Study of electronic properties, stabilities and magnetic quenching of molybdenum-doped germanium clusters: a density functional investigation

Abstract

Evolution of electronic structures, properties and stabilities of neutral and cationic molybdenum encapsulated germanium clusters (Mo@Ge_n, n = 1 to 20) has been investigated using the linear combination of atomic orbital density functional theory method with effective core potential. From the variation of different thermodynamic and chemical parameters of the ground state clusters during the growth process, the stability and electronic structures of the clusters is explained. From the study of the distance-dependent nucleus-independent chemical shifts (NICS), we found that Mo@Ge₁₂ with hexagonal prism-like structure is the most stable isomer and possesses strong aromatic character. Density of states (DOS) plots of different clusters is then discussed to explain the role of d-orbitals of the Mo atom in hybridization. Quenching of the magnetic moment of the Mo atom with increase in the size of the cluster is also discussed. Finally, the validity of the 18-electron counting rule is applied to further explain the stability of the metallo-inorganic magic cluster Mo@Ge₁₂ and the possibility of Mo-based cluster-assembled materials is discussed.

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1. Introduction

The number of electrons involved in the growth of nanoclusters and cluster-assembled materials by formation of chemical bonds is the fundamental concept used to explain and understand the electronic properties and stabilities of nanomaterials. In the last few decades, searching for stable hybrid nanoclusters, particularly transition metal-doped semiconductor nanoclusters, is an extremely active area of research due to their potential applications in nanoscience and nanotechnology. One of the challenges in the computational materials design or synthesis of such materials is to find the clusters that are likely to retain their properties and structural reliability during the formation of cluster assembled materials.¹ Among these materials, those in transition metal-doped semiconductor clusters and cluster-assembled materials are interesting, and it is important to understand the physical and chemical processes taking place at the metal–semiconductor interface for their application as nano-devices.² Pure semiconductor nanoclusters are not really stable, and it is a challenging job to make them stable. Among the different possibilities of stabilizing semiconductor nanoclusters, encapsulation of a transition metal (TM) in a pure semiconductor cage is one of the most effective methods. Many insights into the transition metal-doped Si and Ge clusters were reported in the previously studied reports and also explanations of their stabilities on the basis of electron counting rules.^3–11 The existence of several stable transition metal-doped semiconductor nanoclusters has already been experimentally verified by Beck et al.^12,13 using laser vaporization techniques. Recently, Atobe et al.¹⁴ investigated the electronic properties of transition metal- and lanthanide metal-doped M@Ge_n (M = Sc, Ti, V, Y, Zr, Nb, Lu, Hf, or Ta) and M@Sn_n (M = Sc, Ti, Y, Zr, or Hf) by anion photoelectron spectroscopy and explained the stability of the clusters using electron counting rules. In a theoretical study Hiura et al.¹⁵ argued that the magic nature of a W@Si₁₂ cluster is because of the 18-electron filled shell structure, assuming each silicon atom donates one valence electron to the encapsulated transition metal, which is donating six valence electrons to hybridization. Wang and Han¹⁶ found that the encapsulation of a Zn atom in a germanium cage starts from n = 10, whereas ZnGe₁₂ is the most stable species that is not an 18-electron cluster. In another study, Guo et al.¹⁷ explained the stability of M@Si_n (M = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn; n = 8–16) nanoclusters using a shell filling model, where the d-shell of the transition metals plays an important roll in hybridization to make a closed shell structure. In this context, more correct information was reported by Reveles and Khanna.¹⁸ They considered that the valence electrons in TM-Si clusters to behave like a nearly free-electron gas and that one needs to invoke the Wigner–Witmer (WW) spin conservation rule¹⁹ when calculating the embedding energy of the clusters to explain their stability. It is worth mentioning here that the one-electron levels in spherically confined free-electron gas follow the sequence 1S²1P⁶1D¹⁰2S²… thus, 2, 8, 18, 20, etc., are the shell filling numbers and clusters having these numbers of valence electrons attain enhanced stability. However, in some cases this theory is not valid. For example, by applying the Wigner–Witmer (WW) spin conservation rule¹⁹ and without applying it, Reveles and Khanna¹⁸ found that CrSi₁₂ and FeCr₁₂ in neutral state exhibit the highest binding energy, whereas anionic MnSi₁₂, VSi₁₂ and CoSi₁₂ show maximum embedding energy, which is one of the most important parameter needed to understand the stability of nanoclusters. Therefore, both 18- and 20-electron counting rules are valid for different clusters in different charged states for explaining stability. Experiments also supported the validity of these electron-counting rules in some of the charged clusters. Koyasu et al.²⁰ studied the electronic and geometric structures of transition metal (Ti, V and Sc) doped silicon clusters in neutral and different charged states by mass spectroscopy and anion photoelectron spectroscopy. They found that neutral Ti@Si₁₆, cationic V@Si₁₆ and anionic Sc@Si₁₆ clusters were produced in great abundance, which follows the 20-electron counting rule. In summary, it was found that most of the researched transition metal-doped semiconductor clusters show maximum stability in closed-shell electron configurations with 18 and 20 valence electrons in the cluster by taking into account the fact that each germanium or silicon atom contributes one electron for bonding with the transition metal atom. In the present study, we make an effort to explain the enhanced stability of MoGe₁₂ in Mo@Ge_n (n = 1–20) by following the behavior of different physical and chemical parameters of the ground state clusters of each size using density functional theory (DFT). Detailed studies on this system are important to understand the science behind the cluster stability and its electronic properties. DOS plots of different clusters are also discussed to explain the role of d-orbitals of Mo atom in the hybridization and in the quenching of magnetic moment of Mo atom in the germanium cluster. In addition, to understand the enhanced stability of the MoGe₁₂ isomer, distance dependence nucleus-independent chemical-shift (NICS), which is the measure of the aromaticity of the cluster, is calculated and its role in stability is discussed. Finally, the electron-counting rule is applied to understand the stability of the Mo@Ge₁₂ cluster and the possibility of Mo-based cluster assembled materials.

2. Theoretical method and computational details

All calculations were performed within the framework of linear combination of atomic orbital's density functional theory (DFT). The exchange–correlation potential contributions were incorporated into the calculation using the spin-polarized generalized gradient approximation (GGA) proposed by Lee, Yang and Parr, popularly known as B3LYP.²¹ Different basis sets were used for germanium and molybdenum with effective core potential using a Gaussian’03 (ref. 22) program package. The standard LanL2DZdp and LanL2DZ basis sets were used for germanium and molybdenum to express molecular-orbitals (MOs) of all atoms as linear combinations of atom-centered basis functions. LanL2DZdp is a double-ζ, 18-valence electron basis set with a LANL effective core potential (ECP) and with polarization function.^23,24 All geometry optimizations were performed with no symmetry constraints. During optimization, it is always possible that a cluster with a particular guess geometry is trapped in a local minimum of the potential energy surface. To avoid this, we used a global search method using USPEX²⁵ and VASP^26,27 to get all possible optimized geometric isomers for each size, from n = 5 to 20. The optimized geometries were then optimized again in the Gaussian'03 (ref. 22) program using different basis sets, as mentioned above, to understand the electronic structures. In order to check the validity of the present methodology, a trial calculation was carried out on Ge–Ge, Ge–Mo and Mo–Mo dimers using different methods and basis sets. Detailed results of the outputs are presented in Table 1. The bond length of a germanium dimer at triplet spin state (ground state) was found to be 2.44 Å (with a lowest frequency of 250 cm⁻¹) in the present calculation, which is within the range of the values obtained theoretically as well as experimentally by several groups (Table 1). The bond length and the lowest frequency of the Ge–Mo dimer in the quintet spin state (ground state) were obtained in the present calculation to be 2.50 Å and 207.82 cm⁻¹, respectively. The values reported by other groups are 2.50 Å and 208 cm⁻¹, as shown in Table 1. The optimized electronic structure is obtained by solving the Kohn–Sham equations self-consistently³³ using the default optimization criteria of the Gaussian’03 program.²² Geometry optimizations were carried out to a convergence limit of 10⁻⁷ Hartree in total optimized energy. The optimized geometries as well as the electronic properties of the clusters in each size were obtained from the calculated program output.

Table 1 Bond lengths and lowest frequencies of Ge–Ge, Ge–Mo and Mo–Mo dimers

Dimer	Bond lengths (Å)	Lowest frequencies (cm^−1)
a B3LYP/Lanl2dz-ECP.b B3LYP/aug-cc-pvdz.c B3LYP/aug-cc-pvtz-pp.d M06/aug-cc-pvtz-pp.e M06/Lanl2dz-ECP.
Ge–Ge	2.44^a, 2.44^b, 2.44^c, 2.39^d, 2.3^e, 2.36–2.42,^28,29 2.46 (ref. 32)	250.63^a, 261^b, 263^c, 282^d, 317.12^e, 258 (ref. 28)
Ge–Mo	2.5^a, 2.48^b, 2.51^c, 2.41^d, 2.43^e, 2.50 (ref. 29)	202.56^a, 218^b, 198^c, 251.78^d, 252.53^e, 287.82 (ref. 29)
Mo–Mo	1.97^a, 2.5^b, 2.5^c, 1.88^d, 1.89^e, 1.98 (ref. 30)	561.79^a, 567.24^b, 570.36^c, 582.1^d, 583.07^e, 562,^30 477 (ref. 31)

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3. Results and discussions

The molybdenum atom, a typical 4d transition metal, has an electronic configuration of [Kr]4d⁵5s¹, where both the ‘d’ and ‘s’ shells are half-filled. Optimized ground state clusters with the point group symmetry are shown in ESI Fig. 1Sa.† As per the growth pattern of Ge_nMo clusters from n = 1 to 7, the Mo is absorbed onto the surface of the Ge_n cluster or replaces a Ge atom from the surface of the Ge_n+1 cluster to form a Ge_nMo cluster, where Mo atoms in all clusters are exposed on the outside. In the next stage of the growth pattern, Mo is absorbed partially by the Ge_n clusters of n = 8 and n = 9. Complete encapsulation starts from n = 10. The low energy structures within the size range n = 10 to 16 are all very well known for most of the transition metal-doped silicon and germanium clusters and are also reported by others.^34–38 The first encapsulated ground state isomer Mo@Ge₁₀ is icosahedral, where the Mo atom hybridizes with all ten germanium atoms in the cage. Addition of one germanium atom on the surface of ground state Mo@Ge₁₀ isomers gives an endohedral Mo-doped Mo@Ge₁₁ cluster. Endohedrally absorbed Mo into the hexagonal prism-like structure of Mo@Ge₁₂ is the ground state isomer at size n = 12. Here Mo is bonded with all twelve germanium atoms in the cage. In this structure, the Mo atom is placed between two parallel benzene-like hexagonal Ge₆ surfaces. The ground state isomer of the Mo@Ge₁₃ structure is a Mo-encapsulated hexagonal-capped bowl kind of structure. The structure can be understood by capping one germanium atom with the hexagonal plane of the n = 12 ground state isomer. The ground state structure of Mo@Ge₁₄ is a combination of three rhombuses and six pentagons, where the rhombuses are connected only with the pentagons. It is a threefold symmetric structure. The other bigger structures can be understood by adding a single Ge or a Ge–Ge dimer to the lower size structures. In all the ground state Ge_nMo clusters from n = 10 to 14, Mo atoms take an interior site in the Ge_n cages and make the cages more symmetric compared with the pure Ge_n cages. This continues up to the end of the size range in the present study. Among all these nanoclusters between 8 ≤ n ≤ 20, the ground state Mo@Ge₁₂ is the most symmetric.

3.1. Electronic structures and stabilities of Mo@Ge_n nanoclusters

We first studied the energetics of pure Ge_n and Mo@Ge_n clusters. Then, we explored the electronic properties and stabilities of the Mo@Ge_n clusters by studying the variation of different thermodynamic parameters of the clusters, such as average binding energy (BE), embedding energy (EE), fragmentation energy (FE) and second order change in energy (Δ₂), with the increase of the cluster size, as per the reported work.^7–11 The average binding energy per atom of Mo@Ge_n clusters is defined here as follows:

BE = (E_Mo + nE_Ge − E_Mo@Gen)/(n + 1)

and by definition it is always positive. The variation of the binding energy of the clusters with the cluster size is presented in Fig. 1. For pure germanium clusters E_Mo in the abovementioned equation is taken as zero and n + 1 is replaced by n. As per the graphs, the binding energy of small-sized clusters in the size range from 1 to 5 increases rapidly. This is an indication of the thermodynamic instability of these clusters (both pure and doped Ge_n). For the sizes n > 5 the binding energy curve increases with a relatively slower rate. Binding energy of the Mo doped clusters is always higher than that of the same size pure germanium cluster for n > 6, which indicates that the doping with transition metal atom helps to increase the stability of the clusters. It is to be noted that there are two local maxima in the binding energy graph at n = 12 and 14. According to the 18- or 20-electron counting rule, the binding energy and other thermodynamic parameters should show a local maxima (or minima) at n = 12 and 14 for neutral clusters, respectively. Other 18- and 20-electron clusters are at n = 13 and 15 in cationic and n = 11 and 13 in anionic states, assuming each germanium atom is contributing one valence electron to hybridization with the Mo, as per our previous work.¹⁰ As per Fig. 1, the behaviour of the neutral and anionic clusters is same, and both of them show a peak at n = 12 in the binding energy graph. However, the cationic cluster shows a peak at n = 13 and it follows the demand of the 18-electron counting rule. Because of the anomalous behaviour of the anionic clusters, in the present study we considered only neutral and cationic clusters. Another important parameter that explains thermodynamic stability of the nanoclusters is embedding energy (EE). In the present study, the embedding energy of a cluster after imposing the Wigner–Witmer spin-conservation rule¹⁹ is defined as follows:

EE^WW = E(^MGe_n) + E(⁰Mo) −E (^MGe_nMo)

or,

EE^WW = E(⁰Ge_n) + E(^MMo) −E (^MGe_nMo)

where M is the total spin of the cluster or the atom in units of h/2π. As per this definition, EE is positive, which means the addition of a transition metal atom to the cluster is favorable. In the abovementioned embedding energy expressions, we have chosen the higher of the resulting two EEs. In the present calculation, ground states for n = 1 and 2 are quintet and triplet, respectively. For n > 2, all ground states are in singlet state. Therefore, to calculate the EE according to the WW spin-conversation rule, pure Ge clusters were taken to be in either the triplet or the singlet state. For cationic Mo@Ge_n clusters the EE can be written as follows:

EE^WW = E(^MGe_n^±) + E(⁰Mo) − E(^MGe_nMo^±)

or,

EE^WW = E(⁰Ge_n) + E(^MMo^±) − E(^MGe_nMo^±)

Fig. 1 Variation of average binding energy of the clusters with the cluster size (n).

Variation of EE and ionization potential with the size of cluster is shown in Fig. 2a. Both neutral and cationic clusters show maxima at n = 12 and 13, respectively. Both the clusters are 18-electron clusters. To check whether the neutral and cationic clusters are following the 20-electron counting rule, we studied the BE and EE values at n = 14 and 15. In the BE graph at n = 14, there is no relative maxima. At n = 14, EE shows a local minimum. Hence, it clearly shows that the n = 14 ground state cluster does not follow the 20-electron counting rule. To further check the stability of the clusters during the growth process by adding germanium atoms one by one to the Ge–Mo dimer, the fragmentation energy (FE or Δ(n, n−1)) and 2nd order difference in energy (Δ₂ or stability), are calculated following the relations given below:

Δ(n, n − 1) = −(E_Gen−1Mo + E_Ge − E_GenMo)

Δ₂(n) = −(E_GenMo + E_Gen−1Mo − 2E_GenMo)

which means that higher positive values of these parameters indicate the higher stability of the clusters compared to its surrounding clusters during the growth process. Variations of fragmentation energy and stability with size for neutral and cationic clusters are shown in Fig. 2b and c, respectively. The sharp rise in FE from n = 11 to 12 and sharp drop in the next step from n = 12 to 13 during the growth process indicates that in the neutral state the Mo@Ge₁₂ size is favorable compared to its neighboring sizes. The same is true for cationic clusters at n = 13. This is an indication of the higher stability of neutral Ge₁₂Mo and cationic Ge₁₃Mo clusters. There is a sharp rise in Δ₂ when ‘n’ changes from 11 to 12 and from 12 to 13 in neutral and cationic states, respectively, as shown in Fig. 2c. This is an indication of the higher stability of the clusters at n = 12 and 13 in neutral and cationic states, respectively. Drastic drops in Δ₂ from n = 12 to 13 in neutral and from n = 13 to 14 in cationic clusters are again indication of the enhanced stability of these clusters. Both of these parameters are again supporting the enhanced stability of ground state neutral n = 12 and cationic n = 13 clusters during the growth process and follow the 18-electron counting rule. The binding energy of the clusters, both in pure Ge_n and Ge_nMo, first increases rapidly and then saturates with a small fluctuation. However, the variation of Δ₂ and Δ is oscillatory in nature. We also measured the gain in energy in pure germanium clusters. The gain in energy (2.83 eV) in a pure Ge₁₃ cluster is higher than that of Ge₁₂ (2.68 eV) and Ge₁₄ (2.80 eV). The gain in energy is even more in doped clusters. For Ge₁₁Mo, Ge₁₂Mo and Ge₁₃Mo, these values are 2.33 eV, 3.13 eV and 2.30 eV, respectively. Though the FE and stability are oscillatory in nature, from the systematic behaviour of these two parameters at n = 12 (neutral) and 13 (cationic) sizes, we can take these two 18-electron clusters as the most stable clusters in the neutral and cationic Mo@Ge_n series. Therefore, it is clear that BE, EE, FE and Δ₂(n) parameters support the relatively higher thermodynamic stability of Mo@Ge₁₂ in neutral and Mo@Ge₁₃ in cationic states, where both the clusters have a closed shell of 18-electron filled structure.

Fig. 2 Variation of (a) embedding energy (EE) and ionization potential (IP), (b) stability, and (c) fragmentation energy (FE) of neutral and cationic Mo@Ge_n clusters with the cluster size (n).

To understand the stability of the Ge₁₂Mo cluster we further studied the charge exchange between the germanium cage and the embedded Mo atom in hybridization during the growth process using Mulliken charge population analysis, shown in ESI Fig. 2S.† Similar to the other thermodynamic parameters, the charge on the Mo and Ge atoms show a global maximum and minimum, respectively, at n = 12. The electronic charge transfer is always from the germanium cage to Mo atom in different Mo@Ge_n clusters. In the figure, the charge on Mo is plotted in units of ‘e’, the electronic charge. Because the average charge per germanium atom and the charge on the molybdenum atom in the Ge₁₂Mo cluster are at a minimum and maximum, respectively, the electrostatic interaction increases and hence improves the stability of the Ge₁₂Mo cluster. The effect of ionization (from neutral atom to cation or anion of n = 12 ground state) that gives redistribution of electronic charge density in the orbitals can be seen from the orbital plot in ESI Fig. 1Sb.† With reference to Fig. 1Sb,† with addition of one electron to a Ge₁₂Mo neutral cluster, the higher order orbitals just shift one step down and the orbitals are held similar to those of the neutral cluster. For example, the HOMO, LUMO and LUMO+1 orbitals of neutral Ge₁₂Mo shifts to HOMO-1, HOMO and LUMO orbitals of anionic Ge₁₂Mo, respectively. However, the HOMO and LUMO orbitals remain unchanged when a neutral Ge₁₂Mo cluster is ionized to a cationic cluster. Details of the natural electronic configuration (NEC) for the Ge₁₂Mo ground state cluster are shown in Table 2. By combining Fig. 2S in ESI† and Table 2, it can be seen that when the charge transfer takes place between the germanium cage and the Mo atom, at the same time there is rearrangement of electronic charge in the 5s, 4p and 4d orbitals in Mo and the 3d, 4s and 4p orbitals of Ge to make the cluster stable. According to Table 2, the main charge contribution in hybridization between Mo and Ge are from d-orbitals of Mo and s, p orbitals of Ge atoms in the ground state Mo@Ge₁₂ cluster. The average charge contribution from s, p and d orbitals of Ge are in the ratio of 1.22 : 1.04 : 0.05, whereas in Mo the ratio is 0.37 : 0.48 : 4.28. In the Ge₁₂Mo cage, the Mo atom gains about 4.0 electronic charges from the cage, whereas average charge contribution from the Ge atoms is 0.34e, which means that the Mo atom behaves as a bigger charge receiver or as a superatom. This enhances the electrostatic interaction between the cage and the Mo atom, which plays an important role in stabilizing the Ge₁₂Mo cage as well as its magnetic moment quenching.

Table 2 Natural electronic configuration (NEC) in Mo@Ge₁₂

Atom	Orbital charge contribution			Total charge	NEC
	s	p	d
Ge	1.220	1.052	10.002	12.274	4s^1.2204p^1.0523d^10.002
Ge	1.218	1.057	10.002	12.277	4s^1.2184p^1.0573d^10.002
Ge	1.219	1.054	10.002	12.275	4s^1.2194p^1.0543d^10.002
Ge	1.219	1.053	10.002	12.274	4s^1.2194p^1.0533d^10.002
Ge	1.221	1.049	10.002	12.271	4s^1.2214p^1.0493d^10.002
Ge	1.217	1.058	10.002	12.277	4s^1.2174p^1.0583d^10.002
Ge	1.219	1.051	10.002	12.272	4s^1.2194p^1.0513d^10.002
Ge	1.221	1.046	10.001	12.268	4s^1.2214p^1.0463d^10.001
Ge	1.219	1.050	10.002	12.271	4s^1.2194p^1.0503d^10.002
Ge	1.219	1.053	10.002	12.273	4s^1.2194p^1.0533d^10.002
Ge	1.217	1.054	10.002	12.273	4s^1.2174p^1.0543d^10.002
Ge	1.221	1.045	10.001	12.268	3d^1.2214s^1.0454p^10.001
Mo	0.375	0.484	4.273	5.132	5s^0.3754p^0.4844d^4.273

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We obtained similar information from the total density of states plot with s-, p-, and d-site projected density of state contribution of the Mo atom in different clusters in the size range n = 10 to 14 and in different charged states (ESI Fig. 3S†). The PDOS is calculated using the Mulliken population analysis. The DOS illustrates the presence of an electronic shell structure in Ge₁₂Mo, where the shapes of the single electron molecular orbitals (MOs) can be compared with the wave functions of a free electron in a spherically symmetric potential. The broadening in DOS occurs due to the high coordination of the central Mo atom. The phenomenological shell model in a simple way assumes that the valence electrons in a cluster are usually delocalized over the surface of the entire cluster, whereas the nuclei and core electrons can be replaced by their effective mean-field potential. Therefore, the molecular orbitals (MOs) have shapes similar to those of the s, p, d, etc., atomic orbitals which are labeled as S, P, D, etc.

Enhanced stability of the clusters is expected if the number of delocalized electrons corresponds to a closed electronic shell structure. The sequence of the electronic shells depends on the shape of the confining potential. For a spherical cluster with a square well potential, the orbital sequence is 1S²; 1P⁶; 1D¹⁰; 2S²; 1F¹⁴; 2P⁶; 1G¹⁸; 2D¹⁰; 1H²²;… corresponding to shell closure at 2, 8, 18, 20, 34, 40, 58, 68, 90, … roaming electrons. There are 54 valence electrons in Ge₁₂Mo. By comparing the wave functions, the level sequence of the occupied electronic states in Ge₁₂Mo can be described as 1S²; 1P⁶; 1D⁸ (1D_I⁸ + 1D_II²); 1F¹⁰ (1F_I⁶ + 1F_II² + 1F_III² + 1F_IV²); 2S²; 1G²; 2P⁶ (2P_I² + 2P_II⁴); 3P⁶ (3P_I² + 3P_II²); 2D². Their positions in the DOS plot are shown in Fig. 3. Due to crystal field splitting, which is related to the non-spherical or distorted spherical symmetry of the cluster, some of the orbitals with higher angular momentum lifted up.³⁹ For example, 2P orbital of the Ge₁₂Mo cluster split in two, as mentioned above. The most important difference with the energy level sequence of free electrons in a square well potential is the lowering of the 2D level. Examination of the 2D molecular orbitals show that they are mainly composed of the Mo 3d AOs, representing the strong hybridization between the central Mo with the Ge cage. The strong hybridization of the Mo 4d electrons with the Ge valence electrons (as evidenced by the PDOS shown in ESI Fig. 2S†) has implications for the quenching of the magnetic moment of Mo. According to Hund's rule, the electronic configuration in molybdenum is ([Kr] 5s¹ 4d⁵). As per this arrangement, Mo should pose a very high value of magnetic moment equal to 6 μ_B. The local magnetic moment of Mo in Ge₁₂Mo is zero, as well as in the all the ground state isomers (except for the quintet ground state of the Ge–Mo dimer and triplet ground state of Ge₂Mo). The quenched magnetic moment can be attributed to the charge transfer and the strong hybridization between the Mo 4d orbitals and Ge 4s, 4p orbitals. Mixing of the d-orbital of the transition metal is the main cause of stability enhancement in this cluster. Though contribution of the Mo d-orbital in the Ge₁₂Mo cluster is dominating, close to the Fermi energy level there is hardly any DOS or any contribution from the Mo d-orbital. This explains the presence of the HOMO–LUMO gap in the cluster and the less reactive nature of the cluster. This is also true for the ground state clusters for n = 10 and 11. From the DOS picture, it is clear that for n = 10, 11, 12 and 13 ground state clusters the HOMO–LUMO gap is comparable. The DOS of the anionic Ge₁₁Mo, which is an 18-electron cluster, shows the presence of considerable fraction of DOS on the Fermi level. Therefore, there is a possibility for the anionic Ge₁₁Mo cluster to form a ligand and be at the higher charged states by combining with other species to make a more stable species, which is an indication of the possibility of making cluster-assembled materials. To get an idea of how the magnetic moment of the clusters are changing and reducing to zero from the Ge–Mo dimer with increase of cluster size, we have studied the site of projected magnetic moment of the small-sized neutral and cationic clusters (up to n = 3), as per the work reported by Hou et al.⁵ The Ge–Ge dimer is in the triplet state with ferromagnetic coupling between the germanium atoms with total magnetic moment of 2 μ_B. On the other hand, in the Mo–Mo dimer, though the individual moment of the Mo atoms are very high, the interaction between them is antiferromagnetic, and hence the magnetic moment of Mo-dimer is reduced to zero. Detailed results of the variation of magnetic moments are given in ESI Fig. 1Sc.† The ground states of neutral and cationic Ge–Mo dimers in quintet and quartet spin states have cluster magnetic moments of 4μ and 3μ, respectively. The interaction between the Ge–Mo clusters in the ground state is antiferromagnetic with a bond length of 2.50 Å. In the cationic cluster the bond length reduces to 2.67 Å along with the presence of antiferromagnetic interactions between Ge and Mo. When the same dimer is in triplet and septet spin states, the magnetic interaction changes from antiferromagnetic to ferromagnetic, and the bond length changes from 2.34 Å to 2.73 Å, respectively. Following the electronic configuration of 10 (4 from Ge and 6 from Mo) valance electrons (triplet: σs² σs² πp² πp² σs¹ πp¹; quintet: σs² σs² πp² πp¹ σs¹ πp¹ πp¹; septet: σs² σs² πp¹ πp¹ σs¹ πp¹ πp¹ πp¹) and corresponding orbitals (ESI Fig. 1Sd†), it can be seen that while shifting from the triplet to quintet state, a beta electron from πp² state shifted to α-πp¹ state, which is at considerably lower position compared to the α-HOMO orbital. In the entire rearrangement of the orbitals due to this spin flip, the α-HOMO orbital of the triplet state moves to α-HOMO orbital of the quintet spin state with a small difference in energy of 0.08 eV and with the same antiferromagnetic interaction between the two atomic spins. It is also important to mention that in the quintet state, the local spin of Mo increases, whereas the same in Ge decreases, compared with the spins in the triplet state. Due to the transition from quintet to septet, the πp¹ (β-HOMO) shifted to the α-HOMO of energy difference of 1.10 eV compared with that of the β-HOMO in the quintet state. The magnetic interaction also changes from antiferromagnetic to ferromagnetic. In triplet and septet states, the optimized energies of the clusters are 0.25 eV and 0.57 eV, respectively, which are more compared with that of the quintet ground state. Hence the dimer Ge–Mo is found to be more stable in the quintet spin state. After addition of one germanium atom to the Ge–Mo dimer, the ground state is found to be in the triplet spin state. In the Ge₂Mo ground state cluster in triplet spin state, the interactions between the Mo and the two germanium atoms are antiferromagnetic with spin magnetic moments of 3.34 μ_B, −0.67 μ_B and −0.67 μ_B and with different bond lengths (ESI Fig. 1Sc†). Due to the antiferromagnetic bonding between the Mo and two Ge atoms, the magnetic moment reduces to 2 μ_B in the Ge₂Mo ground state cluster. The two germanium atoms are connected by π-bonding, as shown in the filled α-HOMO orbital (ESI Fig. 1Sd†). The other two low energy clusters are in singlet spin states. From the electronic configuration of 14 (4 from each Ge atoms and 6 from Mo) valance electrons (triplet: (3a₁)² 2(b₂)² 4(a₁)² 2(b₁)² 5(a₁)² 1(a₂)² 3(b₂)¹ 6(a₁)¹; quintet: (3a₁)² 2(b₂)² 4(a₁)² 2(b₁)² 5(a₁)² 1(a₂)¹ 3(b₂)¹ 6(a₁)¹ 3(b₁)¹) and corresponding orbitals (ESI Fig. 1Sd†) in Ge₂Mo, it can be seen that the β-HOMO electron from 1(a₂)² in the triplet state is transferred to α-HOMO in the quintet spin state of Ge₂Mo cluster, which is +0.93 eV higher compare to triplet α-HOMO level. During this transition the overall ground state energy change is +0.56 eV. Therefore, addition of one Ge atom to the Ge–Mo dimer in the quintet state reduces the magnetic moment, and as a result the Ge₂Mo cluster in the triplet spin state is the ground state. It is also interesting to study the charge or the orbital distributions in the β-HOMO triplet and α-HOMO quintet states of the Ge₂Mo cluster. The orbital distributions indicate the presence of electrons distributed along the bond between the Ge–Mo dimers; hence, the bonding nature is strong and the spin magnetic moment of Mo therefore reduces to 3.34 μ_B. In the same state, there is hardly any orbital distribution along the Ge–Ge bond. When it switches to the septet state, the bonding between Ge–Mo has increased and has reduced in Ge–Ge. Therefore, the spin of Mo has increased. The magnetic moment vanishes in the Ge₃Mo ground state cluster completely with no non-zero on-site spin values for the atoms. With reference to the work reported by Khanna et al.,⁴⁰ when a 3d transition atom makes bonds with a Si cluster in a Si_nTM, there always exists a strong hybridization between the 3d orbital of the TM with 3s and 3p of the Si atoms. The present investigation, as discussed above, follows the same reported by Khanna et al.⁴⁰ and is one of the strongest evidence of the quenching of spin magnetic moment of the Mo atom. The strong hybridization of 4d⁵ of Mo with the 4s²4p² of Ge atom results in the magnetic moment of Mo being quenched with no leftover part to hold its spin moment in the Ge₃Mo ground state cluster. In this context, it is also worth mentioning the work of Janssens et al.⁴¹ on the quenching of magnetic moment of Mn in Ag₁₀ cage where they suggested that the valence electrons of silver atoms in the cage can be considered as forming a spin-compensating electron cloud surrounding the magnetic impurity, which is conceptually very similar to the Kondo effect in larger systems and may be applied in our system also.

Fig. 3 Density of states of ground state Ge₁₂Mo cluster and its orbitals with their position in DOS.

To get an idea about the kinetic stability of the clusters in chemical reactions, the HOMO–LUMO gap (ΔE), ionization potential (IP), electron affinity (EA), chemical potential (μ), and chemical hardness (η) were calculated. In general, with the increase of HOMO–LUMO gap, the reactivity of the cluster decreases. Variation of HOMO–LUMO gaps of neutral and cationic Mo@Ge_n clusters is plotted and is shown in the ESI Fig. 4S.† The variation of the HOMO–LUMO gap is oscillatory. Overall there is a large variation in HOMO–LUMO gap in the entire size range from 1.5 to 3.30 eV with a local maxima at n = 12 and at n = 13 in neutral and cationic clusters, respectively. This is again an indication of enhanced stability of 18-electron clusters. The large HOMO–LUMO gap (2.25 eV) of Mo@Ge₁₂ could make this cluster a possible candidate as luminescent material in the blue region. In the neutral state the sizes n = 8, 10, 12, 14, and 18 are magical in nature, which means they have higher relative stabilities. Variation of HOMO–LUMO gap in different clusters around the Fermi level can be useful for device applications. The variation of ionization energy shown in Fig. 2a, with a sharp peak at n = 12 with a value of 7.16 eV, similar to other parameters, supports the higher stability of the Ge₁₂Mo cluster. According to the electron shell model, whenever a new shell starts filling for the first time, its IP drops sharply. De Heer⁴² has reported that in the Li_n series, the Li₂₀ cluster is a filled shell configuration and there is a sharp drop in IP when the cluster grows from Li₂₀ to Li₂₁. This is one of the most important evidence that support Ge₁₂Mo as an 18-electron cluster. There is a local peak in the IP graph at n = 12, followed by a sharp drop in IP at n = 13. The drop in IP could be the strongest indication of the assumed nearly free-electron gas inside the Ge₁₂Mo cage cluster. Following the other parameters, one may demand that the Ge₁₄Mo cluster is following the 20-electron counting rule, but we did not accept it, because the IP at n = 14 does not show a local maximum. From the abovementioned discussion, it is clear that the neutral hexagonal D_6h structure of Ge₁₂Mo, with a large fragmentation energy, average atomic binding energy and IP, is suitable as the new building block of self-assembled cluster materials. This indicates that the stability of the pure germanium cluster is evidently strengthened when the Mo atom is enclosed in its Ge_n frames. Hence, it can be expected that the enhanced stability of Mo@Ge₁₂ contributes to the initial model to develop a new type of Mo-doped germanium superatom, as well as Mo–Ge based cluster assembled materials. Further, to verify the chemical stability of Ge_nMo clusters, chemical potential (μ) and chemical hardness (η) of the ground state isomers were calculated. In practice, chemical potential and chemical hardness can be expressed in terms of electron affinity (EA) and ionization potential (IP). In terms of total energy consideration, if E_n is the energy of the n electron system, then the energy of the system containing n + Δn electrons where Δn ≪ n can be expressed as follows:

Then, μ and η can be defined as:

and

Since IP = E_n−1 − E_n and EA = E_n − E_n+1.

By setting Δn = 1, μ and η are related to IP and EA via the following relations:

and

Now, consider two interacting systems with μ_i and η_i (i = 1, 2) where some amount of electronic charge (Δq) transfers from one system to the other. The quantity Δq and the resultant energy change (ΔE) due to the charge transfer can be determined by the following explanation:

If E_n+Δq is the energy of the system after charge transfer, then it can be expressed for the two different systems 1 and 2 in the following way:

E_1n1+Δq = E_1n1 + μ₁(Δq) + η₁(Δq)²

and E_2n2−Δq = E_2n2 − μ₂(Δq) + η₂(Δq)²

Corresponding chemical potential becomes

and to first order in Δq after the charge transfer. In chemical equilibrium, μ^′₁ = μ^′₂ which gives the following expressions:

and

In the expression, energy is gained by the total system (1 and 2) due to exclusive alignment of chemical potential of the two systems at the same value. From the abovementioned expressions for easier charge transfer from one system to the other, it is necessary to have a large difference in μ together with low η₁ and η₂. Therefore, Δq and ΔE can be taken as the measuring factors to get an idea about the reaction affinity between the two systems. Because they are a function of the chemical potential and chemical hardness related to the system, it is important to calculate these parameters for a system to know about its chemical stability in a particular environment. Keeping these in mind, chemical potential (μ) and chemical hardness (η) for Mo-doped Ge_n clusters were calculated. A dip at n = 12 in the chemical potential plot (Fig. 4a) actually indicates a stable chemical species, and hence the low affinity of the system to take part in chemical reactions in a particular environment. Again at n = 12, the presence of a local peak in the chemical hardness plot also supports the result of low chemical affection of the Mo@Ge₁₂ cluster. The plot of the ratio of these two parameters in positive sense shows a peak, and hence indicates a low chemical affinity. Because n = 12 is an 18-electron cluster, it is clear that this cluster should also show low affinity in chemical reactions, and this indication of stability is in agreement with the other parameters.

Fig. 4 Variation of (a) chemical potential and chemical hardness and (b) polarizability and electrostatic dipole moments of Mo@Ge_n clusters with the cluster size.

4. Polarizability

It is known that the static polarizability is a measure of the distortion of the electronic density and sensitivity to the delocalization of valence electrons.⁴³ Hence, it is the measure of asymmetry in three-dimensional structures and orbital distributions. It gives information about the response of the system under the effect of an external electrostatic electric field. The average static polarizability is defined as follows:
in terms of the principle axis, which is a function of a basis set used in the optimization of the clusters.^44,45 In the current work, the variation of polarizability and the electrostatic dipole moment of the clusters are shown in Fig. 4b. Variation of the exact polarizability with the size of the cluster is shown in ESI Fig. 5S.† As exhibited in Fig. 4b, one can find that the polarizability of the cluster increases as a function of the cluster size ‘n’, which is nearly linear with a local dip at n = 12. At this size the electrostatic dipole moment is also at a minimum. This trend of variation of polarizability with cluster size for Mo@Ge_n clusters is similar to that of the water clusters reported by Ghanty and Ghosh.⁴⁵

5. Nucleus-independent chemical shift (NICS)

The most widely employed method to analyze the aromaticity of different species is the NICS index descriptor proposed by Schleyer et al.⁴⁶ The NICS index is defined as the negative value of the absolute shielding computed at a ring center or at some other point of the system, which can describe the system efficiently, for example, the symmetry point at the center of a hexagon. The rings with more negative NICS values are considered to be more aromatic species. On the other hand, zero (or close to zero) and positive NICS values are indicative of non-aromatic and anti-aromatic species. The NICS is usually computed at ring centers or at a distance on both sides of the ring center. The NICS obtained at 1 Å above the molecular plane⁴⁷ is usually considered to better reflect the p-electron effects than NICS (0). Because we are interested in studying the aromaticity of the overall ground state isomer Mo@Ge₁₂, which is a hexagonal prism-like structure with Mo-doped at the center, we have measured NICS values at the position of Mo and then along the symmetry axis perpendicular to the hexagonal plane surface. The NICS calculations have been performed based on the magnetic shielding using the GIAO-B3LYP level of theory by placing a ghost atom at certain points along the symmetry axis. Variation of NICS value with the distance from the center of the system is shown in Fig. 5. The nature of the variation of the NICS indicates the aromatic behavior of the cluster with a maximum negative value of −96.033 ppm at the center of the hexagonal surface and with a distance of 1.5 Å from the center of the cluster. Aromaticity of hexagonal structures (such as benzene) is an important indication of its stability. Therefore, in the present calculations the NICS behavior of Mo@Ge₁₂ also supports the stability of the cluster.

Fig. 5 NICS plot of Mo@Ge₁₂ cluster.

6. Conclusion

In summary, a report on the study of geometric and electronic properties of neutral and cationic Mo-doped Ge_n (n = 1–20) clusters within the framework of density functional theory is presented. Identification of the stable species and variation of chemical properties with the size of Mo@Ge_n clusters help to understand the science of Ge–Mo based clusters and superatoms that can be future building blocks for cluster-assembled designer materials and could open up a new field in the electronic industry. The present work is the preliminary step in this direction and will be followed by more detailed studies on these systems in the near future. On the basis of the results, the following conclusions have been drawn:

1. The growth pattern of Ge_nMo clusters can be grouped mainly into two categories. In the smaller size range, i.e., before encapsulation of Mo atom, Mo or Ge atoms are directly added to the Ge_n or Ge_n−1Mo, respectively, to form Ge_nMo clusters. At the early stage, the binding energy of the clusters increases at a considerably faster rate than that of the bigger clusters. After encapsulation of Mo atom by the Ge_n cluster for n > 9, the size of the Ge_nMo clusters tend to increase by absorbing Ge atoms one by one on their surfaces, keeping the Mo atom inside the cage.

2. It is favorable to attach a Mo atom to germanium clusters of all sizes, as the EE turns out to be positive in every case. Clusters containing more than nine germanium atoms are able to absorb a Mo atom endohedrally into a germanium cage, both in pure and cationic states. In all Mo-doped clusters beyond n > 2, the spin magnetic moment of the Mo atom is quenched in expense of stability. As measured by the BE, EE, HOMO–LUMO gap, FE, stability and other parameters both for neutral and cationic clusters, it was found that those having 18 valence electrons show enhanced stability, which is in agreement with shell model predictions. This also shows up in the IP values of the Ge_nMo clusters, as there is a sharp drop in IP when cluster size changes from n = 12 to 13. Validity of the nearly free-electron shell model is similar to that of transition metal-doped silicon clusters. Although the signature of stability is not so sharp in the HOMO–LUMO gaps of these clusters, there is still a local maximum at n = 12 for the neutral clusters, indicating enhanced stability of an 18-electron cluster, whereas this signature is very clear in the cationic Ge₁₃Mo cluster. Variation in the HOMO–LUMO gap between different sized clusters could be useful for device applications. The large HOMO–LUMO gap (2.25 eV) of Mo@Ge₁₂ could make this cluster a possible candidate as luminescent material in the blue region.

3. Major contribution of the charge from the d-orbital of Mo in hybridization and its dominating contribution in DOS indicate that the d-orbitals of Mo atoms are mainly responsible for the hybridization and stability of the cluster. Presence of the dominating contribution of the Mo d-orbital close to the Fermi level in DOS is also significant for ligand formation and a strong indication of the possibility to make stable cluster-assembled materials.

4. Computations and detailed orbital analysis of the clusters confirmed the rapid quenching of the magnetic moment of Mo in Ge_n host clusters when increasing the size from n = 1 to 3. Beyond n = 2, all hybrid clusters are in the singlet state with zero magnetic moment. Following the overall shape of the delocalized molecular orbitals of Ge₁₂Mo cage-like clusters (Fig. 3), the valence electrons of the Ge₁₂ cage can be considered as forming a spin-compensating electron cloud surrounding the magnetic Mo atom such as a screening electron cloud surrounding Mo that is similar to the magnetic element-doped bulk materials. Therefore, the system may be interpreted as very similar to that of a finite-sized Kondo system.

5. Variation of calculated NICS values with the distance from the center of the cluster clearly indicates that the cluster is aromatic in nature and the aromaticity of the cluster is one of the main reasons for its stability.

Acknowledgements

R.T., K.D. and D.B. gratefully acknowledge Dr Biman Bandyopadhyay, Department of Chemistry, IEM, Kolkata, INDIA for valuable discussions. A part of the calculation is done at the cluster computing facility, Harish-Chandra Research Institute, Allahabad, UP, India (http://www.hri.res.in/cluster/).

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Footnote

† Electronic supplementary information (ESI) available: Electronic supplementary information includes the calculated low energy isomers, variation of different thermodynamic parameters with cluster size, DOS, results of additional calculations using M06 functional, and details of bonding and anti-bonding in small-sized clusters obtained from the Gaussian outputs. See DOI: 10.1039/c4ra11825a

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