	4.399	2.655	2.820
		c-C1	2.769	0.495	4.996	6.039	1.696	4.343	2.737	2.802
		d-C3v	2.837	0.913	2.999	6.891	1.846	5.046	2.494	2.816
	7	a-C3v	2.822	0.591	1.001	6.446	2.017	4.429	2.586	2.805
		b-Cs	2.905	0.842	3.000	6.666	1.936	4.730	2.713	2.837
		c-Cs	2.777	0.294	2.999	6.861	2.569	4.292	2.765	2.717
		d-C2v	2.823	0.596	0.999	6.806	2.321	4.485	2.734	2.597
	8	a-Cs	2.906	0.616	3.000	6.632	2.200	4.432	2.757	2.731
		b-C2v	2.765	0.393	2.998	6.790	2.620	4.170	2.663	2.591
		c-C2v	2.738	0.575	2.999	6.923	2.637	4.286	2.649	2.717
		d-C2v	2.966	0.570	1.000	7.043	2.727	4.316	2.758	2.809
	9	a-Cs	2.952	0.833	2.999	6.643	2.029	4.614	2.797	2.685
		b-C4v	2.826	0.485	4.899	6.185	2.092	4.093	2.702	2.675
		c-C1	2.953	0.537	2.999	6.648	2.459	4.189	2.712	2.786
		d-C1	2.884	0.444	3.000	6.372	2.339	4.032	2.786	2.755
		e-Cs	2.996	0.651	3.000	6.663	2.244	4.419	2.768	2.773
		f-C1	2.992	0.621	1.002	6.910	2.868	4.042	2.765	2.809
	10	a-C1	2.924	0.564	3.000	6.451	2.189	4.262	2.775	2.695
		b-C1	2.977	0.492	3.000	6.321	2.311	4.010	2.786	2.714
		c-C2v	3.058	0.609	1.000	6.727	2.603	4.124	2.827	2.766
		d-C1	2.913	0.402	3.001	6.172	2.357	3.815	2.736	2.779
	11	a-C2v	3.024	0.446	2.991	6.560	2.817	3.744	2.689	2.920
		b-Cs	2.951	0.831	4.999	5.981	2.010	3.971	2.660	2.630
		c-Cs	2.992	0.698	1.010	6.526	2.789	3.737	2.750	2.871
		d-C1	2.984	0.676	2.999	6.378	2.214	4.164	2.750	2.812
		e-C2v	2.977	0.513	3.000	6.452	2.483	3.969	2.804	2.721
	12	a-C2v	3.018	0.560	3.003	6.948	3.152	3.796	2.772	2.914
		b-C1	3.036	0.733	3.000	6.739	2.615	4.124	2.775	2.873
		c-C1	3.034	0.674	2.999	6.801	2.728	4.073	2.790	2.811
		d-Cs	3.108	0.712	2.999	6.816	2.685	4.131	2.792	2.879
		e-D4h	3.071	0.711	3.000	6.919	2.632	4.287	2.747	2.902
		f-D3d	3.127	0.813	2.999	6.919	2.693	4.226	2.612	2.899
	13	a-C6v	3.131	1.001	1.000	6.763	2.366	4.397	2.695	2.889
		b-C2v	3.029	0.675	2.999	6.445	2.439	4.006	2.765	2.814
		c-C1	3.047	0.629	1.000	6.494	2.828	3.666	2.739	2.845
		d-C1	3.019	0.386	2.999	6.333	2.649	3.684	2.763	2.840
	14	a-Oh	3.183	1.495	2.999	7.120	2.297	4.823	2.693	2.913
		b-D6h	3.100	0.642	2.999	6.488	2.491	3.997	2.780	2.920
		c-C1	3.027	0.320	3.000	6.023	2.544	3.479	2.692	2.901
		d-Cs	3.110	0.333	3.000	6.494	2.925	3.569	2.625	2.944
	15	a-C1	3.099	0.519	3.000	6.491	2.784	3.707	2.778	2.878
		b-C1	3.026	0.530	3.000	6.490	2.762	3.728	2.781	2.682
		c-Cs	3.179	0.469	1.000	6.532	2.760	3.772	2.755	3.005
		d-C1	3.067	0.701	3.000	6.376	2.660	3.716	2.755	2.890
		e-C1	3.062	0.557	3.000	6.383	2.645	3.738	2.776	2.918
	16	a-D2h	3.052	0.442	1.000	6.674	3.047	3.627	2.749	3.025
		b-C1	3.024	0.464	2.999	6.358	2.817	3.541	2.746	2.831
		c-C1	3.028	0.383	3.000	6.331	2.918	3.413	2.814	2.729
		d-C1	3.092	0.630	1.000	6.388	2.838	3.550	2.799	2.888
	17	a-C1	3.099	0.485	2.999	6.250	2.770	3.480	2.755	2.823
		b-C1	3.033	0.375	3.004	6.210	2.834	3.376	2.808	2.801
		c-C1	3.078	0.560	2.999	6.371	2.791	3.580	2.775	2.822
		d-C1	3.043	0.537	2.999	6.227	2.718	3.509	2.755	2.867
		e-C1	3.085	0.630	2.999	6.193	2.599	3.594	2.747	2.876
	18	a-C2v	3.056	0.478	2.999	6.492	3.053	3.439	2.786	2.795
		b-C1	3.104	0.543	3.000	6.349	2.914	3.435	2.776	2.875
		c-C6v	3.016	0.433	3.000	6.351	2.906	3.445	2.612	2.922
		d-Cs	3.036	0.614	4.999	6.029	2.641	3.388	2.801	2.722
		e-C1	3.107	0.682	2.999	6.413	2.728	3.685	2.723	3.041
	19	a-Cs	3.049	0.339	3.000	6.284	2.932	3.352	2.769	2.701
		b-C1	3.081	0.540	2.999	6.207	2.699	3.508	2.771	2.764
		c-C1	3.116	0.552	2.995	6.293	3.092	3.201	2.714	2.836

Acknowledgement

FR thanks the Pôle Scientifique de Modélisation Numérique (PSMN) at Lyon, France, and the GENCI-IDRIS (Grant i2016086864) center for generous allocation of computational time.

VGe12-e D4h VGe5-d C1 VGe6-a C2v VGe6-b C2v VGe6-c C1 VGe6-d C3v VGe7-a C3v VGe7-b Cs VGe7-c Cs VGe8-a Cs VGe8-b C2v VGe7-d C2v VGe8-c C2v VGe8-d C2v VGe9-b C4v VGe9-d C1 VGe9-c C1 VGe9-a Cs VGe11-a C2v VGe11-c Cs VGe11-b Cs VGe11-d C1 VGe11-e C2v VGe9-e Cs VGe9-f C1 VGe10-a C1 VGe10-d C1 VGe10-b C1 VGe10-c C2v c Fig02. Continued. VGe17-e C1 VGe15 -b C1 VGe15-c Cs VGe15-d C1 VGe15-e C1 VGe16-a D2h VGe18-c C6v VGe18-d Cs VGe18-e C1 VGe19-a Cs VGe19-b C1 VGe14-a Oh VGe14-b D6h VGe14-c C1 VGe14-d Cs VGe15-a C1 VGe17-c C1 VGe17-d C1 VGe18-a C2v VGe18-b C1 VGe13-a C6v VGe13-b C1 VGe13-c C1 VGe13-d C1 VGe13 (C1) d VGe12-f D3d VGe16-d C1 VGe16-c C1 VGe16-b C1 VGe17-b C1 VGe17-a C1 VGe19-c C1 VGe19-d C6v VGe19-

VGe 14