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Structures and Stabilities of (CaO) Nanoclusters Mingyang Chen, Kanchana Sahan Thanthiriwatte, and David A Dixon J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b09062 • Publication Date (Web): 26 Sep 2017 Downloaded from http://pubs.acs.org on September 27, 2017

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The Journal of Physical Chemistry

Structures and Stabilities of (CaO)n Nanoclusters Mingyang Chen,1,* K. Sahan Thanthiriwatte,2 and David A. Dixon*,2,† 1

Beijing Computational Science Research Center, Beijing 100193, China.

2

Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, AL, 35487-

0336, USA Abstract The global energy minima structures for (CaO)n for n ≤ 40 were predicted using density functional theory. The cubic structures are found to be the lowest energy isomers for most (CaO)n, n ≥ 4. A fragment-based structure-energy relationship model gave an excellent fit for the calculated total energy. Based on the fitting results, the bulk limit for the normalized clustering energy for (CaO)n particles was predicted to be 157.8 kcal/mol for the enthalpy at 298 K, in good agreement with the experimental/computational bulk value of 156.5 kcal/mol. A (CaO)n nanoparticle with a size of 10 nm is predicted to have a CaO binding energy close to that of the bulk crystal. The infinite chain limit for normalized clustering energy for various 1-D cubic nanoparticle series was also obtained. The surface energy densities were predicted to be 62 kcal/mol per CaO for the 3-coordinate corner fragment, 30 kcal/mol per CaO for the 4coordinate edge fragment, and 11 kcal/mol per CaO for the 5-coordinate face fragment. On the basis of the values of the parity sum for the atom counts for the cube’s edges in three dimensions, cubic (CaO)n nanoclusters can be classified into three types with different geometries. Although no significant difference in stability was found for different types of cubic (CaO)n, several electronic properties of the cubic (CaO)n are related to the parity sum at small n. The type-2 (CaO)n clusters and ultra-small particles in the shape of the odd×odd×even cube, with a parity

†

Email: [email protected]

1 ACS Paragon Plus Environment


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sum of 2, exhibit unique electronic properties. The type-2 3×3×m 1-D nanoparticle series has the lowest HOMO-LUMO excitation energy among all of the 1-D nanoparticle series at all particle sizes: 2.51 eV for the 3×3×4 cube, and 0.70 eV for the 3×3×18 cube. The patterns for the variation of Eg during the 1-D layer-wise growth of (CaO)n were analyzed.


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Introduction There is substantial interest in the oxides and carbonates of Ca2+. CaO in its mineral form is known as quicklime and has the NaCl structure (“rock-salt”). It readily reacts with the CO2 in air to from CaCO3. CaO is a prime component in the production of non-hydraulic cement. Bulk CaO is an insulator with a wide optical gap (~ 8 eV), wide valence band (~ 8.5 eV), and a high dielectric constant of 11.8,

1,2,3,4,5

which are likely to be different for molecular clusters and

ultra-small nanoparticles (USNPs). The optical gaps of the clusters and USNPs are expected to be significantly smaller due to the surface electronic states; the valence band is likely to be narrower due to the quantum confinement effects. It is important to explore the electronic properties of (CaO)n clusters and USNPs, not only because of the potential applications of these particles as semiconductor materials, but also because (CaO)n with a simple structural evolution is ideal for understanding size-effects on its chemical and electronic properties. Batra et al. studied the geometry evolutions of small alkali earth metal oxides (XO)n (X=Mg, Ca, Sr, and Ba, n ≤ 6) at the density functional theory with the PW91 and PWC functionals, and showed that, at n = 6, all of the (XO)n’s have the shape of a cubic slab except for (MgO)6 which is a hexagonal prism. 6 Escher et al. searched the global minima structure of (BaO)n, n = 4-18 and 24 using an evolutionary algorithm, and predicted magic number clusters at n = 4, 6, 8, 10 and 16, all of which are cubic structures. 7 We have previously systematically investigated the properties of (MgO)n using a combination of correlated molecular orbital theory and density functional theory. 8 The nanocluster structures were generated by using a tree growthhybrid genetic algorithm (TG-HGA) developed by us. 9 Cubic structures and their variations dominate for (MgO)n for n > 20, and, for n < 20, hexagonal tubular structures are dominant. We predicted heats of formation for the clusters on the basis of the normalized clustering energies 3 ACS Paragon Plus Environment


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and accurately calculated heat of formation of MgO 10 using the Feller-Peterson-Dixon (FPD) approach. 11,12,13 We have shown that the normalized clustering energy (related to the total energy per formula unit) of a cubic or tubular MgO 1-D nanoparticle series (i.e. a set of nanoparticle products that result from the 1-D growth via a translational symmetry) is a linear function of n-1. (MgO)n clusters with an aspect ratio close to 1:1:1 are proposed to be the most energetically favorable isomers based on our results. These conclusions were later proved by a fragment-based structure-energy relationship model developed by us. 14 The fragment-based model also proves the generality of the conclusions derived from the (MgO)n cluster study. Therefore, the geometries for the “rock-salt”-like oxide clusters are mostly known once the size beyond which the cubic clusters become dominant is determined. Thus, it is important to understand the evolution of the energy and electronic properties that arise from the structural evolution of (CaO)n, the focus of the current work. We have previously reported high-level calculations10 using the FPD approach of the heat of formation of gas phase CaO showing that there are issues with the available experimental heats of formation. 15,16,

17

This value allows us to accurately

predict the energetics of the (CaO)n clusters required for developing structure-energy relationships. Computational Methods The initial geometries for the low energy (CaO)n clusters were taken from our prior work on MgO nanoclusters using the TG-HGA developed by us.9, 8. The resulting geometries were optimized using density functional theory with the B3LYP 18,19 exchange-correlation functional and the DFT optimized (local spin density functional) polarized valence double-ζ DZVP basis set. 20 This basis set was chosen as a large number of structures needed to be studied. In order to survey a broader range of structures, higher energy cubic and tubular isomers (including 4 ACS Paragon Plus Environment

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hexagonal, octagonal, and double hexagonal tubular isomers) were constructed and optimized. Single point energy calculations were carried out at CCSD(T) and at the DFT level with additional functionals for the optimized geometries for selected n. The NCE(n) can be calculated with equation (1) for (CaO)n. NCE = E(CaO) – E((CaO)n)/n

(1)

For (CaO)n, n = 2 - 8, the NCE(n) was calculated at the CCSD(T)/cc-pVTZ, 21,22,23,24,25 level and at the MP2/cc-pVDZ and MP2/cc-pVTZ levels. 26, M06,

28

M11, 29 B3LYP,

30 , 31

PW91,

32 , 33

27

In addition, DFT calculations with the

, BP86, 34 , 35 CAM-B3LYP, 36 and ωB97XD 37

functionals with the cc-pVDZ and cc-PVTZ basis sets were also performed. In addition, the NCE was extrapolated to the complete basis set limit using equation (2) 38 with cc-pVNZ basis sets, N = D, T, Q, was done for n = 2- 6 at the CCSD(T) level. 𝐸𝐸(𝑁𝑁) = 𝐴𝐴 ∗ 𝐸𝐸𝐶𝐶𝐶𝐶𝐶𝐶 + 𝐵𝐵 ∗ 𝑒𝑒 −(𝑁𝑁−1) + 𝐶𝐶 ∗ 𝑒𝑒 −(𝑁𝑁−1)

2

(2)

The sequential addition reaction energy for Eq. (3) (CaO)n-1 + CaO → (CaO)n

(3)

were also calculated to show the presence of any fluctuations in the stability of (CaO)n clusters as n increases. The heats of formation of (CaO)n can be estimated using the reaction enthalpies at 298K from Eq. (1) and the gas-phase heat of formation of CaO (Eq. (4)): 𝛥𝛥𝛥𝛥𝑓𝑓298 ((𝐶𝐶𝐶𝐶𝐶𝐶)𝑛𝑛 ) = 𝑛𝑛 ∗ (𝛥𝛥𝐻𝐻𝑓𝑓298 (𝐶𝐶𝐶𝐶𝐶𝐶, 𝑔𝑔𝑔𝑔𝑔𝑔) − 𝑁𝑁𝑁𝑁𝑁𝑁(𝑛𝑛)298 )

(4)

The DFT calculations were done with the GAUSSIAN 09 39 program. The CCSD(T) calculations were performed with the MOLPRO2012 40,41 package of ab initio programs. Results and Discussion Geometry evolution The lowest energy isomers for the (CaO)n clusters at the B3LYP/DZVP level are shown in Figure 1. The most stable isomers for (CaO)2 and (CaO)3 are planar rings. For 5 ACS Paragon Plus Environment


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(CaO)n, n > 3, the most stable isomers are all predicted to be 3-dimensional. The low energy 3-D structures are mainly cuboids, tubes, and polyhedrons, or their derivatives. The relative stabilities of these isomers changes as n increases, and 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 (CaO)n, and the low energy clusters are defected or deformed cubic structures. The most stable (CaO)n nanoclusters are dominated by cubic structures. The lowest energy structures for each n are essentially the same as previously found8 for (MgO)n except for n = 6, 8, 12, 15, 21, and 28 where cubic structures are preferred over the non-cubic or more distorted structures found for (MgO)n. Thus the (CaO)n structures prefer to be mostly cubic where this is possible. Figure 2 shows the NCE of the cubic and tubular isomers for (CaO)n (n < 40) at the ωB97XD/cc-pVDZ level (see the discussion of the benchmark results in the next section) as a function of n (Figure 2(a)) and as a function of n-1 (Figure 2(b)). A larger value of the NCE at the same n indicates greater stability for that isomer. For the structures in each cubic and tubular series, Figure 2(b) shows that the NCE increases as n increases, and is essentially a linear function of n-1, which is consistent with the conclusion derived from the 1-D scenario of the fragment-based structure-energy model in our prior work on TiO2 nanoclusters.14 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 indicated by the decreased slope (as n increases) of NCE(n) curve for the cubic structures with same lengths in two dimensions (Figure 2(a)). 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 6 ACS Paragon Plus Environment

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Figure 1. Geometries at the B3LYP/DZVP level for the lowest energy (CaO)n isomers. 8 ACS Paragon Plus Environment

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Figure 2. Normalized clustering energies in kcal/mol as a function of (a) n and (b) n-1 for the cubic and tubular (CaO)n structures. Hex = hexagonal tubes and oct = octagonal tubes.

comparable) is expected to be the more stable cubic structure. This can be proven by using the Gibbs-Wulff theorem 42 and is also found in our fragment-based structure-energy relationship model for 3-D oxides.14 The compact cubic structures are expected to be more stable than other regular and irregular structures (such as the tubular structures). However, not every n can be factorized into three nearly equal integers. Instead, for large (CaO)n clusters where n is not a prime number (which means that cubic isomers can exist), the lowest energy isomers are not necessarily cubic if the cubic structures are not the most compact ones (e.g., the 2×2×m and 10 ACS Paragon Plus Environment

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2×3×m cubes). In contrast to (MgO)n, the cubic structures for (CaO)n are predicted to be at least as stable as the tubular structures for essentially all n. This is probably due to the larger ionic radius of Ca2+ (1.06 Å) vs Mg2+ (0.78 Å). 43 The average Ca-O bond distance and coordination number both increase as n increases (Table 1). The average Ca-O bond distance is slowly converging at n = 40 (2.34 Å) to the bulk value (2.40 Å). 44 The average coordination numbers of the clusters are converging to the bulk value (CNbulk = 6), as the condensed cubic clusters which are dominant for higher n become large enough. The average CN is predicted to be 4.6 for the lowest energy (CaO)40 isomer, which has only 12 6-coordinate interior atoms and 68 surface atoms with lower coordination numbers, even though it has the same cubic structure as the bulk. The pair distribution functions (PDFs) for (CaO)n are given in the Figure S3 (Supporting Information) for comparison to experiment when such data may become available. These can be compared to the PDF for the bulk and may be useful for determining the shapes and local structures of CaO nanoparticles when experimental values become available. The calculated DFT infrared spectra for the lowest energy (CaO)n, n ≤ 40 (Supporting Information, Figure S7) are provided to aid in the experimental assignment of spectra when such data becomes available. Normalized Clustering Energies The normalized clustering energies NCE(n) for the lowest energy (CaO)n, n = 2 – 8, were calculated at the CCSD(T)/CBS, MP2/cc-pVNZ, B3LYP/ccpVNZ, CAM-B3LYP/cc-pVNZ, ωB97XD/cc-pVNZ, M06/cc-pVNZ, M11/cc-pVNZ, PW91/ccpVNZ, and BP86/cc-pVNZ (n = D and T) levels (Table 2 and Figure 3). The CCSD(T)/cc-pVTZ results are converged to better than 1 kcal/mol as compared to the CBS level for n = 2 – 6. The ωB97XD/cc-pVDZ and CAM-B3LYP/cc-pVDZ values show the smallest deviations (ΔNCE =



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Table 1. Average Ca-O Bond Distances (R) Coordination Number (CN), NCE (Eq. (1) from electronic energy at 0K and from entropy at 298K), and Sequential Reaction Energies (Eq. (3)) in kcal/mol for (CaO)n at the B3LYP and M06/DZVP level.a

n

R(Å)

CN

NCE(n), electronic, Eq (1) ωB97XD

NCE(n), 0K enthalpy, 298K -ΔH(298) Eq (3) Eq (1) ωB97XD ωB97XD

ΔHf(298)

ΔHf(298)/n

ωB97XD

ωB97XD

4.7b (4.7)

4.7 (4.7)

1

1.818

1

0.0 (0.0)

0.0 (0.0)

2

2.049

2

61.2 (61.3)

60.8 (60.9)

121.6

-112.2 (-112.4)

-56.1 (-56.2)

3

2.055

2

76.9 (75.5)

76.3 (74.9)

107.3

-214.8 (-210.7)

-71.6 (-70.2)

4

2.177

3

97.7 (96.8)

96.9 (96.1)

158.8

-368.8 (-365.5)

-92.2 (-91.4)

5

2.168

2.8

97.2 ()

96.4 ()

94.2

-458.5 ()

-91.7 ()

6

2.224

3.33

107.9 (106.5)

107.1 (105.7)

160.4

-614.4 (-605.9)

-102.4 (-101.0)

7

2.184

3

107.5 (106.4)

106.7 (105.6)

104.5

-714.0 (-706.0)

-102.0 (-100.9)

8

2.242

3.5

113.2 (112.2)

112.3 (111.3)

151.7

-860.8 (-853.1)

-107.6 (-106.6)

9

2.264

3.67

116.4

115.6

141.5

-998.1

-110.9

10

2.252

3.6

116.3

115.4

114.0

-1107.0

-110.7

11

2.254

3.55

117.3

116.4

126.8

-1228.7

-111.7


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12

2.277

3.83

121.2

120.3

162.5

-1387.2

-115.6

13

2.307

3.92

120.1

119.2

106.3

-1488.5

-114.5

14

2.269

3.71

121.1

120.2

133.5

-1617.0

-115.5

15

2.286

3.93

124.0

123.1

162.9

-1776.0

-118.4

16

2.291

4

125.6

124.6

148.3

-1918.4

-119.9

17

2.281

3.82

123.5

122.5

88.9

-2002.6

-117.8

18

2.310

4.17

127.9

126.9

201.6

-2199.6

-122.2

19

2.276

3.79

124.9

124.0

70.7

-2266.7

-119.3

20

2.299

4.1

128.2

127.2

188.8

-2450.0

-122.5

21

2.301

4.05

127.8

126.9

119.6

-2566.2

-122.2

22

2.302

4.05

128.6

127.6

144.0

-2703.8

-122.9

23

2.321

4.26

129.5

128.6

150.5

-2849.7

-123.9

24

2.320

4.33

132.1

131.2

190.0

-3036.0

-126.5

25

2.307

4.2

130.5

129.6

90.5

-3122.5

-124.9

26

2.322

4.31

131.1

130.2

145.0

-3263.0

-125.5

27

2.322

4.33

131.8

130.9

149.2

-3407.4

-126.2

28

2.308

4.21

131.1

130.2

111.3

-3514.0

-125.5



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29

2.311

4.21

130.8

129.9

121.3

-3630.8

-125.2

30

2.328

4.43

134.5

133.6

241.4

-3867.0

-128.9

32

2.331

4.5

136.0

135.1

-4172.8

-130.4

36

2.331

4.5

136.2

135.2

-4698.0

-130.5

40

2.338

4.6

138.0

137.1

-5296.0

-132.4

Bulk (∞)

2.401

6

156.5c

-151.8 ±0.2d

a

Energies are in kcal/mol. Best CCSD(T) values are in parenthesis, i.e., CCSD(T)/CBS for n=2-6, CCSD(T)/D for n = 7 and 8.

b

Calculated ΔHf(CaO(g),298) using the FPD method.10

NCE for the bulk is calculated from experimental ΔHf(CaO(s),298)= -151.8 ±0.2 kcal/mol,15 and the calculated ΔHf(CaO(g),298)= 4.7 kcal/mol.10 c

d

Experimental ΔHf(CaO(s),298).15


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Table 2. Benchmarks for the Normalized Clustering Energies in kcal/mol for (CaO)n, n = 2 - 8.

n

2 3 4 5 6 7 8

CCSD(T) CCSD(T) CCSD(T) CCSD(T) ΔMP2 ΔM11 ΔM06 ΔB3LYP ΔBP86 ΔPW91 ΔCAM/Da /T /Q /CBS /Db /D /D /D /D /D B3LYP /D 61.5 -3.2 3.8 -1.9 -2.8 -6.8 -5.6 1.1 64.5 61.3 61.3 76.3 -2.0 4.8 -2.2 -4.0 -9.7 -8.5 1.1 79.8 75.8 75.5 97.3 -3.9 8.0 -1.7 -5.1 -10.9 -9.2 1.4 102.0 96.9 96.8 96.3 -3.3 7.8 -1.7 -5.1 -11.1 -9.4 1.4 101.9 95.8 95.6 107 -3.4 9.6 -1.8 -5.9 -12.4 -10.6 1.4 111.8 106.6 106.5 106.4 -3.3 8.5 -1.7 -5.7 -12.4 -10.6 1.4 111.5 112.2 -3.2 10.1 -1.8 -6.4 -13.2 -11.3 1.3 117.6

a

ΔωB97XD /D -0.3 0.6 0.4 0.9 0.9 1.1 1.0

For DFT and MP2 calculations, ‘D’ denotes cc-pVDZ on Ca and O. For CCSD(T) calculations, ‘D’, ‘T’ and ‘Q’ denotes cc-pVDZ, cc-pVTZ and cc-pVQZ on Ca (aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ on O), respectively. b ΔNCELevel = NCELevel - NCECCSD(T)/T



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Figure 3. Deviation of NCEs calculated at different theoretical levels with cc-pVDZ basis sets from the CCSD(T)/T values for (CaO)n, n = 2-8.

~1 kcal/mol and ~1.3 kcal/mol) from the CCSD(T)/cc-pVTZ values for the NCEs. The M06/ccpVDZ values show the next best agreement with a deviation of ~2 kcal/mol. The MP2/cc-pVDZ


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values were found to be ~3 kcal/mol too small as compared to the CCSD(T)/cc-pVTZ values. Deviations converge quickly for the ωB97XD, CAM-B3LYP, M06 and MP2/cc-pVDZ values, especially for CAM-B3LYP, for which ΔNCE is mostly invariant to n. The errors in the NCE(n) for B3LYP, PW91, BP86 and M11/cc-pVDZ are much larger, and increase as n increases, so these functionals were not used further. The DFT/cc-pVTZ values appear to have worse agreement with the CCSD(T) values than do the DFT/cc-pVDZ values, even though cc-pVTZ is a larger basis set. The results suggest that the ωB97XD and CAM-B3LYP/cc-pVDZ values are reasonable for the NCEs and can be used for the NCE(n) calculations for larger (CaO)n nanoclusters where CCSD(T) calculations are not yet computationally feasible. The calculated ∆H(n)’s for the (CaO)n, n = 2 – 40, at the M06/cc-pVDZ levels are shown in Table 1, and the related NCE vs. n plot is given in Figure 4(a) and the NCE vs. n-1/3 plot is in Figure 4(b). The value of NCE(40) is predicted to be 136.2 kcal/mol at the ωB97XD/cc-pVDZ level, as compared to the bulk value ∆E(∞) = 156.5 kcal/mol (obtained from the difference between the gas phase10 ∆Hf(CaO(g),298) = 4.7 kcal/mol and the crystalline15 ∆Hf(CaO(s),298) = -151.8 ± 0.2 kcal/mol of CaO). The NCE vs. n-1/3 plot 45 is nearly linear for n > 8, so we exclude the smaller clusters from the curve fitting. 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 NCE(n) as the amount of energy change as one monomer is added is spread over a smaller number of monomers. The y-intercept for the linear plot for (n>8) for ωB97XD, the projected NCEelectronic(n) value at n → ∞, is predicted to be 172 kcal/mol (without the inclusion of ∆EZPE). The ZPE, thermal and Gibbs energy corrections (at 298k) to NCEelectronic are shown in the Supporting Information. The ZPE corrections and thermal corrections at 298K are found to be almost invariant to the size of the cluster, predicted to be -1.4 17 ACS Paragon Plus Environment


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and 0.4 kcal/mol, respectively. The Gibbs free energy correction to NCEenthalpy,298 at 298 K (TΔS) increases as n increases, and is found to converge quickly as n-1/3 decreases (which is projected to be -13 kcal/mol at n → ∞). The NCE(n→∞) is estimated to be 171 kcal/mol for ΔH298, as compared to the bulk value of 156.5 kcal/mol from the literature.10,15


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Figure 4. Normalized clustering energies vs. (a) n and (b) n-1/3 in kcal/mol for the lowest energy isomers of (CaO)n at ωB97XD/cc-pVDZ level. The NCE(n), n ≥ 9 is fitted with a linear function of n-1/3: NCE(n) = -120.2 n-1/3 + 172.2. (c) Predicted normalized clustering energies as a function of n-1/3 for a×a×a (CaO)n cubes (NCE(n) = -26.1n-1 – 57.0n-2/3 – 52.8n-1/3 + 158.8) and a×a×a (MgO)n cubes (NCE(n) = -68.5n-1 – 69.8n-2/3 – 29.4n-1/3 + 168.0). (d) Comparison between the NCE values predicted using fragment-based mode (‘F.M.’) and NCE values predicted using direct DFT calculation (‘calc’, at ωB97XD) for different 1-D (CaO)n nanoparticle series. The fit numerical NCE equation are: NCE2×2×m = -129.3n-1 + 128.9; NCE2×3×m = -167.0n-1 + 135.2; NCE2×4×m = -204.8n-1 + 138.4; NCE2×5×m = -242.5n-1 + 140.3; NCE3×3×m = -215.7n-1 + 140.6; NCE3×4×m = -264.4n-1 + 143.3; NCE3×5×m = -313.1n-1 + 145.0; NCE4×4×m = -324.1n-1 + 145.8; and NCE4×5×m = -383.8n-1 + 147.3.


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The extrapolated value is sensitive to the data selection. For more quantitative predictions, energetic parameters including NCE(n→∞) can be calculated using a fragment-based structureenergy model developed by us.14 We can write the total electronic energy of a cubic (CaO)n as the sum of the fragment energies (Eq. (5)): 𝐸𝐸𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = ∑𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ∗ 𝐸𝐸𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓

(5)

A cubic (CaO)n in the shape of an a×b×c grid is composed of 𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 corner Ca1O1 fragments,

𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 edge Ca1O1 fragments, 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 face Ca1O1 fragments and 𝑛𝑛𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 bulk Ca1O1 fragments, in terms of the position of Ca: 𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 4

(6a)

𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = (𝑎𝑎 − 2)(𝑏𝑏 − 2) + (𝑏𝑏 − 2)(𝑐𝑐 − 2) + (𝑐𝑐 − 2)(𝑎𝑎 − 2)

(6c)

𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 = 2 ∗ ((𝑎𝑎 − 2) + (𝑏𝑏 − 2) + (𝑐𝑐 − 2))

(6b)

𝑛𝑛𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 = (𝑎𝑎 − 2)(𝑏𝑏 − 2)(𝑐𝑐 − 2)/2

(6d)

𝑛𝑛 = 𝑛𝑛𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 + 𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝑎𝑎𝑎𝑎𝑎𝑎/2

(6e)

with the constraint (Eq. (6e)):

The fragment energy for each fragment type can be solved for (or fit by a least-squares

method) from a set of linear equations that represents the energy partition of the cubic nanoparticles. The predicted total energy (in Hartree), NCE (in kcal/mol), and surface energy density (SED, in kcal/mol per CaO) for the bulk, face, edge and corner fragments are shown in Table 3. According to the fragment-based model, NCEelectronic(n→∞) is equal to NCEbulk, which is found to be 158.8 kcal/mol. NCEenthalpy,298(n→∞) is then estimated to be 157.8 kcal/mol, which is in excellent agreement with the bulk value of 156.5 kcal/mol. The surface energy densities (SEDs, in kcal/mol per formula unit) were predicted to be ~62 kcal/mol for the 3coordinate corner fragment, ~ 30 kcal/mol for the 4-coordinate edge fragment, and ~11 kcal/mol 21 ACS Paragon Plus Environment


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Table 3. Calculated electronic energy (in Hartree), NCE (in kcal/mol) and SED (in kcal/mol per CaO) for fragment types in the cubic (CaO)n and (MgO)n fragment

corner

edge

Face

bulk

Cubic (CaO)n Eelectronica

-752.911957

-752.963465

-752.993530

-753.011007

NCEb

96.6

128.9

147.8

158.8

SEDc

62.2

29.8

11.0

0.0

Cubic (MgO)n Eelectronicd

-275.384838

-275.450931

-275.475787

-275.485474

NCEe

104.8

146.3

161.9

168.0

SED

63.2

21.7

6.1

0.0

a

For (CaO)n, energies are at the ωB97XD/cc-pVDZ level.

b

Fragment normalized clustering energy NCEfragment = Eelectronic(monomer) - Eelectronic(fragment). Eelectronic(monomer) is the calculated electronic energy for the CaO monomer at ωB97XD/ccpVDZ level, -752.7579885 Hartree. c

Fragment surface energy density SED(fragment) = Eelectronic(fragment) - Eelectronic(bulk).

d

For (MgO)n, energies are at the B3LYP/DZVP level.

e

Fragment normalized clustering energy NCEfragment = Eelectronic(monomer) - Eelectronic(fragment). Eelectronic(monomer) is the calculated electronic energy for the MgO monomer at B3LYP/DZVP level, -275.217786 Hartree.

for the 5-coordinate face fragment. These can be directly compared to the predicted SEDs for the corner, edge, and face fragments for (MgO)n of ~63, ~22, and 6 kcal/mol, respectively. This suggests that the 4-coordinate and 5-coordinate Ca sites are less relaxed in (CaO)n than the


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analogous Mg sites in (MgO)n. The high surface energy densities of the corner fragments of (CaO)n and (MgO)n suggest that these sites are highly reactive. The NCE for the cubic a×b×c (CaO)n has the following relationship with fragment NCEs and SEDs (Eq. (7)): 𝑁𝑁𝑁𝑁𝑁𝑁(𝑛𝑛) = 𝑁𝑁𝑁𝑁𝐸𝐸𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 −

𝑛𝑛𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛

𝑆𝑆𝑆𝑆𝐷𝐷𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 −

𝑛𝑛𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑛𝑛

𝑆𝑆𝑆𝑆𝐷𝐷𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 −

𝑛𝑛𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑛𝑛

𝑆𝑆𝑆𝑆𝐷𝐷𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐

(7)

The NCE for the most energetically favorable a×a×a (CaO)n can be fit by a cubic function n-1/3 (Eq. (8a)) in kcal/mol: 2

1

𝑁𝑁𝑁𝑁𝐸𝐸𝐶𝐶𝐶𝐶𝐶𝐶,𝑎𝑎×𝑎𝑎×𝑎𝑎 (𝑛𝑛) = −26.1𝑛𝑛−1 − 56.0𝑛𝑛−3 − 52.8𝑛𝑛−3 + 158.8

(8a)

as compared to the fit NCE for the most energetically favorable a×a×a (MgO)n (Eq. (8b)): 2

1

𝑁𝑁𝑁𝑁𝑁𝑁𝑀𝑀𝑀𝑀𝑀𝑀,𝑎𝑎×𝑎𝑎×𝑎𝑎 (𝑛𝑛) = −68.5𝑛𝑛−1 – 69.8𝑛𝑛−3 – 29.4𝑛𝑛−3 + 168.0

(8b)

The coefficient for n-1/3 in Eq. (8a) is about twice the coefficient for n-1/3 in Eq. (8b) showing that the slope for NCE for (CaO)n is approximately twice that for the NCE for (MgO)n slope at small n-1/3 (large n) in the NCE vs. n-1/3 plot. This suggests that, at the same n, (MgO)n is likely to be more stable relative to the bulk than (CaO)n relative to the bulk as (MgO)n is closer to the bulk than is (CaO)n. The plot for 𝑁𝑁𝑁𝑁𝐸𝐸𝑎𝑎×𝑎𝑎×𝑎𝑎 (𝑛𝑛) as a function of n-1/3 is shown in Figure 4(c). It shows

that the NCE for a cubic CaO/MgO nanoparticle with a diameter of 10 nm is very close to the bulk limit. The numerical NCE equations for different 1-D (CaO)n nanoparticles series (Figure 2) were also obtained by being fit as cubic functions of n-1/3 using data points generated using Eq. (7), and the resulting equations are plotted in Figure 4(d). Figure 4(d) shows excellent agreement

between the numerical NCE equations and the direct calculated NCE values. In general, the NCE predicted by the numerical equations are within 0.2 kcal/mol of the directly calculated values for n > 10. The only exception is the 3×3×m case, where the numerical equation predicts NCEs



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that are larger by 0.8, 0.9 and 0.6 kcal/mol for the 3×3×4, 3×3×6, and 3×3×8 cube, respectively. The reason is that the two faces perpendicular to the C4 axis have Ca:O ratios of 4:5 and 5:4, and the local overpopulation on the surface gives rise to the relative instability. Such an effect is not observed for the 3×5×m nanoparticles series, where the corresponding Ca:O ratios are closer to 1 (7:8 and 8:7). In addition, the coefficients of the n-1/3 and n-2/3 terms are both close to zero (|coeff| < 10-5) in all of the numerical equations, and the NCE for the 1-D (CaO)n nanoparticle series is effectively a linear function of n-1. This is consistent with the results in Figure 2. By using the best available heat of formation of gaseous CaO, the heats of formation of (CaO)n can be calculated from the NCE of enthalpy at 298 K and these are given in Table 1. We expect that the values for n = 2 – 8 to be the most reliable as these are based on the CCSD(T) energies. Electronic Structures We found that several electronic properties including the molecular orbital structure and the HOMO-LUMO gap (highest occupied molecular orbital – lowest unoccupied molecular orbital) of the cubic a×b×c (CaO)n are dependent on the parity of number of atoms in each dimension (a, b, and c). Let parity (P) of an edge be 1 if the number of atoms on that edge is odd, and 0 if the number of atoms on that edge is even. We define a parity index (PI) for the cubic (CaO)n as the sum of the parities in all three dimensions (Eq. (11)): 𝑃𝑃𝑃𝑃 = 𝑃𝑃(𝑎𝑎) + 𝑃𝑃(𝑏𝑏) + 𝑃𝑃(𝑐𝑐)

(11)

For example, (CaO)32 (4×4×4) has a PI = 0, and (CaO)27 (3×3×6) has a PI =2. For the cubic (CaO)n, PI can only be 0, 1, or 2. PI = 3 is not allowed as it leads to the cubic structure with inequivalent total numbers of Ca and O atoms (CanOn+1 or Can+1On). Different values of the PI


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are associated with structures (Figures 5(a), 5(b), and 5(c)) with different symmetries and local geometries.

Figure 5. Geometry comparison between (a) a type-0 (4×4×4), (b) a type-1 (3×4×4), and (c) a type-2 (3×3×4) cubic (CaO)n.

Therefore, the cubic (CaO)n can be classified into three categories: type-0 with PI = 0, type-1 with PI = 1, and type-2 with PI = 2. The type-0 cube possesses D2d or higher symmetry, and hence has a dipole moment equal to zero. The 4 corner Ca atoms of the type-0 cube form a tetrahedron, as do the 4 corner O atoms. The type-1 cube has C2h symmetry, and also has a dipole moment equal to zero. The 4 corner Ca atoms and 4 corner O atoms of the type-1 cube form two rectangles cutting through the cube. The type-2 cube has C2v or higher symmetry (C4v), with a non-zero dipole moment perpendicular to the odd×odd faces. For the a×b×c cubes with a and b being fixed odd numbers, and c being an even number, the magnitude of the dipole moment scales approximately linearly with c. In the type-2 (CaO)n cube, odd×odd layers have unequal numbers of Ca and O, with Δ= ±1. The 4 corner Ca atoms of the type-2 cube are the 4



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corners of the Ca-rich odd×odd face, and the 4 corner O atoms are the 4 corners of the opposite O-rich odd×odd face. In the type-0 cubic (CaO)n clusters, with (CaO)32 (4×4×4) as an example, the HOMO and LUMO are localized on the corner O atoms and on the corner Ca atoms, respectively (Figure 6(a)). Additional examples of the HOMO and LUMO for (CaO)40 (4×4×5) are given in Figure 6(b), for (CaO)36 (3×3×8) in Figure 6(c) and for hexagonal tubular (CaO)21 in Figure 6(d). Further partial density of states analysis (Figure 7) shows that the HOMO of (CaO)32 is a localized orbital solely contributed by the 2p orbitals of the corner O atoms, whereas the LUMO is a delocalized orbital in which half of its population is contributed by the 4s orbitals of the 4 bulk Ca atoms despite its location (Figures 7(a), 7(b), and 7(c)). The HOMO and LUMO are spatially separated for (CaO)32, suggesting charge migration might occur when the nanocluster generates electronically excited states on the absorption of visible/UV radiation. In addition, the low lying unoccupied MOs near LUMO are found to be multi-center hybridized Ca 4s and 4p orbitals, and the unoccupied MOs with orbital energy > 1 eV are mixtures of Ca 4s, 4p and 3d orbitals. In the type-1 cubic (CaO)n clusters, with (CaO)40 (4×4×5) as example, the HOMO and LUMO are localized on the corner O atoms and on the corner Ca atoms, respectively (Figure 6(b)). The HOMO is a localized MO, dominated by O 2p orbitals or the corner O atoms, whereas the LUMO is a delocalized MO contributed by all of the Ca atoms. The LUMO density on each corner Ca is an sp hybrid orbital, consistent with its composition of ~60% Ca 4p and ~40% Ca 4s. The two lobes (with different phases) of the Ca 4p densities involved in the LUMO are separated: one lobe is localized on the source Ca and the other lobe is delocalized on the corner Ca. The overall DOS of type-1 cubic (CaO)n clusters are very similar to the DOS of the type-0 cubic 26 ACS Paragon Plus Environment

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(CaO)n clusters. The HOMO energy level is found to be approximately -4 eV and the LUMO energy level is ~ -1 eV for the (CaO)n type-0 and type-1 clusters.

Figure 6. Molecular orbital iso-surfaces for the cubic (a) (CaO)32 (4×4×4), (b) (CaO)40 (4×4×5), (c) (CaO)36 (3×3×8), and (d) the hexagonal tubular (CaO)21. 27 ACS Paragon Plus Environment


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Figure 7. Density of states analysis at the B3LYP/cc-pVDZ level. (a) DOS and PDOS projected to AOs for (CaO)32. (b) PDOS and gross MO occupancies for the corner and bulk atoms in (CaO)32. The PDOS (curves) are normalized to show the values per atom; and the gross MO occupancies (vertical lines ∈ [0, 1] in the lower panels) are not normalized. (c) PDOS and gross virtual MO occupancies projected to Ca atomic orbitals for the corner and bulk atoms in (CaO)32. The PDOS and gross virtual MO occupancies (vertical lines in the lower panels) are composed of 4 corner Ca’s, 12 edge Ca’s, 12 face Ca’s and 4 bulk Ca’s. (d) comparison between DOS’s of (CaO)40 and (MgO)40.

In the cubic type-2 (CaO)n clusters, with (CaO)36 (3×3×8, Figure 6(c)) and (CaO)30 (3 ×4×5, Figure S4 in SI) as an example, the HOMO density is mainly localized on the corner O atoms with trace amounts of density on other O atoms of the odd×odd×2 surface double layer , i.e. the odd×odd O-rich face and the layer next to it. A PDOS analysis shows that the HOMO and the low-lying occupied orbital ranging from -3.5 to -4.3 eV are dominated by the O 2p orbitals of the corner O’s (Figure S5 in SI). These MO levels are separated from the rest of the “valence band” (< -4.6 eV); hence the HOMO energy level of the type-2 cubes are higher than the HOMO level of the type-0 and type-1 cubes. The LUMO density is mostly localized on the corner Ca atoms as indicated by the MO isodensity diagram, and very little density is found on other Ca atoms of the odd×odd×2 surface double layer (the Ca-rich face and the layer next to it) as well as on the bulk Ca atoms. The PDOS analysis indicates that the LUMO and the nearby low-lying MOs are delocalized with the density contributed by the 4p orbitals (>70%) of all the Ca’s except for the Ca’s in the opposite double layer (the O-rich face and the layer next to it). The Ca atoms in the odd×odd×2 surface double layer with the Ca-rich face and the bulk Ca 30 ACS Paragon Plus Environment

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atoms contribute most to the LUMO and the lowest-lying MOs in terms of contribution per Ca. The lowest-lying unoccupied MOs are separated from the rest of the “conducting band” (> -0.5 eV); consequently, the LUMOs of the type-2 cubic (CaO)n clusters are lower in energy than the LUMO levels of the type-0 and type-1 cubic (CaO)n clusters. The HOMO-LUMO gaps of the type-2 (CaO)n are expected to be smaller than the HOMO-LUMO gaps of the type-0 and type-1 (CaO)n due to the type-2 having a higher HOMO and lower LUMO. This result is interesting as the three types of cubic clusters have essentially the same thermodynamic stability at the comparable sizes and aspect ratios, yet will have different reactivities. The HOMO for the hexagonal tubular (CaO)n clusters as shown for (CaO)21 is distributed over all of the surface O atoms (Figure 6(d)). It 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 Ca and -2 on O. As expected, the LUMO is essentially composed of 4s orbitals from the Ca atoms on the surface. The differences in the electronic structures of the three types of cubic (CaO)n are also shown by the electrostatic potentials (ESPs) for each type (Figure 8). The dominant electron and

Figure 8. Electrostatic potential isodensities (0.006) for the cubic (CaO)n. 31 ACS Paragon Plus Environment


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hole binding sites are on the corner Ca’s and corner O’s respectively for the type-0 and type-1 (CaO)n. In addition, in the ESP of the type-1 (CaO)n, the two Ca’s and two O’s at the medians of the four odd-edges (also on the mirror plane) are found to be the second-best binding sites for electrons and holes, respectively. The ESP analysis suggests that the oxygen atoms on the corners of the surface of the type-0 and type-1 cubic clusters, which are the lowest coordinated O2- sites, can serve as Lewis base sites. We suggest that these corner sites should be the sites that are protonated by Brönsted acid/base reactions. The Ca atoms at the corners of the type-0 and type-1 cubic clusters 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. The ESP for the type-2 (CaO)n is essentially bipolar and is split into three regions by two hyperbolic planes: a positive region near the surface double layer with the Ca-rich odd×odd face, a negative region near the surface double layer with the O-rich odd× odd face, and a region consisting of the layers in between with alternate positive and negative potentials. It is suggested both odd×odd×2 surface double layers are charge-deficient. In the type-2 cubic clusters, the O sites on the odd×odd×2 surface double layer with the O-rich odd×odd face are suggested to be the Lewis base sites that can be protonated by Brönsted acid/base reactions, and the Ca sites on the odd×odd×2 surface double layer with the Ca-rich odd×odd face can serve as the Lewis acid sites in the Brönsted acid/base reactions. The corner O and Ca sites are plausibly the most active sites due to their high surface energy density. In comparison with cubic (MgO)40 (Figure 7(d)), (CaO)40 has a higher HOMO (-4.66 eV vs. -5.65 eV) and lower LUMO (-0.97 eV vs. -1.77 eV) in terms of orbital energy at the B3LYP/cc-pVDZ level. The HOMO-LUMO gap of (CaO)40 is 0.19 eV smaller than the HOMO32 ACS Paragon Plus Environment

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LUMO gap of (MgO)40. This is consistent with the fact that Ca has lower first and second ionized potentials (IP1 = 6.11 eV and IP2 = 11.87 eV) 46,47 than Mg (IP1 = 7.65 eV and IP2 = 15.04 eV). 48 The calculated HOMO-LUMO gap (Eg) for (CaO)n is DFT-functional-dependent (see the benchmark results in the Supporting Information), whereas the first excitation energy was calculated using TD-DFT 49 , 50 , 51 (TD-B3LYP/D and TD-ωB97XD) is less dependent on the choice of the DFT functional than the calculated HOMO-LUMO gaps at the DFT/D levels. This is consistent with the previous benchmark study by Zhang and Musgrave on determining Eg using the DFT and TD-DFT methods, 52 where the authors show that the first excitation energies from TD-DFT calculations are in good agreement with the experimental Eg values. Among the benchmarked DFT functionals, B3LYP values show reasonable agreement with the TD-DFT results. We thus used B3LYP/cc-pVDZ for the molecular orbital population analyses including the density of states analysis. The calculated first excitation energies for the lowest energy (CaO)n and cubic (CaO)n are shown in Table S3 and Table S4 in SI, respectively. The HOMOLUMO gap for the lowest energy (CaO)n cluster generally decreases as n increases, and the energy gap rises to ~3.3 eV for cubic (CaO)40, as compared to the experimental band gap value of 7.8 eV for the bulk.4,5 One reason for this difference in the computed gaps for the nanocluster and the value for the bulk 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. There exist 3n occupied molecular orbitals from O p orbitals for a given (CaO)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 more stable than the HOMO, so the interior atoms have stabilized orbital energies. The corner O 2p occupied orbitals more resemble lone pairs, and the interior O 2p 33 ACS Paragon Plus Environment


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orbitals more resemble bonding orbitals. As the interior atoms have more stable high energy MOs, the energy level of the valence band edge in the bulk is expected to be lower than the HOMO in the nanoclusters. On the other hand, the LUMO and low-lying unoccupied MOs are delocalized, and the main contributions come from the either the interior Ca atoms or Ca atoms on the surface double layer, depending on the types associated with the PI. The energy level of the conduction band edge in the bulk is expected to be comparable to the low lying unoccupied MOs that are mainly contributed by the bulk Ca atoms in the nanoclusters or slightly lower (due to the widening of the band as cluster size increases). For the small (CaO)n clusters, the analogue of the band gap (of the bulk) is the excitation energy from low-lying Ca-O bonding MOs to delocalized low-lying unoccupied MOs, rather than the first excitation energy from the HOMO (O lone pairs) to the LUMO. The calculated Eg for most of the lowest energy small (CaO)n clusters fall into the ultra-violet region, and it is likely that photochemistry will play a role in the reactivity of the cluster. The plot for the calculated first excitation energies (Eg) for the cubic (CaO)n as a function of n-1 is shown in Figure 9. For the type-0 and type-1 cubic (CaO)n, Eg is essentially invariant to n. If we grow the a×b×c type-0/type-1 cubic nanoparticle layer by fixing a and b of a cube and increasing c only, Eg increases slightly as c increases if only the type-0/type-1 structures in the growth products are considered. The predicted infinite-chain Eg values (at c →∞) for the type-0 and type-1 cubic (CaO)n nanoparticle series range from 3.3 to 3.4 eV, although at the infinitechain limit Eg is mostly meaningless because of the negligible corner/edge ratio. The Eg for the type-2 cubic (CaO)n is found to be much smaller than the Eg of the type-0/type-1 (CaO)n for the small clusters. If we fix the two odd dimensions (let a and b be odd numbers) of the type-2 (CaO)n and increase the even dimension (let c be an even number) only, Eg will decrease as c 34 ACS Paragon Plus Environment

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increases if the odd×odd faces are small enough; Eg becomes essentially a linear function of n-1 when c is not too small. The infinite-chain limit for the Eg of the 3×3×c is estimated to be ~0.1 eV from the linear extrapolation of Eg(n-1), and is found to be the lowest among all of the 1-D cubic CaO nanoparticle series. The infinite-chain limits for the Eg’s of 3×5×c and 5×5×c are estimated to be 2 eV and 3 eV respectively. For the type-2 cubic (CaO)n nanoparticle series with larger odd×odd faces, their Eg’s are expected to be comparable with the Eg’s of the type-0/type1 nanoparticle series, as the charge deficits on the odd×odd faces that result in the smaller HOMO-LUMO gap become smaller.

Figure 9. Calculated first excitation energy (Eg) as a function of n-1 at the TD-B3LYP/cc-pVDZ level for the cubic (CaO)n.



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The variance of Eg during a minimal 1-D (layer-wise) growth step is depicted in Figure 10 for the cubic (CaO)n clusters. A minimal 1-D growth step is defined for the growth step either grow a layer on the smaller particle, or grow two layers if the monolayer growth is forbidden. The 1-layer growth step can be: 0→1, 1→0, 1→2, and 2→1 growth step, in terms of the variation in PI. The 2-layer growth step refers to the 2→2 growth step (i.e., odd×odd×even→even×odd×(even+2)), as the 2→3 1-layer growth (i.e., odd×odd×even→ odd×odd×(even+1)) is forbidden. We found that, for the small clusters, Eg remains approximately the same during the 0→1 and 1→0 1-D minimal growth steps. Eg slightly decreases during a 1→2 growth step, and slightly increases during a 2→1. A 2-layer growth step composed of consecutive 1→2 and 2→1 steps will result in a net change of ~0 in Eg. The 2→2 growth step reduces Eg, and the decrement decreases as particle size increases. Consequently, for different 1-D cubic nanoparticle series, the variation patterns of Eg as the particle grows longer also differ. The growth of an even×even×c 1-D nanoparticle series yields type-0 and type-1 products alternately, and the corresponding alternate 0→1 and 1→0 growth step will result in a smooth, nearly flat Eg vs. n-1 curve. The growth of an even×odd×c 1-D nanoparticle series yields type-1 and type-2 particles alternately; since consecutive 1→2 and 2→1 steps gives approximately zero net change in Eg, the upper bound of Eg vs. n-1 becomes nearly constant and the lower bound of Eg vs. n-1 is damped as n-1 decreases (or as n increases). The growth of an odd×odd×c 1-D nanoparticle series only yields type-2 particles, and Eg decreases monotonically as n-1 decreases (or as n increases). Such conclusions may be used to monitor the bottom-up synthesis of ultra-small cubic CaO nanoparticles, to better understand such processes.


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Figure 10. Variation of first excitation energy (Eg) at TD-B3LYP/cc-pVDZ level during the minimal 1-D growth steps for the cubic (CaO)n. Green digits denote the types of the cubic (CaO)n clusters; values in red are the calculated first excitation energy at the TDB3LYP/cc-pVDZ level; arrows indicate the minimal 1-D growth steps. 37 ACS Paragon Plus Environment


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The Eg’s dependence on PI is also found for (MgO)n, as shown by the EHOMO-LUMO vs. n-1 plot in Figure S6 for (MgO)n. The 3×3×c cubic nanoparticle series of (MgO)n also has the HOMO-LUMO gap monotonically decreasing as c increases. The early decrements in Eg during the 1-D growth of the 3×3×c (MgO)n are found to be greater than the corresponding values for the 3×3×c (CaO)n nanoparticle series. The calculated dipole moments for the 3×3×c (MgO)n and 3×3×c (CaO)n were found to be comparable for a given c; thus, it seems that the charge density of (MgO)n is more affected by the dipole moment than the charge density of (CaO)n. Since the Eg of the 3×3×c (MgO)n decreases more rapidly than does the Eg of the 3×3×c (CaO)n, at the infinite-chain limit, a partial flipping of the occupied and unoccupied orbitals might occur, so some of the low-lying Ca unoccupied orbitals will become occupied and some of the high-lying O occupied orbitals will become unoccupied. It is potentially even more interesting to investigate the electronic structure at the critical point where the flipping starts to occur; the valence electron will migrate from O-rich face to the Ca-rich face or vice versus through the very long nanorod at a very small bias potential. The uniqueness of the HOMO-LUMO gap evolution of the 3×3×c cubic is due to locally overpopulated ions in each layer and the particle’s overall symmetry; therefore, we expect similar phenomena occur for other types of cubic metal oxide nanoparticles. Unlike the cubic (CaO)n cases, where almost all of the 1-D (except for the thin 1-D odd×odd×c series) series converges to an infinite-chain limit in the narrow range from 3.3 to 3.4 eV in terms of Eg, the infinite-chain limits of the 1-D cubic (MgO)n nanoparticle series have a broader range from 3.5 eV to 4.5 eV for the even×odd×c and even×even×c series. The cubic clusters and ultra-small nanoparticles of (MgO)n and (CaO)n exhibit a smaller HOMO-LUMO gap than do the corresponding bulk materials. This may be utilized in photocatalysis processes such as the heterogeneous water splitting. The HOMO and LUMO 38 ACS Paragon Plus Environment

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energy levels for (MgO)n and (CaO)n are examined against the thermodynamic potentials for the H+/H2 reduction and O2/H2O oxidation in the heterogeneous water splitting (Figure 11). Based

Figure 11. Calculated HOMO and LUMO energy levels (in eV) for the cubic (MgO)n (at B3LYP/DZVP) and (CaO)n (at B3LYP/cc-pVDZ) compared to the estimated thermodynamic potentials for heterogeneous water splitting.

on the values of Chen and Wang, 53 the H+/H2 threshold potential is -4.5 eV and the O2/H2O threshold potential is -5.8 eV. Among the examined cubic nanoparticles, very thin 1-D (MgO)n (2×2×c, 2×3×c, and 2×4×c) series have a low enough “VBM” and high enough “CBM” to split 39 ACS Paragon Plus Environment


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H2O by itself, but the efficiency for water splitting is expected to be low, because the “VBM” is very close to the O2/H2O threshold. However, due to the fact that many viable metal oxides for water splitting have a VBM much lower than the O2/H2O threshold and are limited by their CBM close to the H+/H2 threshold, these 1-D (MgO)n nanoparticles may be used with bulk oxides such as TiO2 and WO3 to yield the ideal band structure for photocatalytic water splitting. 54 The majority of the cubic (CaO)n clusters as well as the 3×2×c (MgO)n clusters would then qualify as photocathodes in photocatalytic water splitting, but the performance is expected to be low due to their HOMO-LUMO gap being relatively wide leading to low photon-harvesting efficiency for visible photons. Many members of the 3×3×c (CaO)n series have the desired HOMO-LUMO gap, but they are not qualified photocathodes due to their “VBM” higher than the H+/H2 threshold potential. Among the 3×3×c (MgO)n series, the 3×3×4 cluster appears to be the only decent candidate for the photocathode, with a HOMO-LUMO gap of 2.81 eV and a “VBM” of -5.07 eV. Conclusions We used our prior work on (MgO)n nanoclusters to predict the lowest energy structures for (CaO)n nanoclusters. The most stable (CaO)n nanoclusters (n > 3) are dominated by cubic structures. (CaO)n clusters with an aspect ratio close to 1:1:1 are predicted to be the most energetically favorable isomers when n can be factorized into three nearly equal integers. The fragment-based structure-energy model provided sound fitting results for the total energy of a large gallery of cubic (CaO)n structures, and also predicted the bulk-limit of the energetics in excellent agreement with the experimental data. The surface energy densities for the fragment types in the cubic (CaO)n are predicted to be 62.2 kcal/mol per f. u. for the corner fragment type, 29.8 for the edge fragment type, and 11.0 for the face fragment type. The comparison between


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the fragment surface energy density and NCE convergence of the (CaO)n and (MgO)n suggests (CaO)n is likely to be more reactive than (MgO)n at the same n. Based on the parities of the edges in three dimensions, the cubic nanoparticles are classified into three types. Although there is no distinct difference between the stabilities for the three types of cubic nanoparticles of the comparable sizes and aspect ratios, their electronic properties such as the DOS and Eg may differ. For the clusters and ultra-small nanoparticles, the type-2 cubic (CaO)n and (MgO)n show unique electronic properties; our calculations show that such clusters possess well-separated charge deficient sites and have a much narrower HOMOLUMO gap. Unlike type-0 and type-1 cubic (CaO)n and (MgO)n clusters for which Eg is nearly invariant to the increase of the particle length during the 1-D growth, the Eg for a thin type-2 cubic (CaO)n and (MgO)n decreases as the particle length increases. The type-2 3×3×m cubic nanoparticle series is predicted to have the lowest Eg at the infinite-chain limit (at m → ∞) among all of the 1-D cubic (CaO)n and (MgO)n nanoparticle series. The location of the HOMO and LUMO suggest that Lewis acids will bond to the oxygen and that Lewis bases will bind to the calcium as expected. The most active atoms will be those on the corners. Our results suggest that it would be interesting to synthesize (XO)n nanoclusters such as (CaO)n and (MgO)n 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. Acknowledgement This work was supported by the U. S. Department of Energy (DOE), Office of Basic Energy Sciences Geosciences program under a subcontract from Pacific Northwest National Laboratory. 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 41 ACS Paragon Plus Environment


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Environmental Research, and located at PNNL.

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PNNL is operated for DOE by Battelle

Memorial Institute under Contract # DE-AC06-76RLO-1830. M. C. gratefully acknowledges funding from the Beijing Computational Science Research Center and National Natural Science Foundation of China (Grant No. U1530401) and computational resources from the Beijing Computational Science Research Center. D.A.D. also thanks the Robert Ramsay Chair Fund of The University of Alabama for support. Supporting Information Complete author lists for refs. 39 and 40. Figures: deviation of NCEs calculated at different theoretical levels as compared to CCSD(T)/T results; energy corrections (ZPE, thermal, and Gibbs) to NCE as functions of n-1/3; radial pair distribution function plots for the lowest energy (CaO)n, n ≤ 40; molecular orbital isodensity diagrams for 3×4×5 (CaO)30; PDOS diagrams for type-1 and type-2 cubic (CaO)30 clusters (4×4×5, 3×3×8, and 3×4×5); calculated HOMO-LUMO gap for the cubic (MgO)n at the B3LYP/DZVP level; calculated vibrational spectra for low energy structures for (CaO)n. Tables: benchmarks for NCE at the DFT/D levels with CCSD(T)/T values; comparison of Eg values at various DFT and TDDFT levels for (CaO)8; calculated Eg at the TD-B3LYP/D level for the lowest energy (CaO)n; calculated Eg at the TD-B3LYP/D level for the cubic (CaO)n; and the optimized Cartesian coordinates in angstrom and electronic energies in a.u. at the B3LYP/DZVP level for the lowest energy (CaO)n. This material is available free of charge via the Internet at http://pubs.acs.org.

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