Structures and Stabilities of (MgO) n Nanoclusters

Apr 9, 2014 - Department of Chemistry, The University of Alabama, Shelby Hall, .... those of Haertelt et al.32 The lowest energy isomer for (MgO)2...
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Structures and Stabilities of (MgO)n Nanoclusters Mingyang Chen,† Andrew R. Felmy,‡ and David A. Dixon*,† †

Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, Alabama 35487-0336, United States Fundamental Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99352, United States

‡

S Supporting Information *

ABSTRACT: Global minima for (MgO)n structures were optimized using a tree growth−hybrid genetic algorithm in conjunction with MNDO/MNDO/d semiempirical molecular orbital calculations followed by density functional theory geometry optimizations with the B3LYP functional. New lowest energy isomers were found for a number of (MgO)n clusters. The most stable isomers for (MgO)n (n > 3) are 3-dimensional. For n < 20, hexagonal tubular (MgO)n structures are more favored in energy than the cubic structures. The cubic structures and their variations dominate after n = 20. For the cubic isomers, increasing the size of the cluster in any dimension improves the stability. The effectiveness of increasing the size of the cluster in a specific dimension to improve stability diminishes as the size in that dimension increases. For cubic structures of the same size, the most compact cubic structure is expected to be the more stable cubic structure. The average Mg−O bond distance and coordination number both increase as n increases. The calculated average Mg−O bond distance is 2.055 Å at n = 40, slightly smaller than the bulk value of 2.104 Å. The average coordination number is predicted to be 4.6 for the lowest energy (MgO)40 as compared to the bulk value of 6. As n increases, the normalized clustering energy ΔE(n) for the (MgO)n increases and the slope of the ΔE(n) vs n curve decreases. The value of ΔE(40) is predicted to be 150 kcal/mol, as compared to the bulk value ΔE(∞) = 176 kcal/mol. The electronic properties of the clusters are presented and the reactive sites are predicted to be at the corners.

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INTRODUCTION Recently, interest in the geological sequestration of carbon dioxide,1 the principle anthropogenic greenhouse gas, has focused interest in magnesium-based minerals. Magnesiumbased minerals are an important focus for geological sequestration studies because of their geological abundance (Mg is the seventh most abundant element in the earth’s crust)2 and Mg oxides and hydroxides are very reactive with supercritical CO2.3 In addition, MgO is the only engineered barrier material approved for emplacement in the Waste Isolation Pilot Plant (WIPP) in the U.S.4 MgO has been used as a support for metal and bimetallic clusters for catalytic processes.5−11 MgO also serves as a catalyst itself, and there have been numerous experimental and computational studies of its properties for a range of reactions including oxidation and of its photochemical behavior.12−25 (MgO)n clusters are models of such supporting materials and could also serve as nanosize containers for catalytic reactions. A common mineral form of MgO is periclase with a cubic crystal structure. MgO nanotube bundles of lengths of tens of micrometers have been synthesized by thermal evaporation.26−28 It is of interest to determine the structures of small (MgO)n cluster as an aid to the study of defects in MgO crystals, in the design of metastable noncubic MgO materials for catalytic uses, and for modeling reactions on minerals. Substructures other than cubes, such as tubular hexagons, could exist in low energy isomers in small clusters and may have © 2014 American Chemical Society

different reactivity properties. MgO nanoclusters have been experimentally studied using transmission electron microscopy and were found to have a cubic structure as found in the bulk.29 Gas phase (MgO)n+ clusters have been studied using laserionization time-of-flight mass spectrometry, and optimized structures were predicted using a rigid ion model.30,31 Recently, small neutral gas phase (MgO)n clusters up to n = 16 were studied using a tunable IR-UV two-color ionization scheme to measure the vibrational spectra.32 This group also used density functional theory with the B3LYP functional and a TZVP basis set to optimize the geometries generated by a hybrid genetic algorithm to find the global minimum and to predict the vibrational frequencies for (MgO)n, n = 3−16. In addition, they have also explored the structures of cationic MgO clusters.33 There are a number of additional theoretical studies on the structure and stabilities of small neutral (MgO)n clusters using molecular dynamics and ab initio methods without using global optimization.34−36 A number or other electronic structure studies of MgO clusters have been reported37−39 as have studies using an empirical ion-pair potential.40−48 We have previously reported49 high level calculations of the heat of formation of gas phase MgO using the Feller−Peterson−Dixon Received: December 31, 2013 Revised: April 9, 2014 Published: April 9, 2014 3136

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approach,50−52 showing that there are issues with the available experimental heats of formation.

rigid ion model might be insufficient to predict geometries for the (MgO)n clusters. The lowest energy (MgO)5 isomer (5a) is predicted to be a distorted cube with one edge capped by a MgO group. A Cs isomer of (MgO)5 (5b) is 10.4 higher in energy than the capped cube. The ladder-like isomer (5c) and another planar isomer (5d) of (MgO)5 are ∼20 kcal/mol higher in energy than the most stable isomer 5a. The hexagonal prism isomer (6a) is predicted to be the most stable isomer for (MgO)6, 7.4 kcal/mol lower in energy than the 2 × 2 × 3 cubic (scaffoldlike) isomer (6b). The Mg2O2 edge-capped cubic isomer (6c) and the ladder-like isomer (6d) of (MgO)6 are more than 50 kcal/mol higher in energy as compared to the most stable isomer. The most stable (MgO) 7 isomer (7a) has C 3v symmetry, with two partial cubes (missing a corner atom) connected together. The second lowest energy (MgO)7 isomer (7b) is predicted to be a basal edge-capped (by MgO) hexagonal prism, 9.3 kcal/mol higher in energy than 7a. A MgO-capped scaffold (7c) and a MgO-capped hexagonal prism (7d) are ∼12 and 15 kcal/mol higher in energy than 7a. The most stable isomer of (MgO)8 (8a) has S4 symmetry, and the structure is assembled from four- and six-membered rings. The 2 × 2 × 4 cubic isomer (8b) is only 2.9 kcal/mol higher in energy, and the (MgO)2@(MgO)6 side-capped hexagonal prism (8c) is predicted to be 6.8 kcal/mol higher in energy. The lowest energy isomer for (MgO)9 is calculated to be a hexagonal prism with three layers of (MgO)6 rings (9a), 12.1 kcal/mol more stable than the 2 × 3 × 3 cuboid (9b), and 16.1 kcal/mol more stable than the C3h isomer (9c) composed of four- and six-membered rings. The 2 × 2 × 5 cubic isomer (10a) is predicted to be the most stable isomer for (MgO)10, 2.4 kcal/mol lower in energy than a C2 isomer (10b) that is composed of seven four-membered rings, four six-membered rings, and one eight-membered ring. Structure 10b was previously predicted to be the most stable isomer for (MgO)10 by Haertelt et al.32 The C2 isomer (10c) in the shape of a distorted scaffold is 4.3 kcal/mol higher in energy than 10a. A polyhedron with six square faces and six hexagonal faces with C3 symmetry (10d) is 5.8 kcal/mol higher in energy than 10a. The structure consisting of two deformed hexagonal prisms (10e) is 8.3 kcal/mol higher in energy than 10a. The basal edge capped hexagonal tubular structure (10f) is 11.4 kcal/mol higher in energy than 10a. The calculated DFT vibrational spectrum (Supporting Information) shows that 10a has two major bands at 550 and 750 cm−1, consistent with the two observed major peaks32 at 540 and 730 cm−1. Structure 10b has split peaks at the position of the 540 and 730 cm−1 experiment bands. We cannot rule out the 10b’s contribution to the infrared spectra for (MgO)10 as the peaks at 540 and 730 cm−1 are broad; however, these two experimental peaks are not split, and it is highly possible that 10a provides the major contribution to the spectrum. The lowest energy isomer for (MgO)11 (11a) has Cs symmetry and is a polyhedron with square and hexagonal faces. The second lowest energy isomer is 3.4 kcal/mol higher in energy than 11a and is a derivative of the hexagonal tubular structure (11b). The isomer consisting of 3 2 × 2 × 2 cubes and a hexagonal prism (11c) is 5.3 kcal/mol higher in energy than 11a. The lowest energy isomer of (MgO)12 (12a) is predicted to be a hexagonal tubular structure composed of four parallel Mg3O3 hexagonal rings, followed by a truncated octahedron (i.e., a type of tetradecahedron) in Th symmetry (12b) 4.1 kcal/

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COMPUTATIONAL METHODS The initial geometries for the low energy (MgO)n clusters were generated by using a tree growth-hybrid genetic algorithm (TG-HGA) developed by us53 with the energetics obtained using semiempirical molecular orbital theory with the MNDO/ MNDO/d54−56 parameters. The resulting geometries were optimized using density functional theory with the B3LYP57,58 exchange−correlation functional and the DFT optimized DZVP59 basis set. This basis set was chosen as a large number of structures needed to be studied. To survey a broader range of structures, higher energy cubic and tubular isomers (including hexagonal, octagonal, and double hexagonal tubular isomers) were constructed and optimized. The normalized clustering energy for (MgO)n is calculated with eq 1 ΔE(n) = E(MgO) − E((MgO)n )/n

(1)

For (MgO)n, n = 2−8, the ΔE(n)’s calculated at the B3LYP/ DZVP level were benchmarked with CCSD(T)/cc-pVnZ (n = D, T, and Q)60 values extrapolated to the complete basis set limit using eq 261 E(x) = A CBS + B

exp[ − (x

− 1)] + C

exp[ − (x

− 1)2 ] (2)

For n > 8, the ΔE(n)’s were calculated at the B3LYP/DZVP level. We used the reaction energy for reaction 3 (MgO)n − 1 + MgO → (MgO)n

(3)

at the CCSD(T)/CBS level with ZPEs and thermal corrections at the B3LYP/DZVP level to calculate the heats of formation of (MgO)n, n = 2−8, starting from our reliable value for MgO. The DFT calculations were done with the GAUSSIAN 0962 program. The CCSD(T) calculations were performed with the 63 MOLPRO2010 package of ab initio programs.

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RESULTS AND DISCUSSION (MgO)n Clusters. The geometry optimization results for the (MgO)n clusters at the B3LYP/DZVP level are shown in Figure 1. For (MgO)n, up to n = 16, many of our lowest energy isomers are in agreement with the predictions of Haertelt et al.32 We only note if our most stable structures differ from those of Haertelt et al.32 The lowest energy isomer for (MgO)2 is predicted to be a rhombus (2a), ∼50 kcal/mol lower than the linear isomer (2b) and a flatter rhombic isomer (2c). The (MgO)2 isomers with an O−O bond are high in energy. The most stable (MgO)3 isomer is a six-membered ring (3a), followed by an edge-capped triangular bipyramid (3b) 52.7 kcal/mol higher in energy. Starting with n = 4, the lowest energy isomers for (MgO)n are found to be 3-dimensional structures. The lowest energy (MgO)4 isomer (4a) is cubic, which is the smallest possible neutral (MgO)n cluster with the subunit of bulk MgO. The eight-membered ring isomer (4b) and the ladder-like isomers (4c) are 17.0 and 29.1 kcal/mol higher in energy, respectively. Our prediction for the lowest energy structure for (MgO)4 is consistent with previous predictions32 using DFT and is consistent with the reported infrared spectra. Ziemann et al.31 reported the ring-like isomer (4b) to be the lowest energy isomer with a lower level rigid ion model. This suggests that the 3137

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Figure 1. Geometries and relative energies in kcal/mol at the B3LYP/DZVP level for the low energy (MgO)n isomers.

mol higher in energy. Haertelt et al.32 predicted 12b to be 0.4 kcal/mol lower in energy than 12a at the B3LYP/TZVP level. However, they found that the predicted infrared spectrum for

12a better matched their experimental infrared spectrum of (MgO)12.32 The difference in the energies of the two lowest energy structures between the current work and that of Haertelt 3138

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× 3 × 5 cuboid is 15.8 kcal/mol higher in energy than 15a. The (Mg5O5)3 double hexagonal tubular structure (15e) is 28.5 kcal/mol higher in energy than 15a. The 2 × 4 × 4 cuboid structure (16a) is predicted to be the most stable structure for (MgO)16, with a structure (16b) consisting of a 2 × 2 × 4 cuboid and a (Mg3O3)4 hexagonal tube 3.4 kcal/mol higher in energy. Structure 16b was predicted to be the lowest energy isomer for (MgO)16 by Haertelt et al., and structure 16a was not reported. 32 The (Mg 4 O 4 ) 4 octahedral tubular structure (16c) is 8.3 kcal/mol higher in energy than 16a. The calculated vibrational spectrum (Supporting Information) for 16a shows excellent agreement with the experimental spectrum32 for (MgCO)16. The intense peaks predicted at the B3LYP/DZVP level at 404, 470, 507, 592, 685, and 690 cm−1 match the more intense peaks at 400, 460, 490, 590, and 690 cm−1 in the experimental spectrum. 16b also has bands in the region from 400 to 600 cm−1, but those bands only have moderate intensity as compared to the two strongest bands near 700 cm−1, all of which are consistent with 16a being the most stable isomer for (MgO)16. The lowest energy isomer of (MgO)17 (17a) is a structure joining a 2 × 3 × 3 cuboid and an (Mg4O4)2 octagonal tube. This structure is 4.7 kcal/mol more stable than the isomer with a similar geometry but with the Mg and O swapping positions (17b). The 3 × 3 × 4 cuboid isomer (18a) of (MgO)18, is calculated to be 7.5 kcal/mol lower in energy than the (Mg3O3)6 tubular structure (18b). The isomer consisting of a polyhedron and two (Mg3O3)2 hexagonal tubes (18c) is 32.5 kcal/mol higher in energy than 18a. The lowest energy structure of (MgO)19 (19a) is a combination of a (Mg3O3)5 hexagonal tube and a 2 × 2 × 4 cuboid, and the structure (19b) that joins a polyhedral ball, a 2 × 2 × 2 cube, and two (Mg3O3)2 hexagonal tubes is only 1.6 kcal/mol higher in energy. Two additional structures (19c and 19d), which are similar to structure 19a, are 7.5 and 10.5 kcal/ mol higher in energy than 19a. The 2 × 4 × 5 cuboid (20a) is most stable isomer for (MgO)20, and the structure (20b) joining a 2 × 2 × 5 cuboid and a (Mg3O3)5 hexagonal tube (20b) is 4.4 kcal/mol higher in energy than 20a. The (Mg4O4)5 octagonal tube (20c) is 12.9 kcal/mol higher in energy than 20a. The (Mg5O5)4 double hexagonal tube (20d) is predicted to be 17.5 kcal/mol higher in energy than 20a. The (Mg3O3)7 hexagonal tubular structure (21a) is predicted to be the lowest energy structure for (MgO)21. The structure (21b) consisting of two touching (Mg3O3)3 hexagonal tubes and three 2 × 2 × 3 cuboids is 6.8 kcal/mol higher in energy than 21a. The 2 × 3 × 7 cuboid is 12.9 kcal/mol higher in energy than 21a. The most stable (MgO)22 structure (22a) contains a (Mg3O3)4 hexagonal tube and three 2 × 2 × 4 cuboids and is predicted to be 3.6 kcal/mol more stable than structure (22b) with two (Mg3O3)4 hexagonal tubes and one 2 × 2 × 4 cuboid. The 3 × 3 × 5 cuboid with a missing corner O atom is 23.8 kcal/mol higher in energy than 22a. The lowest energy structure for (MgO)23 (23a) is a 3 × 4 × 4 cuboid with a missing corner of MgO and is predicted to be 23.7 kcal/mol more stable than a more deformed structure (23b). The most stable isomer for (MgO)24 is a 3 × 4 × 4 cuboid (24a), 36 kcal/ mol lower in energy than the 2 × 4 × 6 cuboid (24b), 54.2 kcal/mol lower than the (Mg3O3)8 hexagonal tube (24c), and 54.8 kcal/mol lower than the octagonal tube (24d). The (Mg8O8)4 tubular structure (24e) consisting of two (Mg3O3)4 tubes and two 2 × 2 × 4 cuboids is predicted to be 64.6 kcal/

et al.32 is due to the use of different basis sets as the functional is the same. The current work uses a 4s3p1d/3s2p1d on Mg/O and the Haertelt et al. used a 5s3p/5s3p1d combination on Mg/O. We note that the energy differences between the structures is small and higher level correlated molecular orbital theory calcualtions with large basis sets will be needed to provide definitive energy differences. The 2 × 3 × 4 cubic structure (12c) is 12.3 kcal/mol higher in energy than the most stable isomer, 12a. The 2 × 2 × 6 cubic structure is predicted to be much higher in energy, ∼40 kcal/mol higher than 12a. The (MgO)12 structure that joining a 2 × 2 × 3 cuboid and a (Mg3O3)3 hexagonal tube (12d) is 16.1 kcal/mol higher in energy than 12a, and the (Mg4O4)3 octagonal tubular structure (12e) is 17.3 kcal/mol higher in energy. The calculated vibrational spectrum for 12a (Supporting Information) has multiple bands in the 600−700 cm−1 region, whereas 12b has two strong bands near 700 cm−1. The calculated frequencies of 12a are in better agreement with the experimental infrared bands32 than are those of 12b. The most stable (MgO)13 isomer is a 3 × 3 × 3 cuboid with a face center defect (13a), and a 3 × 3 × 3 cuboid with a corner oxygen missing (13b) is only 1.5 kcal/mol higher in energy. The polyhedral isomer (13c) with seven square faces, six hexagonal faces, and an octagonal face is 5.3 kcal/mol higher in energy than 13a. The polyhedral isomer (13d) with eight hexagonal and five square faces is 6.4 kcal/mol less stable than 13a. The triple hexagonal prism (13e) is predicted to be 24.3 kcal/mol higher in energy than 13a. Haertelt et al.32 predicted structure 13d to be the most stable isomer, and they reported that it provided the best match to the vibrational frequencies against the experiment. They also reported structure 13b to be 7.1 kcal/mol higher in energy than their ground state structure 13d. They did not report structure 13a, and we find that the vibrational frequencies for structure 13a (see Supporting Information) match the experimental infrared spectra for (MgO)13.32 The experimental vibrational spectrum32 for (MgO) 13 contains a large number of strong peaks in the 600−800 cm−1 region, which suggests that several low energy isomers were detected by infrared spectroscopy. The bands near 700 cm−1 can be attributed to 13a and 13b, and the bands at 740 and 780 cm−1 could be attributed to 13c. Structure 13d could also have contributed to the bands between 700 and 740 cm−1. There is also a strong peak near 400 cm−1 in the experimental spectrum that could arise from 13a, 13b, and 13c. The calculated vibrational spectrum (Supporting Information) for 13c is found to have the best overlap with the experimental spectra of (MgO)13. The lowest energy isomer of (MgO)14 is a deformed capped four-layer hexagonal tubular structure (14a). The polyhedral isomer (14b) in pseudo-C2v symmetry (if O and Mg are viewed equivalently) is 3.5 kcal/mol higher in energy. The 2 × 2 × 7 cubic structure (14d) is ∼20 kcal/mol higher in energy than 14a. The hexagonal tubular (Mg3O3)5 structure (15a) is predicted to be the most stable isomer for (MgO)15, with the 2 × 3 × 5 cuboid structure (15b) 12.7 kcal/mol higher in energy. The structure consisting of an interconnecting polyhedral ball (equivalent to structure 12b of (MgO)12) and a hexagonal prism (15c) is 14.7 kcal/mol higher in the energy than the most stable (MgO)15 isomer. The structure (15d) that is the intermediate between the hexagonal tubular (Mg 3 O 3 ) 5 structure (15a) and the 2 × 3 × 5 cubic structure (15b) and is generated by the breaking some of the Mg−O bonds in the 2 3139

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mol higher in energy than 24a. The most stable isomer (25a) for (MgO)25 is predicted to be a 2 × 5 × 5 cuboid. The structure composed of a 3 × 4 × 4 cuboid and a Mg−O (25b) is ∼15 kcal/mol higher in energy than 25a. Two tubular isomers (25c and 25d) that contain hexagonal tubes and cuboids are ∼45 kcal/mol higher in energy than the most stable isomer (25a). The lowest energy structure (26a) for (MgO)26 is predicted to be a deformed 3 × 4 × 4 cuboid plus a Mg2O2 fragment. Two tubular structures (26b and 26c) are found to be approximately 15 kcal/mol higher in energy than 26a. The most stable isomer for (MgO)27 (27a) is predicted to be a deformed 3 × 4 × 4 cuboid plus a Mg3O3 fragment. The 3 × 3 × 6 cuboid (27b) and the 2 × 3 × 9 cuboid are 8.7 and 57.2 kcal/mol higher in energy than 27a. The (Mg3O3)9 hexagonal tube (27d) is predicted to be 44.4 kcal/mol higher in energy than 27a. The tubular isomer containing two hexagonal tubes and a cuboid is ∼40 kcal/mol higher in energy than 27a. The (Mg7O7)4 tubular structure (28a) consisting of two (Mg3O3)4 tubes and three 2 × 2 × 4 cuboids is predicted to be the lowest energy isomer for (MgO)28, 17.1 kcal/mol lower in energy than the 2 × 4 × 7 cuboid (28b). The (Mg4O4)7 octagonal tube is 41.4 kcal/mol higher in energy than 28a. The lowest energy isomer for (MgO)29 (29a) is essentially a 3 × 4 × 5 cubic structure with a MgO defect near the geometric center, and the lowest energy isomer for (MgO)30 (30a) is predicted to be a 3 × 4 × 5 cuboid. The 2 × 5 × 6, 2 × 3 × 10, and 2 × 3 × 15 cuboids are 44.1, 107.4, and 230.8 kcal/mol higher in energy than the 3 × 4 × 5 cuboid (30a). The hexagonal tubular structure, (Mg3O3)10 (30b), is calculated to be 94.0 kcal/mol higher in energy than 30a. The lowest energy structures for (MgO)32, (MgO)36, and (MgO)40 (32a, 36a and 40a)are 4 × 4 × 4, 3 × 4 × 6 and 4 × 4 × 5 cubic structures, respectively. Geometry Evolution. The most stable isomers for (MgO)2 and (MgO)3 are planar and ring-like. For (MgO)n, n > 3, the most stable isomers are all predicted to be 3-dimensional structures. These low energy 3-D structures are mainly cuboids, tubes, and polyhedrons, or their derivatives. The relative stability of these isomers changes as n increases, and also depends on whether n is a prime number. In the cases where n is a prime number, the highly ordered cubic and tubular geometries are not mathematically allowed for (MgO)n, and the low energy clusters are usually defected or deformed cubic structures (e.g., structures 13a, 17a, 23a, and 29a). Figure 2 shows the normalized clustering energies of the cubic and tubular isomers for (MgO)n (n ≤ 40). A larger value of a normalized clustering energy at the same n indicates better stability of an isomer. Figure 2 shows, for the structures in each cubic and tubular series, that the normalized clustering energy, an indicator of the isomer stability, increases as n increases. Vertical differences between the normalized clustering energy vs n curves provide a comparison of the stability of the isomers from different series at the same value of n. It is found that the 3 × 4 × m and 4 × 4 × m cubic structures with m an integer are the most stable. The 2 × 4 × m, 2 × 5 × m, 2 × 6 × m, and 3 × 3 × m cubic structures show comparable stabilities and are less stable than the 3 × 4 × m and 4 × 4 × m cubic structures. Even less stable are the 2 × 3 × m cubic structures, the hexagonal tubular structures, and the octagonal tubular structures. The normalized clustering energy vs n curve for the 2 × 2 × m cubic structure series are below the curves for other cubic and tubular structure series, which suggests that the 2 × 2 × m cubic structure series are the least stable among the series shown in

Figure 2. Normalized clustering energies in kcal/mol vs n for the cubic and tubular structures.

Figure 2. For n < 20, the (Mg3O3)n and (Mg4O4)n/4 tubular isomers are more stable than the cubic isomers, as most of the available cubic isomers are 2 × 2 × m or 2 × 3 × m. For example, the (Mg3O3)2 hexagonal tubular isomer is 7 kcal/mol more stable than the 2 × 2 × 3 cubic structure for (MgO)6. For (MgO)9, (MgO)12, and (MgO)15, the hexagonal tubular structures are ∼12 kcal/mol more stable than the cubic structures. For the cubic isomers, increasing the size of the cluster in any dimension will improve the stability of the cluster. However, the effectiveness of increasing the size of the cluster in a dimension to gain stability diminishes as the size in that dimension increases, as the slopes of all of the normalized clustering energy vs n curves decrease as n increases. Therefore, for cubic structures of the same size (with the same n and volume), the most compact cubic structure (i.e., the structure where the sizes in the three dimensions are most comparable) is expected to be the more stable cubic structure. Therefore, even for large (MgO)n clusters where n is not a prime number (which means that cubic isomers can exist), the lowest energy isomers are not necessarily cubic (e.g., 26a and 27a), when the cubic structures are not the most compact ones (2 × 2 × m and 2 × 3 × m cubic). The average Mg−O bond distance and coordination number (CN) both increase as n increases (Figure 3). The average Mg−O bond distance is slowly converging at n = 40 (2.055 Å) to the bulk value (2.104 at 19.8 K and 2.106 Å at 297 K).64 The average coordination numbers of the clusters also are converging to the bulk value (CNbulk = 6), as the cubic structures start to become the dominant lowest energy isomers. The average CN is predicted to be 4.6 for the lowest energy (MgO)40 isomer, which has 12 6-coordinate interior atoms and 68 surface atoms with lower coordination numbers, even though it has the cubic structure of the bulk. The pair distribution functions (PDFs) for (MgO)n are given in the Supporting Information for comparison to experiment when such data may become available. For the largest nanoclusters, the PDF has the dominant peak just above 2 Å, as expected. The next two peaks are at ∼3.5 and 4.5 Å. Normalized Clustering Energies. The normalized clustering energies ΔE(n) for (MgO)n, n = 2−8, were calculated at the B3LYP/DZVP and the CCSD(T)/CBS(DTQ) levels (Table 1). The benchmark results show that the 3140

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Table 2. Normalized Clustering Energies ΔE(n) in kcal/mol, Averaged Mg−O Bond Distances r in Å, and Coordination Number CN for (MgO)n at the B3LYP/DZVP level

Figure 3. Average coordination number CN and bond distance r for the lowest energy isomers of (MgO)n.

Table 1. Benchmarks for the Normalized Clustering Energies in kcal/mol for (MgO)n, n = 2−9 n

CCSD(T)/D

CCSD(T)/T

CCSD(T)/Q

CCSD(T)/ CBS(DTQ)

B3LYP/ DZVP

2 3 4 5 6 7 8 9

57.6 86.8 103.0 105.9 118.9 119.4 125.2 130.4

62.6 89.9 104.3 107.1 119.5 119.8 125.4 130.1

64.0 90.6 104.7 107.3 119.5 119.7 125.2

64.9 91.1 104.9 107.3 119.5 119.7 125.1

65.4 91.7 104.5 107.2 119.1 119.3 124.7 129.2

ΔE(n)’s calculated at the B3LYP/DZVP level are within 0.5 kcal/mol of the CCSD(T)/CBS predictions for the small (MgO)n clusters (n up to 8). Therefore, B3LYP is a good functional to be used for the ΔE(n) calculations for larger (MgO)n clusters where CCSD(T) calculations are not feasible. The calculated ΔE(n)’s for (MgO)n, n = 2−40, at the B3LYP/ DZVP level are shown in Table 2, and the related ΔE(n) vs n plot is given in Figure 4. As n increases, the ΔE(n) for (MgO)n increases and the slope of the curve decreases. The slope is changing very slowly after n = 30. The value of ΔE(40) is predicted to be 149.6 kcal/mol, as compared to the bulk value ΔE(∞) = 176.0 kcal/mol (obtained from the difference between the gas phase49 ΔHf = 32.3 kcal/mol and the crystalline65 ΔHf = −143.7 kcal/mol of MgO). We plotted ΔE(n) as a function of n−1/3;66 the ΔE(n) vs n−1/3 curve is near linear for n > 4. We do not expect the very small nanoclusters to have the same convergence to the bulk as do the larger structures, as they are simply too small. The smaller clusters exhibit larger changes in the ΔE(n) as the amount of energy change as one monomer is added is spread over a smaller number of monomers. The intercept on the ΔE(n), the projected ΔE(n) value at n → ∞, is predicted to be ∼185 kcal/ mol as we extrapolate the plot (n > 4) by a linear function. The estimated value of ΔE(∞) from the ΔE(n) vs n−1/3 plot is expected to be slightly smaller than the intercept value, as the slope of the ΔE(n) vs n−1/3 plot slightly decreases as n−1/3 increases. The estimated value of ∼185 kcal/mol is in good agreement with the bulk value of 176.9 kcal/mol from the

n

r

CN

ΔE(n) eq 1

1 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 32 36 40 bulk (∞)

1.753 1.883 1.841 1.966 1.945 1.951 1.950 1.943 1.976 2.006 1.935 1.985 2.013 1.981 1.990 2.026 2.013 2.037 2.006 2.032 1.996 2.030 2.040 2.044 2.036 2.042 2.038 2.027 2.036 2.049 2.051 2.052 2.055 2.106

1 2 2 3 2.8 3 3 3 3.33 3.6 3 3.5 3.77 3.43 3.6 4 3.82 4.17 3.74 4.1 3.71 4.05 4.17 4.33 4.2 4.23 4.22 4.07 4.21 4.43 4.5 4.5 4.6 6

0.0 65.4 91.7 104.5 107.2 119.1 119.3 124.7 129.2 128.2 130.6 134.2 132.6 134.1 137.2 137.8 137.4 139.6 138.6 140.6 140.6 141.4 141.5 144.0 143.1 142.6 144.2 144.4 143.6 146.3 147.6 148.0 149.6 176.0

−ΔH(298) eq 3

ΔHf(298)

ΔHf(298)/n

130.8 144.2 142.9 117.9 178.5 120.5 162.8 165.3 119.2 154.1 174.4 113.7 153.3 180.5 146.1 132.0 176.9 121.1 177.2 141.7 157.7 143.1 201.0 122.9 129.8 184.4 151.3 121.0 225.3

32.3 −67.3 −179.7 −291.9 −377.8 −525.1 −614.0 −745.2 −879.3 −966.8 −1088.9 −1232.5 −1314.8 −1436.0 −1585.5 −1700.6 −1800.6 −1946.9 −2035.3 −2182.0 −2291.6 −2417.7 −2529.6 −2699.8 −2791.0 −2888.8 −3041.9 −3161.5 −3251.7 −3446.5

32.3 −33.6 −59.9 −73.0 −75.6 −87.6 −87.8 −93.2 −97.7 −96.7 −99.0 −102.7 −101.1 −102.6 −105.7 −106.3 −105.9 −108.2 −107.1 −109.1 −109.1 −109.9 −110.0 −112.5 −111.6 −111.1 −112.7 −112.9 −112.1 −114.9

−143.7

literature,49,65 especially considering the slow variation in the energy for the larger clusters. Even though the structures of (MgO)n and (TiO2)n are quite different, the convergence of ΔE(n) of (MgO)n clusters is comparable to that for (TiO2)n.53 For example, ΔE(13) of (MgO)13 differs by 43 kcal/mol from that of bulk MgO and for (TiO2)13, the difference is essentially the same, 44 kcal/mol. Although this similarity in convergence may be accidental, it will be useful to compare the convergence of ΔE(n) for other metal oxide clusters as the data becomes available. By using the best available heat of formation of gaseous MgO, the heats of formation of (MgO)n can be calculated from the ΔH(298) of reaction 3 and are given in Table 3. We expect that the values for n = 2−9 to be the most reliable as these are based on the CCSD(T) energies. By n = ∼10, the normalized heat of formation is about −100 kcal/mol, and the normalized heat of formation is slowly converging to the bulk solid state value of −143.7 kcal/mol. Even at n = 30, the normalized heat of formation is only −115 kcal/mol, 29 kcal/mol from the bulk value. Although there are significant differences in the actual value for ΔH for reaction 3, these differences are not large 3141

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Table 3. HOMO LUMO Energies and Energy Gaps (in eV) for the Lowest Energy (MgO)n n

HOMO

LUMO

HOMO−LUMO gap

1 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 32 36 40

−5.75 −5.42 −6.21 −5.98 −5.68 −6.21 −6.02 −6.18 −6.26 −5.95 −6.22 −6.32 −5.75 −6.16 −6.01 −5.95 −5.95 −5.09 −5.97 −5.98 −6.37 −5.33 −5.47 −5.51 −5.95 −5.51 −5.64 −5.65 −5.85 −5.40 −5.60 −5.47 −5.59

3.63 2.63 1.76 2.52 0.38 −0.22 −0.06 0.05 −0.29 −0.20 −0.10 −0.38 −0.31 −0.46 −0.31 −0.10 −0.88 −0.49 −0.49 −0.30 −0.45 −0.74 −0.75 −0.53 −0.39 −1.15 −0.91 −0.71 −1.05 −0.62 −0.48 −0.73 −0.59

9.38 8.06 7.98 8.50 6.06 5.99 5.96 6.23 5.97 5.75 6.12 5.93 5.44 5.70 5.70 5.84 5.06 4.60 5.47 5.68 5.92 4.59 4.72 4.98 5.55 4.36 4.73 4.94 4.80 4.78 5.12 4.74 5.00

Figure 4. Normalized clustering energies vs (a) n and (b) n−1/3 in kcal/mol for the lowest energy isomers of (MgO)n.

enough to really affect the normalized heats of formation in a significant way. Molecular Orbitals and Atomic Charges. The highest occupied molecular orbital (HOMO) for the (MgO)n clusters is an out-of-phase combination of 2p orbitals from the O atoms on the surface, consistent with the formal oxidation states of +2 on Mg and −2 on O. As expected, the lowest unoccupied molecular orbital (LUMO) is essentially composed of 3s orbitals from the Mg atoms on the surface. In the cubic (MgO)n clusters, as a special case, the HOMO and LUMO are localized on the 2p orbitals of the corner O atoms and 3s orbitals from the corner Mg atoms, respectively (Figure 5). There exist 3n occupied molecular orbitals from O p orbitals for a given (MgO)n cluster, among which the orbitals from the surface O atoms are close in energy to the HOMO and the orbitals from the interior O atoms are significantly lower in energy than the HOMO; i.e., the interior atoms have stabilized orbital energies. The corner Mg 3s unoccupied orbitals and the corner O 2p occupied orbitals more resemble lone pairs, whereas the interior Mg 3s orbitals more resemble antibonding orbitals, and the interior O 2p orbitals more resemble bonding orbitals. The HOMO and LUMO energies and HOMO− LUMO gaps for the lowest energy (MgO)n and cubic (MgO)n

Figure 5. Molecular orbital iso-surfaces for the HOMO and LUMO for (MgO)3, (MgO)21, and (MgO)32.

are shown in Tables 3 and 4, respectively. The HOMO− LUMO gap for the lowest energy (MgO)n cluster generally 3142

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HOMO in the nanoclusters and the energy level of the conduction band in the bulk higher than the LUMO in the nanoclusters. Our gap values are consistent with those reported69,70 for nanocrystals and thin films of 5.4 to 6.0 eV as well as those of 5.0 to 6.0 eV reported71 for nanosized MgO particles. There is also a difference between the calculated Mulliken atomic charges for the surface and interior atoms in the (MgO)n clusters. In general, the interior atoms are more ionic than the surface atoms. For small clusters, this effect is not obvious as the interior atoms are close to the surface. For (MgO)40, the corner Mg and O atoms have Mulliken atomic charges of ∼+1 and −1, and the most interior Mg and O atom have Mulliken atomic charges of ∼+1.4 and −1.4. These results suggest that the oxygen atoms on the corners of the surface of the cluster can serve as Lewis base sites. These are lowest coordinated O2− sites, consistent with the observations of Bailly et al.23 We suggest that these corner sites should be the sites that are protonated by Brönsted acid/ base reactions, again consistent with the conclusions of Bailly et al.23 These authors suggest that the actual Brönsted acid/base reactivity is significantly more complicated and depends on the number of low coordination O2− sites as well as the presence of basic OH groups. The Mg atoms at the corners of the cluster are where Lewis bases will bind and where the anion generated in a Brönsted acid/base reaction could attach, in the simplest model, although the actual chemistry is likely to be more complicated.23 The HOMO−LUMO gap is large enough that it is unlikely that photochemistry will play a role in the reactivity of the cluster unless the cluster is bonded to a photoactive species or impurity sites are present.13,15−18 The results for the cluster stabilities suggest that clusters with n > 20 will be more like the bulk structure in reactivity if differences in the number of surface vs bulk MgO units is accounted for. Many of the clusters with n < 20 should have different reactivity from the bulk as the structures may not have a structure similar to the bulk, have essentially all low coordinated surface atoms, and have smaller normalized binding energies.

Table 4. HOMO LUMO Energies and Energy Gaps (in eV) for the Cubic (MgO)n n

dimensions

4 6 8 9 10 12 12 14 15 16 16 18 18 18 20 20 21 22 24 24 24 24 25 26 27 27 28 28 30 30 30 30 32 32 32 33 34 35 36 36 36 36 36 36 38 39 40 40 40 40

2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 3 2 2 2 2 2 3 2 2 4 2 2 2 2 2 2 2 3 3 2 2 2 2 2 4

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

2 2 2 3 2 2 3 2 3 2 4 2 3 3 2 4 3 2 2 3 4 4 5 2 3 3 2 4 2 3 5 4 2 4 4 3 2 5 2 3 4 6 3 4 2 3 2 4 5 4

× × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × ×

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

HOMO

LUMO

HOMO−LUMO gap

−5.99 −5.89 −5.96 −5.88 −5.95 −5.97 −5.98 −5.96 −6.01 −5.97 −5.95 −5.96 −6.01 −5.08 −5.97 −5.99 −6.02 −5.97 −5.97 −6.02 −6.00 −5.50 −5.95 −5.97 −6.02 −4.57 −5.98 −6.01 −5.98 −6.03 −5.97 −5.41 −5.98 −6.01 −5.60 −6.03 −5.97 −5.98 −5.97 −6.03 −6.01 −5.96 −4.22 −5.47 −5.97 −6.02 −5.98 −6.01 −5.98 −5.59

2.52 0.38 0.19 −0.06 −0.20 −0.47 −0.05 −0.66 −0.31 −0.79 −0.10 −0.88 −0.50 −0.48 −0.88 −0.30 −0.66 −0.88 −0.87 −0.77 −0.48 −0.53 −0.39 −0.88 −0.86 −0.79 −0.88 −0.63 −0.87 −0.92 −0.51 −0.62 −0.88 −0.74 −0.48 −0.97 −0.88 −0.64 −0.88 −1.01 −0.82 −0.57 −1.21 −0.73 −0.88 −1.04 −0.88 −0.88 −0.74 −0.59

8.51 6.27 6.15 5.82 5.75 5.50 5.93 5.31 5.70 5.18 5.84 5.09 5.51 4.60 5.10 5.68 5.36 5.09 5.10 5.25 5.52 4.98 5.55 5.10 5.16 3.78 5.10 5.38 5.11 5.10 5.45 4.79 5.10 5.27 5.12 5.05 5.10 5.34 5.09 5.01 5.19 5.39 3.01 4.74 5.10 4.98 5.10 5.13 5.25 5.00

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CONCLUSIONS We used our tree growth−hybrid genetic algorithm in conjunction with semiempirical molecular orbital calculations followed by B3LYP geometry optimizations to predict the global minima for (MgO)n nanoclusters. Our results for n ≤ 16 agree with those of Haertelt et al.32 except for n = 10, 12, 13, and 16 where we predict different global minima. For n = 10 and 12, they predicted that 10b and 12b were the lowest energy structures in contrast to our prediction of 10a and 12a being the lowest energy structures. We note that Haertelt et al.32 predict 12a and 12b to be essentially isoenergetic. For n = 13 and 16, we report new global minima that have not previously been reported. The most stable isomers for (MgO)n (n > 3) have 3-dimensional structures. Hexagonal tubular (MgO)n structures, except for n = 16, are more favored in energy than the cubic structures which more closely resemble the bulk for n < 20. A cuboid structure is predicted to be the most stable structure for n = 16, but structure 16b, consisting of a cuboid combined with a hexagonal tube, is only 3.4 kcal/mol higher in energy. The cubic structures and appropriate variations are the most stable structures for n > 20. Increasing the size of the cubic-based clusters in any dimension leads to a more stable isomer. The improvement in stability due to increasing the size of the cluster in a specific dimension decreases with increasing

decreases as n increases, and the energy gap oscillates near 5 eV for n > 16, as compared to the experimental band gap value of 7.8 eV from the bulk.67,68 One reason for this difference in the gaps is that the number of surface atoms is large compared to the number of interior atoms in the cluster and the opposite is true for the bulk. As the interior atoms have more stable high energy MOs and less stable unoccupied MOs, the energy level of the valence band in the bulk is expected to be lower than the 3143

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(4) Xiong, Y. L. Experimental Determination of Solubility Constant of Hydromagnesite (5424) in NaCl Solutions Up to 4.4 m at Room Temperature. Chem. Geol. 2011, 284 (3−4), 262−269. (5) Lamb, H. H.; Gates, B. C.; Knö z inger, H. Molecular Organometallic Chemistry on Surfaces: Reactivity of Metal Carbonyls on Metal Oxides. Angew. Chem., Int. Ed. 1988, 27, 1127−1144. (6) Kirlin, P. S.; Knözinger, H.; Gates, B. C. Mononuclear, Trinuclear, and Metallic Rhenium Catalysts Supported on MgO: Effects of Structure on Catalyst Performance. J. Phys. Chem. 1990, 94, 8451−8456. (7) Kawi, S.; Chang, J.-R.; Gates, B. C. Cluster Catalysis: Propane Hydrogenolysis Catalyzed by MgO-Supported Tetrairidium. J. Phys. Chem. 1994, 98, 12978−12988. (8) Uzun, A.; Ortalan, V.; Browning, N. D.; Gates, B. C. A SiteIsolated Mononuclear Iridium Complex Catalyst Supported on MgO: Characterization by Spectroscopy and Aberration-Corrected Scanning Transmission Electron Microscopy. J. Catal. 2010, 269, 318−328. (9) Kulkarni, A.; Chi, M.; Ortalan, V.; Browning, N. D.; Gates, B. C. Atomic Resolution of the Structure of a Metal-Suport Interface: Triosmium Clusters on MgO(110). Angew. Chem., Int. Ed. 2010, 49, 10089−10092. (10) Serna, P.; Gates, B. C. Zeolite- and MgO-Supported Rhodium Complexes and Rhodium Clusters: Tuning Catalytic Properties to Control Carbon-Carbon vs. Carbon-Hydrogen Bond Formation Reactions of Ethene in the Presence of H2. J. Catal. 2013, 308, 201−212. (11) Aydin, C.; Kulkarni, A.; Chi, M.; Browning, N. D.; Gates, B. C. Three-Dimensional Structural Analysis of MgO-Supported Osmium Clusters by Electron Microscopy with Single-Atom Sensitivity. Angew. Chem., Int. Ed. 2013, 52, 5262−5265. (12) Lunsford, J. H. The Catalytic Conversion of Methane to Higher Hydrocarbons. Catal. Today 1990, 6, 235−259. (13) Hacquart, R.; Krafft, J.-M.; Costentin, G.; Jupille, J. Evidence for Emission and Transfer of Energy from Excited Edge Sites of MgO Smokes by Photoluminescence Experiments. Surf. Sci. 2005, 595, 172−182. (14) Trevethan, T.; Shluger, A. L. Building Blocks for Molecular Devices: Organic Molecules on the MgO (001) Surface. J. Phys. Chem. C 2007, 111, 15375−15381. (15) Shluger, A. L.; Grimest, R. W.; Catlow, C. R. A.; Itoh, N. SelfTrapping Holes and Excitons in the Bulk and on the (100) Surfaces of MgO. J. Phys.: Condens. Matter 1991, 3, 8027−8036. (16) Beck, K. M.; Joly, A. G.; Diwald, O.; Stankic, S.; Trevisanutto, P. E.; Sushko, P. V.; Shluger, A. L.; Hess, W. P. Energy and Site Selectivity in O-Atom Photodesorption from Nanostructured MgO. Surf. Sci. 2008, 602, 1968−1973. (17) Diwald, O.; Kno1zinger, E. Intermolecular Electron Transfer on the Surface of MgO Nanoparticles. J. Phys. Chem. B 2002, 106, 3495− 3502. (18) Pacchioni, G.; Freund, H. Electron Transfer at Oxide Surfaces. The MgO Paradigm: from Defects to Ultrathin Films. Chem. Rev. 2013, 113, 4035−4072. (19) Florez, E.; Fuentealba, P.; Mondragón, F. Chemical Reactivity of Oxygen Vacancies on the MgO Surface: Reactions with CO2, NO2 and Metals. Catal. Today 2008, 133−135, 216−222. (20) Livraghi, S.; Paganini, M. C.; Giamello, E. SO2 Reactivity on the MgO and CaO Surfaces: A CW-EPR study of Oxo-Sulphur Radical Anions. J. Mol. Catal. A: Chem. 2010, 322, 39−44. (21) Petitjean, H.; Tarasov, K.; Delbecq, F.; Sautet, P.; Krafft, J. M.; Bazin, P.; Paganini, M. C.; Giamello, E.; Che, M.; Lauron-Pernot, H.; Costentin, G. Quantitative Investigation of MgO Brønsted Basicity: DFT, IR, and Calorimetry Study of Methanol Adsorption. J. Phys. Chem. C 2010, 114, 3008−3016. (22) Cornu, D.; Petitjean, H.; Costentin, G.; Guesmi, H.; Krafft, J.M.; Lauron-Pernot, H. Influence of Natural Adsorbates of Magnesium Oxide on Its Reactivity in Basic Catalysis. Phys. Chem. Chem. Phys. 2013, 15, 19870−19878. (23) Bailly, M.-L.; Chizallet, C.; Costentin, G.; Krafft, J.-M.; LauronPernot, H.; Che, M. A Spectroscopy and Catalysis Study of the Nature

that dimension. The most compact cubic structure is expected to be the most stable cubic structure for a given n. The average Mg−O bond distance and coordination number both increase as n increases. The calculated average Mg−O bond distance at n = 40 of is 0.05 Å smaller than the bulk value of 2.104 Å from the crystal. The average coordination number of 4.6 for (MgO)40 is below the bulk value of 6, consistent with the ratio of 68:12 for the number of surface atoms to the number of 6-coordinate interior atoms. As n increases, the normalized clustering energy ΔE(n) for the (MgO)n increases and the slope of the ΔE(n) vs n curve decreases, showing a slow convergence to the bulk. The convergence to the bulk depends on the type of cubic cluster; although there are only two points, the slope for the 4 × 4 × m cube is higher than the others. Even at n = 40, the normalized clustering energy is still 26 kcal/mol below the bulk value. The location of the HOMO and LUMO suggest that Lewis acids will bond to the oxygen and that Lewis bases will bind to the magnesium as expected. The most active atoms will be those on the corners. Our results suggest that it would be interesting to synthesize (MgO)n nanoclusters to better understand their unique substructures, which can be different from the bulk as well as their role in changing the reactivity of substrates bonded to them.

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ASSOCIATED CONTENT

* Supporting Information S

Complete author lists for refs 62 and 63. Calculated vibrational spectrum for low energy structures for (MgO)n. Pair distribution functions (PDFs) for (MgO)n. Optimized Cartesian x, y, z coordinates in angstroms and total energies in au. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*D. A. Dixon: e-mail, [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the U.S. Department of Energy (DOE), Office of Basic Energy Sciences, Geosciences program grant number DE-SC0009362. 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, and located at PNNL. PNNL is operated for DOE by Battelle Memorial Institute under Contract # DE-AC06-76RLO-1830. D.A.D. also thanks the Robert Ramsay Chair Fund of The University of Alabama for support.

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