Structural and Electronic Properties of Lutetium Doped Germanium Clusters LuGe_{n}^{(+/0/)} (n = 6~19): A Density Functional Theory Investigation
English
Structural and Electronic Properties of Lutetium Doped Germanium Clusters LuGe_{n}^{(+/0/)} (n = 6~19): A Density Functional Theory Investigation

1. INTRODUCTION
In the semiconductor industry, germaniumbased lowdimensional nanomaterials with high performance are gradually becoming potential candidates for nonsilicon transistors^{[1, 2]}. How to develop new materials with highperformance properties has been the primary challenge for both theoretical and experimental scientists over the last decades. It has been widely used in a variety of crucial hightech fields, for example, using germanium nanoparticles as a lightabsorbing layer for solar cell^{[3, 4]}, as doping center of luminescent material application to LED device^{[5, 6]}, as a high performance broadband photo detector because of having excellent band gap^{[7]}, and playing an important role in solid state batteries^{[8]}.
Binary germanium cluster, a class of excellent selfassembled nanomaterials, doped by transition metal (TM) or rare earth (RE) metal, have unique salient properties in the field of structural, electronic, magnetic, and optical aspects. As a suitable building block, stability is an essential factor determined by both geometric configurations and electronic structure^{[9, 10]}. In the early study of transition metal doped germanium cluster by Kumar and Kawazoe, the researchers performed DFT calculation to explore series of M@Ge_{n} (n = 14~16; M = Ti, Zr, Hf, Fe, Ru, Os) cluster and further comparatively analyzed the different growth behavior between M@Ge_{n} and M@Si_{n} including several possible cage structures of FrankKasper, fullerenelike, capped decahedral and cubic model^{[11, 12]}. From here on, transition metal doped germanium cluster with various elements has been studied^{[1320]}. On another side of rare earth doped germanium cluster, Tang with his coworker studied the ground state geometry, energy gap, and optical gap of Ge_{12}M (M = Sc~Ni) by using density functional theory, and concluded that Ge_{12}M clusters having magnetic moments from 1 to 5 μ_{B} were regarded as potential magnetic materials with tunable magnetic properties^{[21]}. Borshch's group using density functional theory took the examination of ScGe_{n}^{} (n = 6~16) cluster for optimization of spatial structure and electronic spectra comparably studied with experimental results^{[22]}. Qin et al. combine genetic algorithm with firstprinciples calculation aiming to find out the lowestenergy structure of Ge_{n}M^{+/0} (n = 9, 10; M = Si, Li, Mg, Al, Fe, Mn, Pb, Au, Ag, Yb, Pm, and Dy)^{[23]}. In 2012, Nakajima's group utilized photoelectron spectroscopy to investigate a series of MGe_{n} (M = Sc, Ti, V, Y, Zr, Nb, Lu, Hf, and Ta). The results pointed that Ge_{16} cage with larger cavity to encapsulate RE and TM atom formed anionic core shell structure, for which electronic and geometric closings are uniformly satisfied^{[24]}. Thereby, the RE and TM based germanium nanomaterials possess a number of great potential applications for next generation devices. That's why it is necessary to further study comprehensively.
In this article, on the basis of Nakajima's previous experimental data^{[24]}, we performed the growth behavior of LuGe_{n}^{(+/0/)} (n = 6~19) and simulated photoelectron spectroscopy of LuGe_{n}^{} (n = 6~19) cluster, focusing on the electronic structure of anionic LuGe_{16}^{} as super atom with FrankKasper stable motif^{[25]}. Another way to say is that the goal of this research is to present the theoretical investigation of the effect of Lu (lutetium) metal doping on germanium clusters, while taking predicted optical and electronic properties into account for potential application to optoelectronic materials.
2. COMPUTATIONAL DETAILS
The initial structures are originated from: (1) By using the ABCluster unbiased global search technique, which can be seen as a random generator^{[26, 27]}. When it was combined with Gaussian software package, more than 400 initial guessed geometries for each LuGe_{n}^{(+/0/)} (n = 6~19) were generated and optimized with the TPSSh functional combined with 631G for Ge atom and ECP60WMB for Lu (lutetium) atom by Gaussian 09 software package^{[28, 29]}. The energy change threshold as convergence condition is set to 10^{6} hartree. (2) The substitutional structure schemes were adopted, in which a Lu atom was substituted for a Ge in the most stable structure of Ge_{n+1} cluster. Subsequently, the lowlying isomers are further optimized by DFTTPSSh functional with allelectron ccpVTZ basis set for Ge atom and ECP28MWB basis set for Lu^{[30, 31]}. Vibrational frequency analyses were performed to confirm those isomers are located at true minima on the potential energy surface. The single point energy calculations of the selected isomers were conducted by DFTTPSSh functional with augccpVTZ basis set for Ge and ECP28MWB basis set for Lu. The PES spectra of LuGe_{n}^{} (n = 6~19) were calculated by means of Koopmans' theorem^{[32, 33]}. The density of states (DOS) and partial density of states (PDOS) of LuGe_{16}^{} have been obtained by Vienna Ab initio Simulation Package (VASP)^{[3437]}, with PerdewBurkeErnzerhof generalized gradient approximation (PBEGGA) functional^{[38]}. The projector augmented wave (PAW) was set to explore the inert core electron^{[39, 40]}. To prevent interaction between adjacent clusters, the 40 × 40 × 40 Å edge lengths cubic cells with periodic boundary condition were taken into account. The plane wave cutoff energy was set up to 500 eV. The structures, isosurface maps, PES spectra and orbitals were created by visualization software of VMD and Multiwfn^{[41, 42]}.
3. RESULTS AND DISCUSSION
3.1 Ground state structure of LuGe_{n}^{(+/0/)} (n = 6~19)
The anionic, neutral and cationic clusters with stable geometries and point group are shown in Figs. 1~3, and the corresponding lowlying isomers with different energies are added in Figs. S1~S3, respectively. For anionic clusters, their ground states are predicted to be a singlet. For n = 6, the ground state structure is a pentagonal bipyramid with C_{s} symmetry which is the same as pure Ge_{7} ground state structure where Lu atom replaces the top vertex of pyramid of the Ge atom^{[43]}. For n = 7, the ground state structure is a bicapped tetragonal bipyramid with C_{2v} symmetry where two capped Ge atoms symmetrically adsorbed on its two faces. For n = 8, both 8a1 and 8a2 isomers compete for the ground state structure with each other owing to the fact that both of them are degenerated in energy (the energy difference is only 0.02 eV). 8a1 and 8a2 are both bicapped pentagonal bipyramids, and the difference lies in the two capped Ge atom adsorption sites. The 9a geometry with C_{s} symmetry is based on 8a2 structure by adding a Ge atom. For n = 10~12, the 10a is a link structure where Lu atom works as a linker to link two orthogonal Ge_{5} trigonal bipyramid (TBP), that for 11a is adding one Ge at 10a one side trigonal bipyramid to form two paralleled TBP and one Ge capped TBP, that for 12a is adding one Ge atom on each bilateral side of 10a. For n = 13, the Lu atom is located in the half cage center. The 13a1 and 13a2 are degenerated due to close energy difference of 0.04 eV. The 13a2 is hexagonal antiprism with C_{6v} symmetry where adsorbed one Ge atom on the hexagon surface and the 13a1 can be viewed that one Ge vertex of hexagonal antiprism of 13a2 moves to the triangular surface. For n = 14, the 14a is seen as adding one Ge atom on 13a2 with slight distortion. For n = 15~17, all 15a, 16a and 17a structures are derived from 16a of FrankKasper structure with T_{d} highsymmetry where the Lu atom locates at the cage center. The 15a is viewed as 16a by reducing one Ge atom from the bottom triangle and 17a is also viewed as 16a by adding one Ge atom on the bottom triangle. Both 15a and 17a have some extended distortion. For n = 18~19, these are all cage structures with Lu atom embedded in the center. 18a comes from the cage skeleton of 16a by adding two Ge atoms, with one in the middle layer and the other in the bottom. 19a consists of a sixmembered ring symmetrical on both sides where each face capped one Ge atom, and a fivemembered ring is at the middle symmetry plane to form an endohedral structure.
Figure 1
Figure 2
Figure 3
For neutral clusters, their ground states are calculated to be a doublet. For n = 6, it is the same as that anionic structure of pentagonal bipyramid. The difference is on the Lu atom that replaces the Ge atom site. For n = 7 and 8, there are one and two Ge atom capped LuGe_{5} pentagonal bipyramids, respectively. For n = 9, its structure is equal to the pure Ge_{10} cluster in which the Lu atom replaces a Ge atom on the top site. For n = 10 and 11, the 10n is a tetracapped pentagonal bipyramid and the 11n is on that structure to add one Ge atom. For n = 12, the 12n is regarded as two subclusters of capped trigonal bipyramid linked by Lu atom. For n = 13 and 14, both are linkstructure with Lu as the linker. In such structures, one side is the same as capped tetragonal antiprism and another side is four Ge atoms forming a trigonal pyramid for 13n. Five Ge atoms form a trigonal bipyramid for 14n. For n = 15, the 15n resembles to the 18n structure by removing Ge capped on each side at the ring face and one Ge atom on the symmetry plane. For n = 16, the 16n is distorted FrankKasper, and for n = 17 and 19, such structures are the same as that of the anionic one discussed above. For n = 18, the 18n cage skeleton is consistent with two fivemembered rings as mirror symmetry where two Ge atoms are capped on each side at the ring face and the remaining four Ge atoms, in order to form rhombus, are coplanar on the symmetry plane.
For cationic clusters, their ground states are evaluated to be singlet. For n = 6 and 7, the 6c is like 6n. And 7c is on that structure by adding one Ge atom. For n = 8 and 9, the 8c is a three Ge atom capped tetragonal bipyramid and 9c is based on 8c for more than one Ge atom. For n = 10, 10c is the same as the Ge_{11} pure cluster^{[44]}. For n = 11~15, all of them are seen as link structures where Lu atom serves as the linker. 11c is two Ge_{5} trigonal bipyramids which have jointly shared one Ge atom that is coplanar with Lu linker at the plane of symmetry. 12c is divided into one part of trigonal pyramid (Ge_{3}) and another part of tetragonal antiprism (Ge_{8}) linked by Lu at the center. 13c is based on 12c by adding one Ge atom at the tetragonal antiprism top site and 14c is based on 13c with the addition of one Ge atom at the trigonal pyramid site. 15c is two capped tetragonal antiprisms which share three Ge atoms on the central symmetry plane and Lu atom is on the center of that geometry. For n = 16 and 17, they are analogous to those of the corresponding neutral. For n = 18 and 19, 18c is built as Lu of center atom around two symmetrical sixmembered rings and fivemembered rings, with the remaining Ge atoms situated at three vertices of the triangle. 19c is the geometry for further epitaxial growth of 18c.
In summary, from the above description, it can be concluded that: (1) The LuGe_{n}^{(+/0/)} (n = 6~19) growth to be cage structure are clearly exhibited when n = 16. (2) The anion growth system has a distinct half cage phase. However, the cation and neutral cluster growth are not. (3) The energy surface of LuGe_{n}^{(+/0/)} is so flat, which indicates distinguishing ground state geometries in the experiment needs more attention.
3.2 PES spectra of LuGe_{n}^{} (n = 6~19)
The photoelectron spectra of different ground state isomers of LuGe_{n}^{} (n = 6~19) determined theoretically and compared with the experimental spectra are shown in Fig. 4. For LuGe_{6}^{}, the four peaks (x, a~c) are located at 2.90, 3.31, 3.98 and 4.84 eV which are highly consistent with experimental ones (X, A~C) at 2.95, 3.33, 4.02 and 4.82 eV. For LuGe_{7}^{}, the five peaks (x, a~d) are placed at 2.72, 3.15, 3.77, 4.31 and 4.53 eV, in good agreement with the experimental data (X, A~D) of 2.70, 2.98, 3.45, 4.23 and 4.53 eV. For LuGe_{8}^{}, two isomers of 8a1 and 8a2 have similar energy whose photoelectron spectra are shown. For 8a1, the six main peaks (x, a~e) reside at 2.83, 3.30, 3.63, 4.08, 4.57 and 4.80 eV, and also for 8a2, the six main peaks (x′, a′~e′) are found at 3.05, 3.45, 3.68, 4.20, 4.60 and 5.01 eV, compared with the experimental spectra of six peaks (X, A~E) of 3.06, 3.30, 3.71, 4.20, 4.61 and 5.03 eV. The conclusion can be deduced that such two types of isomers are coexisting because of the similar structures of two capped pentagonal bipyramids and close energy. For LuGe_{9}^{}, the three peaks (x, a, b) of 3.16, 3.89 and 4.72 eV of the theoretical spectra are in excellent agreement with experimental spectra (X, A, B) of 3.23, 3.89 and 4.72 eV. For LuGe_{10}^{}, the three peaks (x, a, b) positioned at 3.79, 4.31 and 4.78 eV are corresponding experimental value (X, A, B) of 4.10, 4.52 and 4.98 eV. The theoretical result has a certain red shift compared with the experimental one. For LuGe_{11}^{}, the simulated spectra of three peaks marked as x, a and b are 3.60, 4.19 and 4.68 eV. In comparison with the experimental X, A and B of 3.85, 4.21 and 4.73 eV, the peaks of a and b agree well with A and B. For LuGe_{12}^{}, the peaks (x, a~c) situated at 3.47, 4.35, 4.77 and 5.05 eV contrast with the experimental values (X, A~C) of 3.67, 4.51, 4.94 and 5.34 eV. The calculation result has a small deviation from the experimental one. For LuGe_{13}^{}, the 13a1 and 13a2 have been presented as similar total energy isomers. The 13a1 and 13a2 have three peaks placed at 3.38, 3.80, 4.27 eV and 3.54, 3.99, 4.87 eV correlated to the experimental peaks of 3.59, 4.03 and 4.83 eV. Owing to similar half cage structures, combined with PES, 13a1 and 13a2 are the coexisting structures. For LuGe_{14}^{}, three distinct peaks (x, a, b) lie in 3.63, 4.12 and 5.11 eV, which coincide with three experimental peaks (X, A, B) of 3.68, 4.13 and 5.14 eV. For LuGe_{15}^{}, simulated peaks are placed at 3.93, 4.66 and 5.23 eV against experimental peaks of 3.98, 4.69 and 5.25 eV. For LuGe_{16}^{}, due to stable FrankKasper structure, the two peaks of x and a conform with X and A at 3.96 and 4.99 eV. In much the same as LuGe_{16}^{}, the LuGe_{17}^{} of simulated and experimental PES exhibits four obvious peaks of 3.47 (3.47), 4.03 (4.08), 4.67 (4.70) and 5.23 (5.29) eV. For LuGe_{18}^{}, its simulated PES spectra have four distinct peaks (x, a, b, c) at 3.68, 4.16, 4.93 and 5.44 eV, respectively. Except for the first peak, the last three peaks (a, b, c) reproduce good experimental peaks (A, B, C) of 4.16, 4.93 and 5.44 eV^{[24]}. The adjacent isomers 17a and 19a have a small bump in the low energy region of the experimental PES spectra^{[24]}, so we infer that 18a should also have this peak. But it is not observed in the experimental PES spectra^{[24]}. This conclusion needs to be further verified by experiment. For LuGe_{19}^{}, the three peaks (x, a, b) of 3.76, 4.15 and 5.33 eV are analogous to the experimental peaks (X, A, B) of 3.67, 4.15 and 5.33 eV.
Figure 4
Theoretical calculation of adiabatic electron affinities (AEAs) and the first vertical detachment energies (VDEs) of LuGe_{n} clusters as well as experimental data are collected in Table 1. The calculated values of AEA and VDE are derived from AEA = E(optimized neutral) – E(optimized anion) and VDE = E(neutral at optimized anion geometry) – E(optimized anion). From Table 1, for both AEA and VDE, the experimental values of 8a2, 13a2, 14a and 17a are in good agreement with the theoretical ones with a minor difference less than 0.20 eV. The largest deviation of AEA is for LuGe_{15}, which is off by 0.55 eV. Although PES is a powerful experimental technique for measuring AEAs and, in principle, is the most accurate scheme to determine the AEAs of the corresponding neutral species, it is difficult to determine accurate AEAs if the recorded PES shows a featureless long and very rounded tail with no clear onset. The PES of LuGe_{15}^{} may be one such example^{[24]}. In this case, theoretical calculation is necessary to help determine accurate AEAs, especially for excellent agreement between the theoretically simulated and experimental PES spectra. Therefore, we dare to predict the AEA of LuGe_{15} is 3.73 eV rather than 3.18 eV. The theoretical VDE values of 6a, 7a, 9a, 12a, 15a, 16a and 19a are all in accord with their experimental ones with energy difference less than 0.20 eV. Besides, the calculated and experimental VDE of 10a, 11a, 12a and 18a do not match well. However, the AEA of simulation and experiment are consistent, which proves the credibility of the results from the other side. In summary, the simulated AEA and VDE outcomes of 8a2 and 13a2 are highly in line with corresponding experimental values, further proving that 8a1, 8a2 and 13a1, 13a2 are coexistence systems. By the way, LuGe_{16}^{} as the most stable FrankKasper structure has the highest value in AEA and VDE no matter for theoretical and experimental results.
Table 1
AEA VDE Cluster Theor. Exp. Theor. Exp. 6a 2.43 2.13 ± 0.05 2.90 2.95 7a 2.34 2.10 ± 0.05 2.72 2.70 8a1 2.49 2.45 ± 0.05 2.83 3.06 8a2 2.47 2.45 ± 0.05 3.05 3.06 9a 2.93 2.59 ± 0.05 3.16 3.23 10a 3.09 3.29 ± 0.05 3.79 4.10 11a 3.11 3.15 ± 0.05 3.60 3.85 12a 3.39 2.97 ± 0.05 3.47 3.67 13a1 3.21 3.08 ± 0.05 3.38 3.59 13a2 3.17 3.08 ± 0.05 3.54 3.59 14a 3.29 3.11 ± 0.05 3.63 3.68 15a 3.73 3.18 ± 0.05 3.93 3.98 16a 3.79 3.32 ± 0.05 3.96 3.98 17a 3.14 3.08 ± 0.05 3.47 3.47 18a 3.34 3.22 ± 0.05 3.68 4.16 19a 3.52 3.25 ± 0.05 3.76 3.67 3.3 Ionization potential
Ionization potential (IP), including vertical ionization potential (VIP) and adiabatic ionization potential (AIP), is the important parameter in both physical and chemical properties. The VIP is described as the difference of total energies where the equation follows VIP = E(cation at optimized neutral geometry) − E(optimized neutral) and AIP defined as the difference of total energies calculated by AIP = E(optimized cation) − E(optimized neutral). The calculation results of VIP and AIP of LuGe_{n} (n = 6~19) are all gathered in Table 2, with no experimental data for comparison. For the sake of observation, point line chart is also presented in Fig. 5, in which the highest value of VIP for LuGe_{15} is 7.08 eV and that of AIP for LuGe_{16} is 6.20 eV. On the contrary, the lowest values of VIP and AIP are 5.85 eV for LuGe_{6} and.61 eV for LuGe_{11}. On the whole of VIP and AIP curves, the three humps are obviously displayed in n = 7, 12 and 15 for VIP and four humps in n = 7, 9, 12 and 16 for AIP.
$ {\text{ABE(LuG}}{{\text{e}}_n}{\text{)}} = \frac{{{\text{[}}E({\text{Lu}}) + nE({\text{Ge)}}  E({\text{LuG}}{{\text{e}}_n}){\text{]}}}}{{(n + {\text{1)}}}} $ (0.1) $ {\text{ABE(LuGe}}_n^{{\text{ + 1/  1}}}{\text{)}} = \frac{{{\text{[}}E{\text{(Lu)}} + {\text{(}}n  {\text{1)}}E{\text{(Ge)}} + E{\text{(G}}{{\text{e}}^{{\text{ + 1/  1}}}}{\text{)}}  E{\text{(LuGe}}_n^{{\text{ + 1/  1}}}{\text{)]}}}}{{{\text{(}}n + {\text{1)}}}} $ (0.2) $ {{{\Delta }}^{\text{2}}}E{\text{(LuGe}}_n^{{\text{ + 1/0/  1}}}{\text{)}} = E{\text{(LuGe}}_{n  1}^{{\text{ + 1/0/  1}}}{\text{)}} + E{\text{(LuGe}}_{n + 1}^{{\text{ + 1/0/  1}}}{\text{)}}  {\text{2}}E{\text{(LuGe}}_n^{{{ + 1/0/  1}}}{\text{)}} $ (0.3) Table 2
Cluster VIP (eV) AIP (eV) Cluster VIP (eV) AIP (eV) LuGe_{6} 5.85 5.68 LuGe_{13} 6.66 5.65 LuGe_{7} 6.47 6.10 LuGe_{14} 6.25 5.79 LuGe_{8} 6.48 5.91 LuGe_{15} 7.08 5.93 LuGe_{9} 6.49 5.94 LuGe_{16} 6.79 6.20 LuGe_{10} 6.05 5.68 LuGe_{17} 6.31 6.01 LuGe_{11} 6.48 5.61 LuGe_{18} 6.34 6.14 LuGe_{12} 6.81 5.76 LuGe_{19} 6.57 6.13 Figure 5
3.4 Stabilities
To investigate the relative stabilities of the lowest energy structure of LuGe_{n}^{(+/0/)} (n = 6~19), the average bonding energy (ABE) and second energy difference (Δ^{2}E) are calculated by equations (1.1)~(1.3) in which E is the total energies related to atom or compound, and HOMOLUMO gap (E_{gap}) is derived from the energy of the lowest unoccupied molecular orbital minus the highest occupied molecular orbital. All are exhibited in Fig. 6.
Figure 6
From Fig. 6a, it is clear to see that the ABE values of the neutral cluster are all lower than the others, which indicates the cation and anion are more stable because both of them have closedshell electronic configuration which has increased their stabilities. The ABE value of LuGe_{16}^{} is 3.54 which is higher than the other anionic and neutral clusters. The reason can be described that not only LuGe_{16}^{} is FrankKasper endohedral motif with high T_{d} symmetry, but also has the closedshell electronic configuration as super atom with ultrastability. Owing to LuGe_{16}^{}, the neutral and cationic LuGe_{16} are also more stable than the other ones of the same type because both those structures are derived from FrankKasper structure with different degrees of distortion.
The secondorder difference in energy of LuGe_{n}^{(+/0/)} (n = 6~19) is shown in Fig. 6b, which is to evaluate the relative stability of such cluster and its two directly adjacent ones. For anion, the three peaks of anionic type are located at LuGe_{9}^{}, LuGe_{12}^{} and LuGe_{16}^{}. For the neutral, the obvious four peaks are situated at LuGe_{9}, LuGe_{11}, LuGe_{13} and LuGe_{16}. For cation, LuGe_{8}^{+}, LuGe_{11}^{+} LuGe_{13}^{+} and LuGe_{17}^{+} have four peaks. That means such clusters are more stable than the adjacent ones.
The E_{gap} is an important physical parameter for semiconductors to evaluate not only the chemical reactivity but also the optical properties. As can be seen in Fig. 6c, the apparent two peaks of LuGe_{10}^{} and LuGe_{16}^{} are about 2.41 and 2.55 eV. As far as we know, such energy gap value is larger enough for the luminescent host from ground states to excited states^{[45]}. Besides the proper energy gap for luminescent materials, that is also suitable for catalysis material. As photo catalyst, it is very important for visible light response, so that the materials with adjustable energy gaps can be achieved^{[46]}. As the LuGe_{19}^{+} has the lowest energy gap of 1.06 eV to LuGe_{16}^{} of 2.55 eV, it can fit within the visible light range. In summary, these clusters as an excellent building block are potentially applied to multifunctional nanomaterials.
3.5 Isochemical shielding surface of LuGe_{16}^{}
To further understand the stability of LuGe_{16}^{} FrankKasper structure, the isochemical shielding surface (ICSS) based on the realspace function which is similarly related to the nucleusindependent chemical shift (NICS) is calculated by gaugeindependent atomic orbital (GIAO) method, and the results are analyzed by Multiwfn code^{[47, 48]}. Compared to NICS, the ICSS can exhibit the entire threedimensional space of chemical shielding against the external magnetic field as well as different isosurfaces drawn at different isovalues to clearly reveal the shielding or deshielding effect from the delocalized electron. From Fig. 7a, the red region is the shielding area with isovalue of 0.05 ppm and the blue region is deshielding area with isovalue of 0.05 ppm. Due to the high symmetry of the LuGe_{16}^{} cluster, the shielding areas are exhibited by three protruding red areas, which indicates that the inner region of the cluster has strong magnetic shielding. Owing to the electron lone pair existence, it presents high electron delocalization and aromaticity. Fig. 7b displays the ICSS value from the distance of 0.4 to 3.0 Å, which reveals that the shielding value of the inner cage is larger than that of the outer surface. The maximum shielding value about 83.97 ppm is located at a distance of 1 Å from the center. In conclusion, the measurement of ICSS can be used for the aromatic properties of the LuGe_{16}^{} cluster, which shows the key factors of its stability.
Figure 7
3.6 Chemical bonding analysis and density of states of the LuGe_{16}^{} cluster
In order to gain insight and understand the better thermosdynamic and chemical stability of the LuGe_{16}^{} cluster with FrankKasper structure, the adaptive natural density partitioning (AdNDP) method, based on the natural bond orbital (NBO) developed by Zubarev and Boldyrev, is used to study the cluster as a multicenter bonding system^{[49]}. For example, nc2e is denoted as n from the range 1 to the number of atoms in the cluster system. According to Fig. 8, the LuGe_{16}^{} cluster of chemical bonding of 68 valence electrons with T_{d} symmetry can be split into three parts: lone pair, 2c2e, and 4c2e. The Ge atom, which is located on each of the four triad axes, has a lone pair. Besides the four Ge atoms on the triad axes, the remaining Ge_{12} cage is attributed to 18 2c2e localized Ge–Ge σ bonds possessing 1.87~1.89 electrons in each bond. The last ones are categorized into 12 delocalized 4c2e σ bonds connecting the cage core atom of Lu to the outer FrankKasper Ge_{16} skeleton, which can stabilize the entire structure and reveals the full encapsulated LuGe_{16}^{} endohedral cluster.
Figure 8
For more clarity regarding the unique electronic structure of LuGe_{16}^{} cluster, the calculated density of states and partial density of states are analyzed and shown in Fig. 9. It can be clearly seen that 6s, 5d, 5p, and 5s orbitals of Lu combined with 4s and 4p orbitals of Ge contribute to the main density of states of LuGe_{16}^{} cluster. The 4f orbitals are isolated at –3~–4 eV region. Near the Fermi level (dot line), the main contribution is from 5d orbital of Lu and 4p orbital of Ge. In other words, the dorbital of Lu is mainly participating in hybridization with Ge orbital to stabilize the cluster. Moreover, the bottom part of the conduction band mainly consists of 5d orbitals of Lu, and 4s, 4p and 4d orbitals of Ge. In the context of the AdNDP and DOS analyses, the orbital sequence of LuGe_{16}^{} can be represented as 1S^{2}1P^{6}1D^{10}1F^{14}2S^{2}1G^{18}2P^{6}2D^{10}, a jellium model for 68 valence electrons.
Figure 9
3.7 UVVis spectra of LuGe_{16}^{} cluster
The UVVis spectra are an essential parameter for luminescent and photocatalytic materials. The most stable LuGe_{16}^{} cluster is selected as a model structure due to its significant stability and appropriate energy gap. Timedependent DFT (TDDFT) calculations are performed at the level of TPSSh/augccpVTZ for Ge and ECP28WMB for Lu in order to analyze transitions in 120 excited states which are enough to describe. The curves are arranged by Gaussian broadening function with the full width at half maximum (FWHM) about 0.20 eV. Fig. 10 shows that the four peaks are positioned at 334, 369, 408 and 540 nm. The peak of 334 nm has a maximum absorption intensity and its 95% contribution can be decomposed into three parts of S_{0} → S_{115} with contribution of 8%, S_{0} → S_{107} with contribution of 72% and S_{0} → S_{104} with contribution of 15%. The peak of 369 nm with 92% comes from the S_{0} → S_{74}. The peak of 408 nm mainly has 12% contribution of S_{0}→S_{58} and 69% of S_{0} → S_{35}. The last peak of 540 nm is comprised of 46% contribution of S_{0} → S_{6} and 53% contribution of S_{0} → S_{12}. The absorption band is in the range of 300~600 nm and most of it falls in the visible light region, which can be excited by natural light. Meanwhile, the strong absorption bands are situated in the blue and nearultraviolet regions. In other words, the LuGen_{16}^{} cluster with high stability can be further explored as possible optoelectronic material.
Figure 10
4. CONCLUSION
The growth behavior, thermodynamic stabilities, chemical activities, and electronic properties of cationic, neutral, and anionic lutetium doped germanium cluster LuGe_{n}^{(+/0/)} (n = 6~19) are calculated by the ABCluster unbiased global search technique with a TPSShhybrid density functional theory. In terms of experimental PES spectra, AEA and VDE compared with the simulations, the growth pattern of anionic LuGe_{n}^{} can be depicted from adsorbed (n = 6~9) to the link structure (n = 10 ~ 12), then to half cage (n = 13~14), finally to the cagelike motif (n = 15~19). For the neutral and cationic LuGe_{n} clusters, the growth characteristics are less evident than anions. The neutral LuGe_{n} forms the link structure when n = 12~14 and when n = 15, those can be formed as cagelike structures. For cations, the link structure begins at n = 11 and lasts until n = 15, then creates cagelike motif. The stability analysis shows the great performance for LuGe_{16}^{}, and the reason can be described not only by its unique FrankKasper framework with high T_{d} symmetry without any imaginary frequency, but also by the specific electronic configurations. To further support that discovery, the aromaticity, density of states and AdNDP analysis are set. The ICSS results show the LuGe_{16}^{} possesses the aromatic nature with a max value of 83.97 ppm. Furthermore, the 3D isosurface of ICSS also proves the strong magnetic shielding effect in the cage inner. The AdNDP results exhibit that the orbital sequence 1S^{2}1P^{6}1D^{10}1F^{14}2S^{2}1G^{18}2P^{6}2D^{10} matches 68 electrons forming as a closedshell superatomic model. Existence of lone pair increases the delocalization of LuGe_{16}^{}, which is also beneficial for the cluster stability. Finally, the density of states presents the 5d orbitals of Lu hybridized with 4p orbital of Ge to further stabilize the cluster of LuGe_{16}^{}. Owing to the excellent stability and proper energy gap of LuGe_{16}^{} and others, the lutetium doped germanium clusters are possible candidates to fabricate a number of assembled optoelectronic materials.


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[1]

Table 1. Theoretical and Experimental AEA, and VDE of LuGe_{n} (n = 6~19)
AEA VDE Cluster Theor. Exp. Theor. Exp. 6a 2.43 2.13 ± 0.05 2.90 2.95 7a 2.34 2.10 ± 0.05 2.72 2.70 8a1 2.49 2.45 ± 0.05 2.83 3.06 8a2 2.47 2.45 ± 0.05 3.05 3.06 9a 2.93 2.59 ± 0.05 3.16 3.23 10a 3.09 3.29 ± 0.05 3.79 4.10 11a 3.11 3.15 ± 0.05 3.60 3.85 12a 3.39 2.97 ± 0.05 3.47 3.67 13a1 3.21 3.08 ± 0.05 3.38 3.59 13a2 3.17 3.08 ± 0.05 3.54 3.59 14a 3.29 3.11 ± 0.05 3.63 3.68 15a 3.73 3.18 ± 0.05 3.93 3.98 16a 3.79 3.32 ± 0.05 3.96 3.98 17a 3.14 3.08 ± 0.05 3.47 3.47 18a 3.34 3.22 ± 0.05 3.68 4.16 19a 3.52 3.25 ± 0.05 3.76 3.67 Table 2. Vertical Ionization Potential (VIP) and Adiabatic Ionization Potential (AIP) of LuGe_{n} (n = 6~19)
Cluster VIP (eV) AIP (eV) Cluster VIP (eV) AIP (eV) LuGe_{6} 5.85 5.68 LuGe_{13} 6.66 5.65 LuGe_{7} 6.47 6.10 LuGe_{14} 6.25 5.79 LuGe_{8} 6.48 5.91 LuGe_{15} 7.08 5.93 LuGe_{9} 6.49 5.94 LuGe_{16} 6.79 6.20 LuGe_{10} 6.05 5.68 LuGe_{17} 6.31 6.01 LuGe_{11} 6.48 5.61 LuGe_{18} 6.34 6.14 LuGe_{12} 6.81 5.76 LuGe_{19} 6.57 6.13
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