Chromium-doped germanium clusters CrGen (n = 1-5): geometry, electronic structure, and topology of chemical bonding

First-row transition metal doped germanium clusters Ge16M: some remarkable superhalogens

A theoretical study of geometric and electronic structures, stability and magnetic properties of both neutral and anionic Ge₁₆M^0/− clusters with M being a first-row 3d transition metal atom, is performed using quantum chemical approaches. Both the isoelectronic Ge₁₆Sc⁻ anion and neutral Ge₁₆Ti that have a perfect Frank–Kasper tetrahedral T_d shape and an electron shell filled with 68 valence electrons, emerge as magic clusters with an enhanced thermodynamic stability. The latter can be rationalized by the simple Jellium model. Geometric distortions from the Frank–Kasper tetrahedron of Ge₁₆M having more or less than 68 valence electrons can be understood by a Jahn–Teller effect. Remarkably, DFT calculations reveal that both neutral Ge₁₆Sc and Ge₁₆Cu can be considered as superhalogens as their electron affinities (≥3.6 eV) exceed the value of the halogen atoms and even that of icosahedral Al₁₃. A detailed view of the magnetic behavior of Ge₁₆M^0/− clusters shows that the magnetic moments of the atomic metals remain large even when they are quenched upon doping. When M goes from Sc to Zn, the total spin magnetic moment of Ge₁₆M^0/− increases steadily and reaches the maximum value of 3 μ_B with M = Mn before decreasing towards the end of the first-row 3d block metals. Furthermore, the IR spectra of some tetrahedral Ge₁₆M are also predicted.

A theoretical study of geometric and electronic structures, stability and magnetic properties of both neutral and anionic Ge₁₆M^0/− clusters with M being a first-row 3d transition metal atom, is performed using quantum chemical approaches.

Three-Dimensional Convolutional Neural Networks Utilizing Molecular Topological Features for Accurate Atomization Energy Predictions

Deep learning methods provide a novel way to establish a correlation between two quantities. In this context, computer vision techniques such as three-dimensional (3D)-convolutional neural networks become a natural choice to associate a molecular property with its structure due to the inherent 3D nature of a molecule. However, traditional 3D input data structures are intrinsically sparse in nature, which tend to induce instabilities during the learning process, which in turn may lead to underfitted results. To address this deficiency, in this project, we propose to use quantum-chemically derived molecular topological features, namely, localized orbital locator and electron localization function, as molecular descriptors, which provide a relatively denser input representation in a 3D space. Such topological features provide a detailed picture of the atomic and electronic configuration and interatomic interactions in the molecule and hence are ideal for predicting properties that are highly dependent on the physical or electronic structure of the molecule. Herein, we demonstrate the efficacy of our proposed model by applying it to the task of predicting atomization energies for the QM9-G4MP2 data set, which contains ∼134k molecules. Furthermore, we incorporated the Δ-machine learning approach into our model, which enabled us to reach beyond benchmark accuracy levels (∼1.0 kJ mol^-1). As a result, we consistently obtain impressive mean absolute errors of the order 0.1 kcal mol^-1 (∼0.42 kJ mol^-1) versus the G4(MP2) theory using relatively modest models, which could potentially be improved further in a systematic manner using additional compute resources.

A joint experimental and theoretical study on structural, electronic, and magnetic properties of MnGen - (n = 3-14) clusters

A systematic structure and property investigation of MnGe_n ^- (n = 3-14) was conducted by means of density functional theory coupled with mass-selected anion photoelectron spectroscopy. This combined theoretical and experimental study allows global minimum and coexistence structures to be identified. It is found that the pentagonal bipyramid shape is the basic framework for the nascent growth process of MnGe_n ^- (n = 3-10), and from n = 10, the endohedral structures can be found. For n = 12, the anion MnGe₁₂ ^- cluster probably includes two isomers: a major isomer with a puckered hexagonal prism geometry and a minor isomer with a distorted icosahedron geometry. Specifically, the puckered hexagonal prism isomer follows the Wade-Mingos rules and can be suggested as a new kind of superatom with the magnetic property. Furthermore, the results of adaptive natural density partitioning and deformation density analyses suggest a polar covalent interaction between Ge and Mn for endohedral clusters of MnGe₁₂ ^-. The spin density and natural population analysis indicate that MnGe_n ^- clusters have high magnetic moments localized on Mn. The density of states diagram visually shows the significant spin polarization for endohedral structures and reveals the weak interaction between the Ge 4p orbital and the 4s, 3d orbitals of Mn.

Interpretation of photoelectron spectra of CoGen- (n = 4, 5) clusters by multiconfigurational RASPT2 calculations

The low-lying electronic states CoGe_n^-/0 (n = 4, 5) have been investigated with density functional theory and the state-of-the-art RASSCF/RASPT2 method to give assignments for the anion photoelectron spectra. The BP86 functional was employed to optimize the geometrical structures of the electronic states, while the RASSCF/RASPT2 was applied to calculate the single-point energies. With the RASSCF/RASPT2 approach, the active spaces are extended to a size of 21 orbitals for CoGe₄^-/0 and 24 orbitals for CoGe₅^-/0. The ground states of CoGe₄^-/0 are determined to be ³A″ and ²A″ of a trigonal bipyramidal structure in which the Co atom is situated at the equatorial corner of the bipyramid. The vertical detachment energies of the transitions from the anionic ground state to the neutral ²A″, 1⁴A″, ²A', 2⁴A″, 3⁴A″, 1⁴A', 2⁴A', and 6⁴A″ states are evaluated to be 2.29, 2.39, 2.60, 2.83, 3.17, 3.24, 3.47, and 4.00 eV. For the CoGe₅^-/0 clusters, the ground states are computed to be ¹A₁ and 1²A₂ of an octahedral structure. The vertical detachment energies of the removal of one electron from the anionic ground state to result in the 1²A₂, 1²A₁, 2²A₁, 1²B₁, 1²B₂, 4²B₁, 4²B₂, and 6²A₂ states are estimated to be 2.16, 2.79, 2.84, 3.06, 3.06, 3.59, 3.59, and 4.22 eV. All features in the photoelectron spectra of CoGe₄^- and CoGe₅^- are interpreted based on the computed electron detachment energies of the anionic ground states.

Thermochemical Properties and Growth Mechanism of the Ag-Doped Germanium Clusters, AgGe n λ with n = 1-13 and λ = -1, 0, and +1

A systematic investigation of the silver-doped germanium clusters AgGe n with n = 1-13 in the neutral, anionic, and cationic states is performed using the unbiased global search technique combined with a double-density functional scheme. The lowest-energy minima of the clusters are identified based on calculated energies and measured photoelectron spectra (PES). Total atomization energies and thermochemical properties such as electron affinity (EA), ionization potential (IP), binding energy, hardness, and highest occupied molecular orbital-lowest unoccupied molecular orbital (HOMO-LUMO) gap are obtained and compared with those of pure germanium clusters. For neutral and anionic clusters, although the most stable structures are inconsistent when n = 7-10, their structure patterns have an exohedral structure except for n = 12, which is a highly symmetrical endohedral configuration. For the cationic state, the most stable structures are attaching structures (in which an Ag atom is adsorbed on the Ge n cluster or a Ge atom is adsorbed on the AgGe n-1 cluster) at n = 1-12, and when n = 13, the cage configuration is formed. The analyses of binding energy indicate that doping of an Ag atom into the neutral and charged Ge n clusters decreases their stability. The theoretical EAs of AgGe n clusters agree with the experimental values. The IP of neutral Ge n clusters is decreasing when doped with an Ag atom. The chemical activity of AgGe n is analyzed through HOMO-LUMO gaps and hardness, and the variant trend of both versus cluster size is slightly different. The accuracy of the theoretical analyses in this paper is demonstrated successfully by the agreement between simulated and experimental results such as PES, IP, EA, and binding energy.

DFT-based investigation of different properties for transition metal-doped germanium TMGen (TM = Ru, Rh; n = 1-20) clusters

The geometries and energetic, electronic, and magnetic features of transition metal-doped germanium (TMGe_n with TM = Ru, Rh; n = 1-20) clusters are systematically studied by means of first principle computations on the basis of the density functional theory (DFT) approach. The doping TM atom largely participates to strengthen the Ge_n cluster stability by increasing the binding energies. A good stability is obtained for RuGe₁₂, RhGe₁₂, and RhGe₁₄ clusters. The various explored isomers of TMGe_n clusters possess a total spin magnetic moment going from 0 to 2μ_B, except for RhGe₂ with 3μ_B. These results open nice perspectives of these good candidate clusters for applications in nanoelectronics and nanotechnologies.

A remarkable mixture of germanium with phosphorus and arsenic atoms making stable pentagonal hetero-prisms [M@Ge5E5]+, E = P, As and M = Fe, Ru, Os

A pentagonal hetero-prismatic structural motif was found for singly transition metal doped M@Ge₅E₅⁺ clusters, where the transition metal atom is located at the centre of a (5/5) Ge₅E₅ prism in which Ge is mixed with either P or As atoms. Structural characterization indicates that each (5/5) Ge₅E₅ prism is established by joining of two Ge₃E₂ and Ge₂E₃ strings in a prismatic fashion rather than two Ge₅ and E₅ strings. Each string results from a remarkable mixture of Ge and E atoms and contains only one E–E connection due to the fact that Ge–E bonds are much stronger than E–E connections. From the donor–acceptor perspective, the Ge₅E₅ tube donates electrons to the M center, which behaves as an acceptor. NBO atomic charge and ELI_D analyses demonstrate such electrostatic interactions of the M dopant with a Ge₅E₅⁺ tube which likely induce thermodynamic stability for the resulting M@Ge₅E₅⁺ cluster. CMO analysis illustrates that the conventional 18 electron count is recovered in the M@Ge₅E₅⁺ cations.

Geometric shape of the lowest-energy structure M@Ge₅E₅⁺.

Transformation between Hexagonal Prism and Antiprism of the Singly and Doubly Cr-Doped Ge12 Clusters

Structural transformation is a unique characteristic of atomic clusters, but it turns out very different from cluster to cluster. This theoretical study proves that the isomeric transformation between hexagonal prism and hexagonal antiprism is found for the doubly doped Cr₂Ge₁₂ cluster but not for singly doped CrGe₁₂ cluster. We confirm that the ground state of CrGe₁₂ is the distorted hexagonal prism C_2h at the ³B_g triplet state instead of various shapes predicted in the previous studies. Upon comparison between the estimation at the B3P86/6-311+G(d) level of theory and the detachment energies measured by photoelectron spectroscopy, hexagonal antiprismatic shape is identified as the most stable isomer of the Cr₂Ge₁₂ cluster and it is easy to transform to the hexagonal prism-a less stable isomer by the rotation of the hexagonal rings. That is the first evidence for the structural transformation between a hexagonal prism and an antiprism of the germanium clusters, referring to the ability of Ge-based clusters in the formation of tubular geometry by doping Cr atoms. All the low-energy isomers of both Cr-doped germanium clusters have high magnetic moments. Interestingly, there is a tuning in magnetic properties of Cr₂Ge₁₂ from the ferromagnetism of the lowest-lying hexagonal antiprism to the ferrimagnetism of the higher-energy hexagonal prism. The stronger Cr-Cr bond and stronger interaction between the Cr₂ moiety and the antiprism cage are accounted for by the higher stability of the hexagonal antiprismatic isomer.

Structure, stability, and electronic properties of niobium-germanium and tantalum-germanium clusters

The structural, electronic and magnetic properties of niobium- and tantalum-doped germanium clusters MGe_n (M = Nb, Ta and n = 1-19) were investigated by first principles calculations within the density functional theory (DFT) approach. Growth pattern behaviors, stabilities, and electronic properties are presented and discussed. Endohedral cage-like structures in which the metal atom is encapsulated are favored for n ≥ 10. The doping metal atom contributes largely to strengthening the stability of the germanium cage-like structures, with binding energy ordered as follows BE(Ge_n + 1) < BE (VGe_n) < BE(NbGe_n) < BE(TaGe_n). Our results highlight the relative high stability of NbGe₁₅, TaGe₁₅ and VGe₁₄. Graphical abstract Structure, stability, and electronic properties of niobium-germanium and tantalum-germanium clusters.

Spin State Energetics of VGe n -/0 (n = 5-7) Clusters and New Assignments of the Anion Photoelectron Spectra

The geometrical structures, electron leading configurations, and relative energies of the low-lying states of VGe _n ^-/0 (n = 5-7) clusters have been investigated with density functional theory and CASSCF/CASPT2 method. The pure GGA BP86 functional gave almost the same relative energy order for the low-lying states as the CASPT2 method. At the BP86 and CASPT2 levels, the ground states of VGe _n ^-/0 (n = 5-7) clusters were proposed to be the ¹ A₁ and ² A₁ of A-VGe₅ ^-/0 , ³ A″ and ² A″, ² A' (² E₂ ) of A-VGe₆ ^-/0 , and ¹ A' and ² A' of A-VGe₇ ^-/0 isomers. The adiabatic and vertical detachment energies (ADEs and VDEs) of the detachments of one electron from several orbitals of the anionic ground states were reported at the CASPT2 level. The calculated ADEs and VDEs corresponded well with the experimental values as observed in the 266 nm anion photoelectron spectra. © 2018 Wiley Periodicals, Inc.

Structural evolution and magnetic properties of anionic clusters Cr2Ge n (n = 3-14): photoelectron spectroscopy and density functional theory computation

The structural, electronic and magnetic properties of dual Cr atoms doped germanium anionic clusters, [Formula: see text] (n  =  3-14), have been investigated by using photoelectron spectroscopy in combination with density-functional theory calculations. The low-lying structures of [Formula: see text] are determined by DFT based genetic algorithm optimization. For [Formula: see text] with n  ⩽  8, the structures are bipyramid-based geometries, while [Formula: see text] cluster has an opening cage-like structure, and the half-encapsulated structure is gradually covered by the additional Ge atoms to form closed-cage configuration with one Cr atom interior for n  =  10 to 14. Meanwhile, the two Cr atoms in [Formula: see text] clusters tend to form a Cr-Cr bond rather than be separated. Interestingly, the magnetic moment of all the anionic clusters considered is 1 μ _B. Almost all clusters exhibit antiferromagnetic Cr-Cr coupling, except for two clusters, [Formula: see text] and [Formula: see text]. To our knowledge, the [Formula: see text] cluster is the first kind of transition-metal doped semiconductor clusters that exhibit relatively stable antiferromagnetism within a wide size range. The experimental/theoretical results suggest high potential to modify the magnetic behavior of semiconductor clusters through introducing different transition-metal dopant atoms.

Structural and magnetic properties of FeGen-/0 (n = 3-12) clusters: Mass-selected anion photoelectron spectroscopy and density functional theory calculations

The structural, electronic, and magnetic properties of FeGe_n^-/0 (n = 3-12) clusters were investigated by using anion photoelectron spectroscopy in combination with density functional theory calculations. For both anionic and neutral FeGe_n (n = 3-12) clusters with n ≤ 7, the dominant structures are exohedral. The FeGe₈^-/0 clusters have half-encapsulated boat-shaped structures, and the opening of the boat-shaped structure is gradually covered by the additional Ge atoms to form Ge_n cage from n = 9 to 11. The structures of FeGe₁₀^-/0 can be viewed as two Ge atoms symmetrically capping the opening of the boat-shaped structure of FeGe₈, and those of FeGe₁₂^-/0 are distorted hexagonal prisms with the Fe atom at the center. Natural population analysis shows that there is an electron transfer from the Ge atoms to the Fe atom at n = 8-12. The total magnetic moment of FeGe_n^-/0 and local magnetic moment of the Fe atom have not been quenched.

Growth Behavior and Electronic Structure of Noble Metal-Doped Germanium Clusters

Structures, energetics, and electronic properties of noble metal-doped germanium (MGe_n with M = Cu, Ag, Au; n = 1-19) clusters are systematically investigated by using the density functional theory (DFT) approach. The endohedral structures in which the metal atom is encapsulated inside of a germanium cage appear at n = 10 when the dopant is Cu and n = 12 for M = Ag and Au. While Cu doping enhances the stability of the corresponding germanium frame, the binding energies of AgGe_n and AuGe_n are always lower than those of pure germanium clusters. Our results highlight the great stability of the CuGe₁₀ cluster in a D_4d structure and, to a lesser extent, that of AgGe₁₅ and AuGe₁₅, which exhibits a hollow cage-like geometry. The sphere-type geometries obtained for n = 10-15 present a peculiar electronic structure in which the valence electrons of the noble metal and Ge atoms are delocalized and exhibit a shell structure associated with the quasi-spherical geometry. It is found that the coinage metal is able to give both s- and d-type electrons to be reorganized together with the valence electrons of Ge atoms through a pooling of electrons. The cluster size dependence of the stability, the frontier orbital energy gap, the vertical ionization potentials, and electron affinities are given.

Computational Investigation of the Geometrical and Electronic Structures of VGen-/0 (n = 1-4) Clusters by Density Functional Theory and Multiconfigurational CASSCF/CASPT2 Method

Density functional theory and the multiconfigurational CASSCF/CASPT2 method have been employed to study the low-lying states of VGe_n^-/0 (n = 1-4) clusters. For VGe^-/0 and VGe₂^-/0 clusters, the relative energies and geometrical structures of the low-lying states are reported at the CASSCF/CASPT2 level. For the VGe₃^-/0 and VGe₄^-/0 clusters, the computational results show that due to the large contribution of the Hartree-Fock exact exchange, the hybrid B3LYP, B3PW91, and PBE0 functionals overestimate the energies of the high-spin states as compared to the pure GGA BP86 and PBE functionals and the CASPT2 method. On the basis of the pure GGA BP86 and PBE functionals and the CASSCF/CASPT2 results, the ground states of anionic and neutral clusters are defined, the relative energies of the excited states are computed, and the electron detachment energies of the anionic clusters are evaluated. The computational results are employed to give new assignments for all features in the photoelectron spectra of VGe₃^- and VGe₄^- clusters.

Insights into Geometric and Electronic Structures of VGe3-/0 Clusters from Anion Photoelectron Spectrum Assignment

The global minima of both neutral and anionic clusters of VGe₃^-/0 were determined using different quantum chemical methods (DFT, RCCSD(T), CASSCF/CASPT2). On the basis of the ground states identified, most excited bands in the anion photoelectron spectrum of VGe₃^- were assigned. The tetrahedral isomers of both charged states are the most stable ones. A singlet state (C_s^, 1A') of the tetrahedral isomer has the globally lowest energy on the potential hypersurface of VGe₃^-. Two states 1²A' and 1²A″ of the neutral tetrahedral isomer are nearly degenerate and identified as the competing ground state of VGe₃. From the anionic ground state, four of five bands in the anion photoelectron spectrum of VGe₃^- were determined to be the consequences of one-electron transitions starting from the anionic ground state ¹A'. Both nearly degenerate neutral ground states are responsible for generation of the first band. Two different transitions from the anionic ground state ¹A' to the first two nearly degenerate excited states (2²A' and 2²A″) of the neutral underlie the second lowest ionization band. Two higher levels of ionization recorded in the spectrum were assigned to the two higher excited states 4²A' and 5²A' of the neutral. Franck-Condon factor simulations of the first band were performed to obtain more insights into the experimental bands of the spectrum.

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

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

Titanium Digermanium: Theoretical Assignment of Electronic Transitions Underlying Its Anion Photoelectron Spectrum

Electronic structures of both the anionic and neutral triatomic species TiGe₂^-/0 were theoretically studied employing single-reference (DFT and RCCSD(T)) and multiconfigurational (CASSCF/CASPT2 and CASSCF/NEVPT2) methods with large basis sets. The ground state of TiGe₂^- (C_2v) was identified to be ⁴B₁, but the ²A₁ state is nearly degenerate, whereas the ³B₁ is clearly the ground state of the neutral TiGe₂ (C_2v). On the basis of the computed ground and excited states of both neutral and anionic structures, all electronic transitions giving rise to experimental anion photoelectron bands in the spectrum of TiGe₂^- can now be assigned. The X band of the anion photoelectron spectrum is attributed to a one-electron transition between two ground states ⁴B₁ → ³B₁. Three neutral excited states 2³A₂, 2⁵B₁, and 3⁵B₁ are energetically responsible for the B band upon one-electron photodetachement from the anionic ground state ⁴B₁. The C band is assigned to the transition ⁴B₁ → 2⁵A₁. A transition from the nearly degenerate ground state ²A₁ of the anion to the low-spin ¹A₁ of the final neutral state can be ascribed to the A band. Furthermore, the first two bands' progressions, whose normal vibrational modes were accessible from CASSCF/CASPT2 calculations, were also simulated by determination of multidimensional Franck-Condon factors.

Structural and magnetic properties of CoGe(n)- (n=2-11) clusters: photoelectron spectroscopy and density functional calculations

A series of cobalt-doped germanium clusters, CoGe(n)(-/0) (n=2-11), are investigated by using anion photoelectron spectroscopy combined with density functional theory calculations. For both anionic and neutral CoGe(n) (n=2-11) clusters, the critical size of the transition from exo- to endohedral structures is n=9. Natural population analysis shows that there is electron transfer from the Ge(n) framework to the Co atom at n=7-11 for both anionic and neutral CoGe(n) clusters. The magnetic moments of the anionic and neutral CoGe(n) clusters decrease to the lowest values at n=10 and 11. The transfer of electrons from the Gen framework to the Co atom and the minimization of the magnetic moments are related to the evolution of CoGe(n) structures from exo- to endohedral.

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

The evolution of electronic structures, properties and stabilities of neutral and cationic molybdenum encapsulated germanium clusters (Mo@Ge_n, n = 1 to 20) has been investigated.

Toward accurate theoretical thermochemistry of first row transition metal complexes

The recently developed correlation consistent Composite Approach for transition metals (ccCA-TM) was utilized to compute the thermochemical properties for a collection of 225 inorganic molecules containing first row (3d) transition metals, ranging from the monohydrides to larger organometallics such as Sc(C(5)H(5))(3) and clusters such as (CrO(3))(3). Ostentatiously large deviations of ccCA-TM predictions stem mainly from aging and unreliable experimental data. For a subset of 70 molecules with reported experimental uncertainties less than or equal to 2.0 kcal mol(-1), regardless of the presence of moderate multireference character in some molecules, ccCA-TM achieves transition metal chemical accuracy of ±3.0 kcal mol(-1) as defined in our earlier work [J. Phys. Chem. A2007, 111, 11269-11277] by giving a mean absolute deviation of 2.90 kcal mol(-1) and a root-mean-square deviation of 3.91 kcal mol(-1). As subsets are constructed with decreasing upper limits of reported experimental uncertainties (5.0, 4.0, 3.0, 2.0, and 1.0 kcal mol(-1)), the ccCA-TM mean absolute deviations were observed to monotonically drop off from 4.35 to 2.37 kcal mol(-1). In contrast, such a trend is missing for DFT methods as exemplified by B3LYP and M06 with mean absolute deviations in the range 12.9-14.1 and 10.5-11.0 kcal mol(-1), respectively. Salient multireference character, as demonstrated by the T(1)/D(1) diagnostics and the weights (C(0)(2)) of leading electron configuration in the complete active self-consistent field wave function, was found in a significant amount of molecules, which can still be accurately described by the single reference ccCA-TM. The ccCA-TM algorithm has been demonstrated as an accurate, robust, and widely applicable model chemistry for 3d transition metal-containing species with versatile bonding features.

A Stochastic Search for the Structures of Small Germanium Clusters and Their Anions: Enhanced Stability by Spherical Aromaticity of the Ge10 and Ge12(2-) Systems

Investigations on germanium clusters in the neutral, anionic, and dianion states Gen(x) (n = 2-12 and x = 0, -1, -2) are performed using quantum chemical calculations with the B3LYP functional and the coupled-cluster singles and doubles [CCSD(T)] methods, in conjunction with the 6-311+G(d) basis set. An improved stochastic method is implemented for searching the low-lying isomers of clusters. Comparison of our results with previous reports on germanium clusters shows the efficiency of the search method. The Ge8 system is presented in detail. The anionic clusters Gen(-/2-) are studied theoretically and systematically for the first time, and their energetics are in good agreement with available experiments. The clusters Ge10, Ge10(2-), and Ge12(2-) are, in their ground state, characterized by large highest occupied molecular orbital-lowest unoccupied molecular orbital gaps, high vertical and adiabatic detachment energies, and substantial average binding energies. The enhanced stability of these magic clusters can consistently be rationalized using the jellium electron shell model and the spherical aromatic character.