# Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations

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## Abstract

**:**

## 1. Introduction

## 2. Computational Methods

#### 2.1. The Embedded-Atom Method

#### 2.2. The Basin-Hopping Method

#### 2.3. DFT Calculations

## 3. Results and Discussion

#### 3.1. Structural Properties

#### 3.2. Energetic Properties

#### 3.3. Electronic Properties

#### 3.4. Growth Patterns

#### 3.5. Vibrational Frequencies

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Sample Availability

## References

- Haslett, T.L.; Bosnick, K.A.; Moskovits, M. Ag
_{5}is a planar trapezoidal molecule. J. Chem. Phys.**1998**, 108, 3453–3457. [Google Scholar] [CrossRef] - Howard, J.A.; Sutcliffe, R.; Mile, B. The geometric and electronic structures of small metal clusters of group 1B metals. Surf. Sci.
**1985**, 156, 214–227. [Google Scholar] [CrossRef] - Xing, X.; Danell, R.M.; Garzón, I.L.; Michaelian, K.; Blom, M.N.; Burns, M.M.; Parks, J.H. Size-dependent fivefold and icosahedral symmetry in silver clusters. Phys. Rev. B
**2005**, 72, 081405. [Google Scholar] [CrossRef] [Green Version] - Blom, M.N.; Schooss, D.; Stairs, J.; Kappes, M.M. Experimental structure determination of silver cluster ions (Ag${}_{n}^{+}$, 19 ≤ n ≤ 79). J. Chem. Phys.
**2006**, 124, 244308. [Google Scholar] [CrossRef] - Handschuh, H.; Cha, C.Y.; Bechthold, P.S.; Ganteför, G.; Eberhardt, W. Electronic shells or molecular orbitals: Photoelectron spectra of Ag${}_{n}^{-}$ clusters. J. Chem. Phys.
**1995**, 102, 6406–6422. [Google Scholar] [CrossRef] [Green Version] - Huda, M.N.; Ray, A.K. Electronic structures and magic numbers of small silver clusters: A many-body perturbation-theoretic study. Phys. Rev. A
**2003**, 67, 013201. [Google Scholar] [CrossRef] - Huda, M.; Ray, A. A correlation study of small silver clusters. Eur. Phys. J. D
**2003**, 22, 217–227. [Google Scholar] [CrossRef] - Bonačić-Koutecký, V.; Češpiva, L.; Fantucci, P.; Koutecký, J. Effective core potential-configuration interaction study of electronic structure and geometry of small neutral and cationic Ag
_{n}clusters: Predictions and interpretation of measured properties. J. Chem. Phys.**1993**, 98, 7981–7994. [Google Scholar] [CrossRef] - Bonačić-Koutecký, V.; Pittner, J.; Boiron, M.; Fantucci, P. An accurate relativistic effective core potential for excited states of Ag atom: An application for studying the absorption spectra of Ag
_{n}and Ag${}_{n}^{+}$ clusters. J. Chem. Phys.**1999**, 110, 3876–3886. [Google Scholar] [CrossRef] - Bonačić-Koutecky, V.; Veyret, V.; Mitrić, R. Ab initio study of the absorption spectra of Ag
_{n}(n = 5–8) clusters. J. Chem. Phys.**2001**, 115, 10450–10460. [Google Scholar] [CrossRef] - Idrobo, J.C.; Öğüt, S.; Jellinek, J. Size dependence of the static polarizabilities and absorption spectra of Ag
_{n}(n = 2–8) clusters. Phys. Rev. B**2005**, 72, 085445. [Google Scholar] [CrossRef] - Yang, M.; Jackson, K.A.; Jellinek, J. First-principles study of intermediate size silver clusters: Shape evolution and its impact on cluster properties. J. Chem. Phys.
**2006**, 125, 144308. [Google Scholar] [CrossRef] [PubMed] - Michaelian, K.; Rendón, N.; Garzón, I.L. Structure and energetics of Ni, Ag, and Au nanoclusters. Phys. Rev. B
**1999**, 60, 2000–2010. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.; Luo, Y.; Wang, G. Tight-binding study of structural and electronic properties of silver clusters. Eur. Phys. J. D
**2001**, 14, 309–316. [Google Scholar] [CrossRef] - Shao, X.; Liu, X.; Cai, W. Structural Optimization of Silver Clusters up to 80 Atoms with Gupta and Sutton-Chen Potentials. J. Chem. Theory Comput.
**2005**, 1, 762–768. [Google Scholar] [CrossRef] - Alamanova, D.; Grigoryan, V.G.; Springborg, M. Theoretical Study of the Structure and Energetics of Silver Clusters. J. Phys. Chem. C
**2007**, 111, 12577–12587. [Google Scholar] [CrossRef] - Baletto, F.; Mottet, C.; Ferrando, R. Reentrant Morphology Transition in the Growth of Free Silver Nanoclusters. Phys. Rev. Lett.
**2000**, 84, 5544–5547. [Google Scholar] [CrossRef] - Poteau, R.; Heully, J.L.; Spiegelmann, F. Structure, stability, and vibrational properties of small silver cluster. Z. Phys. At. Mol. Clust.
**1997**, 40, 479–482. [Google Scholar] [CrossRef] - Wales, D.J.; Doye, J.P.K. Global Optimization by Basin-Hopping and the Lowest Energy Structures of Lennard-Jones Clusters Containing up to 110 Atoms. J. Phys. Chem. A
**1997**, 101, 5111–5116. [Google Scholar] [CrossRef] [Green Version] - Wales, D.J.; Scheraga, H.A. Global Optimization of Clusters, Crystals, and Biomolecules. Science
**1999**, 285, 1368–1372. [Google Scholar] [CrossRef] [Green Version] - Wales, D.J. Energy Landscapes: Applications to Clusters, Biomolecules and Glasses; Cambridge University Press: Cambridge, MA, USA, 2004. [Google Scholar]
- Cleveland, C.L.; Landman, U. The energetics and structure of nickel clusters: Size dependence. J. Chem. Phys.
**1991**, 94, 7376–7396. [Google Scholar] [CrossRef] [Green Version] - Vlachos, D.G.; Schmidt, L.D.; Aris, R. Structures of small metal clusters. I. Low temperature behavior. J. Chem. Phys.
**1992**, 96, 6880–6890. [Google Scholar] [CrossRef] - Montejano-Carrizales, J.M.; Iñiguez, M.P.; Alonso, J.A.; López, M.J. Theoretical study of icosahedral Ni clusters within the embedded-atom method. Phys. Rev. B
**1996**, 54, 5961–5969. [Google Scholar] [CrossRef] [PubMed] - Grigoryan, V.G.; Springborg, M. A theoretical study of the structure of Ni clusters (Ni
_{N}). Phys. Chem. Chem. Phys.**2001**, 3, 5135–5139. [Google Scholar] [CrossRef] - Grigoryan, V.G.; Springborg, M. Structure and energetics of Ni clusters with up to 150 atoms. Chem. Phys. Lett.
**2003**, 375, 219–226. [Google Scholar] [CrossRef] [Green Version] - Rey, C.; Gallego, L.J.; García-Rodeja, J.; Alonso, J.A.; Iñiguez, M.P. Molecular-dynamics study of the binding energy and melting of transition-metal clusters. Phys. Rev. B
**1993**, 48, 8253–8262. [Google Scholar] [CrossRef] - Daw, M.S.; Baskes, M.I. Semiempirical, Quantum Mechanical Calculation of Hydrogen Embrittlement in Metals. Phys. Rev. Lett.
**1983**, 50, 1285–1288. [Google Scholar] [CrossRef] - Daw, M.S.; Baskes, M.I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B
**1984**, 29, 6443–6453. [Google Scholar] [CrossRef] [Green Version] - Foiles, S.M.; Baskes, M.I.; Daw, M.S. Embedded-atom-method functions for the fcc metals Cu, Ag, Au, Ni, Pd, Pt, and their alloys. Phys. Rev. B
**1986**, 33, 7983–7991. [Google Scholar] [CrossRef] - Daw, M.S.; Foiles, S.; Baskes, M.I. The embedded-atom method: A review of theory and applications. Mater. Sci. Rep.
**1993**, 9, 251–310. [Google Scholar] [CrossRef] [Green Version] - Clementi, E.; Roetti, C. Roothaan-Hartree-Fock atomic wavefunctions: Basis functions and their coefficients for ground and certain excited states of neutral and ionized atoms, Z ≤ 54. At. Data Nucl. Data Tables
**1974**, 14, 177–478. [Google Scholar] [CrossRef] - McLean, A.D.; McLean, R.S. Roothaan-Hartree-Fock atomic wave functions Slater basis-set expansions for Z = 55 − 92. At. Data Nucl. Data Tables
**1981**, 26, 197–381. [Google Scholar] [CrossRef] - Daw, M.S. Model of metallic cohesion: The embedded-atom method. Phys. Rev. B
**1989**, 39, 7441–7452. [Google Scholar] [CrossRef] [PubMed] - Niesse, J.A.; Mayne, H.R. Global geometry optimization of atomic clusters using a modified genetic algorithm in space-fixed coordinates. J. Chem. Phys.
**1996**, 105, 4700–4706. [Google Scholar] [CrossRef] [Green Version] - Johnston, R.L.; Mortimer-Jones, T.V.; Roberts, C.; Darby, S.; Manby, F.R. Applications of Evolutionary Computing; Springer: Berlin/Heidelberg, Germany, 2002; Volume 2279, pp. 25–61. [Google Scholar] [CrossRef]
- Kirkpatrick, S.; Gelatt, C.D.J.; Vecchi, M.P. Optimization by Simulated Annealing. Science
**1983**, 220, 671–680. [Google Scholar] [CrossRef] - Stillinger, F.H.; Weber, T.A. Nonlinear optimization simplified by hypersurface deformation. J. Stat. Phys.
**1988**, 52, 1429–1445. [Google Scholar] [CrossRef] - Finnila, A.B.; Gomez, M.A.; Sebenik, C.; Stenson, C.; Doll, J.D. Quantum annealing: A new method for minimizing multidimensional functions. Chem. Phys. Lett.
**1994**, 219, 343–348. [Google Scholar] [CrossRef] - Doye, J.P.K.; Meyer, L. Mapping the Magic Numbers in Binary Lennard-Jones Clusters. Phys. Rev. Lett.
**2005**, 95, 063401. [Google Scholar] [CrossRef] [Green Version] - Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Petersson, G.A.; Nakatsuji, H.; et al. Gaussian 09; Revision D.01; Gaussian, Inc.: Wallingford, CT, USA, 2009. [Google Scholar]
- Vosko, S.H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: A critical analysis. Can. J. Phys.
**1980**, 58, 1200–1211. [Google Scholar] [CrossRef] [Green Version] - Chiodo, S.; Russo, N.; Sicilia, E. LANL2DZ basis sets recontracted in the framework of density functional theory. J. Chem. Phys.
**2006**, 125, 104107. [Google Scholar] [CrossRef] - Al-Odail, F.; Mazher, J.; Abuelela, A.M. A density functional theory study of structural, electronic and magnetic properties of small Pd
_{n}Ag (n = 1–8) clusters. Comput. Theor. Chem.**2018**, 1125, 103–111. [Google Scholar] [CrossRef] - Farshad, M.; Perera, D.C.; Rasaiah, J.C. Theoretical study of the stability, structure, and optical spectra of small silver clusters and their formation using density functional theory. Phys. Chem. Chem. Phys.
**2021**, 23, 25507–25517. [Google Scholar] [CrossRef] [PubMed] - Tsuneda, T. Theoretical investigations on geometrical and electronic structures of silver clusters. J. Comput. Chem.
**2019**, 40, 206–211. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kolandaivel, P.; Nirmala, V. Study of proper and improper hydrogen bonding using Bader’s atoms in molecules (AIM) theory and NBO analysis. J. Mol. Struct.
**2004**, 694, 33–38. [Google Scholar] [CrossRef] - Beutel, V.; Kramer, H.G.; Bhale, G.L.; Kuhn, M.; Weyers, K.; Demtröder, W. High-resolution isotope selective laser spectroscopy of Ag
_{2}molecules. J. Chem. Phys.**1993**, 98, 2699–2708. [Google Scholar] [CrossRef] - Simard, B.; Hackett, P.A.; James, A.M.; Langridge-Smith, P.R.R. The bond length of silver dimer. Chem. Phys. Lett.
**1991**, 186, 415–422. [Google Scholar] [CrossRef] - Wang, M.; Liu, X.; Meng, J.; Wu, Z. Interaction of H
_{2}with transition metal homonuclear dimers Cu_{2}, Ag_{2}, Au_{2}and heteronuclear dimers PdCu, PdAg and PdAu. J. Mol. Struct. Theochem.**2007**, 804, 47–55. [Google Scholar] [CrossRef] - Häkkinen, H.; Moseler, M.; Kostko, O.; Morgner, N.; Hoffmann, M.A.; Issendorff, B.V. Symmetry and Electronic Structure of Noble-Metal Nanoparticles and the Role of Relativity. Phys. Rev. Lett.
**2004**, 93, 093401. [Google Scholar] [CrossRef] [Green Version] - Doye, J.P.K.; Wales, D.J. Global minima for transition metal clusters described by Sutton–Chen potentials. New J. Chem.
**1998**, 22, 733–744. [Google Scholar] [CrossRef] [Green Version] - Grigoryan, V.G.; Springborg, M. Structural and energetic properties of nickel clusters: 2 ≤ N ≤ 150. Phys. Rev. B
**2004**, 70, 205415. [Google Scholar] [CrossRef] [Green Version] - Grigoryan, V.G.; Alamanova, D.; Springborg, M. Structure and energetics of Cu
_{N}clusters with 2 ≤ N ≤ 150: An embedded-atom-method study. Phys. Rev. B**2006**, 73, 115415. [Google Scholar] [CrossRef] - Chaves, A.S.; Piotrowski, M.J.; Da Silva, J.L. Evolution of the structural, energetic, and electronic properties of the 3d, 4d, and 5d transition-metal clusters (30 TM
_{n}systems for n = 2–15): A density functional theory investigation. Phys. Chem. Chem. Phys.**2017**, 19, 15484–15502. [Google Scholar] [CrossRef] [PubMed] - McKee, M.L.; Samokhvalov, A. Density functional study of neutral and charged silver clusters Ag
_{n}with n = 2–22. Evolution of properties and structure. J. Phys. Chem. A**2017**, 121, 5018–5028. [Google Scholar] [CrossRef] [PubMed] - Rekha, T.; Rajkumar, B.J. Density functional theory study on silver clusters using dimers, trimers, and tetramers as building units. Can. J. Phys.
**2015**, 93, 318–325. [Google Scholar] [CrossRef] - Parr, R.G.; Yang, W. Density functional approach to the frontier-electron theory of chemical reactivity. J. Am. Chem. Soc.
**1984**, 106, 4049–4050. [Google Scholar] [CrossRef] - Morell, C.; Grand, A.; Toro-Labbé, A. New dual descriptor for chemical reactivity. J. Phys. Chem. A
**2005**, 109, 205–212. [Google Scholar] [CrossRef] [PubMed] - Lu, T.; Chen, F. Multiwfn: A multifunctional wavefunction analyzer. J. Comput. Chem.
**2012**, 33, 580–592. [Google Scholar] [CrossRef] [PubMed] - Alonso, J.A. Electronic and Atomic Structure, and Magnetism of Transition-Metal Clusters. Chem. Rev.
**2000**, 100, 637–678. [Google Scholar] [CrossRef] - Baletto, F.; Ferrando, R. Structural properties of nanoclusters: Energetic, thermodynamic, and kinetic effects. Rev. Mod. Phys.
**2005**, 77, 371–423. [Google Scholar] [CrossRef] [Green Version] - Grigoryan, V.G.; Springborg, M. Vibrational and thermodynamic properties of metal clusters with up to 150 atoms calculated by the embedded-atom method. Phys. Rev. B
**2011**, 83, 155413. [Google Scholar] [CrossRef] - Grigoryan, V.G.; Springborg, M. Temperature dependence of stability of copper clusters. Z. Phys. Chem.
**2016**, 230, 1037–1055. [Google Scholar] [CrossRef] - Grigoryan, V.G.; Springborg, M. Temperature and isomeric effects in nanoclusters. Phys. Chem. Chem. Phys.
**2019**, 21, 5646–5654. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Optimized geometries of of Ag${}_{n}$ (n = 2–7) clusters as obtained with the DFT calculations. The labels refer to the energetic ordering according to the EAM calculations.

**Figure 2.**Optimized geometries of of Ag${}_{n}$ (n = 8–11) clusters as obtained with the DFT calculations. The labels refer to the energetic ordering according to the EAM calculations.

**Figure 3.**Similarity function for the comparison of silver clusters with copper clusters. Structures of both systems have been defined by the same EAM potential model but global optimization of Ag clusters have been done by BH and those for Cu are the results of Aufbau/Abbau optimizations. The similarity function is based on the radial distances.

**Figure 4.**Similarity function for the comparison of silver clusters with nickel clusters. Structures of both systems have been defined by the same EAM potential model but global optimization of Ag clusters have been done by BH and those for Ni are the results of Aufbau/Abbau optimizations. The similarity function is based on the radial distances.

**Figure 5.**Similarity function for silver clusters calculated using the EAM and the n-body Gupta potential. Also the applied global optimization methods were different: BH for EAM and Aufbau/Abbau method for Gupta potential. The similarity function is based on the radial distances.

**Figure 6.**The stability function (in eV) of the silver clusters versus cluster size calculated using EAM-BH.

**Figure 7.**The difference between energies of energetically first and second lowest lying isomers of Ag clusters versus the number of atoms in the cluster as found in the EAM calculations.

**Figure 8.**The variation of (

**a**) ${E}_{b}$/n, (

**b**) ${\Delta}^{2}{E}_{n}$, (

**c**) $\mu $, and (

**d**) ${E}_{\mathrm{gap}}$ as a function of cluster size (n) as found in the DFT calculations and considering solely the energetically most stable isomer of these calculations. In (

**a**,

**b**) we also show the equivalent results from the EAM calculations (the red curves).

**Figure 9.**The dual descriptor ($\Delta f\left(\overrightarrow{r}\right)$) of the most stable Ag${}_{n}$ (n = 2–11) clusters according to the DFT calculations. Green and blue regions correspond to positive and negative regions, respectively.

**Figure 10.**Molecular electrostatic (MEP) of the most stable Ag${}_{n}$ (n = 2–11) clusters according to the DFT calculations. The potential is shown on a surface of constant electron density (0.0004 a.u.) and the scales in each panel are given in a.u.

**Figure 11.**Similarity function of n atomic clusters when comparing with the clusters with n − 1 atoms. This can be used in identifying the structural changes due to a growth. The similarity function is based on the interatomic distances.

**Figure 13.**Maxima and minima of vibrational frequencies for the optimized Ag${}_{n}$ clusters. The dashed line shows the maximum value of the vibrational frequencies for bulk silver.

**Table 1.**Point groups of the optimized Ag${}_{n}$ clusters as obtained with the DFT calculations. $n.1$, $n.2$, and $n.3$ represents the results for the energetically lowest, 2nd lowest, and 3rd lowest isomer, respectively, from the EAM—and not the DFT—calculations.

n | $\mathit{n}.1$ | $\mathit{n}.2$ | $\mathit{n}.3$ |
---|---|---|---|

2 | ${D}_{\infty h}$ | ||

3 | ${C}_{2v}$ | ||

4 | ${D}_{2h}$ | ||

5 | ${C}_{2v}$ | ||

6 | ${C}_{5v}$ | ${C}_{5v}$ | |

7 | ${D}_{5h}$ | ${C}_{3v}$ | ${D}_{5h}$ |

8 | ${D}_{2d}$ | ${C}_{s}$ | ${C}_{2v}$ |

9 | ${C}_{2v}$ | ${C}_{1}$ | ${C}_{s}$ |

10 | ${C}_{1}$ | ${C}_{2}$ | ${C}_{2v}$ |

11 | ${C}_{2v}$ | ${C}_{2v}$ | ${C}_{2v}$ |

**Table 2.**Point groups of the optimized Ag${}_{n}$ clusters from the EAM calculations. $n.1$, $n.2$, and $n.3$ represents the results for the energetically lowest, 2nd lowest, and 3rd lowest isomer, respectively.

n | $\mathit{n}.1$ | $\mathit{n}.2$ | $\mathit{n}.3$ | n | $\mathit{n}.1$ | $\mathit{n}.2$ | $\mathit{n}.3$ | n | $\mathit{n}.1$ | $\mathit{n}.2$ | $\mathit{n}.3$ |
---|---|---|---|---|---|---|---|---|---|---|---|

2 | ${D}_{\infty h}$ | 35 | ${C}_{s}$ | ${D}_{3}$ | ${C}_{2v}$ | 68 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{s}$ | ||

3 | ${D}_{3h}$ | 36 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{2}$ | 69 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ | ||

4 | ${T}_{d}$ | 37 | ${C}_{s}$ | ${C}_{3v}$ | ${C}_{2}$ | 70 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ | ||

5 | ${D}_{3h}$ | 38 | ${O}_{h}$ | ${C}_{5v}$ | ${C}_{s}$ | 71 | ${C}_{2v}$ | ${C}_{5}$ | ${C}_{5v}$ | ||

6 | ${O}_{h}$ | ${C}_{2v}$ | 39 | ${C}_{5v}$ | ${C}_{5}$ | ${C}_{4v}$ | 72 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ | |

7 | ${D}_{5h}$ | ${C}_{3v}$ | ${C}_{2}$ | 40 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 73 | ${C}_{2v}$ | ${C}_{s}$ | ${C}_{S}$ |

8 | ${D}_{2d}$ | ${C}_{s}$ | ${D}_{3d}$ | 41 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 74 | ${C}_{5v}$ | ${C}_{1}$ | ${C}_{s}$ |

9 | ${C}_{2v}$ | ${D}_{3h}$ | ${C}_{2v}$ | 42 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{2v}$ | 75 | ${D}_{5h}$ | ${C}_{s}$ | ${C}_{s}$ |

10 | ${C}_{3v}$ | ${C}_{2}$ | ${C}_{2v}$ | 43 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 76 | ${C}_{1}$ | ${C}_{s}$ | ${C}_{1}$ |

11 | ${C}_{2v}$ | ${C}_{2}$ | ${C}_{2}$ | 44 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{s}$ | 77 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{2v}$ |

12 | ${C}_{5v}$ | ${D}_{2d}$ | ${C}_{1}$ | 45 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ | 78 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{s}$ |

13 | ${I}_{h}$ | ${C}_{s}$ | ${C}_{s}$ | 46 | ${C}_{2v}$ | ${C}_{s}$ | ${C}_{1}$ | 79 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

14 | ${C}_{3v}$ | ${C}_{2v}$ | ${C}_{1}$ | 47 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ | 80 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{s}$ |

15 | ${C}_{2v}$ | ${D}_{6d}$ | ${C}_{2v}$ | 48 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 81 | ${C}_{1}$ | ${C}_{s}$ | ${C}_{1}$ |

16 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 49 | ${C}_{3v}$ | ${C}_{s}$ | ${C}_{s}$ | 82 | ${C}_{1}$ | ${C}_{s}$ | ${C}_{s}$ |

17 | ${C}_{2}$ | ${C}_{s}$ | ${C}_{s}$ | 50 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{s}$ | 83 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

18 | ${C}_{s}$ | ${C}_{5v}$ | ${C}_{s}$ | 51 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ | 84 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{s}$ |

19 | ${D}_{5h}$ | ${C}_{1}$ | ${C}_{s}$ | 52 | ${C}_{2v}$ | ${C}_{3v}$ | ${C}_{s}$ | 85 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ |

20 | ${C}_{2v}$ | ${C}_{s}$ | ${D}_{3d}$ | 53 | ${C}_{2v}$ | ${D}_{5d}$ | ${C}_{2v}$ | 86 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

21 | ${C}_{1}$ | ${C}_{2v}$ | ${C}_{s}$ | 54 | ${C}_{5v}$ | ${I}_{h}$ | ${C}_{2v}$ | 87 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{2}$ |

22 | ${C}_{1}$ | ${C}_{s}$ | ${C}_{s}$ | 55 | ${I}_{h}$ | ${C}_{s}$ | ${C}_{1}$ | 88 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

23 | ${D}_{3h}$ | ${D}_{3h}$ | ${C}_{2}$ | 56 | ${C}_{3v}$ | ${C}_{s}$ | ${C}_{s}$ | 89 | ${C}_{3v}$ | ${C}_{s}$ | ${C}_{1}$ |

24 | ${C}_{2v}$ | ${C}_{s}$ | ${D}_{3}$ | 57 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{s}$ | 90 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

25 | ${C}_{3}$ | ${C}_{s}$ | ${C}_{1}$ | 58 | ${C}_{3v}$ | ${C}_{s}$ | ${C}_{1}$ | 91 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{1}$ |

26 | ${C}_{1}$ | ${T}_{d}$ | ${C}_{2v}$ | 59 | ${C}_{2v}$ | ${C}_{1}$ | ${C}_{1}$ | 92 | ${C}_{3v}$ | ${C}_{1}$ | ${C}_{1}$ |

27 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{2}$ | 60 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{s}$ | 93 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ |

28 | T | ${C}_{1}$ | ${C}_{3v}$ | 61 | ${C}_{2v}$ | ${C}_{1}$ | ${C}_{1}$ | 94 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ |

29 | ${C}_{3}$ | ${C}_{2v}$ | ${C}_{2}$ | 62 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ | 95 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ |

30 | ${C}_{s}$ | ${C}_{2v}$ | ${C}_{1}$ | 63 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{s}$ | 96 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{s}$ |

31 | ${C}_{3}$ | ${C}_{2v}$ | ${C}_{s}$ | 64 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ | 97 | ${C}_{1}$ | ${C}_{1}$ | ${C}_{1}$ |

32 | ${C}_{2v}$ | ${D}_{3}$ | ${C}_{1}$ | 65 | ${C}_{2v}$ | ${C}_{1}$ | ${C}_{s}$ | 98 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ |

33 | ${C}_{2}$ | ${C}_{s}$ | ${C}_{s}$ | 66 | ${C}_{s}$ | ${C}_{1}$ | ${C}_{1}$ | 99 | ${C}_{s}$ | ${C}_{2v}$ | ${C}_{1}$ |

34 | ${C}_{s}$ | ${C}_{s}$ | ${C}_{s}$ | 67 | ${C}_{2v}$ | ${C}_{s}$ | ${C}_{s}$ | 100 | ${C}_{5v}$ | ${C}_{1}$ | ${C}_{s}$ |

**Table 3.**Calculated values of VIE, VEA, $\mu $, the energy of HOMO, that of LUMO, and E${}_{\mathrm{gap}}$ of Ag${}_{n}$ (n = 2–11) clusters. All quantities are given in eV.

Cluster Size | VIE | VEA | $\mathit{\mu}$ | ${\mathit{E}}_{\mathbf{HOMO}}$ | ${\mathit{E}}_{\mathbf{LUMO}}$ | ${\mathit{E}}_{\mathbf{gap}}$ |
---|---|---|---|---|---|---|

2 | 8.02 | 1.08 | −4.55 | −5.26 | −3.18 | 2.08 |

3 | 6.11 | 0.56 | −3.34 | −3.90 | −3.49 | 0.40 |

4 | 6.67 | 1.73 | −4.20 | −4.51 | −3.66 | 0.85 |

5 | 6.30 | 1.86 | −4.08 | −4.23 | −3.87 | 0.36 |

6 | 7.01 | 1.44 | −4.23 | −5.01 | −3.13 | 1.89 |

7 | 6.15 | 2.00 | −4.08 | −4.22 | −2.95 | 1.26 |

8 | 6.80 | 1.51 | −4.16 | −4.90 | −3.14 | 1.75 |

9 | 5.57 | 1.91 | −3.74 | −3.86 | −3.42 | 0.44 |

10 | 6.13 | 1.72 | −3.93 | −4.41 | −3.29 | 1.13 |

11 | 5.92 | 2.38 | −4.15 | −4.26 | −3.42 | 0.84 |

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## Share and Cite

**MDPI and ACS Style**

Garg, S.; Kaur, N.; Goel, N.; Molayem, M.; Grigoryan, V.G.; Springborg, M.
Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations. *Molecules* **2023**, *28*, 3266.
https://doi.org/10.3390/molecules28073266

**AMA Style**

Garg S, Kaur N, Goel N, Molayem M, Grigoryan VG, Springborg M.
Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations. *Molecules*. 2023; 28(7):3266.
https://doi.org/10.3390/molecules28073266

**Chicago/Turabian Style**

Garg, Shivangi, Navjot Kaur, Neetu Goel, Mohammad Molayem, Valeri G. Grigoryan, and Michael Springborg.
2023. "Properties of Naked Silver Clusters with Up to 100 Atoms as Found with Embedded-Atom and Density-Functional Calculations" *Molecules* 28, no. 7: 3266.
https://doi.org/10.3390/molecules28073266