A DFT Study Combined with the

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C: Physical Processes in Nanomaterials and Nanostructures 2

Physical and Chemical Properties of Unsupported (MO)n Clusters for M = Ti, Zr, Ce, and n = 1 - 15: A DFT Study Combined with the Tree-Growth and Euclidean Similarity Distance Algorithm Larissa Zibordi-Besse, Yohanna Seminovski, Israel Rosalino, Diego Guedes-Sobrinho, and Juarez L. F. Da Silva J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b08299 • Publication Date (Web): 06 Nov 2018 Downloaded from http://pubs.acs.org on November 7, 2018

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The Journal of Physical Chemistry

Physical and Chemical Properties of Unsupported (MO2)n Clusters for M = Ti, Zr, Ce, and n = 1-15: A DFT Study Combined with the Tree-Growth Scheme and Euclidean Similarity Distance Algorithm Larissa Zibordi-Besse,† Yohanna Seminovski,† Israel Rosalino,† Diego Guedes-Sobrinho,†,‡ and Juarez L. F. Da Silva∗,† †São Carlos Institute of Chemistry, University of São Paulo, PO Box 780, 13560-970, São Carlos, SP, Brazil ‡Department of Physics, Technological Institute of Aeronautics, 12228-900, São José dos Campos, São Paulo, Brazil E-mail: juarez [email protected]

Phone: +55 16 3373 6641. Fax: +55 16 3373 9952

Abstract

the differences between the asymptotic (MO2 )n values and the bulk values are due to the surface and relaxation effects. We found very similar increasing in the binding energy with increased n for both systems, in particular for large n values, which is associated with an increasing in coordination of the core-atoms towards the bulk values, while the magnitude of the binding energy is largely determined by the ionic contribution due to the charge transfer among the cation and oxygen atoms. From the relative stability function, the most stable clusters pGMC pGMC pGMC are the (TiO2 )6 , (ZrO2 )8 , and (CeO2 )10 . As expected, from the density of states, we found discrete energy levels for smaller n, which forms the valence and conduction bands separated by an energy gap for large n values, and hence, the evolution of the HOMO-LUMO energy separation was obtained for the studied metal-oxides.

Metal-oxide clusters, (MO2 )n , have been widely studied along the years by experimental and theoretical techniques, however, our atomistic knowledge is still far from satisfactory for systems such as ZrO2 and CeO2 , which plays a crucial role in nanocatalysis. Thus, with the aim to improve our atomistic understanding of the physical and chemical properties of the metal-oxide clusters as a function of size, n, we performed a systematic ab initio density functional theory (DFT) study of the (MO2 )n clusters, where M = Ti, Zr, Ce, n = 1 − 15. In this work, the trial atomic configurations were obtained by a tree-growth (TG) scheme combined with the Euclidean similarity distance (ESD) algorithm. Using the (TiO2 )n clusters, we validated the TG-ESD algorithm, which found the same putative global minimum configurations (pGMCs) reported in the literature for most of the (TiO2 )n systems, and in few cases, there are lower energy configurations than previous data. From our analyses, the structural parameters of the (MO2 )n clusters show an asymptotic behavior towards the values obtained from the non-optimized bulk fragments, and hence,

Abbreviations DFT, PBE, KS, FHI-aims, pGMC

ACS Paragon Plus Environment

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I Introduction

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and trimmed cubes were proposed as one of the ideal morphologies for the ceria nanoparticles. 22 In addition, it was found a direct proportion between non-stoichiometric geometries and an increasing in average coordination number of the Ce atoms, for large nanoparticles. 23–25 Reduced CeIII were found at low-coordinated sites in stable cuboctahedral nanoparticles, within the clusters’ morphology being confirmed by experimental findings. 10–12 The energy required for the formation of oxygen vacancies in the surface was compared with respect their growing in a particle, 25,26 showing that in the reduced structure, unlike of the surface, O vacancies are localized at lower coordinated sites with the formation energy smaller for larger cluster sizes. The Ti, Zr, and Ce atoms belong to the 3d, 4d, and 4 f electrons systems, and hence, a comparative study of the (MO2 )n clusters can contribute to improve our atomistic understanding of the mechanisms that drive the structural formation of the those clusters, however, it is a challenge as the atomic structure of small metal-oxide clusters have not been well known in the literature, except for (TiO2 )n . Therefore, with the aim to contribute to this field, we will report a theoretical study based on ab initio density functional theory (DFT) calculations for the (MO2 )n clusters, with n = 1 − 15 and M = Ti, Zr, and Ce. Thus, we studied the evolution the physical and chemical properties as a function of size, which opens the possibility to understand the behavior of the properties and the effects that arise at this size-limit. To achieve those goals, our trial atomic configurations were based on the tree-growth scheme combined with the Euclidean similarity distance algorithm. 27

Particles of metal-oxides, MOx , with diameters from subnano up to several nanometers, have been extensively studied along the years by experimental and theoretical techniques with the aim to obtain an atomistic understanding of the mechanisms that drive the structural formation of MOx clusters, 1–5 which plays a crucial role in the physical and chemical properties ox metal-oxides at the nanosize regime. 5–7 Among a large number of metaloxides, the TiO2 , 1,8,9 ZrO2 , 2,3,5 and CeO2 , 10–13 systems have attracted great interest, which can be explained by their wide use in technological applications, e.g., support for transition-metal (TM) particles in catalysis, 12–14 candidates for economic and environmental important chemical reactions such as water splitting, 15,16 CO2 conversions, 14,17 etc. Although metal-oxides have been widely studied, our atomistic understanding of the structural formation of their respective clusters is far from satisfactory, which will be discussed below. Titania clusters have been widely studied from the theoretical and experimental point of view. 1,6,8,9 Infrared spectroscopy data reveal that partially reduced cluster structures are close to the rutile formation, with a strong band of vibration around 740 cm−1 . 8 Anionic (TiO2 )n clusters generated from photo-electron spectroscopy are compared with theoretical calculations, wherein the charge induces a different geometrical stability between anionic and neutral clusters. 9 Furthermore, there is a tendency for the surfaces of the most stable titania clusters to present mono-coordinated bonds (Ti – O), while high-energy isomers have higher-coordinated oxygen atoms. 1,6,18,19 Even knowing the importance of different oxides, especially those involving group IV, studies on zirconia and ceria clusters are still far from achieving a well-established knowledge of these materials as achieved for (TiO2 )n . Using inter-atomic potentials, it was found that zirconia clusters present structures previously reported for titania, 3 which can be associated with the bulks that are stable for ZrO2 , namely, monoclinic baddeleyite, cubic fluorite, and tetragonal. 20 The inter-atomic potentials were also used for the description of larger clusters of ceria, n = 50, which achieved the fluorite structure, 21 while a force field was recently designed

II Theoretical Approach and Computational Details A

Total Energy Calculations

Our ab initio calculations were based on spinpolarized DFT within the semi-local exchangecorrelation (XC) energy functional formulated by Perdew–Burke–Ernzerhof (PBE). 28 For the stoichiometric CeO2 systems, which includes bulk, surfaces, nanoparticles, and clusters, the occupied and


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unoccupied Ce4+ f -states show a delocalized behavior (itinerant electrons), which can be described by the semilocal PBE functional. 29 However, it is not the case for partially reduced CeO2 systems, where the occupied Ce4+ f -states change their behavior from delocalized to localized one, i.e., the oxidation state changes to Ce3+ . The Ce3+ cations can occur due to the presence of oxygen vacancies 30,31 or even by the presence of transition-metal adatoms on CeO2 (111). 32,33 Here, we will consider only stoichiometric (MO2 )n clusters, and hence, all the DFT calculations were performed using the semilocal PBE functional. 28–31 To solve the Kohn–Sham (KS) equations, we employed the all-electron and full-potential Fritz– Haber Institute ab initio molecular simulations (FHI-aims) package, 34,35 from which the electronic solution is obtained through the atomic scalarrelativistic framework within the zeroth-order relativistic approximation (atomic-ZORA). 36 At the FHI-aims package, the KS orbitals are expanded in numerical atom-centered orbitals, which were hierarchically built from minimal basis set on up to meV accuracy. Here, we employed the second improvement level, namely, the light-tier2 basis set (using the FHI-aims terminology). 34,35 The electronic self-consistency was achieved once the total energy and atomic forces differences between 3 consecutive cycles reached values smaller than 10−6 eV and 10−3 eV/Å, respectively. Also, with the aim to avoid fractional occupation of the electronic states near the highest occupied molecular orbitals (HOMO), we used a Gaussian broadening parameter of 0.010 eV. The equilibrium geometries were obtained once the atomic forces on each atom were smaller than 10−2 eV/Å using the modified Broyden–Fletcher–Goldfarb– Shanno (BFGS) algorithm as implemented at the FHI-aims package. 37,38 For analyses of the oxides clusters, we calculated the vibrational frequencies using finite displacements of ±0.0025 Å in the x,y,z directions for each atom to built up a Hessian matrix, and hence, the zero-point energy was also obtained. The vibrational calculations were performed mainly for the the putative global minimum configurations (pGMCs), and hence, the zero-point energy correction was not included in the study of the relative stability (total energy) among the atomic config-

urations. For particular cases, e.g., (TiO2 )10 , the zero-point energy was considered for few isomers, and from that, we found that configurations with relative total energies smaller than about 5 meV/f.u. are degenerated in energy.

B Atomic Structure Configurations To obtain a reliable description of the physical and chemical properties of the (MO2 )n clusters, M = Ti, Zr, Ce and n = 1 − 15, we employed two schemes to generate the trial configurations for the DFT-PBE calculations, namely, (i) an approach inspired in the tree-growth (TG) scheme 6 combined with the Euclidean similarity distance (ESD) algorithm, 27 (ii) and a second approach based on carved bulklike fragments, which takes directly into account the bulk-like structural features. One of the advantages of the TG-ESD algorithm implemented in our group from scratch, as indicated in Figure 1(a), is the possibility to use previous optimized atomic structures to generate additional trial configurations instead to generate random structures for every atomic size as in the Basin Hopping Monte Carlo, 39,40 however, it has the disadvantage to build up a large sequence of structures to obtain the pGMC for a particular given size, n. 1 Tree-Growth Scheme Combined with the Euclidean Similarity Distance Our implementation of the TG-ESD algorithm is composed by a set of steps, as defined below: 1. A seed, which is a pGMC composed by n pGMC formula units, (MO2 )n , is frozen with its geometric center at the Cartesian origin (0, 0, 0), and a molecular cluster unit, n = 1, is pGMC randomly allocated near to the (MO2 )n particle at a distance of R + dM – O , where R is the cluster radius and dM−O is the M – O bond length. We can generate from 1 up to several 108 configurations for the (MO2 )n+1 cluster. 2. All the generated configurations differ only on the orientation and surrounding position of the molecular fragment, e.g., n = 1, with pGMC respect to the (MO2 )n cluster, and hence, many of the generated configurations might


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Figure 1: (a) Schematic diagram of the tree-growth (TG) scheme combined with the Euclidean similarity distance (ESD), which compose the TG-ESD algorithm employed in this work. (b) Schematic representation of symmetric bulk fragments for the rutile, anatase, baddeleyite, and fluorite structures. The carved clusters are shown by the highlighted atoms. Blue, dark green, light green and red atoms represent titanium, zirconium, cerium and oxygen species, respectively. cept a configuration β when S (α, β) is larger than a given cutoff parameter, otherwise, β is discarded because it is assumed similar enough to α. At the end, the ESD yields a smaller set of configurations, e.g., it can reduces from (n + 1)107 to (n + 1)10 configurations, where n = 1 − 15.

be very similar, which can leads to the same local minimum configurations once they are optimized using a local optimization algorithm such as the BFGS. 37,38 Furthermore, it is prohibitive to perform DFT-PBE calculations for 108 configurations even using the most powerful supercomputers. Thus, to reduce the number of initial configurations for a smaller set, e.g., about 10n, we employed the ESD algorithm, in which every (MO2 )n+1 cluster is converted into a 3 × (n + 1)-dimensional vector (3 atoms in each formula unit and (n + 1) formula units), where the components of the vectors, xi , are the distances of each atom to the geometric center of the (MO2 )n+1 cluster. The ESD algorithm is defined as, N P

S (α, β) =

xi,α − xi,β

i=1 N P i=1

3. Then, all the selected (MO2 )n+1 configurations are optimized using DFT-PBE within local optimization algorithms such as the BFGS. At the end, we have a set of local minimum configurations, while one of the them has the lowest total energy, i.e., it is the pGMC. 4. Once the pGMC is identified for a given n, a new set of structures are generated using the TG scheme, and the steps described above are applied. This process was repeated up to reach n = 15.

2 ,

(1)

2 + x2 xi,α i,β

2 Bulk-Like Fragments

where two configurations, α and β, are compared by 3×(n+1)-dimension vectors, which include the atomic positions of α and β configurations, respectively. Therefore, we ac-

Experimental results have shown that titania nanostructures tend to form anatase-like configurations due to the lower surface free-energy and enthalpy compared to the rutile structure. 41–43 Furthermore,


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instead of the baddeleyite ZrO2 structure (monoclinic), zirconia nanoparticles show tetragonallike features. 44–46 Therefore, to improve the exploration of the potential energy surface of the (MO2 )n clusters, we selected fragments from the bulk structures, namely, rutile, anatase, baddeleyite, and fluorite, as complementary trial configurations, Figure 1(b). Furthermore, we considered also the crossover among the carved (MO2 )n structures and their pGMCs. Additionally, we considered reported configurations in the literature for titania, 6,18,19,47 zirconia, 3,48 and ceria 49 for directly comparisons.

tions for selected trial structures. Thus, due to the large number of calculated configurations, we selected only a set of representative structures using the Euclidean similarity distance to show in Figure 2, while part of the optimized structures (x,y,z coordinates) are reported within the Supporting Information. Several analyses were performed to improve our understanding of the structural, energetics, and electronic properties as a function of the number of formula units, n. For most of the properties, the analyses were performed for all optimized structures and not only for the pGMCs, which is crucial to define the range of values for particular properties, and hence, it allows further comparison with experimental results.

III Performance of the TGESD Algorithm

A

To evaluate the performance of the TG-ESD algorithm without the addition of the trial configurations obtained from bulk-like fragments, we compared the pGMCs obtained for the (TiO2 )n clusters with results reported in the literature, 1,6,18,19,47,50–52 as (TiO2 )n is one of the most studied systems among the selected oxides. We found that the TG-ESD algorithm reached identical pGMCs for n = 1 − 9 and 12. 6,19 For the particular case of n = 11, 13, and 15, the TG-ESD algorithm found new pGMCs, which are lower in energy by 15, 31, and 334 meV/f.u., respectively, than previous results. 6,19 Our strategy could not identify an alternative pGMC for (TiO2 )10 47 and (TiO2 )14 , 19 however, we identified degenerated structures for n = 10 and 14 within our relative total energy accuracy, i.e., 5 meV/f.u., due to missing zero-point energy correction for all configurations. Thus, the TG-ESD algorithm is capable to identify even new pGMCs, which is a challenge task even for sophisticated global optimization algorithms such as the BHMC. 39,40 Further details of the implementation of the TG-ESD algorithm will be discussed elsewhere along with a detailed comparison with the revised BHMC algorithm implemented in our group. 40

Structural Properties

To improve our atomistic understanding of the structural properties, we calculated several geometric parameters, namely, average bond lengths, dav , coordination number, CN, chemical order parameter, σ, using an approach based on a cutoff parameter estimated from the atomic radius for each chemical specie. Additional to that, the average cluster radius was calculated. The results obtained from all optimized structures are shown in Figure 3. Furthermore, the radial distribution function was pGMC calculated for the (MO2 )n clusters, Figure 4, which is a key information to identify the radial position of the cations and anions within the particles. The analyses were performed for all the calculated structures, as well as for the non-optimized fragments carved from the crystalline structures, which helps to identify effects derived from the relaxation of the clusters. Below, all the results are discussed. 1 Average Bond Length and Coordination Number Analyses Based on the ionic radius of the Ti, Zr, Ce, and O atoms, namely, 0.61, 0.78, 0.97, and 1.40 Å, 53,54 we expect an increase in the average bond lengths, dav , from TiO2 , ZrO2 , and CeO2 , which is obtained for all calculated pGMC clusters, Figure 3. It is well known that the average bond length increases by increasing the coordination as the same number of electrons are shared by a larger number of bonds, which is in fact observed for all three oxides. For

IV Results and Discussion Along the DFT-PBE calculations for the (MO2 )n clusters, n = 1 − 15 and M = Ti, Zr, and Ce, we performed about six thousands geometric optimiza-


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flect their macroscopic geometrical properties. The difference between dav of pGMCs and the reference values obtained from fragments can be explained by the relaxation effects, which contract the bonds lengths in the surface by a larger magnitude compared with the M – O within the cluster. As expected, the coordination number of the cation atoms increase by increasing the number of formula units, n, e.g., CN M = 2 for n = 1, and increases steadily by adding further units up to reach values nearly the same as the bulk fragments, which are lower than the cation coordination numbers calculated for the bulk systems due to the surface effects, i.e., the cation species closer to the surface has lower coordination numbers, which affects the average results. 2 Chemical Order Parameter The chemical order parameter, σ, yields local information on the chemical species distribution, 55,56 which is defined as follows, σ=

N M−M + N O−O − N M−O , N M−M + N O−O + N M−O

(2)

where N M−M and N O−O are the number of homogeneous bonds of metals and oxygens, while N M−O is the total number of M – O bonds. Considering only the formation of M – O bonds among the cations and anions in the MO2 oxides, we obtain σ = −1, however, it is not the case for the bulk systems as the distance between the O – O atoms is smaller than the cutoff parameter based on the atomic radius of the cation and anion species, e.g., σ = 0.20 for ceria fluorite. Hence, there is a shift up of the σ values closer to zero, Figure 3. For CeO2 , the values of σ increases with n and moves towards the bulk fragments, which indicates that the chemical specie distribution follows similar patters as in the bulk even that the structures have geometrical differences. However, it is not the case of TiO2 , where the values of σ deviates substantially from the bulk fragments (unrelaxed), and hence, the pGMCs have features that are not present in the bulk fragments. The results obtained for ZrO2 are between both mentioned systems, i.e., there is relative large differences.

Figure 2: Representative set of structures for the (MO2 )n clusters for M = Ti, Zr, and Ce, n = 1 − 15. The numbers above the structures are the number of formula units, n, and the relative total energy (in parentheses), ∆Etot , in meV/n, wherein pGMC the (MO2 )n structures present ∆Etot = 0 meV/n. Blue, dark green, light green and red atoms represent titanium, zirconium, cerium and oxygen species, respectively. example, dav increases by increasing the number of formula units, n, and it reaches a nearly asymptotic behavior for larger values of n, which are lower than the reference bulk values obtained directly from the nono-ptimized bulk fragments. The non-optimized fragments are bulk clippings, and hence, we understand that those structures re-


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pGMC

(TiO2)n

Anatase

davO (Å)

2.7

(ZrO2)n

(CeO2)n

pGMC

Rutile

pGMC

Baddeleyite

Fluorite

Frag.Nonopt.

Frag.Nonopt.

Frag.Nonopt.

Frag.Nonopt.

Anatase Frag.Opt.

Rutile Frag.Opt.

Baddeleyite Frag.Opt.

Frag.Opt.

Fluorite

2.4 2.1 1.8

2.5

davM (Å)

2.2 1.9 1.6 2.5

dav (Å)

2.2 1.6 8 6 4 2

CN

O

1.9

CN

M

6 4 2 8

CN

6 4

σ

2

0.0 -0.3 -0.6 -0.9

Radius (Å)

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


12 9 6 3 0

1

3

5

7 9 n (f.u.)

11

13

15

1

3

5

7 9 n (f.u.)

11

13

15

1

3

5

7 9 n (f.u.)

11

13

15

O , d M , and d , and Figure 3: Structural parameters of (MO2 )n clusters, namely, average bond length, dav av av coordination number, CNO , CN M , and CN, for oxygen, metallic species, and total, respectively; chemical order parameter, σ; and average cluster radius. Red lines and black triangles, respectively, show the data for the pGMC and disperse values for high-energy isomers. Anatase, rutile, baddeleyite, and fluorite fragments within their respective values for non-optimized and optimized structures are also presented.

3 Cluster Radius Analysis and Radial Distribution Function

ical species within the clusters, we calculated the radial distribution function (RDF) for the pGMCs measured with respect to the geometric center of the respective clusters, Figure 4. We found that the oxygen atoms are distributed along all the cluster, however, for all systems and atomic sizes, the surfaces are oxygen-rich, i.e., there is a higher incidence of oxygen atoms occupying the surface region, which can be explained by the lower surface energy of oxygen-rich surfaces of those systems compared with cation-rich surfaces. 57,58 From our results, it looks like that TiO2 has a higher prefer-

The cluster radii were estimated on the largest distance between the atoms and the largest distance of an atom to the center of gravity. We could not identify no direct correlation between the pGMCs and the values of cluster radius, however, most of the pGMCs have cluster radius smaller than most of the configurations, i.e., there is a preference for compact configurations, which is an important information. With the aim to identify the location of the chem-


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pGMC

Figure 4: The radial distribution function, g(r), obtained for (MO2 )n clusters with respect to their center of gravity. Red, black, blue, and green lines are related with oxygen, titanium, zirconium, and cerium species. ence of oxygen atoms in the core region, however, these structural differences can be explained by the coordination of the cation atoms, which are larger than for the remaining systems.

B

we observe that the neutral titania clusters still do not contain the vibrational characteristics of stoichiometric rutile or anatase bulks. In general, the IR modes are at the range of 37−1022 cm−1 for Ti containing clusters, while for Zr and Ce cases the ranges are 30 cm−1 to 906 cm−1 and 11 cm−1 to 772 cm−1 , respectively. These differences are governed by the reduced mass of the system and the force constant in the harmonic approximation. Hence, even for similar configurations, there is a red shift in the wavenumber for ceria clusters compared to those of zirconia and titania due to the atomic number of Ce being larger than that of Zr and Ti atoms. Furthermore, we found different fingerprint regions for the M – O symmetrical stretching, which is the highest wave-number for each system with mono-coordinated bonds, which are at 983 cm−1 to 1022 cm−1 for Ti – O, 883 cm−1 to 906 cm−1 for Zr – O, and 748 cm−1 to 772 cm−1 for Ce – O.

Vibrational Properties

Taking into account that infrared (IR) spectroscopy is among the most important experimental techniques for structure characterization of systems containing few atoms, 4 the IR spectra with the vibrapGMC tional frequencies were obtained for (MO2 )n , Figure 5. The pGMCs are true minima in the potential energy surface as no imaginary frequency was found. The available experimental and theoretical literature data for neutral clusters is related to titania clusters for n = 1 − 2, from which we found small deviations up to 2.7 % in the wavenumbers compared to our IR spectra. 59,60 Furthermore, the experimental results for titania rutile (anatase) present characteristic bands at 365, 437, and 740 cm−1 (345 and 588 cm−1 ), 8,61 from where


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The Journal of Physical Chemistry (TiO2)n

pGMC

(ZrO2)n

pGMC

(a)

(CeO2)n

n=2

Bulk TiO2 Rutile Bulk TiO2 Anatase (TiO2)npGMC

-15 -18 -21

Eb (eV/f.u.)

n=3 n=4 n=5

Bulk ZrO2 Baddeleyite (ZrO2)npGMC

-15 -18 -21 -24 -15

Eb (eV/f.u.)

n=6 n=7 n=8

Bulk CeO2 Fluorite (CeO2)npGMC

-17 -19 -21

n=9

1

n = 10

(b) ∆2E (eV)

n = 11 n = 12 n = 13 n = 14 n = 15 0

250

1000

500 750 -1 Wavenumber (cm )

Figure 5: Infrared spectra of pGMCs. Black, pGMC red, and blue lines are related with (TiO2 )n , pGMC pGMC (ZrO2 )n , and (CeO2 )n clusters, respectively.

C

-12

Eb (eV/f.u.)

n=1

∆2E (eV)

10 5 0 10 0 10 0 15 0 20 0 20 0 25 0 25 0 25 0 20 0 20 0 25 0 40 0 40 0 40 0

∆2E (eV)

2

2

Intensity (D /Å )

pGMC

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0.6 0.3 0.0 -0.3 -0.6 -0.9

7 9 n (f.u.)

5

11

13

15

(TiO2)npGMC

(ZrO2)npGMC

2.0 1.0 0.0 -1.0 -2.0 -3.0 0.9 0.6 0.3 0.0 -0.3 -0.6

(CeO2)npGMC

2

4

6

8 10 n (f.u.)

12

14

Figure 6: (a) The binding energy, Eb (in eV/f.u.), for (MO2 )n and n = 1 − 15, considering high-energy isomers (black triangles), and the pGMCs (red triangles). The respective bulk cohesive energies are plotted in dashed lines. (b) The stability function pGMC for (MO2 )n clusters.

Binding Energy and Stability Function

The binding energy, Eb , which measures the bond strength, can be calculated as the following: (MO2 )n Etot M O Eb = − [Etot + 2Etot ], n

3

(3)

of uncoordinated atoms decreases by increasing the cluster size. For n = 15, the Eb reached values that are 3.61 %, 6.52 %, and 7.22 % smaller than the cohesive energy for the bulk TiO2 , ZrO2 , and CeO2 phases, respectively. Although, the binding energy was calculated for all cluster sizes and shows asymptotic behavior as a function of size, n, it does not measure the relative stability among the clusters, i.e., some of the

(MO2 )n where Etot is the total energy for a given cluster M O are the total energies for the size, n. Etot and Etot metal and the oxygen free-atoms, respectively. The binding energy results and their respective bulk values are shown in Figure 6(a). The Eb values for the pGMC increase in magnitude by increasing the value of n, which is expected as the number


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The Journal of Physical Chemistry 0.9

clusters might have higher stability that others. 62,63 To identify those clusters, we calculated the relative stability function, ∆2 E(n), which measures the stability of a particular clusters, n, with respects to its neighbors at left, n − 1, and right n + 1, and it can be calculated as the following: n+1 n−1 n ∆2 E(n) = Etot + Etot − 2 × Etot ,

(CeO2)npGMC

0.8

(4)

0.7

Bulk ZrO2 Baddeleyite Bulk CeO2 Fluorite

0.6

n+1 , E n−1 , and E n are the total energies of where Etot tot tot the pGMCs with n + 1, n − 1, and n formula units, respectively. The results are shown in Figure 6(b). We found that the most stable clusters occur for n = 6, 8 and 10, respectively for titania, zirconia, and ceria. In the case of (TiO2 )n systems, there is an decrease in the relative stability following n = 6 > 8 ≈ 10 > 12 ≈ 13, within their atomic distribution closer to the rutile-anatase formation. For zirconia clusters, the most stable systems occur for n = 8, followed by 10 and 13; and for ceria, the stability decreases in order of n = 10, 14, and 12 ≈ 8. Also, with the aim to understand the chemical environment that stabilizes those systems, we correlated the ∆2 E(n) with the σ values. In the case of titania, the σ values for pGMCs do not present a clear tendency, which makes difficult this association. However, we observe that higher ∆2 E(n) values for pGMC (ZrO2 )n are related with a specific range for σ, pGMC varying from −0.36 to −0.37, and for (CeO)n σ = −0.33 up to −0.32 are related with the most stable clusters, pointing to a higher stability for structures with oxygens high-coordinated.

D

(TiO2)npGMC (ZrO2)npGMC

∆QHM (e /atom)

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 17

Bulk TiO2 Rutile

0.5

1

3

5

7 9 n (f.u.)

11

13

15

Figure 7: The Hirshfeld charges of metal cations, M , given in number of electrons per atom, are ∆QH pGMC shown for (MO2 )n and the most stable oxide bulk phases. Hirshfeld are shown in Figure 7 for the oxide clusters and bulk reference systems. We confirmed our expectations based on the Pauling’s electronegativity concept for TiO2 , which has the smallest effective cation charge among the studied systems. However, the results are not confirmed for CeO2 , i.e., the largest effective charge occurs for ZrO2 , and hence, the expected trends is not obtained for all systems. Thus, those concepts should employed with caution. Furthermore, we found a correlation of the magnitude of the charge transfer with the magnitude of the binding energy of the clusters, which indicates that the binding energy is dominated by the Coulomb interactions among the cations and anions.

Electronic Properties

2 Density of States

1 Hirshfeld Charge Analysis

With the aim to understand the behavior of the electronic states as function of size, we calculated the projected density of states (pDOS), which are shown in Figure 8 for an energy range that includes the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). As expected from previous cluster studies, 1,51 for smaller clusters, e.g., n = 1 − 3, the number of electronic states are smaller and with several energy gaps, however, by increasing the number of formula units in the clusters, the number of states increases and the energy gaps decrease up to form a well defined occupied valence band and

Based on Pauling’s electronegativity concept, the magnitude of the charge transfer among the chemical species depends on the electronegativity difference among the chemical species, namely, M and O. The Pauling’s electronegativities for Ti, Zr, Ce, and O atoms are 1.54, 1.33, 1.12, and 3.44, 64 respectively. Therefore, the largest charge transfer is expected among the Ce and O atoms, while the smallest one is expected among Ti and O. Thus, to verify our analysis, we employed the Hirshfeld charge analysis, 65 which yields the Hirshfeld effective charges on each atom. The cation effective


10

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Figure 8: Atom decompose projected density of states (pDOS) for the description of the electronic structure pGMC pGMC pGMC of (TiO2 )n (right), (ZrO2 )n (center), and (CeO2 )n (left). The respective bulk phases were plotted for a direct comparison of electronic structure. A Gaussian broadening of 0.10 eV was taken into the analysis. unoccupied conduction band for n = 15, which are separated by the HOMO-LUMO energy gap. The valence states are dominated mainly by the electronic states derived from the O 2p-states with a small contribution from the M d-states for the TiO2 and ZrO2 clusters, while it is derived from the Ce 4 f -states for the CeO2 clusters. The conduction states (unoccupied states) are formed by the M dstates contribution for TiO2 and ZrO2 , while for the CeO2 clusters, the main contribution is derived from Ce 4 f -states. Furthermore, we calculated the HOMO-LUMO energy separation, Eg , as a function of n, which is shown in Figure 9 for all optimized (MO2 )n configurations. As expected, the results obtained for Eg depends on the structures, chemical species, etc, and hence, there are a wide range of calculated values for Eg . In the case of titania, although few exceptions, we found that the pGMCs have

the largest HOMO-LUMO separations, however, in the case of zirconia and ceria clusters, the majority of Eg values for pGMCs are smaller than their respectively reference bulks values. Hence, there is a correlation between stability and magnitude of the HOMO-LUMO energy separations, which can be explained through the minimization of the Coulomb interactions. We found smaller band gaps compared to the respective bulks as the molecular orbitals tend to be closer to each other for structures containing few atoms, which decreases the energetic distance between HOMO and LUMO. Hence, we found a large range of structures with smaller band gaps than their respective bulk phases, wherein few configurations present a metallic behavior (Eg = 0 eV). However, a small set of structures, including some pGMCs, present larger HOMO-LUMO than the bulk, which correlates to the high stability of


11

The Journal of Physical Chemistry 5 4

Bulk TiO2 Anatase Bulk TiO2 Rutile

(TiO2)n

Bulk ZrO2 Baddeleyite

(ZrO2)n

Bulk CeO2 Fluorite

(CeO2)n

in the PES from the vibrational analysis, with mono-coordinated bonds as the fingerprint being 983 cm−1 to 1022 cm−1 , 883 cm−1 to 906 cm−1 , and 748 cm−1 to 772 cm−1 for Ti – O, Zr – O, and Ce – O, respectively. The atomic distribution in the clusters are similar to their bulk, since the geometrical properties, e.g., CN and dav for the pGMCs tend to the same values of optimized carved fragments. Also, we found a preference of oxygen-rich surface formation due to its smaller energy when compared to a cation-rich surface. For the energetic parameters, we found an increasing in the binding energy with the increase in cluster size, which is associated with the increase in the CN of the systems. Thus, as more compact the tendency of the cluster formation, larger is the distance between the asymptotic pGMC curve and the cohesion energy of the respective bulk. Besides, the magnitude of Eb is determined by the magnitude of the charge transfer from the cation to oxygen, which follows the order TiO2 < CeO2 < ZrO2 . The most stable clusters are found to be pGMC pGMC pGMC (TiO2 )6 , (ZrO2 )8 , and (CeO2 )10 from the relative stability analysis. From the DOS, we observe that the discrete levels, smaller n, tend to form a valence and a conduction bands separated by a gap, HOMO-LUMO gap, with the increase in size. Therefore, the clusters are insulators as their bulk phases, within the valence band mainly formed by O 2p−states, and conduction band being M d−states for TiO2 and ZrO2 systems, while the Ce 4 f −states constitute the conduction band of CeO2 systems. From an analysis of DOS includes the behavior of the HOMO-LUMO gap, we observed a balance between the structural formation and the Coulomb repulsion, that stabilizes pGMCs values larger than the bulk values, in some cases, pointing to their high stability.

3 2 1 HOMO-LUMO gap (eV)

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 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

0 4 3 2 1 0 4 3 2 1 0

1

3

5

7 9 n (f.u.)

11

13

Page 12 of 17

15

Figure 9: The HOMO-LUMO energy gap for the (MO2 )n , n = 1 − 15, systems as a function of size and considering high-energy isomers (black triangles) and the pGMCs (red triangles). A direct comparison is revealed for rutile and anatase (TiO2 ), baddeleyite (ZrO2 ), and fluorite (CeO2 ), which are shown in the dashed lines. those systems. We found an odd-even curve for pGMC (ZrO2 )n , even there is no magnetic moment involved in those systems. Also, the high values of Eg occur for (MO2 )10 , which collaborates to the relative stability analysis, pointing to the the high stability of this structure.

V Conclusions With the aim to understand the geometric, energetic, and electronic properties of the oxide (MO2 )n clusters, with n = 1 − 15, and M = Ti, Zr, and Ce, we performed about six thousands geometric optimizations for selected trial structures, which were generated by the TG-ESD algorithm. The performance of the TG-ESD algorithm was evaluated using the (TiO2 )n clusters as reference, as those systems have been widely studied. We found that our implementation of the TG-ESD algorithm, implemented from scratch, can yields excellent results for the studies of oxide clusters. We found that our pGMCs are true minima

Supporting Information Available: The supporting information contains figures (S1– S9) with part of (MO2 )n structures, n = 1 − 15, and also the x, y, z coordinates for all non-identical structures that were optimized with DFT-PBE. This material is available free of charge via the Internet at http://pubs.acs.org/. Acknowledgement The authors gratefully acknowledge support from FAPESP (São Paulo Research Foundation, Grant


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Number 2017/11631-2), Shell and the strategic importance of the support given by ANP (Brazils National Oil, Natural Gas and Biofuels Agency) through the R&D levy regulation. This study was financed in part by the Coordenaça˜ o de Aperfeiçoamento de Pessoal de N´ıvel Superior Brasil (CAPES) - Finance Code 001 and the National Counsel of Technological and Scientific Development (CNPq). The authors also acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC resources of the SDumont supercomputer, which have contributed to the research results reported within this paper. URL: http://sdumont.lncc.br. Authors thank also the infrastructure provided to our computer cluster by the Department of Information Technology − Campus São Carlos. Research developed with the help of HPC resources provided by the Information Technology Superintendence of the University of São Paulo.

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Normalized Binding Energy and Charge Transfer

IR Vibrational Spectra

Page 17 ofThe 17 Journal of Physical Chemistry

Charge Transfer

Binding Energy

nn== 11

n=4

1 2 nn = 1 3= 1 4 5 6 7

n = 15 n = 15 700

Increase in Size

800 900 -1 Wavenumber (cm ) n = 14

ACS Paragon Plus Environment n = 15 Increase in Size

1000

A DFT Study Combined with the

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