Ab initio DFT simulation of electronic and magnetic properties of Tin+1 and FeTin clusters

We report a computational investigation of the electronic and magnetic properties of neutral Tin+1 and FeTin (n = 1–10) clusters using ab initio calculations based on density functional theory (DFT) within the generalized gradient approximation (GGA). The best structures for Tin+1 and FeTin clusters are planar for size n < 5, while from n = 5, they showed a compact three-dimensional cage structure. For the best structures of the FeTin clusters, the Fe atoms favor the peripheral position with the highest coordination with the neighboring Ti atoms. The evolution as a function of the size of the average binding energies (Eb/atom) and HOMO–LUMO gaps of Tin+1 and FeTin (n = 1–10) clusters are studied. The stability results show that the Tin+1 clusters have relatively higher stability than the FeTin cluster with the same size. In addition, the vertical ionization potentials and electron affinities, chemical hardness, and atomic magnetic moment of Tin+1 and FeTin (n = 1–10) clusters are also investigated.


Introduction
During the last 4 decades, the semiconductor and transition metal clusters intensively studied because of their specific properties and their great potential use in optoelectronic materials and other nanotechnology areas. The properties of titanium clusters have been investigated because they are interesting for the fine processing and the synthesis of novel materials and the exploration of possibilities of finding novel species and phenomena. Due to the particularity of their properties, the titanium material is applied in many fields especially in medical and different industrial processes. In the literature data, less attention has been allowed to the small titanium clusters. In order to explore and understand the structural, electronic, and magnetic behavior of the pure and doped titanium clusters, different authors have recently carried out many experimental and theoretical works.
Sakurai et al. [11] using a TOF mass spectrometry of Ti n clusters with up to 30 atoms found that Ti 7 , Ti 13 , Ti 15 , Ti 19 , and Ti 25 were the magic numbers. With anion photoelectron spectra, Wu et al. [13] show that the 3d band appears at n = 8 then widens and evolves towards the bulk band in the charged Ti n − (n = 3-65) clusters. Using resonant tow-photon ionization technique, Doverstål et al. [14] studied Ti dimer and found a bond length as 1.9429 ± 0. 0013 Å. Neukermans et al. [20,21], basing on the photo-fragmentation experiments, investigated the stability of Au n X + clusters doped with a 3d atoms from Sc to Ni and Au n X m clusters (X = Sc, Ti, Cr, Fe). Kim et al. [22] studied nanoparticles of Ti-Cr using the electrical wire explosion of electrode-posted metal wires. Koyasu et al. [23] proved experimentally by using mass and photoelectron spectroscopies the high stability of MSi 16 clusters, while Furuse et al. [24] studied the MSi 16 − , MGe 16 − , MSn 16 − , and MPb 16 − by experimental and theoretical characterization, and they confirmed the exceptional stability for MSi 16 .
In the theoretical aspect, Anderson [25] investigates with very early Hükel molecular orbital calculations the properties of Ti 2-6 clusters. Wei et al. [26], by using DFT and LSDA approach, investigated the properties of Ti n (n = 2-10) clusters. They found that Ti 7 with good stability is a magic number. Zhao et al. [3] studied the properties of Ti n (n = 2-14, 19, and 55) by using the plane wave ultra-soft pseudopotential method and GGA approach. They found a pentagonal growth models for the clusters and a rapid convergence towards bulk bands for electron density of state. Using GGA, Castro [5] investigated the properties of small Ti n − and Ti n (n = 3-8, and 13) clusters. The structures of small Ti n (n = 2-5) were also studied by Duet al. [4]. Salazar-Villanueva [27] investigated the structure of Ti n (n = 2-15) clusters, and they identified that they are three magic number clusters n = 7, 13, and 15. Lee et al. [28] studied the stability of titanium clusters taking into account the spin polarization and structural distortion. The magnetic properties of Ti n (n = 2-13) clusters have been studied by Medina et al. [29]. By using genetic algorithm, Lazauskas et al. [30] investigate the potential energy surface (PES) for small Ti n (n = 2-32) clusters. Sun et al. [1] studied by fully self-consistent DFTbased calculation the evolution as a function of the size of the electronic properties of Ti n (n = 2-20). The stabilization mechanism of Ti n clusters with n = 3, 4, 5, 7, 13, 15, and 19 have also investigated by Sun et al. [2]. The same author and their co-workers have also studied the magnetic and structural properties of Ti 12 M clusters (M = Sc to Zn) [8] and Ti n-x Al x (n = 2-8, 13, x = 0-n) [15], by using the DFT approach. Some other authors are also interested to the properties of titanium clusters such as Ni x Ti y (x + y ≤ 5) [16], Ti n P (n = 1-12) [17], Au n Ti (n = 1-9) [18], Ti-Ni clusters [6,7], Pt x-y M y (M = Ti, V) [19], and Ga n Ti n (0, ±1) (n = 1-10) [31]. The purpose of this study is to investigate by using the ab initio and DFT approach of the different properties of the small-sized titanium clusters doped by iron atom. In the literature data, to our knowledge, there are no studies on the small neutral and iron-doped titanium clusters until now. In addition, many recent studies have reported the importance of Fe atom in the stabilization of metallic and semiconductor structures and that Ti-Fe alloys have good physical properties [32] and significant photocatalytic properties [33]. Moreover, Fe-Ti alloys are very important in the treatment of pollution by some chemical elements and their ecological damage, such as antimony Sb [34]. Therefore, the geometries, stabilities, electronic, and magnetic properties of Ti n and FeTi n (n = 1-10) clusters will be studied. We hoped that our results will provide powerful guidelines for further studies and would be helpful to understand the effect of the introduction of transition metal impurity on the metallic and semiconducting cages and its properties. The article is organized as follows. In the "Methods" section, we give the theoretical methods and simulation parameters which are used in this study. Presentation and the analysis of the obtained results are presented in the "Results and discussion" section, finally we give the conclusions of this work in the "Conclusion" section.

Methods
In this work, all of the calculations on the geometry optimizations of FeTi n clusters were performed using DFT approach [35] in the generalized gradient approximation (GGA) with the Perdew, Burke, and Ernzerhof (PBE) parameterization [36] for the exchange-correlation term, implemented in the SIESTA package [37]. In order to obtain the best energy structure, a large number of possible initial structures have been considered in geometry optimizations. All geometry optimizations are done with any symmetry constraints. Using the conjugate gradient scheme, the geometries are relaxed without any symmetry constraints. A large supercell with side-length of 40 Å is used to avoid the different interaction between the neighboring systems. The single gamma-point (Γ) was used in the k grid integration. Double zeta basis with polarization functions (DZP) was used for both Fe and Ti species. Self-consistent flied electronic calculations are done with a convergence condition of 10 −4 a.u. of the total energy. For the geometry optimization, the convergence criteria is 10 −3 eV/Å for the forces. The validation of this current computational was performed by the calculations tests on Ti 2 and Fe 2 dimers. The obtained results are given in Table 1. It shows the reliability of current computational method to study the small FeTi n clusters.

Geometrical structures of pure and Fe-doped Ti n clusters
In the first time, we describe the equilibrium structures of pure titanium clusters. The obtained lowest energy structures and their first close isomers are shown in Fig. 1. Their other physical parameters are given in Table 2. For Ti 2 the dimer with D ∞h symmetry, the obtained bond length Ti-Ti is 2.059 Å which is in good agreement with the previous theoretical and experimental values which are given in Table 1. For the trimer Ti 3 (a), the ground state isomers are triangular structure with C 2v symmetry. The average bond length is obtained to be 2.647 Å and an apex angle of 76.911°. For Ti 5 the geometries exploration of the ground state isomers revels that the rhombus pyramid (C 4v ) with 2.226 eV/atom is the lowest energy structures. The best energy structure for Ti 6 is a trigonal antiprism (C 2v ) with 2.536 eV/atom and average bond length Ti-Ti of 2.784 Å. For Ti 7 , the caped trigonal antiprism Ti 7 (a) with C 1 symmetry is considered as the most stable structure with 2.622 eV/atom. In the case of Ti 8 , the regular cubic structure with high symmetry O h (a) is obtained as the lowest energy structure with average binding energy of 2.554 eV/atom. For Ti 9 , the lowest energy isomer is a capped cubic structure Ti 9 (a) with average binding energy of 2.808 eV/atom and C 1 symmetry. Bi-capped square antiprism Ti 10 (a) with symmetry C s and binding energy of 2.946 eV/atom is found the lowest energy isomers for n = 10. Like spherical compact structure with one core atom, bending energy of 3.069 eV/atom and C 1 symmetry obtained as the best structure for Ti 11 .
For doped FeTi n clusters, the most stables structures and their corresponding isomers are shown in Fig. 2. For FeTi dimer, the bond length Fe-Ti is 2.941 Å much larger than their corresponding Ti 2 dimer. As in the case of Ti 3 cluster, the best isomer of FeTi 2 is triangular structure with Fe-Ti bond length of 2.887 Å and C 2v symmetry. For FeTi 3 , the lowest energy structure is triangular-based pyramid with binding energy 1.752 eV/atom and C 3v symmetry. In the case of FeTi 4 cluster, a bi-capped tetrahedron with binding energy of 1.949 eV/atom and C 3v for symmetry is found as the best isomer. For FeTi 5 , the best isomer is a distorted bi-capped tetragonal structure with 2.322 eV/atom and C s symmetry. In the case of FeTi 6 , our calculations show that the tri-capped triangular base pyramid structure is the most stable one with Cs symmetry. The Fe atom is located at the surface of the cluster cage and highly coordinated to all of the other Ti atoms in the system. The lowest energy isomer for FeTi 7 is by the composition of rectangular and tetragonal bi-capped structures with O h symmetry and binding energy of 2.576 eV/atom. This structure is only 0.041 eV/atom more stable than their first isomer FeTi 7 (b) with D 4h symmetry. For FeTi 8 , the most stable structure is composed by two distorted tetragonal-based pyramid with binding energy of 2.786 eV/atom and high symmetry D 2d . The lowest energy isomer for FeTi 9 is a like-spherical compact structure with Fe atom occupied a peripheral position (C 1 ) and binding energy of 2.792 eV/atom. In the case of FeTi 10 , the lowest energy structure is a Fe centered a bi-capped cubic structure with 2.925 eV/atom and high symmetry D 4h . The centered position and the high coordination number of Fe atom can be the origin of the good stability of this structure. As we see from Table 2 to Table 3, the average bond length of both Ti n+1 and FeTi n (n = 1-10) show an increasing tendency as the cluster size increases. In addition, in the most of the best structure of FeTi n clusters, the Fe atom occupies the highest coordinated site in the clusters, and this behavior increases as the size of cluster increases. These two geometrical parameters can have direct consequences on hybridization between Fe and Ti atoms. It turns out a considerable changing of the clusters electronic and magnetic properties.

Relative stability and electronic properties
In this section, the relative stability of the Ti n+1 and FeTi n (n = 1-10) clusters will be analyzed and discussed by using the different values of the averaged binding energy, second-order energy difference, HOMO-LUMO gaps (the  [29] b From Ref [15] c From Ref [27] d From Ref [8] e From Ref [1] f From Ref [35] g From Ref [36] h From Ref [14] i From Ref [37] j From Ref [38] k From Ref [39] l From Ref [40] m From Ref [41] n From Ref [42] o From Ref [43] Dimer difference in energy between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO)), the vertical ionization potential (VIP), and the vertical electron affinity (VEA). The averaged binding energy (Eb/atom) is useful for quantity to study the stability of small clusters. This parameter can be calculated for Ti n+1 and FeTi n as follows: where E b Ti n+1 and E b FeTi n represent, respectively, the total energy of Ti n+1 and FeTi n clusters, and E(Ti) and  Tables 2 and 3. Their evolutions as a function of the size for the most stable clusters of each size are plotted in Fig. 3. We note the monotonically increasing of binding energy with the increasing of cluster size, which means that these clusters will get energy during their growth. Since, the overall stability (binding energy) is supposed to increase and reach the bulk value of titanium, which is experimentally reported to be 4.85 eV [38], where a cluster requires up a very large number of atoms [39]. We can also see that the binding energy for Ti n+1 clusters is larger than those of FeTi n , which means that the doping Fe atom does not enhance the stability of host Ti n clusters at small size.
However, we observe, for n = 7 and n = 8, very close equal values of the binding energy of FeTi n cluster and those of corresponding pure Ti n+1 . We notice local peaks in the curve of Ti n+1 at size n = 2 and n = 6 implying that Ti 3 and Ti 7 are more stable than their neighbors where Ti 7 is reported as a magic number in the previous studies [1,7,11]. In addition, FeTi 3 , FeTi 5 , and FeTi 8 show high stabilities compared to their other neighboring clusters. The second-order energy difference (ΔE 2 ) is calculated for the best structures of each size by using:  (eV), average bond length a 0 (Å), vertical ionization potential VIP (eV), electronic affinity VEA (eV), chemical hardness η (eV), and atomic magnetic moment μ (μ B /atom) of Ti n+1 ( n = 1-10) clusters Size (n) Point group E b (eV/atom) E g (eV) (eV) a 0 (Å) VIP (eV) VEA (eV) η (eV) μ (μ B /atom) where E represents the total energy for corresponding cluster. In cluster sciences, this quantity Δ 2 E is an interesting quantity that reflects the relative stability of clusters. The systems with positive values are more stable than those with negative values of Δ 2 E. The calculated value of Δ 2 E for the most stable Ti n+1 and FeTi n clusters are plotted in Fig. 4.
We notice the pronounced positive value of Δ 2 E at Ti 7 for pure Ti n+1 indicating that this cluster has a special stability what has already well-mentioned at binding energy (Fig. 3).
In addition, this number refers to the magic number cluster observed during the experimental studies [10,11]. Although HOMO-LUMO gap curve shows a dip at this size, meaning the enhanced stability is due to geometric effects instead of electronic effects [1,40,41]. As we can see, FeTi 6 is less stable than Ti 7 , but a prominent maximum is found for FeTi 3 , FeTi 5 , and FeTi 8 indicating that they are more stable than the other clusters. The measurement of the HOMO-LUMO gap, which depends on the eigenvalues of the HOMO and LUMO energy levels, is important to characterize the electronic properties in clusters. Its knowledge is useful to examine the chemical reactivity and the chemical stability of clusters. A large HOMO-LUMO energy gap indicates a weaker chemical reactivity, and high strength required perturbing the electronic structure and bigger HOMO-LUMO gap signifies a higher stability. Clusters with very large gap have a very low chemical reactivity and a high chemical stability [8,42]. In Fig. 5, the obtained HOMO-LUMO gaps are plotted as a function of cluster size for pure Ti n+1 and doped FeTi n (n = 1-11). It is interesting to note that the obtained HOMO-LUMO gaps varies between 0.164 and 0.991 eV for FeTi n clusters and 0.022-0.843 for Ti n+1 clusters, which indicate that the metallic behavior can be observed for these two systems. Thus, these clusters can be used in catalytic reactions applications. The behaviors show a general decrease of the HOMO-LUMO gap as the cluster size increase for both Ti n+1 and FeTi n clusters. Also, the FeTi n clusters have generally a smallest HOMO-LUMO gap comparing to a pure Ti n+1 , indicating that the Fe doping atom weakens the chemical stability and enhances the metallic behavior for the FeTi n clusters. We observe that the clusters FeTi and Ti 4 have the largest value of HOMO-LUMO gaps suggesting that these clusters have a higher chemical stability and a smaller chemical reactivity than their neighbors. Table 3 Point group, binding energy per atom E b (eV/atom), HOMO-LUMO gap E g (eV), average bond length a 0 (Å), vertical ionization potential VIP (eV), electronic affinity VEA (eV), Chemical hardness η (eV) and atomic magnetic moment μ (μ B /atom) of FeTi n ( n = 1-10) clusters Size (n) Point group E b (eV/atom) E g (eV) a 0 (Å) VIP(eV) VEA(eV) η(eV) μ (μ B /atom) In cluster physics, VIP and VEA are considered as important measurement that can reflect the chemical stability of the small clusters. The VIP is calculated by the difference in energy between the cationic and neutral clusters with the same geometry of the neutral structure, and the VEA is the energy difference between the neutral and anionic cluster with the same geometry of the neutral structure. In addition, these two parameters are instructive to examine another quantity, which is the global chemical hardness. The higher value of VIP indicates that they need more energy for the FeTi n cluster to lose one electron, and the smaller value of VEA indicates that the cluster is more difficult to accept an electron, meanwhile the neutral cluster is high chemically stable. The VIP and VEA are defined for pure and doped titanium clusters as: E is the total energy of the considered neutral cluster, while E Ti + n+1 /E FeTi + n and E(Ti − n+1 )/E(FeTi − n ) are, respectively, the total energies of the cationic and anionic clusters. The obtained results of VIP and VEA are given in Tables 2 and 3, plotted, respectively, in Figs. 6 and 7. As we see from Fig. 6, the VIP increase monotonically for pure titanium clusters and show an oscillating behavior with a trend to increase with increasing size for iron-doped titanium clusters. In addition, the VIP exhibits obvious odd-even oscillations from the size 2 to 7. From Fig. 7, we observe that the VEA increases with the increasing size n with an obvious increasing from n = 6. The largest values of VEA are in general observed for the large sized clusters, which mean that we need more energy to add an electron to the systems, indicating the growing of their stability. Among the Ti n+1 and FeTi n clusters, FeTi 6 , FeTi 9 , and FeTi 10 show a pronounced peak for VIP and VEA parameters which indicates their high stability compared to the neighboring clusters. By using the obtained values of VIP and VEA, we investigated the chemical hardness (η) of Ti n+1 and FeTi n clusters. Chemical hardness is an important parameter that characterizes the resistance to charge transfer and the stability of clusters. A large value of chemical hardness corresponds to a less reactivity and a higher stability. According to the maximum hardness principles (PMH) [43], the chemical hardness is calculated by: The calculated values for the most stables structures of both Ti n+1 and FeTi n clusters are given in Tables 2 and  3 and shown in Fig. 8. As we can see from the figure, the chemical hardness has a decreasing evolution with the increasing of the size for FeTi n clusters. Except for n = 7 and 10 the chemical hardness of FeTi n clusters are higher than the pure Ti n+1 clusters with the same number of atoms. Through the PMH of chemical hardness, this result indicates that the doped clusters with higher value of chemical hardness are more stable than the corresponding pure clusters. Among the observed values of η, a pronounced peak is observed for FeTi 2 and FeTi 5 clusters, which indicate their very less chemical reactivity.

Magnetic properties
The magnetic properties of Ti n+1 and FeTi n are studied under the spin-polarized DFT calculations. These magnetic properties are studied by the evaluation of the total spin magnetic moment, which is calculated by the difference between the Mullikan charge populations for the electrons with spin up and the electrons with spin down. The obtained average atomic of spin magnetic moments (SMM) of the two systems are given in Tables 2 and 3 and Ti n+1 FeTi n presented in Fig. 9. We observe the same variance tendency for both pure Ti n+1 and FeTi n clusters. The average SSM for the two systems decrease with the increase of the size until n = 5 for FeTi n and n = 7 for Ti n clusters. From these sizes, the average SSM for the two systems show an increasing tendency with an oscillating behavior with the increasing of the size. As reported in the previous studies, the bulk titanium is nonmagnetic, but the small Ti n clusters have significant magnetic moment, and it is sensitive to their structures and geometries [7,17,18,26,29]. The same behavior FeTi is observed in our results. The FeTi dimer with 3.7 µ B has the largest average atomic magnetic moment. This is maybe correlated to its linear structure with a long average bond length. For pure Ti n+1 , we notice a local peak at n = 2 and 9 with 3.3 µ B which are the highest in all clusters. In order to understand the relation between the magnetic behavior and structural properties of Ti n+1 and FeTi n clusters, we represent in Fig. 10 the evolution of the average interatomic distances a 0 (Ti-Ti) and a 0 (Fe-Ti) as a function of cluster size. High differences between a 0 (Ti-Ti) and a 0 (Fe-Ti) are observed for very small (n ≤ 2) and large (n ≥ 8) size of clusters. For these clusters, large values of total spin magnetic moment are observed. Between n = 2 and n = 8, where the difference between the a 0 (Ti-Ti) and a 0 (Fe-Ti) is small, the total spin magnetic moment shows reduced values. This result constitutes a direct relationship between the magnetic and structural properties.
To understand the origin of average SMM of Ti n+1 and FeTi n clusters and to evaluate the contribution of different valence orbital (s and d) of the Fe and Ti components, we explore the total and partial densities of states for some low energy structures of the two systems. The obtained results for Ti 3 , Ti 6 , FeTi 1 , FeTi 2 , and FeTi 4 clusters are shown in Fig. 11. The spin up densities is plotted as positive and the spin down as negative. From this figure, we can clearly see that the 3d states of Ti and Fe atoms play an important contribution in the determination of the magnetic behavior of the Ti n and FeTi n clusters. The 4 s states of Ti and Fe atoms contribute little and almost negligible in the FeTi n systems.

Conclusions
In this work, we have systematically investigated the properties of small Ti n+1 and FeTi n (n = 1-10) clusters by using DFT-GGA with PBE parameterization for the exchange-correlation functional calculations. In the doped FeTi n clusters, the Fe atom occupies preferentially the position near or at the surface of the most favorable geometries. The geometrical structures, stabilities, electronic, and magnetic properties of small Ti n+1 and FeTi n are calculated and discussed. The geometric optimizations of Ti n+1 clusters show that for each cluster size, multiple isomers and new structures in addition to other structures are obtained in the previous studies. The analyses of the binding energies and second-order energies differences show enhanced stability of FeTi 3 , FeTi 5 , and FeTi 8 clusters. From the HOMO-LUMO gaps, we found that the FeTi and Ti 4 clusters possess a high chemical stability. The average SSM for the two systems depends on the size of FeTi n and n = 7 for Ti n clusters. The VIP and VEA calculations analysis shows that the clusters with large size exhibit high metallic character. Consequently, they will liberate more energy when they gain one electron. The chemical hardness shows that the clusters with small size are less reactive and more stable. PDOS analysis reveal that high value of the average total spin magnetic moment for Ti n+1 and FeTi n clusters is due to the contribution of the valence orbitals with a large domination of 3d states