Prediction of Structures and Atomization Energies of Small Silver

Article pubs.acs.org/JPCA

Prediction of Structures and Atomization Energies of Small Silver Clusters, (Ag)n, n < 100 Mingyang Chen, Jason E. Dyer, Keijing Li, and David A. Dixon* Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, Alabama 35487-0336, United States S Supporting Information *

ABSTRACT: Neutral silver clusters, Agn, were studied using density functional theory (DFT) followed by high level coupled cluster CCSD(T) calculations to determine the low energy isomers for each cluster size for small clusters. The normalized atomization energy, heats of formation, and average bond lengths were calculated for each of the different isomeric forms of the silver clusters. For n = 2−6, the preferred geometry is planar, and the larger n = 7−8 clusters prefer higher symmetry, three-dimensional geometries. The low spin state is predicted to be the ground state for every cluster size. A number of new low energy isomers for the heptamer and octamer were found. Additional larger Agn structures, n < 100, were initially optimized using a tree growth-hybrid genetic algorithm with an embedded atom method (EAM) potential. For n ≤ 20, DFT was used to optimize the geometries. DFT with benchmarked functionals were used to predict that the normalized atomization energies (⟨AE⟩s) for Agn start to converge slowly to the bulk at n = 55. The ⟨AE⟩ for Ag99 is predicted to be ∼50 kcal/mol.

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INTRODUCTION The study of small atomic clusters is of interest because of their inherent properties as well as the role that they can play in understanding the evolution of particles toward the physical and chemical properties of the bulk. Silver clusters serve as a bridge between the atomic level and the metal1 and have a number of important technological applications including photography and redox catalysis.2,3 A variety of experimental and computational methods have been used to study silver clusters. Computational approaches include the semiempirical diatomics-in-molecules (DIM), extended Huckel theory methods, and ab initio molecular orbital methods including coupled cluster (CCSD) theory, and configuration interaction.4−7 The Ag−Ag bond length in Ag2 has been determined to be re = 2.5310 Å by laser induced fluorescence, 8 with a De of 1.66 ± 0.07 eV and ωe of 192.4 cm−1.9,10 The Agn0/+ clusters for n = 2 to 4 have been studied at the CCSD level with a 5s3p2d contracted basis set.11 The Ag trimer has been the subject of many theoretical studies due to the presence of a conical intersection for the equilateral triangle, which leads to a Jahn− Teller distortion.4,12−15 It has also been studied experimentally using a variety of spectroscopic methods.16−18 For the tetramer, the singlet rhombus structure has previously been shown to be the lowest lying isomer by both theoretical 19−23 and experimental work.24 The structure of the pentamer is generally considered to be a trapezoidal shape composed of interlocking triangles25−27 but an ab initio molecular dynamics study28 predicted the bipyramidal structure to be lower in energy. Small Agn clusters (n > 5) have also been studied by many research groups. Fournier used the local spin density method with the © 2013 American Chemical Society

Vosko−Wilk−Nusair correlation functional to study Agn (n = 2 to 12).22 Lecoultre et al. measured and calculated the ultraviolet−visible absorption for Agn (n = 1 to 9) using the time-dependent density functional theory (TD-DFT) method; their calculated results showed excellent agreement with their experiments.24 Fernández et al. discussed the trends in structure and bonding in Agn for n ≤ 13 and n = 20.29 Lee et al. reported the optimized geometries for Agn, Aun, and AgxAun−x (n up to 13) using the Becke−Perdue exchange and correlation functionals.30 Yang et al. optimized Agn (n = 2−20) using the DFT methods with the PBE functional.31 Shao et al. optimized Agn (n up to 80) with the Gupta and Sutton−Chen potentials.32 Yang et al. optimized Agn (n up to 160) using a modified dynamic lattice searching method with the Gupta potential.33

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COMPUTATIONAL METHODS The initial geometries of the smaller Agn clusters (n ≤ 8) were built using fundamental geometry blocks such as triangles, squares, and tetrahedrons. For the larger clusters with more than 8 Ag atoms, we applied our novel “tree-growth-hybrid genetic algorithm” procedure34 to generate the initial geometries. In the tree-growth algorithm, the clusters are grown from a small seed (an atom or small groups of atoms) to the size of interest in a stepwise fashion. Additional atoms are attached to the smaller cluster from the previous step at each new step by analogy to new leaves grown by a tree. The Received: May 6, 2013 Revised: July 19, 2013 Published: July 24, 2013 8298

dx.doi.org/10.1021/jp404493w | J. Phys. Chem. A 2013, 117, 8298−8313

The Journal of Physical Chemistry A

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Figure 1. Optimized structures of neutral Agn (n = 2−6). Bond lengths in Å. Relative energies (kcal/mol) are shown below illustrations in the order of CCSD(T)/aD, aT, and aQ.

program system38 with the B3LYP exchange-correlation functional39 and the aug-cc-pVDZ-PP (PP = pseudopotential; aug-cc-pVDZ-PP abbreviated as aD) basis set (labeled as aD).40 Zero-point energies (ZPE) were calculated at the B3LYP/aD level. The B3LYP functional was chosen because of our general experience with using it for a broad range of transition metal containing molecules and because it yields reasonable geometries.41 As described in more detail, below, a range of other functionals were used to predict normalized atomization energies and isomerization energies. Single-point calculations were done at the correlated molecular orbital theory R/UCCSD(T) (coupled cluster theory with single and double excitations that included a perturbative triples correction starting from restricted Hartree−Fock orbitals with the spin constraint relaxed in the CCSD(T) calculations

addition of the atoms is controlled by predefined geometry parameters to minimize the search space. At each step, the newly generated structures are evaluated using a simplified energy expression, and those that are of low energy are carried to the next step. The tree-growth algorithm generates the input for the hybrid genetic algorithm module,34 and the latter searches for the global minimum of the cluster. We applied the ‘Universal 6’35 embedded atom method (EAM)36 potential for Ag to evaluate the Agn total energies during the hybrid genetic algorithm steps. The EAM single-point calculations were carried out using the LAMMPS code.37 The other steps were performed using the software package34 developed by our group. The initial electronic structure calculations were done at the density functional theory (DFT) level using the Gaussian 8299



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kcal/mol.9,10 The dimer has a stretching frequency of 179 cm−1 at the B3LYP level as compared to the calculated vibration frequency of 190 cm−1 at the CCSD(T)/aQ level and an experimental value of 192 cm−1, 10 consistent with the longer bond length predicted at the B3LYP level. For the trimer at the B3LYP level, we found two triangles with C2v geometries due to the presence of a conical intersection for the equilateral triangle, which leads to a Jahn−Teller distortion.4,51 The isosceles triangular 2B2 state (3b) with a vertex ∠ = 75.7° is predicted to be ∼1 kcal/mol lower in energy than the isosceles triangular structure (3a) with a vertex ∠ = 54.8° for the 2A1 state at the B3LYP level; the 2A1 state has an imaginary frequency of ∼100i cm−1. We note that calculations at the B3LYP/aD level incorrectly predict a third geometry with a bond angle of 143° for the 2B2 state to be the global minimum by a small amount. This is not consistent with previous studies and with our higher level CCSD(T)/aT calculations as the large bond angle minimum disappears at the higher level. At the CCSD(T) level, the structure with a 143° angle is found to be ∼2 kcal/mol higher in energy than the global minimum and is not a local minimum. The two low-lying isomers of Ag3 at the CCSD(T)/aT level have re = 2.615 Å and θ = 67.7° for the 2B2 state, and re = 2.760 Å and θ = 55.5° for the 2A1 state. At the CCSD(T)/aQ//CCSD(T)/aT level, the energies of the 2A1 and 2B2 states are within 0.3 kcal/mol of each other with the 2B2 being the minimum. In contrast at the CCSD(T)//B3LYP level, the 2A1 is predicted to be lower in energy by less than ∼0.3 kcal/mol, showing that the relative energies for these two states depend on the actual bond distances. The 2B2 state is predicted to be 0.4 kcal/mol lower in energy than the 2A1 state at the CCSD(T)/CBS(aDTQ) and CCSD(T)/CBS(aQ5) levels at the CCSD(T) geometries. The ⟨AE⟩ for the 2B2 and 2A1 states are 19.1 and 19.0 kcal/mol at both the CCSD(T)/CBS(aDTQ) and the CCSD(T)/CBS(aQ5) levels with the CCSD(T) geometries. The trimer bond lengths are slightly longer than in the dimer. The calculated CCSD(T)/CBS ⟨AE⟩s fall between the 22.6 kcal/mol that Bonacic-Koutecky et al.11 calculated at the CCSD level/ 5s3p2d-ECP level and the 17.3 kcal/mol calculated using the semiempirical DIM method.4 The dissociation energy for reaction Ag3 → Ag2 + Ag is calculated to be 18.5 kcal/mol at the CCSD(T)/CBS(aQ5) level. The 2A1 state is predicted to have a1 vibrational modes at 119 cm−1 and 187 cm−1, and an imaginary b2 frequency of 71i cm−1 at the CCSD(T)/aT level. The 2B2 state is predicted to have a1 frequencies at 70 cm−1 and 180 cm−1 and a b2 frequency at 120 cm−1 at the CCSD(T)/aT level, which are the bending, symmetric stretching, and the antisymmetric stretching modes, respectively. Experimental values of a stretch at 184 cm−1 and a bend at 67 cm−1 have been reported by Ellis et al.,17 and a band at 121 cm−1 is attributed to the overtone of the bending mode. Fielicke et al. observed Ag3 vibrational bands at 113 and 183 cm−1 using far-infrared spectroscopy.52 The CCSD(T) values are in reasonable agreement with these experimental values. The actual calculation of the spectra is complicated by the presence of the conical intersection and the very low frequencies with high anharmonicity. For the tetramer, the D2h rhombus structure (4a) is predicted to be the most stable isomer as expected from previous calculations.22,24,53 A range of calculations on different geometries and higher spin states were performed in order to determine the energetically low lying isomers of the tetramer. The singlet “T-shaped” C2v structure (4b) is 7.4 kcal/mol

for open shell molecules) level42−46 with the effective core potential (ECP) and augmented correlation consistent basis sets, aug-cc-pVnZ (n = D, T, Q) using the MOLPRO-2009 program system.47 For n = 1−6, calculations were done with the above three basis sets and extrapolated to the complete basis set (CBS) limit using equation (1) 48 E(x) = A CBS + Bexp[−(x − 1)] + Cexp[−(x − 1)2 ]

(1)

where x is 2, 3, and 4 for the aVnZ basis sets with n = D, T, and Q, respectively. For the CBS energy calculation for Agn, n = 2− 4, we also used the Q5 extrapolation to the CBS limit with equation (2)49,50 E(l) = ECBS + B /l 3

(2)

with l = 4 and 5. For the Ag7 and Ag8 clusters, however, the CBS extrapolation was not performed because the clusters were too large to perform all of the necessary calculations. The normalized atomization energy is defined by eq 3 AE = (nE(Ag) − E(Ag n))/n

(3)

and corresponds to the cohesive energy of the bulk with this definition. The heats of formation (ΔHf) for Agn at 0 K are calculated by using eq 4 ΔHf (Ag n)calcd = n(ΔHf (Ag)exptl − ⟨AE⟩calcd )

(4)

The reaction energy for the nucleation reaction Agx + Agy → Agx+y is given by Erxn = E(Agx + y) − E(Agx) − E(Ag y)

(5)

For x = 1, this corresponds to a possible definition of the cohesive energy of the cluster, and we choose to use normalized atomization energy and nucleation reaction energies to avoid confusion. The smaller calculations were performed on the Dense Memory Cluster (DMC) in the Alabama Supercomputing Center, the University of Alabama High Performance Computer (UAHPC), and a Dell Cluster. The larger MOLPRO calculations were performed on the massively parallel HP Linux cluster in the Molecular Science Computing Facility in the William R. Wiley Environmental Molecular Sciences Laboratory (EMSL) located at Pacific Northwest National Laboratory.

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RESULTS AND DISCUSSION Geometries and Isomer Stabilities for Agn, n = 2−8. The optimized structures for n = 2−6 are shown in Figure 1 for the low energy isomers. For the dimer (2a), a bond length of 2.585 Å was calculated at the DFT/B3LYP level, which is ∼0.05 Å longer than the experimental distance8 of re = 2.53350 ± 0.00048 Å; the corresponding CCSD value is 2.52 Å.11 We reoptimized the Ag−Ag bond distance at the CCSD(T) level with the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets giving bond lengths of 2.565, 2.533, and 2.532 Å, respectively, with the latter two in excellent agreement with the experimental value. The extrapolated CBS(aDTQ) and CBS(aQ5) ⟨AE⟩s for Ag2 at the B3LYP geometry or at the optimized CCSD(T) geometry are all within 0.2 kcal/mol, showing that the precise bond distance to within 0.03 Å is not needed for an accurate bond energy calculation consistent with the low vibrational frequency. The calculated CCSD(T)/CBS ⟨AE⟩s are consistent with the experimental value of 19.0 ± 0.3 8300



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higher in energy than the rhombus at the CCSD(T)/ CBS(aDTQ) level. The next two states (4c and 4d) are triplets. The lowest energy triplet (4c) has a D2h rhombic structure and is ∼16 kcal/mol higher than the singlet ground state at the B3LYP/aD level; this energy difference increases to 18.0 kcal/mol at the CCSD(T)/CBS(aDTQ) level using the B3LYP geometries. The linear triplet isomer (4d) is predicted to be ∼20 kcal/mol higher in energy than the lowest energy D2h triplet. At the B3LYP/aD level, the D2h rhombic structure (4a) has three vibrational bands (over 100 cm−1) at 101, 151, and 176 cm−1, This is essentially consistent with previous experimental observation, by Fielicke et al., at 163 and 196 cm−1 for Ag4 using far-infrared spectroscopy.52 The pentamer prefers a planar geometry as does the n = 4 structure. The five lowest energy isomers are calculated to be doublets. A C2v 2-D trapezoidal shape (5a) composed of interlocking triangles is the lowest energy isomer followed by a 3-D C2v bipyramidal isomer (5b), and then by a D2h ‘bowtie’ isomer (5c). The bipyramidal (5b) and bowtie (5c) isomers are 8.6 and 10.2 kcal/mol higher in energy than the lowest in energy trapezoidal structure at the CCSD(T)/CBS(aDTQ) level, respectively. An edge-capped tetrahedron (5d) in C2v symmetry is predicted to be ∼12 kcal/mol above the lowest energy trapezoidal structure (5a). The twisted bowtie structure (5e) in C2v symmetry is ∼20 kcal/mol higher in energy than 5a. At n = 5, the 3-D structures are beginning to become closer in energy to the 2-D planar structures. Our results are consistent with those of other workers.24,22 The hexamer is predicted to be the largest cluster that still prefers a 2-D planar structure. A D3h planar trigonal isomer (6a) with an average bond length of 2.756 Å is found to be the lowest energy isomer with a 3-D pentagonal pyramid isomer (6b) ∼4 kcal/mol higher in energy. The average bond length for the lowest energy hexamer (6a) is nearly the same as that of the lowest energy pentamer (5a). A 2-D planar C2v structure (6c) with a ‘shield’ shape is predicted to be 7 kcal/mol higher than the lowest energy planar D3h isomer (6a). The ⟨AE⟩ for the planar trigonal isomer (6a) is 32.6 kcal/mol at the CCSD(T)/CBS(aDTQ) level. The two triplets are a planar structure tiled by triangles (6e) and an off center pyramid (6d). For the triplets, the 3-D pyramid and the planar geometry are, respectively, 19.3 and 27.6 kcal/mol higher in energy than the ground state singlet. Unlike Agn, n ≤ 6, Ag7 has a 3-D, densely packed structure as the most stable isomer (Figure 2). It is generally agreed29 that this cluster size is the size where the silver nanocluster begins to prefer a densely packed arrangement, consistent with the cubic close packed structure of bulk Ag in the crystal.54 Higher symmetry geometries are predicted to dominate the heptamer structure with the lowest lying isomer having a D5h pentagonal bipyramid geometry (7a), which was previously observed in the Ne matrix using ESR by Bach et al.55 The C3v tetrahedral type structure (7b), previously observed in the Ar matrix by resonance Raman spectroscopy,56 is ∼4 kcal/mol higher in energy than 7a at the CCSD(T)/aD level. Previous computational studies also predicted these two isomers to be the two lowest isomers with the planar hexagonal geometry being discussed as well.29 Together with these previously discussed structures, a number of other energetically low lying geometries were predicted including a 3-edge-capped planar square (7f), a planar structure with a hexagonal pattern (7g), structures with off center pyramids with an extra Ag (7e), and planar structures composed of triangles (7i and 7j). These latter structures are

Figure 2. Optimized Ag7 geometries. For three-dimensional geometries, the overhead view and side view are shown from top to bottom. Relative energies (kcal/mol) are at the CCSD(T)/aD level with symmetries and spin states listed last.

the least stable doublet states that we found. The B3LYP calculations overestimate the stabilities of the low-lying 2-D structures and predict the capped planar square, the planar hexagon, and the structure with a trigonal tiling pattern to be less than ∼5 kcal/mol above the lowest energy isomer as compared to CCSD(T), which predicted energies of at least 15 kcal/mol above the ground state for these structures. There are four quartet states within 40 kcal/mol of the ground state, three geometries that are densely packed (7l, 7m, and 7n) and one planar geometry consisting of triangles (7o). This latter structure is found for both spin states, and the quartet is ∼17 kcal/mol higher than the doublet. Only R/UCCSD(T)/aD calculations (378 basis functions) were able to be performed on all of the heptamer geometries due to the size of the basis set and the open shell nature of the calculations. Continuing the pattern of forming densely packed 3-D structures as the cluster grows, the lowest energy isomer for Ag8 is a Td structure (8a) followed very closely by a D2d structure (8b). Our prediction for the most stable isomer agrees well with most of the previous studies,24,29,22 and molecular dynamic studies using force field potentials predicted the D2d structure (8b) to be the most stable.32,33 In fact, the eight lowest energy isomers for the octamer are all three-dimensional (8a−8h in Figure 3). The lowest energy 2-D planar isomer (8i), an edge-capped hexagon tiled with triangles, is the ninth lowest energy isomer, and is >20 kcal/mol higher in energy than the lowest energy structure (8a) at the CCSD(T) level; DFT predicted a smaller energy difference of 10 kcal/mol between the 2-D and 3-D structures. D2d pseudoplanar (8j) and D4h planar (not shown, but similar to 8j) structures, which are edge-capped squares, are ∼23 kcal/mol above the ground state (8a) at the CCSD(T)/aD and aT levels. Other planar 1Ag8 isomers consisting of smaller triangles are ca. 30−40 kcal/mol higher in energy than the lowest energy isomer (8a). These planar structures have similar structures to the ground states for the smaller Agn (n = 3−6) clusters and the higher energy planar states for the heptamers. We predicted new triplet state geometries as well. The energy of the two low energy triplet 8301



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Figure 3. Optimized Ag8 geometries. Lowest energies structures shown from left to right. Relative energies (kcal/mol) are shown below in order of CCSD(T)/aD and CCSD(T)/aT with symmetries and spin states listed last. B3LYP/aD relative energies are in parenthesis when CCSD(T) values are missing.

Table 1. Ag7 Isomer Relative Energies at the CCSD(T) and DFT Levels with the aD Basis Set cluster

CCSD(T)

B3LYP

M06

PBE

PW91

SVWN5

TPSS

ωB97XD

7a 7b 7c 7d 7e 7f 7g 7h 7i

0.0 4.0 8.9 14.1 14.8 16.6 16.6 16.7 16.8

0.0 1.7 3.3 6.0 7.2 4.0 4.8 6.1 2.7

0.0 4.0 8.0 12.9 12.9 16.6 13.8 15.8 14.6

0.0 3.2 6.7 10.8 11.6 13.4 10.6 13.0 10.8

0.0 3.2 6.7 10.8 11.5 13.3 10.5 12.9 10.8

0.0 5.1 11.0 17.4 17.5 25.5 18.6 21.7 21.6

0.0 4.1 8.8 13.9 15.0 19.2 14.8 17.0 15.8

0.0 2.0 4.9 8.7 11.0 9.0 10.5 9.8 8.0

cluster increases beyond n = 6. DFT with the B3LYP functional tends to overstabilize the 2-D structures for Agn (n = 2−8) as compared to the CCSD(T) energies, especially for the Ag7 and Ag8, but it still predicts 3-D structures to be the lowest energy ones for n = 7 and 8. All of the lowest energy structures for Agn (n = 2−8) are low-spin, and the high-spin structures are all exited states much higher in energy. As a result, we calculated the energies for Agn (n > 8) as low spin singlets (n even) or doublets (n odd). Benchmarking DFT Functionals for Isomer Relative Energies. For Agn (n > 8), the energies of less stable isomers are only calculated at the DFT level due to the high computational cost of the CCSD(T) method. In order to further examine the quality of the DFT results, we benchmarked a range of DFT functionals, B3LYP, 39 SVWN5,57 M06,58 PBE,59 PW91,60 TPSS,61 and ωB97XD62

states (8k and 8q) were able to be calculated at the CCSD(T)/ aT level, but for the remainder of the structures, this was not possible. The first triplet state (8k) is actually lower in energy than some of the higher singlet states, ∼25 kcal/mol higher in energy than the singlet ground state. The rest of the triplet states are mostly planar consisting of smaller triangles as found for the smaller clusters. We now summarize the results for Agn, n ≤ 8. For n = 4, there are no 3-D structures within 20 kcal/mol of the 2-D most stable isomer. The lowest 3-D isomers are ∼10 and ∼5 kcal/ mol above the lowest 2-D isomers for the pentamer and hexamer, respectively. For the heptamer, the lowest isomer is a 3-D structure, but the lowest energy 2-D structure is closer in energy to the ground state as compared to the octamer. This shows that the densely packed 3-D geometries gain stability and start to become the lowest isomers as the size of the silver 8302



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Geometries and Isomer Stabilities for Agn, n = 9−20. The geometries at the B3LYP/aD level for the low lying isomers of Agn (n = 9−20) are shown in Figure 4, and the relative energies calculated at the B3LYP, TPSS, and M06 levels with the aD basis set are shown in Table 3. The starting structures were generated using the TG-HGA method with the EAM potential or taken from prior work. B3LYP and M06 functionals predicted that the lowest energy Ag9 isomer is a Cs structure (9a), which can be obtained by capping a face of the

for the isomer relative energies for Ag7 calculated at the CCSD(T)/aD level (Table 1). The TPSS and M06 functionals gave the best agreement with the CCSD(T) results for Ag7, giving isomer relative energies within ∼1 kcal/mol of the CCSD(T) values. The B3LYP, PBE, PW91, and ωB97XD functionals predicted isomer relative energies smaller than the CCSD(T) energy differences, and the local SVWN5 functional gives larger energy differences. Table 2 shows the isomer relative energies for Agn (n = 3−8) calculated at the CCSD(T) level and at the DFT level with the Table 2. Isomer Relative Energies for Agn, n = 3−8 at the CCSD(T), B3LYP, TPSS, and M06 Levels with the aD Basis Set cluster

spin

CCSD(T)

B3LYP

TPSS

M06

3a 3b 4a 4b 4c 4d 5a 5b 5c 5d 5e 6a 6b 6c 6d 6e 7a 7b 7c 7d 7e 7f 7g 7h 7i 8a 8b 8c 8d 8e 8f 8g 8h 8i 8j

2 2 1 1 3 3 2 2 2 2 2 1 1 1 3 3 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1

0.0 0.1 0.0 6.6 17.5 40.2 0.0 9.8 10.7 11.9 19.8 0.0 4.5 6.7 19.0 26.8 0.0 4.0 8.9 14.1 14.8 16.6 16.6 16.7 16.8 0.0 0.8 4.6 7.4 8.4 10.2 16.7 17.1 22.3 22.5

1.0 0.0 0.0 2.7 15.7 31.5 0.0 14.9 7.3 15.4 18.1 0.0 7.3 7.1 26.6 25.6 0.0 1.7 3.3 6.0 7.2 4.0 4.8 6.1 2.7 0.0 2.0 3.3 5.6 7.3 6.7 8.1 10.2 9.5 6.8

0.0 0.0 0.0 5.7 16.6 38.1 0.0 10.7 9.9 13.2 26.3 0.0 4.9 6.5 18.6 25.6 0.0 4.1 8.8 13.9 15.0 19.2 14.8 17.0 15.8 0.0 1.2 4.5 9.0 8.6 10.2 16.4 17.4 20.0 22.5

0.5 0.0 0.0 5.5 17.2 39.0 0.0 10.6 10.6 13.1 20.9 0.0 3.9 6.7 20.5 26.9 0.0 4.0 8.0 12.9 12.9 16.6 13.8 15.8 14.6 0.9 0.0 3.8 5.0 8.4 10.0 15.5 16.0 20.3 21.5

B3LYP, TPSS, and M06 functionals with the aD basis set. We found that all three DFT functionals predicted the same lowest energy isomer as CCSD(T)/aD predicted if the isomer energy differences are greater than 1 kcal/mol. If the isomer energy differences are ≤1kcal/mol with respect to the lowest energy isomer, then DFT may predict a different lowest energy isomer. This is not unexpected as the methods are certainly no more accurate than ±1 kcal/mol. The TPSS and M06 functionals predict more accurate relative energies for the isomers than does the B3LYP functional.

Figure 4. Geometries and relative energies (in kcal/mol) at B3LYP/ aD level for Agn (n = 9 to 20). 8303



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Table 3. Isomer Relative Energies for Agn, n = 9−20 at the B3LYP, TPSS, and M06 Levels with the aD Basis Set cluster

B3LYP

TPSS

M06

9a 9b 9c 9d 10a 10b 10c 11a 11b 11c 12a 12b 12c 13a 13b 13c 13d 14a 14b 15a 15b 15c 16a 16b 17a 17b 17c 18a 18b 18c 19a 19b 19c 20a 20b 20c

0.0 0.9 1.8 3.2 0.0 5.2 9.9 0.0 3.2 4.8 0.0 11.4 25.4 0.0 0.6 11.3 18.3 0.0 1.6 0.0 1.4 3.7 0.0 2.2 0.0 2.5 3.0 0.0 1.4 5.3 0.0 4.4 17.3 0.0 21.8 27.8

1.0 0.5 0.0 4.0 2.9 7.4 0.0 0.0 0.0 3.4 0.0 5.9 21.4 0.0 0.3 13.1 22.8 0.0 12.7 0.0 4.9 9.4 5.9 0.0 0.0 0.8 2.2 0.0 7.1 6.1 0.0 8.7 11.8 0.0 1.2 2.8

0.0 2.2 0.9 4.5 1.1 1.9 0.0 0.0 2.7 6.3 0.0 11.0 23.2 0.0 0.7 13.7 20.6 0.0 9.0 0.0 3.1 4.2 0.5 0.0 0.0 4.6 2.9 0.0 2.0 5.8 0.0 4.7 12.9 0.0 4.9 8.7

Agn cluster such as Ag9, which cannot have all of the bulk connectivity. Despite this, the TG-HGA(EAM) approach is capable of providing reasonable structures for these small nanoclusters, as the lowest energy Ag9 structure (9c) from TGHGA(EAM) is found to be the most stable isomer at the TPSS level and is found to be close in energy to the most stable Ag9 structure (9a) at the B3LYP and M06 levels. At the B3LYP level, the most stable Ag10 isomer has Cs symmetry (10a), and two C2v (10b) and Cs (10c) symmetry isomers are predicted to be 5.2 and 9.9 kcal/mol higher in energy. TPSS/aD predicted 10c to be the most stable isomer, with 10a and 10b being 2.9 and 7.4 kcal/mol higher in energy. M06/aD predicted 10c to be the lowest energy isomer, and 10a and 10b are within 2 kcal/mol of 10c. A C2v structure (11a) is predicted to be the lowest energy isomer for Ag11 using the B3LYP, TPSS, and M06 functionals. Structure 11b in C1 symmetry is predicted to be very close in energy to 11a at the TPSS level and is calculated to be ∼3 kcal/ mol less stable than 11a at the B3LYP and M06 levels. Another C1 isomer, 11c, is found to be ∼5 kcal/mol higher in energy than the most stable structure 11a. The lowest lying isomer for Ag12 (12a) has Cs symmetry, and the next most stable isomer (12b) with D2d symmetry is ∼10 kcal/mol higher in energy. A C2 isomer (13a) is predicted to be most stable one for Ag13, with a Cs isomer (13b) less than 1 kcal/mol higher in energy, by all of the three DFT functionals. The most stable isomer for Ag13 is predicted to be ∼ 20 kcal/ mol more stable than the next most stable isomer (13d), which has D2h symmetry. An icosahedral structure in Ih symmetry was predicted to be most stable by TG-HGA and also by Shao et al. and Yang et al.32,33 The icosahedral structure optimizes to a distorted icosahedron with Ci symmetry at the DFT level. The icosahedron and the distorted icosahedron are 52.5 and 28.0 kcal/mol higher in energy than the most stable C2 isomer at the B3LYP/aD level. The lowest energy isomer for Ag14 (14a) has a cubic arrangement of atoms similar to the structure of diamond and is 1.6 kcal/mol more stable than a Cs isomer (14b), at the B3LYP level. Structure 14a is calculated to be 12.7 and 9.0 kcal/mol more stable than 14b at the TPSS and M06 levels. The lowest energy Ag15 isomer has Cs symmetry (15a), with a Cs isomer (15b) 1.4 kcal/mol higher in energy and a D2 isomer (15c) 3.7 kcal/mol higher in energy at the B3LYP level. The M06 functional predicted the same energy order and similar relative energies as B3LYP for Ag15. At the TPSS level, 15a is also predicted to be the lowest energy isomer, with 15b 4.9 kcal/ mol higher in energy and 15c 9.4 kcal/mol higher in energy than 15a. A C1 symmetry structure (16a) is predicted to be the lowest energy Ag16 isomer at the B3LYP level, with a Cs isomer 2.2 kcal/mol higher in energy. At the M06 level, 16a and 16b are predicted to be very close in energy (within 1 kcal/mol). At the TPSS level, 16b is found to be 5.9 kcal/mol lower in energy than 16a. A C2 isomer (17a) is predicted to be the lowest energy structure for Ag17 by all of the three DFT functionals. Another C2 isomer (17b), which is derived from the Ag13 icosahedron, is predicted to be the second lowest energy isomer by B3LYP and TPSS, 2.5 and 0.8 kcal/mol higher in energy than 17a, respectively. A Cs isomer is also within 3 kcal/mol of 17a and is predicted to be the second lowest energy isomer at the M06 level. The lowest energy isomers for Ag18 (18a) is a derivative of the Ag13 icosahedron with Cs symmetry. The B3LYP and M06 functionals predicted a Cs isomer (18b) to be

second lowest energy Ag8 with an Ag atom. This Cs structure has been previously reported as the most stable isomer for Ag9 by Lecoultre et al.24 A C2v isomer (9b) is predicted to be 0.9 kcal/mol higher in energy than 9a at the B3LYP level, and it is basically the lowest Ag7 structure with two faces capped by two Ag atoms. The M06 functional predicted 9b to be 2.2 kcal/mol less stable than 9a, whereas the TPSS functional predicted 9b to be 0.5 kcal/mol more stable than 9a. This structure (9b) was previously reported as the lowest energy structure for Ag9 by Fournier22 and was not found by the TG-HGA method using the EAM potential. The lowest energy structure (9c in C2v symmetry) found by the TG-HGA method is calculated to be the most stable isomer at the TPSS/aD level, 1.0 kcal/mol lower than 9a. Structure 9c is calculated to be 1.8 kcal/mol higher in energy than 9a at the B3LYP/aD level, and 0.9 kcal/ mol higher at the M06/aD level. Shao et al. and Yang et al. also predicted this structure to be the lowest energy structure with a Gupta potential.32,33 Another low energy Cs isomer (9d) found by the TG-HGA method is ∼4 kcal/mol higher in energy than the most stable isomer. It is noted that the EAM potential used in the TG-HGA method is derived from the properties of bulk silver and thus may not be optimal for describing a very small 8304



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Figure 5. TG-HGA(EAM) geometries for Agn, n = 21−99.

∼2 kcal/mol higher in energy and a C1 isomer (18c) to be ∼5 kcal/mol higher in energy than 18a. Structure 18c is also derived from a distorted Ag13 icosahedron. At the TPSS level, 18a is predicted to be 6.1 and 7.1 kcal/mol more stable than 18c and 18b. The lowest energy isomer for Ag19 (19a) is a derivative of the Ag13 icosahedron with Cs symmetry. A C1 isomer (19b) is ∼5 kcal/mol higher in energy at the B3LYP and M06 levels and ∼9 kcal/mol higher at the TPSS level. The double icosahedral isomer (19c) is calculated to be ∼10 kcal/ mol less stable than 19a at the TPSS and M06 levels and is calculated to be 17 kcal/mol less stable than 19a at the B3LYP level. This is consistent with previous results by Yang et al. that the double icosahedron (19c) is ∼10 kcal/mol less stable than a Ag19 variation of the Ag13 icosahedron.31 The lowest energy structure (20a) for Ag20 was predicted to be a tetrahedron (Td symmetry) with the B3LYP, TPSS, and M06 functionals. Structure 20a was previously reported to be a low energy isomer but not the most stable isomer by Fernández et al. using the DFT PBE functional.29 Yang et al. assigned the lowest energy Ag20 isomer to a icosahedra-based isomer also

using the DFT PBE functional.31 Structure 20a is a fragment of a hexagonal-close packing (hcp) crystal lattice as compared to the cubic-close packing (ccp) lattice found in the bulk silver.63 This Td structure serves as a building block for larger Ag clusters. The next two lowest energy isomers (20b and 20c) in C1 and Cs symmetry were generated by the TG-HGA method. Structure 20b is a variation of the Ag13 icosahedron, and 20c can be obtained by adding a Ag to the Ag19 double icosahedron (19c). Although all of the three DFT functionals, B3LYP, TPSS, and M06, agree that 20a is the lowest energy isomer for Ag20 and predicted the same energy order for the three isomers for Ag20, the isomer relative energies are found to be very different for the three functionals. At the B3LYP level, 20b and 20c are predicted to be much higher in energy than 20a, by 21.8 and 27.8 kcal/mol. At the TPSS level, 20b and 20c are only ∼1 and ∼3 kcal/mol higher in energy than 20a. The prediction at the M06 level lies between the predictions at the B3LYP and TPSS levels, and 20a is predicted to be ∼5 and ∼9 kcal/mol more stable than 20b and 20c. This suggests that the hcp tetrahedron structures can be missed by the TG8305



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HGA(EAM) method due to the use of the EAM potential, even though as the size of the cluster increases, the clusters should tend to have ccp units as in the bulk silver. The silver clusters that potentially can be tetrahedrons after Ag20 include Ag35, Ag56, and Ag84. The three DFT functionals B3LYP, TPSS, and M06 agree well on the prediction of the lowest energy isomers for most Agn clusters for n = 9−20. However, the calculated relative energies for the higher energy isomers can be very different for the three DFT functionals. Somewhat surprisingly, the TPSS and M06 functionals predicted very different relative energies for some of the Agn (n = 9−20) clusters, even though they agree well on the relative energies for Ag7. Additionally, as the size of Agn increases, the number of low energy isomers that are close in energy also increases. It is very difficult to determine the ground state isomers at the DFT level, but the DFT predictions for the low energy isomers are still useful in predicting trends in how the geometry evolves. Geometries and Isomer Stabilities for Agn, n = 21−99. After Ag20, it was no longer possible to optimize the cluster geometries at the DFT level. The TG-HGA geometries for Ag21−Ag99 are shown in Figure 5. Most structures are derivatives of the icosahedron structure with a few exceptions, and we discuss only a few structures as examples. Agn, n = 21− 33 are mostly an icosahedral Ag13 core with Ag atoms bonded external to the core. Ag34 is a Cs structure with a fragment of the structure in the ccp lattice form. Ag38 with Oh symmetry is the first structure with the ccp lattice arrangement. Ag55 is predicted to be a perfect 2-layer icosahedron in Ih symmetry with an icosahedral Ag13 core, consistent with the Ag55 geometry predicted by Shao et al. and Yang et al.32,33 and also similar to the Cu55 Ih structure obtained by Kabir et al. using a tightbinding, molecular dynamics method.64 The 20 faces on the second layer of Ag55 are composed of equilateral triangles with a Ag at each edge center. The magic number for a perfect icosahedron increases to 147 and 309 after 13 and 55, as the number of atoms on each of the layers are 12, 42, 92, 162, etc., from inside to outside. Ag147 is expected to be constructed from 20 hcp Ag20 tetrahedrons that share faces with each neighboring Ag20, which is predicted to be the low energy structure by Yang et al.33 The structure for Agn, n = 39−54, are mostly partial icosahedrons, fragments of Ag55. Ag56−Ag99 have mostly two types of structure, additives to Ag55 and structures composed of ccp units. Beyond n = 99, derivatives of the icosahedral structure and derivatives of the ccp structures are expected to dominate. Another possible type of structure is derived from joining structures of two or more icosahedrons due to the hcp tetrahedral building blocks inherent in the icosahedron. Table S1 (in the Supporting Information) lists the average bond length and coordination number for the most stable isomer for n = 21−99, and the values for the smaller clusters are given in Table 4. Figure 6 is a plot of the average bond length and coordination number versus n for the most stable isomer of Agn. The bond distance converges to a value of ∼2.9 Å for the optimized structures up to n = 20, which is essentially the bulk value (2.889 Å).63 The EAM geometry for Ag21 has an average bond length of 2.84 Å, slightly lower but still comparable to the average bond length of 2.89 Å of the B3LYP geometry for Ag20. This shows that the EAM potentials can predict reasonable bond lengths for the larger Agn. The embedded atom method potential results for the geometry of the larger clusters converges to a slightly shorter bond distance, but this is expected based on comparing to the DFT optimized structures.

Table 4. Average Bond Length and Coordination Number (CN) of Agn (n = 2−20) at the B3LYP/aD Levela

a

n

r (Å)

CN

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 bulk (CCP)

2.585 2.853 2.745 2.757 2.756 2.846 2.833 2.855 2.837 2.847 2.863 2.889 2.892 2.928 2.893 2.902 2.902 2.895 2.863 2.889

1 2 2.5 2.8 3.0 4.57 4.50 4.89 4.20 4.91 5.17 5.69 5.57 6.27 5.75 6.12 6.33 6.32 5.40 12

For bonds that are shorter than 3.3 Å.

Figure 6. Average coordination number (CN) and average bond length (r) vs number of Ag atoms (n) for Agn, n = 2 to 99.

The average coordination number (CN) continues to increase and is almost 9 for n = 99 as compared to the ccp value of 12 for the bulk.63 This lower value is not surprising as the clusters still has substantial surface sites as compared to the bulk structure with no edges. The vibrational frequencies of the lowest energy Agn clusters for up to n = 20 are shown in the Supporting Information. The values do not show any real variation from the values for the smaller clusters briefly discussed above. Heats of Formation. The calculated heats of formations for a range of levels for the lowest energy structures for Agn (n = 2−8) are given in Table 5 based on a value of ΔHf(Ag) = 68.0 ± 0.2 kcal/mol at 298 K. 65 The value at 0 K for ΔHf(Ag) = 68.1 ± 0.2 kcal/mol differs from the 298 K value by about 0.1 kcal/mol. 66 Since the aQ and aT calculations for the heptamer and aQ calculation for the octamer are too large for current computational resources, we performed calculations for n = 1− 6 without augmented diffuse functions in the basis sets to 8306



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Table 5. Normalized Atomization Energies ⟨AE⟩ and Calculated Heats of Formation of ΔHf (Agn) (n = 1−8) Clusters at 0 K in kcal/mola n

basis set

2

aug-cc-pVnZ cc-pVnZ aug-cc-pVnZb cc-pVnZb aug-cc-pVnZc cc-pVnZc aug-cc-pVnZ cc-pVnZ aug-cc-pVnZ cc-pVnZ aug-cc-pVnZ cc-pVnZ aug-cc-pVnZ aug-cc-pVnZ

3

4 5 6 7 8 a

⟨AE⟩ D ⟨AE⟩ T 17.8 17.0 17.4 16.9 17.4 16.8 24.3 23.7 26.4 25.8 30.6 29.9 31.9 34.3

⟨AE⟩ Q

⟨AE⟩ 5

18.8 18.0 18.6 17.6 18.5 17.6 25.8 24.7 28.0 26.7 32.3 30.9

18.9 18.4 18.7 18.1 18.6 18.1 26.0 25.3

18.5 17.5 18.2 17.0 18.1 17.0 25.3 23.9 27.5 25.9 31.8 30.0

⟨AE⟩ CBS DTQ

⟨AE⟩ CBS Q5

19.1 18.2 18.7 17.9 18.6 17.8 26.1 24.9 28.1 27.0 32.4 31.3

19.0 18.9 18.9 18.7 18.8 18.7 26.2 26.0

33.5d 36.4d

35.5

ΔHf B3LYP/ aD

ΔHf D

ΔHf T

ΔHf Q

ΔHf 5

100.0 100.0 149.7 149.7 153.0 153.0 171.6 171.6 216.5 216.5 237.0 237.0 282.1 304.0

100.4 102.0 151.8 153.3 151.8 153.6 174.8 177.2 208.0 211.0 223.2 228.6 252.7 269.6

99.0 101.0 149.4 153.0 149.7 153.0 170.8 176.4 202.5 210.5 217.2 228.0

98.4 100.0 148.2 151.2 148.5 151.2 168.8 173.2 200.0 206.5 214.2 222.6

98.2 99.2 147.9 149.7 148.2 149.7 168.0 170.8

ΔHf CBS DTQ

ΔHf CBS Q5

97.8 99.6 147.9 150.3 148.2 150.6 167.6 172.4 199.5 205.0 213.6 220.2

98.0 98.2 147.3 147.9 147.6 147.9 167.2 168.0

240.7d 253.2d

260.0

Experimental ΔHf (Ag) = 68.1 ± 0.2 kcal/mol from ref 65. bVertex angle = 55°. cVertex angle =75°. dCBS estimate from nucleation reactions.

Table 6. Nucleation Reaction Energies in kcal/mol Ag + Ag → Ag2 Ag + Ag2 → Ag3 Ag + Ag3 → Ag4 Ag2 + Ag2 → Ag4 Ag + Ag4 → Ag5 Ag2 + Ag3 → Ag5 Ag + Ag5 → Ag6 Ag2 + Ag4 → Ag6 Ag3 + Ag3 → Ag6 Ag + Ag6 → Ag7 Ag2 + Ag5 → Ag7 Ag3 + Ag4 → Ag7 Ag + Ag7 → Ag8 Ag2 + Ag6 → Ag8 Ag3 + Ag5 → Ag8 Ag4 + Ag4 → Ag8

aD

aT

aQ

a5

CBS(aDTQ)

CBS(aQ5)

−35.5 −16.7 −45.0 −26.2 −34.9 −44.4 −51.5 −50.9 −79.2 −39.6 −55.6 −73.8 −51.0 −55.1 −89.9 −79.8

−36.9 −17.7 −46.8 −27.5 −36.0 −45.9 −53.2 −52.2 −81.3

−37.6 −18.2 −47.5 −28.1 −36.6 −46.5 −53.9 −52.8 −82.1

−37.8 −18.4 −47.8 −28.3

−38.2 −18.0 −48.4 −28.2 −35.8 −46.0 −53.8 −51.4 −81.8

−38.1 −18.5 −48.0 −28.5

−56.6 −92.0 −81.2

determine if the same degree of accuracy could be obtained in the results. For all of the clusters where the unaugmented basis set was used (Table 5), the energies are in reasonable agreement with the augmented basis values. When extrapolated to the CBS using aDTQ (CBS(DTQ)), the ⟨AE⟩ only increased by ∼1 kcal/mol from the CCSD(T)/aT values and by less than 2 kcal/mol from the CCSD(T)/aD values for Agn (n = 2−6). Inclusion of the cc-pv5Z-PP results in extrapolating to the CBS(Q5) limit slightly improves the results, but they are still not as close as would be desired as the values are divided by n. A small error in ⟨AE⟩ for Agn will lead to a large error in ΔHf by a factor of n. To reduce the error in ΔHf, we estimate the ΔHf values for the larger clusters using the nucleation reaction energies calculated with a smaller aD or aT basis set and ΔHf of the smaller clusters, based on the nucleation reaction (6)

Agx + Ag y → Agx + y

ΔHf (Agx + y)est.CBS = ΔErxn + ΔHf (Agx)CBS + ΔHf (Ag y)CBS

(7)

where ΔErxn is the calculated nucleation reaction energy defined by eq 5 with smaller basis sets, and ΔHf(Agn)CBS is the CBS extrapolated heat of formation for the smaller clusters Agx and Agy. To benchmark our approach to predict these estimated ΔHf values, we compared the CBS ΔHf values for Ag4, Ag5, and Ag6 with the predicted heats of formation from the nucleation reaction energies (Table 6 for n = 2 to 8) at the CCSD(T)/aD and aT levels. The estimated ΔHf values for Ag4 using eq 7 are 169.6 kcal/mol with the aD basis set and 168.5 kcal/mol with aT basis set; both values are within 3 kcal/mol of the calculated ΔHf of 167.2 kcal/mol at the CCSD(T)/CBS(aQ5) level for Ag4. The estimated ΔHfs for Ag5 are 200.9 kcal/mol with the aD basis set and 199.4 kcal/mol with the aT basis set, which agrees well with the calculated ΔHf of 199.5 kcal/mol at the CCSD(T)/CBS(aDTQ) level. The estimated ΔHfs for Ag6 are 215.4 kcal/mol with the aD basis set and 213.3 kcal/mol with the aT basis set, as compared to the CBS(aDTQ) ΔHf of 213.6

(6)

as the errors in the reaction energies are likely to be smaller than those in ΔHf due to the error cancellation in the reaction energies. A value for ΔHf close to the CBS limit can be estimated using eq 7 8307



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expensive, and CCSD(T) can be used to calculate the closedshell clusters with even number of Ags only with small basis sets. The ⟨AE⟩s for Agn (n = 1−8 and n = 10, 12, 14, 16, and 20) were calculated at the CCSD(T)/cc-pVDZ-PP (denoted as ‘D’) level (see Table 7). The ⟨AE⟩s at the CCSD(T)/D level for Agn (n = 2−8) are ∼3 kcal/mol less than the ⟨AE⟩s predicted at the CCSD(T)/CBS level. The ⟨AE⟩ for the largest cluster Ag20 at the CCSD(T)/D level was calculated to be 40.9 kcal/mol. To calculate the energetics for Agn for n > 20, DFT methods need to be used. The ⟨AE⟩s for Agn, n = 2−99, were calculated using the B3LYP, M06, ωB97XD, and PW91 exchangecorrelation functionals with the LANL2DZ67 basis set and effective core potential on Ag. Figure 8 shows the comparison between ⟨AE⟩s calculated by using the four DFT functionals with the LANL2DZ basis set and at the CCSD(T)/cc-pVDZPP level up to n = 20. The DFT results using the PW91 and ωB97XD functionals were found to have the best agreement with the CCSD(T)/D results. The predicted ⟨AE⟩ values at the M06 level are a few kcal/mol greater than the predictions at the CCSD(T)/D level, and the M06 functional predicted a similar ⟨AE⟩ vs n plot as do the PW91 and ωB97XD functionals. For Agn (n ≤ 8), the ⟨AE⟩ values predicted using the CCSD(T)/ccpVDZ-PP level are actually ca. 2−3 kcal/mol smaller than the more accurate ⟨AE⟩ predictions at the CCSD(T)/CBS level, and thus, the ⟨AE⟩s at the M06 level are only ∼2 kcal/mol greater than ⟨AE⟩s predicted at the CCSD(T)/CBS level. Thus, M06 is also a good functional for estimating the ⟨AE⟩ for the larger Agn clusters. B3LYP significantly underestimates the ⟨AE⟩s for Agn, as compared to the ⟨AE⟩ prediction at the CCSD(T)/cc-pVDZ-PP level. B3LYP will tend to get worse in the prediction of the energetics as the cluster grows and approaches the bulk metal as B3LYP contains a fractional Hartree−Fock contribution, and Hartree−Fock cannot give a finite density of states at the Fermi level. However, the ⟨AE⟩s show that we are far from bulk metal in terms of the normalized atomization energy (the cohesive energy of the bulk), so that the B3LYP results can still be used for qualitative comparisons. Figure 9a shows a plot of the ⟨AE⟩ vs n for Agn, n = 2−99 at the PW91/LANL2DZ and ωB97XD/LANL2DZ levels. The ⟨AE⟩ predicted by PW91 is ∼2 kcal/mol greater than the ⟨AE⟩ predicted by ωB97XD for Agn, n < 40, and the energy difference decreases as n increases. From Ag20 to Ag21, the ⟨AE⟩ dropped by ∼1 kcal/mol at the PW91 level and by ∼3 kcal/mol at the ωB97XD level. This is most likely due to the use of TGHGA(EAM) geometries for Agn, n > 20. At n = 99, PW91 and ωB97XD predicted essentially the same value for the ⟨AE⟩, ∼48 kcal/mol. On the basis of the comparison of the DFT results for smaller clusters with the CCSD(T) results, the ⟨AE⟩ for Ag99 should be ∼50 kcal/mol. For 55 < n < 100, the ⟨AE⟩ vs n plot is essentially flat, which indicates a slow convergence of the ⟨AE⟩ toward the ⟨AE⟩ bulk limit of 68 kcal/mol.65 Figure 9b shows a plot of ⟨AE⟩ vs n−1/3, which effectively is a plot of the average atomization energy vs the inverse of the geometric size of the cluster. The plot is approximately linear as expected. Fitting all of the ⟨AE⟩s excluding the dimer to a straight line yields an intercept of 60 kcal/mol, which would correspond to the cohesive energy of the bulk. This value is expected to be low as compared to the CCSD(T)/CBS intercept value by at least 5 kcal/mol, which would give us a best estimate of the intercept of 65 kcal/mol as compared to the bulk value of 68 kcal/mol. This is quite good agreement considering the modest size of the clusters, the use of approximate geometries, and our

kcal/mol. These benchmark results show that the estimated ΔHf values from the aT basis set are in excellent agreement with the calculated ΔHf at the CCSD(T)/CBS level and that the estimated ΔHf values from the aD basis are also of high quality, with errors of about 1 to 2 kcal/mol greater than the errors of the aT estimates. Therefore, we can apply eq 7 to estimate the ΔHf of Ag7 and Ag8. The ΔHf for Ag7 is estimated to be 240.7 kcal/mol with the aD basis set and eq 7, which is ∼10 kcal/mol lower in value than the ΔHf calculated from the atomization energy at the CCSD(T)/aD level. The ΔHf for Ag8 is estimated to 253.2 kcal/mol with the aD basis set and eq 7, ∼15 kcal/mol lower than the calculated ΔHf from the atomization energy at the CCSD(T)/aD level and ∼5 kcal/ mol lower than the calculated ΔHf from the atomization energy at the CCSD(T)/aT level. Using the estimated ΔHfs, the ⟨AE⟩s for Ag7 and Ag8 are predicted to be 33.5 and 36.4 kcal/mol, respectively. Normalized Atomization Energies. The normalized atomization energies (⟨AE⟩s) and heats of formation were calculated at different levels depending on the size of the cluster. The ⟨AE⟩s for the smaller clusters up to n = 8 are given in Table 5. The ⟨AE⟩ for each structure size increases as the level of calculation increases. The ⟨AE⟩s at the CCSD(T)/CBS level improve on the CCSD(T)/aD values by ∼2 kcal/mol and by ∼1 kcal/mol on the CCSD(T)/aT values. Figure 7

Figure 7. Average atomization energy (⟨AE⟩) vs. the number of cluster atoms (n) for Agn, n = 2−8.

illustrates the dependence of the ⟨AE⟩ on the number of cluster atoms for the most stable structures up to n = 8. The odd−even variation in the ⟨AE⟩s noted previously22,29 is clearly apparent. At the B3LYP level, the closed shell structures with an even number of atoms are predicted to be more stable than the open shell structures with an odd number of electrons. At the CCSD(T) level, the ⟨AE⟩ increases for each Ag that is added, but the difference between an even n and n + 1 is smaller than the difference between n − 1 and n, which shows that the odd n do not fall quite on the same line as the even n. The differences in the even and odd ⟨AE⟩s has been attributed to an electron pairing effect.28 The ⟨AE⟩s increase at n = 8 at the CCSD(T) level, which agrees well with size-dependent properties of clusters. For Agn, 8 < n ≤ 20, the CCSD(T) calculations for the openshell clusters with odd number of Ags are computationally too 8308



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Table 7. ⟨AE⟩ for Agn, n = 2−99 as a Function of Computational Levela n

CCSD(T) CBS

CCSD(T) D

B3LYP aD

B3LYP LANL2DZ

M06 LANL2DZ

ωB97XD LANL2DZ

PW91 LANL2DZ

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62

19.0 18.9 26.2 28.1 32.4 33.6 36.5

17.0 15.9 23.7 25.8 29.9 31.6 34.0

18.2 17.1 22.8 24.9 28.7 28.0 30.3 29.2 29.9 30.7 31.5 31.7 31.2 31.3 32.6 33.1 34.1 34.2 35.3

17.9 16.7 21.9 23.9 27.6 26.8 29.0 27.9 28.6 29.2 29.9 30.1 29.7 29.8 30.9 31.3 32.3 32.4 33.4 30.6 30.9 30.6 30.2 30.2 31.5 31.0 31.0 31.6 31.7 31.8 32.4 32.5 32.9 32.9 32.8 32.9 33.0 33.3 33.1 33.4 33.7 33.6 33.6 33.7 33.8 34.1 34.0 34.2 34.2 34.2 34.5 34.5 34.8 35.1 35.1 35.2 35.2 35.2 35.2 35.2 35.2

21.2 21.1 28.0 30.3 34.5 35.6 37.9 37.5 37.4 39.2 40.2 40.7 40.4 41.0 42.2 43.0 44.1 44.3 44.7

20.5 20.3 26.6 29.0 32.9 33.5 35.8 35.2 35.2 36.7 37.6 38.1 37.6 38.2 39.2 40.0 41.0 41.3 41.8 40.6 40.9 40.8 40.4 40.6 41.8 41.4 41.5 42.1 42.3 42.5 43.0 43.3 43.5 43.7 43.6 43.8 43.8 44.3 44.2 44.5 44.7 44.7 44.7 44.9 45.1 45.3 45.2 45.6 45.4 45.6 45.8 46.0 46.1 46.7 46.6 46.7 46.8 46.7 46.8 46.8 46.8

18.7 18.0 24.3 26.3 30.6 30.6 33.4 32.5 33.2 34.4 35.3 35.6 35.7 36.0 37.1 37.8 38.9 39.2 40.1 37.9 38.3 38.4 38.1 38.1 39.5 39.0 39.4 39.8 39.7 40.4 41.1 41.4 41.5 41.3 41.8 42.0 42.0 42.7 42.4 42.9 43.2 43.2 43.2 43.5 43.6 44.1 43.9 44.3 44.4 44.4 44.8 44.9 45.2 45.8 45.8 45.8 45.9 45.9 45.9 45.9 45.9

33.2 36.3 36.4 38.5

40.9

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Table 7. continued n

CCSD(T) CBS

CCSD(T) D

B3LYP aD

ωB97XD LANL2DZ

PW91 LANL2DZ

35.1 35.1 35.1 35.0 35.1 35.1 34.9 34.9 35.2 34.9 35.1 34.9 34.9 35.0 35.1 35.2 35.2 35.5 35.6 35.6 35.6 35.6 35.4 35.6 35.8

46.7 46.8 46.8

36.0 36.0 36.1 36.2 36.1 36.1 36.1 35.8 35.8 36.1 36.2

48.0 48.1 48.1 48.2 48.2 48.2 48.2 47.9 48.0

45.9 45.9 45.8 45.9 45.9 45.9 45.8 45.9 46.5 45.9 46.1 45.7 46.0 46.1 46.2 46.1 46.3 46.9 47.1 47.2 47.2 47.2 46.7 46.9 47.4 47.5 47.7 47.8 47.9 48.0 47.9 48.0 48.0 47.4 47.5 47.9 48.2

B3LYP LANL2DZ

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

M06 LANL2DZ

46.8 46.8 46.6 46.7 47.0 46.7 46.8

46.9 47.0 47.0 47.1 47.5 47.5 47.6 47.4 47.5 47.8

48.4

a

Geometries for Agn, n = 2−20, were optimized using B3LYP/aD. Geometries for Agn, n = 21−99, were optimized using the TG-HGA method with the EAM potential.

is calculated to be −51.0 kcal/mol, both at the CCSD(T)/aD level. These results suggest a faster convergence to the bulk value than the convergence of the ⟨AE⟩ for the smaller clusters. The reaction energies for adding a Ag to the Agn cluster at the PW91/LANL2DZ and ωB97XD/LANL2DZ levels are listed in the Supporting Information. Because of the use of the geometries at the TG-HGA(EAM) level, there are errors in the calculated total energies for each cluster, as the TGHGA(EAM) geometry may not all have the same error with respect to the DFT minimum. Error cancellation could reduce the error in the calculated reaction energy, but the results in the Supporting Information clearly show that this is not the case. The calculated reaction energies show significant variations, and all that can be observed is a very qualitative trend that the energy for adding Ag to a Agn cluster increases as n increases.

estimates of the errors in the binding energies. If we extrapolate the values for n > 70, the intercept at the ωB97XD level is 68 kcal/mol and that with the PW91 functional is 62 kcal/mol, showing improved agreement with the bulk value. Another approach to compare to experiment is to look at the energy to add an Ag atom to a cluster. The energetics for the addition of an Ag atom to a cluster Agn to form Agn+1 for the small clusters up to n = 8 is given in Table 6. For the even and odd silver clusters separately, the reaction for adding a silver atom to the silver cluster becomes more exothermic as the size of the cluster increases. The formation of the even Ag2m cluster is more exothermic than the formation of the Ag2m−1 and Ag2m+1 and the energy differences decrease as the cluster size increases. The reaction energy for reaction 8 Ag5 + Ag → Ag6

■

(8)

CONCLUSIONS The geometries of Agn (n = 2−8) were optimized to determine the lowest energy isomers for each cluster size. For n ≤ 6, planar 2-D geometries with compact, nonlinear structures

is calculated to be −53.8 kcal/mol and that for reaction 9 Ag 7 + Ag → Ag8

(9) 8310



Article

Geometries of Agn, n = 9−20, were obtained by DFT geometry optimizations starting from the initial structures obtained using a TG-HGA method using the EAM potential during the HGA step. The TG-HGA method using the EAM potential was used to predict a set of geometries for n > 20 in order to generate geometries for use in predicting the ⟨AE⟩s with density functional theory. The EAM potential is derived from the properties of bulk silver and thus may not be optimal for describing small Agn clusters. However, it is currently not possible to use modern search algorithms as used by us to search geometry space with much more expensive computational methods such as the DFT methods used by us. Thus, the geometries we report for n > 20 are the global minimum on the EAM potential surface, but they could well not be the global minimum at higher computational levels. In fact, the results for n = 20 show this. However, the EAM geometries have reasonable bond distances and proper connectivities so they can be used to estimate the ⟨AE⟩ as this value is normalized by the number of atoms, which reduces the error due to the use of an approximate geometry that could correspond to a higher energy isomer. The ⟨AE⟩ for Agn, n = 2−99, were predicted at different levels depending on the size of the Agn. The PW91 and ωB97XD functionals with the LANL2DZ basis set predicted ⟨AE⟩s in good agreement with the ⟨AE⟩s predicted at the CCSD(T)/D level for up to n = 20, which are expected to be a few kcal/mol smaller than the CCSD(T)/CBS predictions for ⟨AE⟩. The M06 functional also gives good estimation for ⟨AE⟩s, and its prediction values for ⟨AE⟩s are expected to be a few kcal/mol larger than the ⟨AE⟩ predictions at the CCSD(T)/CBS level. For n < 20, the ⟨AE⟩ increases significantly as n increases. The ⟨AE⟩ for Agn starts to converge very slowly at n = 55. The ⟨AE⟩ for Ag99 was predicted to be ∼50 kcal/mol, which is 18 kcal/mol less than the ⟨AE⟩ bulk limit.

Figure 8. ⟨AE⟩ vs n for Agn, n = 2−20 at the CCSD(T)/cc-pVDZ-PP and DFT/LANL2DZ levels with different exchange-correlation functionals.

■

ASSOCIATED CONTENT

S Supporting Information *

Complete citations for refs 38 and 47b. The average bond lengths (R) and coordination numbers (CN) for Agn, n = 21− 99, using TG-HGA(EAM). Vibrational frequencies (>60 cm−1 for n = 1−10; >100 cm−1 for n = 11−20) for Agn (n = 2−8). Reaction energies for Agn+1 → Agn + Ag at the DFT/ LANL2DZ levels in kcal/mol. Cartesian coordinates and total energies for Agn (n = 1−99). This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*(D.A.D.) E-mail: [email protected].

Figure 9. ⟨AE⟩ vs (a) n and (b) n−1/3 for Agn, n = 2−99 at the PW91 and ωB97XD/LANL2DZ levels.

Notes

The authors declare no competing financial interest.

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composed of triangles are preferred. For n = 7 and 8, the clusters favor 3-D densely packed, high symmetry geometries. The DFT functionals, B3LYP, PBE, PW91, SVWN5, TPSS, M06, and ωB97XD were benchmarked against the CCSD(T)/ aug-cc-pVDZ-PP results for the Ag7 relative isomer energies. The relative isomer energies obtained using the TPSS and M06 functionals agree best with the CCSD(T) predictions. The B3LYP, PBE, PW91, and ωB97XD predicted smaller relative energies than does the CCSD(T) for most Ag7 isomers, whereas the local SVWN5 predicted larger relative energies. The lowest energy structure always involves a low spin state.

ACKNOWLEDGMENTS This work was supported by the Argonne National Laboratory (ANL) LDRD program. DAD also thanks the Robert Ramsay Chair Fund of The University of Alabama and Argonne National Laboratory for support. Some of the computational work was performed at the Molecular Science Computing Facility, William R. Wiley Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the Department of Energy’s DOE Office of Biological and Environmental Research, located at PNNL. PNNL is operated 8311



Article

for DOE by Battelle Memorial Institute under Contract # DEAC06-76RLO-1830.

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Prediction of Structures and Atomization Energies of Small Silver

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