Geometrical Evolution Pattern and Spectroscopic Properties of Terbium-Doped Germanium Anionic TbGe_n (n = 6–17) Nanoclusters: From Tb-Lined to Tb-Encapsulated Structures

Abstract

Developing advanced materials with enhanced performance through the doping of nanoclusters is a promising strategy. However, there remains an insufficient understanding of the specific effects induced by such doped nanoclusters, particularly regarding the structural evolution pattern after doping with rare-earth elements and their impact on performance. To solve this problem, we used first-principles calculation to study the structural evolution pattern and spectroscopic properties of anionic TbGe_n (n = 6–17) nanoclusters through the ABCluster global search technique coupled with the mPW2PLYP double-hybrid density functional theory. The results revealed that the geometrical evolution pattern is from the typical Tb-linked structures (for n = 10–13, in which Tb acts as a linker connecting two germanium sub-clusters) to Tb-centered cage configurations (for n = 14–17). The simulated photoelectron spectroscopy of anionic TbGe₁₆ agrees well with its experimental counterpart. Furthermore, we calculated properties such as infrared spectroscopy, Raman spectroscopy, ultraviolet–visible (UV–vis) spectra, magnetism, charge transfer, the HOMO-LUMO gap, and relative stability. The results suggest that TbGe₁₂⁻ and TbGe₁₆⁻ clusters, with their remarkable stability and tunable photothermal properties, can serve as ideal building blocks for developing novel functional nanomaterials. These clusters demonstrate promising applications in solar photothermal conversion, photoelectric conversion, and infrared imaging technologies through their distinct one- and three-dimensional architectures, respectively.

1. Introduction

Doping nanoclusters is one of the promising strategies in materials science for developing advanced materials with enhanced performance, particularly for incorporating rare-earth atoms due to the fact that rare-earth elements possess unfilled 4f electrons, which endow them with diverse properties such as excellent magnetic, optical, electronic and catalytic capabilities. Germanium, as a key semiconductor material, is widely used in electronics and optics owing to its high electron mobility, infrared transparency, and high refractive index. Its unique properties make it particularly irreplaceable in infrared technology and specialized electronic devices. As fundamental building blocks of nanomaterials, clusters provide a unique platform for studying the evolution of structural and physical properties across different scales. Consequently, researchers have conducted extensive theoretical [1,2,3,4,5,6,7,8,9,10] and experimental [11,12,13,14,15,16] investigations on both pure germanium clusters and transition-metal-doped germanium clusters.

Doping with a single rare-earth atom (REA) provides distinctive advantages in terms of controllability and directional design for optimizing the physicochemical properties of clusters [16,17,18,19,20,21,22,23,24,25,26,27,28,29]. In terms of experiments, Atobe et al. [16] detected the electron binding energy and stability of TMGe_n (TM = Sc, Y, Lu, Tb, n = 8–20) by using anion photoelectron spectroscopy (PES) and the adsorption reactivity of the transition metals to H₂O. They found that the larger cavity of a Ge₁₆ can result in the metal encapsulation configuration of Tb@Ge₁₆⁻, Y@Ge₁₆⁻, and Lu@Ge₁₆⁻ nanoclusters. Stimulated by the experimental observations, some theoretical simulation calculations by means of single- or double-hybrid density functional theory were performed for REA-doped Ge_n clusters such as ScGe_n (n = 6–17) [6,17,18], YGe_n (n = 6–20) [19], LuGe_n (n = 5–17) [20,21,22,23], and their anionic clusters. In addition, Li et al. [24] systematically studied the structural evolution, relative stability, and electron binding energy of LaGe_n⁻ (n = 3–14) clusters through single-hybrid density functional theory. They found that the global minimum (GM) of LaGe₁₁⁻ and LaGe₁₄⁻ clusters is linked structures analogous to their Sc-, Y-, and Lu-doped counterparts [17,18,19,20]. Trivedi et al. [25] investigated the stability and magnetism of EuGe_n (n = 1–20) clusters under the framework of single-hybrid density functional theory and reported that the ground-state structures were Eu-encapsulated structures in Ge_n cages with n = 14–20. Recently, our group investigated the structural growth patterns and electronic properties of REAs (REA = Sc, Y, Lu, Ce, Eu, Gd) incorporated into anionic Ge_n⁻ (n ≤ 20) nanoclusters by means of double-hybrid density functional theory. The results revealed that their growth patterns transition from REA-linked to REA-encapsulated configurations as the cluster size n increases [18,19,20,26,28,29]. Although all REA-doped clusters adopt linked structures, their ground-state geometries vary significantly for a given cluster size, as demonstrated in this study. In this work, we selected terbium-doped anionic germanium clusters as the research focus, not only to extend our previous work but also to leverage terbium’s unique electronic structure, which enables its application in functional materials such as sensitizing agents, magnetic distortion materials, and magneto-optical storage devices. Incorporating a Tb atom into an anionic Ge_n cluster not only enhances its stability but also introduces novel magnetic and photoelectric activity, positioning it as a promising multifunctional fundamental unit for downsized semiconductor materials. The aim of this work is to elucidate the geometric stability and growth pattern of the nanocluster GM structures, quantify thermochemical parameters, simulate spectral properties, and provide critical insights for advancing theoretical and experimental studies on terbium-doped (or other metal-doped) semiconductor nanoclusters.

2. Computational Details

The detailed calculation method has been described elsewhere [18,19,20,26,27,28], and here is a brief description. Three techniques were applied to search for the initial structures of anionic TbGe_n (n = 6–17) nanoclusters. Firstly, with the ABCluster (artificial bee colony (ABC) for cluster) global search technique [30,31,32] coupled with the Gaussian 09 software package [33], larger than 300 geometries for each anionic TbGe_n clusters were optimized by means of single-hybrid TPSSh density functional theory [34] with the pseudopotential ECP54MWB basis set [35,36] for Tb atoms and the ECP28MWB basis set for Ge atoms [37]. Secondly, “substitutional structure” tactics were employed, where a Tb atom substitutes for a Ge atom in the GM configuration of neutral Ge_n+1 and/or anionic Ge_n+1⁻ clusters. Thirdly, GM spatial structures already presented in earlier articles [16,17,18,19,20,21,22,23,24,25,26,27,28,29] were adopted. The gained low-lying geometries were reoptimized through the single-hybrid density functional TPSSh coupled with the def2-TZVP basis set [38] for Tb atoms and the cc-pVTZ-PP basis set [39,40] for Ge atoms. All TbGe_n⁻ (n = 6–17) stationary point geometries were interrogated by the prediction of their harmonic vibrational frequencies at the same level. After settlement of the primary isomer optimization through TPSSh, we picked the low-lying structural candidates and reoptimized them through a double-hybrid functional mPW2PLYP [41] with the def2-TZVP basis set [38] for Tb atoms and the cc-pVTZ-PP basis set [39,40] for Ge atoms (the frequency calculation was not executed at the mPW2PLYP level of theory). At length, to further refine the energy, single-point energy calculation was performed by means of the mPW2PLYP functional with the def2-TZVP basis set [38] for Tb atoms and the aug-cc-pVTZ basis set [42] for Ge atoms (denoted as LARGE-BS). The PES spectra of TbGe_n⁻ (n = 6–17) clusters were simulated based on the OVGF (outer-valence Green function) scheme [43] in combination with the def2-TZVP basis set [38] for Tb atoms and the cc-pVTZ basis set [42] for Ge atoms and compared with the effective experimental data. To comprehend in depth the interaction between Tb atoms and germanium clusters, natural population analyses (NPAs) were also performed by the mPW2PLYP functional with LARGE-BS. The ultraviolet–visible (UV–vis) spectra were simulated through time-dependent density functional theory with the PBE0 functional [44,45] and LARGE-BS. The visualization results were achieved through Multifwn (Ver. 3.8) [46,47] and VMD 1.9.3 software [48]. Furthermore, the spin multiplicities of septet and nonet states were considered for anionic TbGe_n⁻ (n = 6–17) nanoclusters. The reliability of our calculation has been tested previously [18,19,20,49,50]. All calculations were performed using the Gaussian 09 software package [33].

3. Result and Discussion

3.1. GM Configurations of Anionic TbGe_n (n = 6–17) Nanoclusters

The GM configurations and selected low-lying isomers for anionic TbGe_n (n = 6–17) nanoclusters obtained at the mPW2PLYP level of theory are shown in Figure 1. Each nanocluster is named nA-x, in which n is number of Ge atoms, A is the anion, and x is the number of isomers. The electronic states for anionic TbGe_n (n = 6–17) were all evaluated to septet states. For n = 6, three isomers are presented. The C_s-symmetry 6A-1 is predicted to be the GM configuration, which differs from that of REAGe₆⁻ (REA = Sc, Y, Lu, Ce, Gd) species [18,19,20,26,27]. Their GM structures are analogous to 6A-3. It is also different than that of the SmGe₆⁻ cluster, which is analogous to the 6A-2 isomer. The 6A-2 and 6A-3 geometries are less stable in energy than the geometry of the 6A-1 GM structure by 0.54 and 0.93 eV, respectively. For n = 7, three geometries are reported. The C₁-symmetry 7A-1 is evaluated to be the GM structure. It differs from those of the REAGe₆⁻ (REA = Y, Lu, Gd) species [19,20,27]. Their GM geometries are analogous to the 7A-3 isomer. The 7A-2 and 7A-3 isomers are less stable in energy than the isomer of 7A-1 by 0.95 and 1.90 eV, respectively. For n = 8, four isomers are displayed. The GM configuration is calculated to be C₁-symmetry 8A-1, analogous to the GM of anionic GdGe₈ [27]. The C₂-symmetry 8A-2 is less stable in energy than that of 8A-1 by 0.36 eV. The isomer 8A-3 has approximately C_s-symmetry, but there is no symmetry constraint during the calculation. It is analogous to the GM structure of the ScGe₈⁻ and LuGe₈⁻ clusters [18,20]. The C_2v-symmetry 8A-4 isomer is analogous to the GM geometry of SmGe₈⁻ and EuGe₈⁻ clusters [28,29]. Energetically, the 8A-3 and 8A-4 isomers are higher than the isomer of 8A-1 by 0.76 and 1.19 eV, respectively. For n = 9, two isomers are presented. The C_3v-symmetry 9A-1 is predicted to be the GM configuration. It differs from that of REAGe₉⁻ (REA = Sc, Y, Lu, Ce, Gd, Sm, Eu) [18,19,20,26,27,28,29], of which the GM structures are analogous to that of the 9A-2 isomer. Energetically, 9A-2 is higher than 9A-1 by 0.40 eV. For n = 10, three isomers are shown. The GM configuration is calculated to be C₂-symmetry 10A-1, analogous to the GM of anionic REAGe₁₀ (REA = Sc, Y, Lu, Ce, Gd) [18,19,20,26,27]. It is a linked structure, in which the Tb atom links two Ge₅ trigonal bipyramids. The C_s-symmetry 10A-2 isomer is less stable in energy than that of 10A-1 by 0.53 eV. The C_s-symmetry 10A-3 isomer is analogous to the GM structure of the SmGe₁₀⁻ and EuGe₁₀⁻ clusters. Energetically, it is predicted to be 0.79 eV above the 10A-1 structure. For n = 11, four isomers are presented. The C₁-symmetry 11A-1 is predicted to be the GM structure. It belongs to the linked structure, in which the Tb atom links two subgroups of the Ge₅ trigonal bipyramid and Ge₆ capped trigonal bipyramid. It is analogous to the GM configurations of anionic REAGe₁₁ (REA = Sc, Y, Lu, Ce, Gd) [18,19,20,26,27]. It is different from the GM structures of the SmGe₁₁⁻ and EuGe₁₁⁻ clusters because their GM belongs to the adsorption structure [28,29]. The 11A-2, 11A-3, and 11A-4 isomers in the C_s-symmetry are calculated to 0.64, 0.72, and 2.46 eV above GM 11A-1, respectively. For n = 12, two isomers are displayed. The GM structure is predicted to be D_2d-symmetry-linked geometry 12A-1, in which the Tb atom links two Ge₆ capped trigonal bipyramids. It is analogous to the GM structure of the YGe₁₂⁻, LuGe₁₂⁻ and GdGe₁₂⁻ clusters [19,20,27]. The C_s-symmetry 12A-2 is analogous to the GM geometry of the SmGe₁₂⁻ and EuGe₁₂⁻ clusters [28,29]. Energetically, it is evaluated to be 1.67 eV above the 12A1 GM. For n = 13, two isomers are presented. The C_s-symmetry 13A-1 is predicted to be the GM structure. It can be seen as a linked structure, in which the Tb atom links two subgroups of the Ge₅ approximate planar five-membered ring and the Ge₈ square antiprism. It differs from those of the YGe₁₃⁻, LuGe₁₃⁻, GdGe₁₃⁻, SmGe₁₃⁻, and EuGe₁₃⁻ clusters [19,20,27,28,29], of which the GM structure is analogous to the 13A-2 isomers. The 13A-2 isomer is also a linked structure, in which the Tb atom links two subgroups of the Ge₄ approximate planar four-membered ring and the Ge₉ tricapped trigonal prism (TTP). Energetically, it is less stable than that of 13A-1 by 1.10 eV. For n = 14, four isomers are shown. The C_s-symmetry 14A-1, an incomplete cage structure, is predicted to be the GM configuration. It is more stable in energy than the linked structures of 14A-2 in the C₂-symmetry and 14A-4 in the C_s-symmetry by 0.55 and 1.52 eV, respectively. The 14A-3 structure belongs to the substitutional structure. It is predicted to 0.56 eV above the 14A-1 cage-like structure. For n = 15, three isomers are reported. The GM structure is predicted to be Tb-centered in the Ge-cage configuration (C_s-symmetry 15A-1), of which the cage can be viewed as being derived from fullerene of Ge₁₆ by removing a Ge atom analogous to that of the GdGe₁₅⁻ cluster [27]. The C_s-symmetry 15A-2 belongs to the linked structure, in which the Tb links two subgroups of a Ge₆ capped trigonal bipyramid and Ge₉ TTP. The C₂-symmetry 15A-3 belongs to the substitutional structure, in which the Tb atom substitutes for a Ge atom in the GM configuration of neutral Ge₁₆ clusters [1,2,3]. Energetically, the 15A-2 and 15A-3 isomers are calculated to 0.67 and 2.25 eV above that of the 15A-1 structure, respectively. For n = 16, its GM structure (T_d-symmetry 16A-1) is predicted to be Tb-centered in the Frank–Kasper (FK) cage of Ge₁₆. This result is consistent with the experimental observations [16]. For n = 17, the GM structure is evaluated to be the Tb-centered Ge-cage configuration with C_4v-symmetry (17A-1).

After discussing the GM configuration, we concentrate on the equilibrium geometrical evolution of the anionic TbGe_n (n = 6–17) nanocluster at present. According to the structural traits of the determined GM configuration, the equilibrium geometrical evolution patterns are inclined to the Tb-linked configuration, in which the Tb atom links two Ge sub-clusters beginning from n = 10, and the Tb-centered Ge cage polyhedron is preferred when n reaches 14. In fact, since Kumar and coworkers first reported the formation of linked clusters and endohedral cages in Y-doped anionic silicon clusters [51], it has been found that REA-doped anionic semiconductor clusters do exhibit a transition pattern from the linked structure to the cage configuration. Different rare-earth dopants and semiconductor clusters will form different linked structures and/or endohedral cages. This is because the GM configuration form depends on the diameter and electronic structure of dopants and host atoms. For example, the atomic radii of Sm and Eu are larger than that of Tb. Therefore, the minimum cluster size required to form a cage-like structure is n = 14 for Tb-doped anionic Ge nanoclusters (Tb@Ge_n⁻), compared to n = 20 for Sm-doped clusters [28]. In contrast, no cage-like structure is observed in the Eu-doped anionic germanium clusters even at n = 20 [29], and their minimum cage-forming size is approximately n = 26. The case of n = 9 serves as the most illustrative example of how electronic structures influence GM configurations. Ce and Gd exhibit an oxidation state of +3 in the CeGe₉⁻ and GdGe₉⁻ clusters [26,27], originating from their 5d¹6s² electron configuration, whereas Tb adopts the +4 oxidation state in TbGe₉⁻ derived from its 4f²6s² configuration (see below). These distinct electronic structures lead to different structures: the TbGe₉⁻ exhibits C_3v-symmetry geometry, while the CeGe₉⁻ and GdGe₉⁻ clusters adopt the C_4v-symmetry configuration.

3.2. The Charge Transfer and Magnetic Moment of TbGe_n⁻ (n = 6–17) Nanoclusters

To better understand the interaction between Tb atoms and Ge clusters or the influence of electronic structure on structure and property, we then conducted a natural population analysis (NPA) of the anionic TbGe_n⁻ (n = 6–17) clusters. The data including charge, valence electron configuration, and magnetic moments of each orbital and the total configuration of anionic TbGe_n⁻ (n = 6–17) clusters are listed in

Table 1. The NPA valence electron configuration, charge of the Tb atom, each valence orbital, and total magnetic moments of the GM structure for TbGe_n⁻ (n = 6–17) calculated with the mPW2PLYP scheme.

Clusters	Charge (a.u.)	Electron Configuration	Magnetic Moment of Tb Atom						Molecule (μB)
			6s	4f	5d	6p	7s	Total
TbGe6−	0.07	[core]6s0.824f7.365d2.216p0.317s0.24	−0.50	6.59	0.28	0.01	−0.04	6.34	6
TbGe7−	0.17	[core]6s0.754f7.365d2.176p0.297s0.25	−0.52	6.44	0.26	0.01	−0.03	6.16	6
TbGe8−	0.19	[core]6s0.784f7.385d2.086p0.367s0.21	−0.54	6.58	0.26	0.01	−0.03	6.28	6
TbGe9−	0.39	[core]6s1.284f7.005d1.736p0.437s0.17	0.46	6.96	−0.50	−0.27	−0.04	6.61	6
TbGe10−	0.32	[core]6s1.014f7.015d1.966p0.587s0.11	−0.74	6.95	0.04	0.02	0.01	6.28	6
TbGe11−	0.12	[core]6s0.714f7.365d1.886p0.687s0.25	−0.53	6.60	0.15	0.02	−0.07	6.17	6
TbGe12−	−0.27	[core]6s0.714f7.355d2.156p0.817s0.24	−0.54	6.61	0.15	0.01	−0.06	6.17	6
TbGe13−	0.10	[core]6s0.734f7.345d2.356p0.397s0.13	−0.57	6.62	0.26	−0.01	−0.01	6.29	6
TbGe14−	−2.94	[core]6s0.704f7.355d4.176p1.377s0.33	−0.71	6.94	0.15	0.02	−0.01	6.16	6
TbGe15−	−3.94	[core]6s0.824f6.995d5.066p1.747s0.27	−0.60	6.95	0.26	0.02	−0.05	6.58	6
TbGe16−	−4.76	[core]6s0.664f7.355d5.286p2.007s0.41	−0.43	6.61	0.20	0.01	−0.18	6.21	6
TbGe17−	−3.93	[core]6s0.674f7.355d4.816p1.687s0.37	−0.46	6.61	0.19	0.01	−0.15	6.20	6

. It can be seen from Table 1 that (i) the valence configuration of Tb in the anionic TbGe_n nanoclusters is 6s^0.66–1.284f^6.99–7.385d^1.73–5.286p^0.29–2.007s^0.10–0.41. This means that the 4f electrons of the Tb ([Xe]4f⁹6s²) atom are involved in bonding. The way to involve in bonding is initially transferring 4f electrons to the 5d and/or 6p orbital, and then the 5d and/or 6p orbital participates in bonding with valence electrons of the Ge_n nanocluster. To be precise, in addition to the 4f electron transition to the 5d orbital, 6s electrons also transfer to the 5d and/or 6p orbitals, leading to hybridization between the 6s and 5d (and/or 6p) orbitals. (ii) For the n = 6–8,11–13,14,16,17 clusters, one 4f orbital electron transition to that of 5d reveals that the oxidation state of the Tb atom in these cluster is three. For clusters of n = 9,10,15, their oxidation numbers are four because there are two 4f orbital electrons that transfer to 5d orbitals. (iii) For clusters with n = 6–8,10,11,13, the charge of Tb in TbGe_n⁻ is within +0.07 to +0.39 a.u. These results imply that terbium serves as an electron donor, thereby suggesting that ionic bonding dominates between Tb and the germanium network. For clusters of n = 9,12,14–17, the atomic charge of Tb ranges from −0.27 to −4.76 a.u., demonstrating its role as an electron acceptor. Furthermore, the bonding between Tb and the germanium cluster framework is predominantly metallic in nature. (iv) The total magnetic moments of TbGe_n⁻ (n = 6–17) were kept at a fixed value of 6 μ_B and were contributed by the Tb atom.

3.3. The Evaluated Photoelectron Spectroscopy of TbGe_n⁻ (n = 6–17)

Owing to the multiple isomers and states of nanoclusters, there are no experimental methods for directly measuring the GM of nanoclusters so far. Photoelectron spectroscopy is highly sensitive to changes in both the electronic and geometric structure of anionic clusters. Therefore, PES is one of the significant techniques for indirectly verifying the spatial geometric shape and electronic structure of nanoclusters. Given the crucial importance of the PES characterization of nanoclusters, we simulated the photoelectron spectra of the clusters. The simulated energy range was set between 0 and 5.5 eV, with a full width at half maximum (FWHM) of 0.50 eV using a Gaussian function. The first simulated peak was defined as X (the vertical detachment energy (VDE)), followed by subsequent peaks labeled as A, B, C, and so on corresponding to ascending energy positions. The results of the simulated spectra are summarized in Figure 2. When n = 6, three consecutive peaks can be observed in the spectrum, located at 2.38 eV (X), 3.18 eV (A), and 4.41 eV (B), respectively. When n = 7, four identifiable peaks, X, A, B and C, are present at 2.92, 3.63, 4.44, and 5.27 eV, respectively. For n = 8, the spectrum exhibits three prominent peaks, A (3.48 eV), B (4.24 eV), and C (5.31 eV), along with a minor shoulder peak X (3.02 eV) preceding peak A in the lower-energy region. When n = 9, the PES features a relatively isolated peak X (2.53 eV) and peak C (4.44 eV), accompanied by two shoulder peaks A (3.80 eV) and B (4.44 eV) on the lower-energy side of peak C. The number of peaks in the PES of 7A-1, 8A-1, and 9A-1 is the same, but they can be distinguished by differences in peak shape and position. When n = 10, the spectrum exhibits a distinct X peak at 4.64 eV, accompanied by peak A at 5.33 eV. For n = 11, two weakly resolved peaks, X (4.60 eV) and A (5.49 eV), can be observed within the fitted energy range. When n = 12, the PES shows three distinct peaks: an isolated X peak at 3.99 eV, a prominent B peak at 5.49 eV, and peak A (4.77 eV) on the lower-energy shoulder of peak B. Due to the structural similarity of 10A-1, 11A-1, and 12A-1, their photoelectron spectra are quite similar, especially between 10A-1 and 11A-1, where the peak positions and the number of peaks display minimal differences. Therefore, careful distinction is required in the experimental analyses. When n = 13, four distinct peaks, X (3.01 eV), B (4.07 eV), C (4.68 eV), and D (5.33 eV), along with a shoulder peak A (3.63 eV) adjacent to B, are observed in the PES. When n = 14, the spectra exhibit two peaks, X (4.38 eV) and A (5.30 eV). For n = 15, only a single peak, X (4.94 eV), is observed within the investigated energy range. For TbGe₁₆⁻, the simulated PES was compared with the experimental spectrum. The simulated spectrum shows two peaks: X at 4.22 eV and A at 5.03 eV. The experimental PES also detected two peaks, located at 4.23 eV and 4.86 eV [16]. It is worth noting that the simulated PES shows excellent consistency with the experimental PES in terms of peak shape, the number of peaks, and peak positions. When n = 17, within the 0–5.5 eV range, the spectrum exhibits an isolated peak X at 3.71 eV, followed by two consecutive peaks, A (4.82 eV) and B (5.44 eV). Overall, the consistency between the theoretical and experimental PES spectra of the TbGe₁₆⁻ cluster validates the reliability of our predicted GM structure. The simulated spectra can provide an important theoretical reference and guidance for future related experimental studies.

3.4. The Simulated Infrared and Raman Spectra of TbGe_n⁻ (n = 6–17)

In addition to PES, infrared (IR) and Raman spectroscopy serve as critical techniques for indirectly validating the GM structures of clusters. By integrating computational simulations with the spectral analysis of IR and Raman data for TbGe_n⁻ clusters (n = 6–17), we systematically characterize the vibrational signatures of their GM configurations, thereby establishing a predictive framework for the experimental determination of cluster geometries. All simulations were performed at the TPSSh level, employing the def2-TZVP basis set for Tb atoms and the aug-cc-pVTZ basis set for Ge atoms. The calculated spectra, presented in Figure 3, exhibit no imaginary frequencies, confirming the thermodynamic stability of these configurations. For the TbGe₆⁻ cluster, the IR spectrum exhibits three prominent peaks accompanied by several minor ones. The most intense vibrational peak at 272 cm⁻¹ is attributed to the Ge-Ge stretching vibration, with the Tb atom remaining stationary. The secondary peak at 126 cm⁻¹ arises from the scissoring vibration of the Ge-Tb-Ge moiety, while the peak at 208 cm⁻¹ corresponds to an in-plane rocking vibrational mode. In the Raman activity spectrum, a dominant peak emerges at 204 cm⁻¹, which belongs to the symmetric stretching vibration. For the TbGe₇⁻, its IR spectrum demonstrates the strongest vibration at 261 cm⁻¹, originating from asymmetric stretching vibrations. The second most intense peak at 184 cm⁻¹ is associated with in-plane deformation vibrations, whereas the feature peak at 95 cm⁻¹ results from coupled scissoring and symmetric stretching motions. The Raman spectrum shows a characteristic intense peak at 197 cm⁻¹, which arises from the degeneracy of two distinct stretching vibrational modes. For TbGe₈⁻, the strongest peak in the IR spectrum corresponds to multiple stretching vibrations at 192 cm⁻¹. The second strongest peak, located at 242 cm⁻¹, is attributed to the breathing vibration of the trigonal pyramidal Ge₄ cluster. In the Raman spectrum, the most intense characteristic peak appears at 184 cm⁻¹, corresponding to a deformation vibration. For TbGe₉⁻, three prominent peaks, ranked by intensity, appear at 166 cm⁻¹, 180 cm⁻¹, and 139 cm⁻¹, which are attributed to multiple out-of-plane rocking vibrations, the breathing vibration of the trigonal pyramidal TbGe₃ unit, and anti-symmetric stretching vibrations, respectively. The most intense characteristic peak in the Raman spectrum is observed at 224 cm⁻¹, corresponding to the breathing vibration of the Ge₉ sub-cluster in a TTP structure, with the Tb atom remaining relatively stationary. The second strongest peak, located at 180 cm⁻¹, arises from symmetric stretching vibrations. In the IR spectrum of TbGe₁₀⁻, the strongest peak, located at 173 cm⁻¹, is caused by anti-symmetric stretching involving two Ge-Tb-Ge units. The second characteristic peak, appearing at 309 cm⁻¹, originates from the coupled scissoring vibrations of two Ge₅ sub-clusters. The Raman spectrum exhibits only one prominent characteristic peak, attributed to the breathing vibrations of the two Ge₅ sub-clusters, with the Tb atom remaining relatively stationary. For TbGe₁₁⁻, the IR spectrum shows that the stronger characteristic peaks are concentrated, with the most intense peak at 178 cm⁻¹, resulting from a combination of stretching and scissoring vibrations. The second strongest peak, located at 214 cm⁻¹, is attributed to in-plane rocking vibrations of the Ge-Tb-Ge units and symmetric stretching vibrations. In contrast to the IR spectrum, the characteristic peaks in the Raman spectrum are more dispersed, with three strong peaks appearing at 104 cm⁻¹, 166 cm⁻¹, and 213 cm⁻¹, which correspond to in-plane rocking, scissoring, and stretching vibrations, respectively. The IR spectrum of TbGe₁₂⁻ exhibits three distinct peaks, with the strongest peaks at 182 cm⁻¹ and 214 cm⁻¹, both attributed to symmetric stretching vibrations, while the peak at 155 cm⁻¹ arises from scissoring vibrations. The Raman spectrum features a single strong characteristic peak at 251 cm⁻¹, originating from two degenerate breathing vibrations. In the IR spectrum of TbGe₁₃⁻, four peaks of nearly equal intensity are observed, with the peak at 47 cm⁻¹ arising from out-of-plane rocking vibrations, the peak at 103 cm⁻¹ attributed to double stretching vibrations, and the peaks at 192 cm⁻¹ and 241 cm⁻¹ originating from symmetric stretching motions involving different atoms. The Raman spectrum exhibits significantly fewer characteristic peaks than the IR spectrum, with only one prominent peak at 175 cm⁻¹, which is attributed to symmetric stretching vibrations. The most prominent feature in the infrared spectrum of the cage-like TbGe₁₄⁻ cluster is attributed to a dual bending vibration, appearing at 215 cm⁻¹. In contrast, the Raman spectrum exhibits two significant peaks: one at 183 cm⁻¹ arising from a mixed vibrational mode involving in-plane wagging and stretching motions and another at 170 cm⁻¹ due to symmetric deformation vibrations. The IR and Raman spectra of TbGe₁₅⁻ both exhibit a single prominent characteristic peak, corresponding to the triply degenerate bending vibration at 217 cm⁻¹ and the symmetric deformation vibration at 168 cm⁻¹, respectively. Similar to TbGe₁₅⁻, the TbGe₁₆⁻ cluster with T_d symmetry also displays a dominant strongest peak in both its IR and Raman spectra. These are attributed to the triply degenerate bending vibration at 221 cm⁻¹ and the stretching vibration of the Ge₁₆ cage at 167 cm⁻¹, respectively, where the Tb atom remains stationary. The infrared spectrum is highly active, while the Raman spectrum is inactive due to the structure having a center of symmetry, which follows the centrosymmetric rule. The analysis of the infrared and Raman spectra reveals that before the Tb atom is encapsulated by Ge, both spectra exhibit numerous characteristic peaks, particularly in the IR spectrum. However, upon the formation of a cage-like cluster structure, the number of significant peaks in both vibrational spectra is notably reduced and becomes more distinct. These simulated spectra provide robust evidence to support future qualitative experiments on such clusters. Moreover, since all the observed peaks are located in the far-infrared region, these ground-state clusters hold promising potential for application in the development of infrared sensing devices.

3.5. The Relative and Chemical Stabilities of TbGe_n⁻ (n = 6–17)

The assessment of cluster stability is a critical focus in nanocluster research. In this study, the stability of clusters is investigated by calculating the average binding energy (ABE), the second-order energy difference (Δ²E), and the HOMO-LUMO energy gap (E_gap). The higher the value of the ABE is, the more stable the clusters of TbGe_n⁻ (n = 6–17) is, and Δ²E reflects the relative stability between clusters of adjacent sizes in a system. E_gap, to some extent, indicates the photochemical stability of the clusters. The formulas for calculating the three parameters are as follows:

$$ABE({\mathrm{TbGe}}_n^{-})={\displaystyle \frac{(n-1)E(\mathrm{Ge})+E({\mathrm{Ge}}^{-})+E(\mathrm{Tb})-E({\mathrm{TbGe}}_n^{-})}{n+1}}$$

(1)

$${\Delta }^{2}E({\mathrm{TbGe}}_n^{-})=E({\mathrm{TbGe}}_{n-1}^{-})+E({\mathrm{TbGe}}_{n+1}^{-})-2E({\mathrm{TbGe}}_n^{-})$$

(2)

$$E_{gap}=E(\mathrm{LUMO})-E(\mathrm{HOMO})$$

(3)

Figure 4a illustrates the trend of ABE variation with cluster size. The ABE generally increases as the size of the cluster increases. When the ΔABE increment diminishes, a local maximum point will be formed, and the cluster corresponding to this local maximum point is relatively stable. Distinct stability minima are observed at n = 9 and 13, manifested by a slight decrease in ABE values. The stability progression reaches its maximum at n = 16, where the ABE peaks at 4.57 eV within the studied size range. Another intriguing feature observed in the ABE curve suggests potential periodicity in stability patterns, where clusters with sizes corresponding to multiples of four (n = 4k, k is a natural number) exhibit enhanced structural stability. Figure 4b shows the Δ²E analysis. It can be clearly seen from Figure 4b that clusters n = 8, 12, 14, and 16 are locally stable. These observational results indicate that clusters n = 8, 12, and 16 will be highly prominent in their mass spectrometry distributions. Figure 4c presents the E_gap values for clusters of different sizes. As evident from Figure 4c, the TbGe_n⁻ (n = 6–17) nanoclusters with n = 16 exhibit the largest E_gap (4.60 eV), followed by clusters n = 9–12 (their Egap values differ from each other, varying within the range of 4.27–4.36 eV), while cluster n = 6 demonstrates the smallest E_gap (2.90 eV). These results correlate with chemical stability trends, where the n = 16 cluster displays optimal stability, follow by clusters with n = 9–12. The E_gap calculated using the pure density functional theory is smaller than that calculated by the hybrid DFT. This occurs because the Kohn–Sham molecular orbital approximations predict similar energy upshifts for both the HOMO and LUMO, whereas the Hartree–Fock (HF) method shifts the LUMO to significantly higher energy levels compared to the HOMO [52]. For example, for the YGe_n⁻ (n = 6–20) cluster, the E_gap predicted by mPW2PLYP is larger than that of the PEB by 2.33 eV [19]. In summary, clusters with n = 12 and 16 exhibit not only superior thermal stability but also optimal chemical stability, making them prime candidates as building blocks for evolving into novel functional nanomaterials with tunable one- and three-dimensional architectures.

3.6. UV–Vis Spectra of the TbGe₁₂⁻ and TbGe₁₆⁻ Clusters

As discussed above, clusters with n = 12 and 16 not only have superior thermal and chemical stability but also represent prototypical examples of Tb-linked and Tb-centered caged architectures, respectively. Consequently, these two clusters were selected for further investigation of their ultraviolet absorption (UV–vis) spectra. Time-dependent density functional theory (TD-DFT) calculations were performed, with 120 excited states computed for each cluster to ensure a comprehensive description of their optical properties. The maximum half-peak width of the spectrum was set to 0.25 eV. The simulated UV–vis spectra of the TbGe₁₂⁻ and TbGe₁₆⁻ clusters are shown in Figure 4 and Figure 5, respectively. Figure 5a displays the total UV–vis spectrum of TbGe₁₂⁻ across a broad wavelength range of 400–7000 nm. For enhanced clarity, the spectrum is subdivided into two regions for detailed analysis: 400–1100 nm and 1100–7000 nm. As shown in Figure 5a, TbGe₁₂⁻ exhibits five distinct absorption peaks spanning the visible (380–780 nm), near-infrared (780–2500 nm), and mid-infrared (2500–25,000 nm) regions. The two most intense absorption peaks occur in the visible region at 727 nm and 630 nm. Two weaker peaks are observed in the near-infrared region at 899 nm and 1610 nm, while the mid-infrared region features a broad absorption peak with maximum intensity at 2839 nm. The strongest absorption peak at 727 nm arises predominantly from the S₀→S₆₈ (72.91%) and S₀→S₆₃ (16.91%) transitions, where “S” denotes a septet. The secondary peak at 630 nm is primarily attributed to S₀→S₈₃ (60.89%) and S₀→S₁₀₃ (30.44%) transitions. The third strongest peak at 2839 nm results from contributions by S₀→S₁₂ (59.95%) and S₀→S₇ (40.4%). The minor peak at 899 nm comprises S₀→S₃₉ (83.90%) and S₀→S₄₃ (12.59%), while the broad weak peak at 1610 nm is almost exclusively due to S₀→S₂₀ (99.30%). A quantitative analysis reveals that 19.32% of the TbGe₁₂⁻ absorption spectrum lies in the visible range, 20.35% in the near-infrared, and 61.08% in the mid-infrared region. For TbGe₁₆⁻, its simulated UV–vis spectrum is shown in Figure 6. From it we can see that the spectrum exhibits a prominent absorption peak accompanied by two weaker peaks. The most intense absorption in the visible-light region occurs at 609 nm, primarily arising from the transitions S₀→S₈₀ (42.86%), S₀→S₈₆ (23.00%), and S₀→S₆₉ (20.93%). A second peak, located at the boundary between the visible and near-infrared regions, reaches its maximum at 788 nm, dominated by the transitions S₀→S₃₁ (66.49%) and S₀→S₂₀ (29.52%). In the near-infrared region, the adsorption peak at 956 nm is predominantly attributed to the S₀→S₁₂ (97.36%) transition. The UV–vis spectral coverage shows that the visible-light region accounts for 80.12% of the total spectral area, while the near-infrared region constitutes 19.85%. These results highlight the TbGe₁₆⁻ and TbGe₁₂⁻ clusters—highly symmetric structures with excellent stability—as a material with exceptional photoresponse properties. These findings suggest that the TbGe₁₆⁻ and TbGe₁₂⁻ clusters hold promise for applications in solar photothermal conversion, photoelectric conversion, and mid-infrared imaging technologies.

4. Conclusions

The geometrical evolution patterns and spectroscopic properties of Tb-doped germanium anionic nanoclusters (TbGe_n⁻, n = 6–17) were systematically investigated using the ABCluster global search technique coupled with the mPW2PLYP double-hybrid density functional theory. The results demonstrate that the equilibrium geometries of the global minimum structures evolve toward the Tb-linked configuration, in which the Tb atom links two Ge sub-clusters starting from n = 10, while the Tb-centered Ge cage polyhedrons become energetically favored at n = 14. The photoelectron spectra of TbGe_n⁻ (n = 6–17) nanoclusters were simulated, and vertical detachment energies were reported. The strong agreement between the theoretical and experimental photoelectron spectra for the TbGe₁₆⁻ cluster validates the reliability of the predicted global minimum structures. Additionally, infrared and Raman spectra were computationally simulated. The stability analysis revealed that clusters with n = 12 and 16 exhibit not only superior thermal stability but also optimal chemical stability. A natural population analysis indicated that the total magnetic moments of TbGe_n⁻ (n = 6–17) remain constant at 6 μ_B, primarily originating from the Tb atom. Although 4f electrons participate in bonding, their antiparallel spin configuration persists, resulting in the magnetic moments of the TbGe_n⁻ (n = 6–17) nanoclusters matching that of an isolated Tb atom. The UV–Vis spectra of the TbGe₁₂⁻ and TbGe₁₆⁻ clusters not only overlap with the solar visible spectrum but also exhibit strong absorption peaks in the near-infrared and mid-infrared regions, demonstrating exceptional light-harvesting capabilities. With their remarkable stability and tunable photothermal properties, TbGe₁₂⁻ and TbGe₁₆⁻ clusters can serve as ideal building blocks for developing novel functional nanomaterials, showing promising applications in solar photothermal conversion, photoelectric conversion, and infrared imaging technologies through their distinct one- and three-dimensional architectures, respectively.