Embedded Cluster Model for Al2O3 and AlPO4 Surfaces Using Point

Article pubs.acs.org/JPCC

Embedded Cluster Model for Al2O3 and AlPO4 Surfaces Using Point Charges and Periodic Electrostatic Potential Masafuyu Matsui† and Shigeyoshi Sakaki*,†,‡ †

Elements Strategy Initiative for Catalysts and Batteries (ESICB), Kyoto University, 1-30 Goryo-Ohara, Nishikyo-ku, Kyoto 615-8245, Japan ‡ Fukui Institute for Fundamental Chemistry, Kyoto University, Nishi-hiraki-cho 34-4, Takano, Sakyo-ku, Kyoto 606-8103, Japan S Supporting Information *

ABSTRACT: An embedded cluster model with either a large number of point charges (PCs) or periodic electrostatic (PE) potential was proposed to incorporate the electrostatic effects by the bulk surface and applied to non-transition-metal oxide supports such as Al2O3 and AlPO4. A large number of PCs are placed on several layers of surface. The PC values were taken to be the same as the Bader charges obtained from periodic DFT calculation of slab model. The PE potential was derived so as to consider the infinite three-dimensional PC distribution obtained by the calculation of slab model. One electron integral of the PE potential in the Gaussian basis function was evaluated using Poisson’s equation, Fourier transformation within a supercell approach, and the Ewald summation method. These embedded cluster models were applied to Rh2-adsorbed Al2O3 and AlPO4. A bare cluster model with neither PC nor the PE potential presented very poor computational results for the interaction energies of Rh2 with Al2O3 and AlPO4 surfaces, density of states, projected density of states, frontier orbital features, and spin density distribution. In contrast, the embedded cluster model successfully reproduced those properties when either a large number of PCs or the PE potential was employed. These results indicate that the embedded cluster models proposed here are useful for investigating theoretically non-transition-metal oxide surface using the hybrid DFT functional.

1. INTRODUCTION

because large unit cells must be employed in the calculations; if not, the coverage of the surface becomes unreasonably high. In the case of the cluster model, on the other hand, hybrid DFT functional and post-Hartree−Fock methods can be used with reasonable computation cost, as discussed in a recent review.20 However, the bulk effects must be considered in such cluster models. To incorporate bulk effects in cluster models, embedding potential methods have been proposed21−28 and successfully applied to the O2 reaction on Al(111),29 H2 dissociation on the Au nanoparticle,30,31 and semiconductor defect states.32 Those successful results strongly suggest that the use of an embedded cluster model with bulk effects is a promising method for theoretical study of heterogeneous catalysts. Many heterogeneous catalysts are complex systems consisting of metal cluster/particle and metal oxide supports such as Al2O3, AlPO4, CeO2, and ZrO2. To investigate catalytic reactions such as CO−NO, CO−O2, hydrocarbon−O2, and O2 cleavage reactions on those heterogeneous catalysts,2,3 we need a simple but robust cluster model that can be applied to realistic catalyst models. Because these metal oxide supports are polarized material consisting of positively charged metal atoms

Heterogeneous catalysts play crucial role(s) in many industrial chemical reactions,1 depollution of automotive exhaust gas,2 fuel cells,3 and so on. Because the experimental technique to investigate such heterogeneous catalysts is limited even today, theoretical knowledge is indispensable for understanding and predicting heterogeneous catalysts and their reactions. In the theoretical studies of heterogeneous catalysts, periodic DFT calculation with plane wave basis sets has been carried out as one of the standard methods using the slab model. The use of a hybrid functional such as B3LYP is desired for DFT calculations of chemical reaction and band gap. However, the use of the hybrid functional in the slab calculation is still timeconsuming, and its application has been limited to several pioneering works.4−7 Considering that open-shell molecules such as the dioxygen molecule and nitrogen oxide (NO) participate in many heterogeneous catalytic reactions, we want to use the hybrid functional and also the post-Hartree−Fock methods such as MP2 to MP4, CCSD(T), and CASPT2 in theoretical studies of heterogeneous catalysts using a realistic surface size. Though MP2, CCSD(T), and full-CI methods have been applied to theoretical study of real solid systems,8 the examples have been limited to systems with small unit cells.9−19 It is not easy to apply these methods to heterogeneous catalyst consisting of metal cluster/particle and metal-oxide support © XXXX American Chemical Society

Received: May 2, 2017 Revised: August 11, 2017 Published: August 21, 2017 A

DOI: 10.1021/acs.jpcc.7b04126 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C and negatively charged oxygen atoms, the cluster model must be calculated by incorporating correctly electrostatic interaction by the bulk system at least. In this regard, the cluster model with point charge distribution is considered a simple but useful method. The idea of the embedded cluster model with point charges (PCs) is old, and several different approaches have been proposed so far.33−36 For instance, an electrostatic embedding scheme for infinite periodic PC distribution was proposed and applied to the theoretical study of defects in CaF2 and CeO2.37−39 The embedded cluster model with PCs was also applied to theoretical studies of the band gap of the MgO crystal40 and adsorption of a small gas molecule to the Nisubstituted MgO surface.7 However, the application of an embedded cluster model with PCs has been limited to highly ionic and rather simple metal oxides such as MgO, CaF2, and CeO2. Al2O3 is used as a support in many heterogeneous catalysts. However, it is not easy to construct a cluster model of Al2O3 because Al2O3 has a more complex structure than MgO, etc.; for instance, each Al is bound with four or six O atoms in Al2O3.41 AlPO4, which was recently reported to be a good support of Rh cluster/particle for the de-NOx reaction of automotive exhaust gas,42−45 has a more covalent P−O bond than the Al−O bond and a more complex electronic structure than Al2O3. In this regard, good cluster models of Al2O3 and AlPO4 are worthy of investigating for theoretical study of their catalytic reactions. Though theoretical study of the surface is necessary for understanding heterogeneous catalysts, the embedded cluster model with PCs has not been applied to surface phenomena except for the above-mentioned several pioneering theoretical studies of gas molecule adsorption to surfaces.7,29−31 Therefore, the systematic comparison between the embedded cluster model with PCs and the usual slab model is still necessary for Al2O3, AlPO4, and related metal oxides. In this work, we propose embedded cluster models in which the bulk effect is incorporated by employing either a large number of point charges (PCs) or periodic electrostatic (PE) potential. PCs were not placed in a three-dimensional space to mimic the slab model but placed on several layers of the surface to avoid an unreasonable presence of electrostatic potential above the surface. The PE potential was evaluated so as to reproduce three-dimensional PC distribution of the slab model, as will be described in the next section. We applied this embedded cluster model to Rh2-adsorbed Al2O3 and AlPO4 surface systems. Our purposes here are (i) to investigate how much the cluster model is improved by placing PCs or the PE potential and (ii) to check if these embedded cluster models can reproduce computational results by the slab model such as interaction energy between metal cluster and metal oxide surface, density of states (DOS), projected density of states (pDOS), and frontier orbital energies, their shapes, and spin density distribution by making comparison with the recently reported slab calculations.46,47 Because metal cluster/particle on the Al2O3 surface has been an important target for theoretical study of heterogeneous catalyst,46−68 good embedded cluster model for Al2O3 and related materials are useful for performing further theoretical studies of these catalysts using hybrid DFT functional and post-Hartree−Fock method.

Scheme 1. Schematic Representation of (a) Slab Model, (b) Bare Cluster Model, (c) Embedded Cluster Model with Large Number of Point Charges, and (d) Embedded Cluster Model with Periodic Electrostatic Potential

2. METHOD, COMPUTATIONAL DETAILS, AND MODELS 2.1. Embedded Cluster Model with Point Charges. In the slab calculation, infinite three-dimensional charge distribution can be considered, as shown in Scheme 1a. In the bare cluster model, however, neither distribution of PCs nor PE potential is considered, as shown in Scheme 1b. The direct way to incorporate the electrostatic potential provided by bulk surface is to place a large number of PCs at atomic positions in several layers of surface to mimic electrostatic potential by the surface, as shown in Scheme 1c; in this model, the PCs are not placed above the surface, because the PC distribution is absent above the surface in a real system. The Bader charges evaluated by the slab model were employed as PCs, which will be described in detail in section 2.3. 2.2. Embedded Cluster Model with Periodic Electrostatic (PE) Potential. Another method is to use periodic electrostatic (PE) potential based on periodic PC distribution obtained by slab calculation, using Poisson’s equation and Fourier transformation. Essentially the same idea has been proposed previously as a periodic electrostatic embedded cluster method (PEECM)37−39 implemented in the TURBOMOLE program,69,70 in which the periodic electrostatic potential is incorporated into a one-electron integral using an expansion of a product of Gaussian basis functions in regular solid harmonics. We employed here the supercell approach,71 where each cluster was positioned in one unit cell with very large vectors and centers of Gaussian basis functions were placed in the cell. Under these conditions, one electron integral of the PE potential (VES(r)) for primitive Gaussian basis functions can be represented as eq 1 ES

⟨gAa|V̂ |gBb⟩ =

∑ V ES(G) ∫ G

=

all

drgAa(r)gBb(r)eiG·r

⎛ π ⎞3/2 ⎛ 2⎞ ES ⎜⎜ ⎟⎟ FAaBbe−G·rP exp⎜⎜ − |G| ⎟⎟ V ( G ) ∑ ⎝ αp ⎠ ⎝ 4αp ⎠ G ES

(1)

ES

where V (G) is a Fourier transform of V (r), G is a reciprocal lattice vector, and gAa(r) is a primitive Gaussian function at rA with exponent αa: ⎛ ⎞ αα αp = αa + αb , FAaBb = exp⎜ − a b |rA − rB|2 ⎟ , ⎝ αa + αb ⎠ αa rA + αb rB and rp = αa + αb

Detailed derivation is presented in the Supporting Information. Though Fourier transform of the product of two Cartesian B


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The Journal of Physical Chemistry C

Figure 1. Structures of (A) slab model, (B) bare cluster model, and (C) embedded cluster model for Rh2/Al2O3, and (D) slab model, (E) bare cluster model, and (F) embedded cluster model for Rh2/AlPO4: The green, yellow, red, and black balls indicate Al, P, O, and Rh atoms, respectively. The layers below the first and second layers are represented by sticks in slab models. The atoms in external region of the cluster are represented by pale colors in embedded cluster models. These geometries were drawn using the computational results of ref 47.

Å × 15 Å, and 900 Å × 900 Å × 15 Å, respectively, where PCs in eight surface layers are considered; these point charges are named “very small number of point charges” (VSP charges), “small number of point charges” (SP charges), “medium number of point charges” (MP charges), “large number of point charges” (LP charges), and “very large number of point charges” (VLP charges), respectively, hereafter. In the embedded cluster model of AlPO4 with PCs, we employed 2934, 8310, 7.551 × 105, 4.115 × 105, and 1.016 × 106 PCs. These PCs correspond to the inclusion of electrostatic potential involved in the region of 53 Å × 50 Å × 15 Å, 88 Å × 84 Å × 15 Å, 270 Å × 250 Å × 15 Å, 620 Å × 580 Å × 15 Å, and 970 Å × 920 Å × 15 Å, respectively, where PCs in seven surface layers are considered; they are named in the same manner as those of Al2O3. The region used for the AlPO4 is moderately larger than that for the Al2O3, because it was not easy to obtain the SCF convergence of the Rh2/AlPO4 without enough number of PCs when the PBE functional was employed, as will be discussed below. We briefly discuss here the differences of the cluster model with PCs or the PE potential from the slab model: In the cluster model with the PE potential, the infinite threedimensional PC distribution is considered, which is schematically shown in Scheme 1d. This electrostatic potential is constructed so as to mimic that of the slab model. In the cluster model with PCs, PCs are placed on several layers of the surface model but not placed above the surface, as shown in Scheme 1c. In other words, the electrostatic potential above the surface is considered in the PE potential but not in the cluster model with PCs in this work; note that the two-dimensional (2D) Ewald summation method is also useful for cluster model but it was not used here. This is one important difference of the cluster model with PCs from that with PE potential. Another difference is that the infinite three-dimensional PC distribution is considered in the cluster model with PE but a large number of PCs surrounding a quantum mechanics (QM) region are considered in the cluster model with PCs. 2.4. Computational Details. The geometries of the Rh2/ Al2O3 and Rh2/AlPO4 were taken from our previous study47 without further geometry optimization. In the study, the slab model was calculated by the plane-wave density functional

Gaussian functions cannot be provided as a simple form, it can be represented as a recursion formula within a binomial expansion; see the Supporting Information. In general, the Fourier expansion of eq 1 is not complete owing to the high-frequency components of the nuclear pointcharge potential in reciprocal space. To avoid this problem, the Ewald summation method was applied to evaluation of electrostatic interaction between PCs and one-electron orbitals represented by the Gaussian basis functions in the periodic boundary condition; see also the Supporting Information. In this way, the electrostatic interaction between the periodic PC distribution and the one-electron orbital of the embedded cluster model can be correctly evaluated in reasonable computational cost, where no approximation is employed except for very large lattice vectors for supercells. The determination of cutoff energy in the Ewald summation method and the effects of the supercell size on computational results are discussed in the Supporting Information. 2.3. Models Employed for Calculations. We investigated here the adsorption of rhodium dimer (Rh2) on the Al2O3 and AlPO4 surfaces (named Rh2/Al2O3 and Rh2/AlPO4 hereafter). Geometries of the Rh2/Al2O3 and Rh2/AlPO4 were optimized by the slab model in our previous work,47 as shown in Figure 1A,D. To keep charge neutrality, stoichiometric cluster models were constructed from the optimized slab models, as follows: In the Al2O3, we constructed an (Al2O3)12 cluster model using 16 Al and 18 O atoms in the first layer and eight Al atoms and 18 O atoms in the second layer, as shown in Figure 1B. In the AlPO4, we constructed an (AlPO4)15 cluster model using eight Al, eight P, and 32 O atoms in the first layer and seven Al, seven P, and 28 O atoms in the second layer, as shown in Figure 1E. These models are named “bare cluster model”, hereafter. The effects of cluster size were investigated using larger models, (Al2O3)18 and (AlPO4)19, named Al2O3-L and AlPO4-L, respectively. In the embedded cluster model of Al2O3 with PCs, we employed 1060, 11940, 1.079 × 105, 5.879 × 105, and 1.452 × 106 PCs, where these PCs were employed so as to keep charge neutrality of the system. They correspond to the inclusion of electrostatic potential involved in the region of 24 Å × 28 Å × 15 Å, 83 Å × 83 Å × 15 Å, 250 Å × 250 Å × 15 Å, 590 Å × 590 C


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The Journal of Physical Chemistry C Table 1. Interaction Energy (Eint; in eV) of Rh2/Al2O3 and Rh2/AlPO4

a Numbers of point charges are 1060 (24 × 28 × 15 Å3), 11940 (83 × 83 × 15 Å3), 1.079 × 105 (250 × 250 × 15 Å3), 5.879 × 105 (590 × 590 × 15 Å3), and 1.452 × 106 (920 × 920 × 15 Å3) for the Al2O3 embedded models with very small, small, middle, large, and very large numbers of point charges (VSP, SP, MP, LP, and VLP), respectively. Those are 2934 (53 × 50 × 15 Å3), 8310 (84 × 84 × 15 Å3), 7.551 × 105 (270 × 250 × 15 Å3), 4.115 × 105 (620 × 580 × 15 Å3), and 1.016 × 106 (970 × 920 × 15 Å3) for the AlPO4’s with VSP, SP, MP, LP, and VLP, respectively. bPE indicates periodic electrostatic potential. cRh2/Al2O3 and Rh2/Al2O3-L mean Rh2/(Al2O3)12 and Rh2/(Al2O3)18. Rh2/AlPO4 and Rh2/AlPO4-L mean Rh2/ (AlPO4)15 and Rh2/(AlPO4)19. dIn parentheses, the Eint with BSSE correction are presented. eB3LYP calculation cannot be performed with the slab model in VASP. fSCF calculation cannot be converged in the case of PBE functional.

used for Al, P, and Rh. For each of Al and P, one d-polarization function taken from the 6-31G(d) basis sets85 was added. For the O atom, the Hujinaga−Dunning basis set was used;86 see Table S3 in the Supporting Information for the basis set effects. Structures, orbitals, and spin densities were drawn with XCrysDen.87

theory (DFT) with the Perdew−Burke−Ernzerhof generalized gradient approximation of the exchange-correlation functional (PBE)72,73 and the projector augmented wave (PAW) potentials,74,75 using the Vienna Ab initio Simulation Package (VASP).76−78 Though the geometry optimization is important in the theoretical method and it can be performed in the QM/MM approach, we focus here on the interaction energy (Eint) defined by eq 2, as well as the DOS and p-DOS, as the first step to propose the cluster models with PCs and PE potential

3. RESULTS AND DISCUSSION A cluster model for heterogeneous catalyst must satisfy several conditions described below, at least: (i) the interaction energy between the surface and the metal cluster is similar to that calculated by the slab model, (ii) the frontier orbitals of the cluster model, which correspond to top of valence band and bottom of conduction band, have similar shape and similar orbital energy to those of the slab model, and (iii) density of states (DOS) around the Fermi level is similar between the cluster model and the slab model. We wish to discuss these properties in this work. 3.1. Interaction Energy (Eint) Evaluated with Embedded Cluster Model. Table 1 shows the interaction energy (Eint) of the Rh2 with the Al2O3 surface calculated with cluster and slab models. PBE-calculated Eint (−7.18 eV) of the bare cluster model is significantly larger than that of the slab model (−5.44 eV), where a negative value means stabilization energy. Even though the basis sets used are different between the slab and the cluster models, this significantly large difference in Eint is unreasonable. Apparently, the bare cluster model cannot be used for evaluating the interaction energy between metal cluster and Al2O3 surface. When SP and MP charges were employed in the cluster model, the PBE-calculated Eint was −6.44 eV and −6.04 eV, respectively, indicating that the Eint was improved very much by employing the SP and MP charges. The Eint value further increased to −5.99 eV when the LP and the VLP charges were employed and to −6.02 eV when the PE potential was employed. It is noted that the Eint value converges to about −6.0 eV in the cluster model as the number of PCs increases. This is about 0.6 eV larger than that of the slab model.

E int = Et(Rh 2/cluster model of surface)opt − [Et(Rh 2)distorted + Et(cluster model of surface)distorted ] (2)

where Et represents total energy and the geometries of Rh2/ cluster model, Rh2, and surface model were taken to be the same as those in the Rh2/Al2O3 and Rh2/AlPO4 optimized by the slab calculation; the subscript “opt” means the optimized geometry of slab model and “distorted” represents the geometries of Rh2 and surface model taken from the optimized structure of slab model; because these geometries of Rh2 and surface model are not equilibrium ones, they are named “distorted” geometry, hereafter. The binding energy (EBE) is represented by the distortion energies (Edists) of surface and Rh2 cluster and the interaction energy (Eint); see ref 79. Several choices are possible for atomic charge. Compared to the formal charge, the use of the Bader charge provides better results about the Eint of the Rh2/Al2O3, while the Eint of the Rh2/AlPO4 differs little between the formal and the Bader charges; see Table S2 in the Supporting Information. Here, we employed the Bader charge obtained from the slab calculation; the Bader charges (in |e| unit) used are +2.34 to +2.45 for Al and −1.68 to −1.53 for O in Al2O3 and +2.39 to +2.49 for Al, + 3.53 to +3.78 for P, and −1.58 to −1.40 for O in AlPO4.80 The DFT calculations of the cluster model with PCs were performed using the Gausian09 program.81 The embedding method with the PE potential was implemented in the GAMESS program package.82 LANL2DZ basis sets83,84 were D


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It should be noted here that the Eint is somewhat different between the VLP and the PE in the case of the Rh2/AlPO4, though it is similar in the case of Rh2/Al2O3. As discussed above, the cluster model with the VLP is not exactly the same as that with PE, because the electrostatic potential above the surface is considered in the cluster model with the PE but not in the cluster model with the VLP. This difference is larger in the Rh2/AlPO4 than in Rh2/Al2O3 because the Al−O bond is less sensitive to the electrostatic potential than the P−O bond, as described above. The difference between the VLP and the PE potential is moderately larger in the PBE calculation than in the B3LYP, but the reason is not clear. Further examination about the combination of functional and the way to incorporate electrostatic potential is still necessary. The comparison between the Rh2/Al2O3 and Rh2/AlPO4 systems should be made correctly for discussing the support effects of these materials because the AlPO4 more strongly interacts with Rh cluster/particle than does the Al2O3 to improve the catalytic activity of Rh cluster/particle for de-NOx reaction.42−45 In the slab model, the Eint of the Rh2/Al2O3 is moderately larger than that of the Rh2/AlPO4 by 0.15 eV. In the embedded cluster model, the PBE-calculated Eint of the Rh2/Al2O3 is larger than that of the Rh2/AlPO4 in the LP, VLP, and PE cases; the difference in Eint between Rh2/Al2O3 and Rh2/AlPO4 is 0.44 eV for the VLP and 0.31 eV for the PE. It decreases to 0.11 eV for the larger cluster models, the Rh2/ Al2O3-L and Rh2/AlPO4-L with the VLP, while the difference is negligibly small for the larger model with the PE. Because the distortion energy of the Al2O3 surface is considerably larger than that of the AlPO4 by about 1.24 eV (slab calculation with the PBE),47 the binding energy of the Rh2 with the AlPO4 surface is estimated to be larger than that with the Al2O3, using the above-mentioned Eint values of the cluster model.89 This is consistent with the experimental findings.42−45 The similar discussion can be presented for the B3LYP computational results.90 On the basis of the above results, it is concluded that (i) the bare cluster model cannot be used for theoretical study, as expected, (ii) the embedded cluster model with either a large number of PCs or the PE potential provides similar Eint value to that of the slab model, (iii) the hybrid functional can be easily used in embedded cluster model with reasonable computation cost when either a large number of PCs or the PE potential is employed in the cluster model, but (iv) further examination on the number of PCs, size of cluster, functional, and so on, is still needed because the Eint is somewhat dependent on computational levels. 3.2. Charge Distribution and HOMO and LUMO Energies of Al2O3 and AlPO4 Surfaces. One of the weak points of the cluster model with charge distribution is the unreasonable polarization of the QM moiety near positive PCs. To suppress it, the pseudopotential method has been proposed by Sauer and co-workers.39 To investigate the effects of pseudopotential, NBO charge of the O atom is compared between the edge (E1 to E4 nearby the positive PC) and the inside positions (B1 and B2) in Figure 2, where the pseudopotentials of LANL2DZ83 for Al and P were employed for calculations; hereafter, (Al2O3)12 and (AlPO4)15 clusters are used for discussion because it is likely that the orbital features do not vary much in the larger cluster models (see Table S4 in the Supporting Information for the orbital energy of large cluster models). Apparently, the O atoms at the edge are more negatively charged than those at the inside, indicating that the

The size of cluster model is one important factor to be checked. As shown in Table 1, the increase of the cluster size leads to moderate decrease in the Eint by 0.27 eV (with VLP charges and the PE potential) using the large Rh2/Al2O3-L cluster model. These results suggest the Eint by the larger cluster model becomes closer to that of the slab model but still somewhat larger. The correction of basis set superposition error (BSSE) is another important issue in discussing Eint. The Eint decreases by about 1 eV by the BSSE correction with counterpoise method,88 as shown in Table 1 (see values in parentheses); the BSSE corrected Eint value is named Eint(noBSSE) hereafter. The Eint of the cluster model with the PE potential was calculated to be moderately larger but the Eint(noBSSE) is moderately smaller than that of the slab model; see also Table S3 for basis set effects on Eint and Eint(no-BSSE). Interestingly, the Eint is somewhat different between the Rh2/ Al2O3 and Rh2/Al2O3-L but the Eint(no-BSSE) does not differ so much, suggesting that the size effect mainly arises from BSSE. The B3LYP-calculated Eint (−9.79 eV) of the bare cluster model was very large compared with the PBE-calculated Eint (−5.44 eV) of the slab model (Table 1). However, it was significantly improved to −5.56 eV and −5.55 eV by employing SP and VLP charges and finally to −5.47 eV by employing the PE potential. The B3LYP calculation shows the same trend in both of BSSE correction and cluster size effect as the PBE calculation but their extents are smaller than those of PBE computational results. The B3LYP-calculated Eint is moderately smaller than the PBE-calculated value, suggesting that the geometry optimization with the cluster model at the B3LYP or other functional is worthy of investigation. In the Rh2/AlPO4 cluster model, the SCF convergence could not be obtained with the PBE functional when the VSP to the MP charges were employed, suggesting that the electronic structure of the Rh2/AlPO4 more easily fluctuates than that of the Rh2/Al2O3 during the SCF calculation. When LP and VLP charges were employed, the SCF calculation converged and the Eint was calculated to be −5.55 eV. When the PE potential was employed, it was −5.71 eV. In the B3LYP calculation, the SCF convergence could be obtained even without PC, suggesting that the B3LYP increases HOMO−LUMO energy gap, as will be discussed below, leading to the easier SCF convergence. Though the B3LYP-calculated Eint value (−12.36 eV) of the bare cluster model is abnormally large, it is improved very much by employing the LP and the VLP charges and the PE potential. The dependency of Eint on the cluster size is marginal (Table 1). The Eint(no-BSSE) is smaller than the Eint by about 0.5 eV in both of PBE and B3LYP calculations, indicating that BSSE is smaller in the Rh2/AlPO4 than in the Rh2/Al2O3. The PBE-calculated Eint(no-BSSE) with the VLP and the PE agrees with the Eint of the slab model. The smaller BSSE and small dependency of Eint on the cluster size suggest that the cluster model can be more easily constructed for AlPO4 than for Al2O3. Apparently, the effect of the point charges on SCF convergence is much larger in the Rh2/AlPO4 than in the Rh2/Al2O3, indicating that the electronic structure of the AlPO4 surface is more sensitive to the surrounding electrostatic potential than that of the Al2O3 surface. This is reasonable because the strongly polarized Al−O bond is less flexible to the surrounding electrostatic potential than the P−O bond; remember that the difference in electronegativity between P and O atoms is smaller than that between Al and O atoms. E


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Table 2. Frontier Orbital Energies (εHOMO and εLUMO Relative to Vacuum Level;a eV) and Band Gaps of Distorted Al2O3 and AlPO4 Surfacesb

a

In the slab model, the electrostatic potential at the middle point between two surfaces is taken as a standard (vacuum level). bThese geometries were taken to be the same as the corresponding moiety of Rh2/Al2O3 and Rh2/AlPO4 optimized by the slab calculations. c Numbers of point charges are 1060 (24 × 28 × 15 Å3), 11940 (83 × 83 × 15 Å3), 1.079 × 105 (250 × 250 × 15 Å3), 5.879 × 105 (590 × 590 × 15 Å3), and 1.452 × 106 (920 × 920 × 15 Å3) for the Al2O3 embedded models with very small, small, middle, large, and very large number of point charges (VSP, SP, MP, LP, and VLP), respectively. Those are 2934 (53 × 50 × 15 Å3), 8310 (88 × 84 × 15 Å3), 7.551 × 105 (270 × 250 × 15 Å3), 4.115 × 105 (620 × 580 × 15 Å3), and 1.016 × 106 (970 × 920 × 15 Å3) for the AlPO4’s with VSP, SP, MP, LP, and VLP, respectively. dPE indicates periodic electrostatic potential. eBand gap = εLUMO − εHOMO unless no caution is presented. fB3LYP calculation cannot be performed with the slab model in VASP. g Frontier orbitals close to HOMO or LUMO. hSCF calculation cannot be converged in PBE calculations. iHOMO−2. jεLUMO − εHOMO−2. kLUMO+1. lHOMO−4. mεLUMO+1 − εHOMO−4

Figure 2. Comparison of interaction energy (Eint; eV) and natural atomic charge (|e|) of (A) Rh2/(Al2O3)12 and (B) Rh2/(AlPO4)15 with VLP charges between with ECP and without ECP at the edge. PBE functional was used. “Pseudo” means the presence of pseudopotential. Eint with BSSE correction is shown in parentheses.

artificial polarization is induced by the PCs. However, the difference in negative charge between the edge and inside O atoms becomes smaller by the presence of the pseudopotential. This means that the presence of pseudopotential suppresses the artificial accumulation of negative charge at the edge O atom, indicating that the pseudopotential is useful to present reasonable electron distribution. In the case of the Rh2/ Al2O3, the discrepancy in Eint from that of the slab model becomes smaller by the presence of pseudopotential, while it changes little in the case of the Rh2/AlPO4. These results indicate that the use of the pseudopotential is recommended for the cluster model with PCs. The highest energy occupied (HO) and lowest energy unoccupied (LU) bands of the Al2O3 and AlPO4 surfaces play important roles in interacting with metal cluster/particle. The HOMO and LUMO energies of the embedded cluster model are compared with those of the slab model in Table 2.91 In the bare cluster model of the Rh2/Al2O3, the PBE-calculated LUMO energy is much lower than in the slab model, though the HOMO energy is similar to that of the slab model. When the VSP charges were employed in the calculation, the deviation of HOMO and LUMO energies from those of the slab model increased very much and the HOMO−LUMO energy gap was not improved at all. When the SP to VLP charges were employed, the HOMO−LUMO energy gap was improved and the HOMO and LUMO energies seem to converge as going from the SP to the VLP charges. However, the deviation of the HOMO and LUMO energies from those of the slab model was not improved at all even when either the VLP charges or the PE potential was employed. Essentially the same results were observed in the B3LYP calculation. These results indicate that the orbital energies are more sensitive to the surrounding PCs than the Eint.

In the Rh2/AlPO4, the PBE-calculated HOMO−LUMO energy gap of the cluster model moderately differs from that of the slab model but the HOMO and LUMO energies significantly differ from those of the slab model even when either the VLP charges or the PE potential was employed, like in the Rh2/Al2O3. Notably, the B3LYP-calcualted HOMO and LUMO energies of the Rh2/AlPO4 fluctuate more than those of the Rh2/Al2O3 as the number of PCs increases. This result indicates that the electronic structure of the Rh2/AlPO4 is more sensitive to the charge distribution than that of the Rh2/Al2O3, as was found in the Eint. It is noted that the HOMO−LUMO energy gap is considerably larger in the B3LYP calculation than in the PBE (Table 2), as expected. In general, the semilocal functional tends to present too small HOMO−LUMO energy gap. It is likely concluded that the use of the hybrid functional with the embedded cluster model in the presence of either the VLP charges or the PE potential is recommended for discussing the band gap. Another important result here is that the HOMO and LUMO energies are considerably different between the embedded cluster model with the VLP charges and that with the PE potential. One reason could be the difference in PC distribution between the embedded cluster model with the PE potential and that with PCs. The PE potential is determined so F


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Figure 3. PBE-calculated frontier orbitals of distorted surfaces of Rh2/Al2O3 and Rh2/AlPO4 for (A) slab model, (B) bare cluster model, and (C) embedded cluster models. Numbers of point charges are 1060 (24 × 28 × 15 Å3), 1.079 × 105 (250 × 250 × 15 Å3), and 1.452 × 106 (900 × 900 × 15 Å3) for the Al2O3 embedded models with very small, middle, large, and very large number of point charges (VSP, MP, and VLP), respectively. Those are 2934 (53 × 50 × 15 Å3), 7.551 × 105 (270 × 250 × 15 Å3), and 1.016 × 106 (970 × 920 × 15 Å3) for the AlPO4’s with VSP, MP, and VLP, respectively. PE means periodic electrostatic potential.

at the important site for the Rh2 adsorption. We tried to find orbitals similar to those of the HO and LU bands of the slab model but could not. It is concluded that the bare cluster model cannot be used even for qualitative discussion of HOMO and LUMO shapes. Because the HOMO and LUMO at the edge were calculated at similar energy, the HOMO−LUMO energy gap was very small in the bare cluster model. These unreasonable features of HOMO and LUMO were not improved by employing the VSP and the SP charges. When the MP charges were employed in the cluster model, the HOMO and LUMO were localized at the Rh2 adsorption site (Figure 3C). When either the LP/VLP charges or the PE potential was employed, their shapes became almost the same as those of the slab model. Based on these results, it is concluded that in the cluster model the use of either the LP/ VLP charges or the PE potential is necessary to discuss the HOMO and LUMO features. The spin distribution deeply relates to the interaction between the Rh2 and the surface. In the slab model, the spin density was localized around the Rh2 moiety, as shown in Figure 4A. In the bare cluster model, however, considerably large spin density was observed on the edge of the cluster (Figure 4B). This unreasonable spin distribution comes from the presences of HOMO and LUMO at the edge of the bare cluster model (remember Figure 3B). When either the LP/VLP

as to mimic the infinite three-dimensional PC distribution obtained by the slab calculation. This means that the charge distribution above the surface is considered in the cluster model with the PE potential (Scheme 1d). In the cluster model with PCs, on the other hand, the PC distribution above the surface is not considered but a large number of PCs in several layers of surface are considered (Scheme 1c), which is different from the cluster model with the PE potential. It is concluded that the cluster model with either PCs or the PE potential is useful for discussing the HOMO−LUMO energy gap but HOMO and LUMO energies are considerably different between these two cluster models. 3.3. HOMO and LUMO Features of Al2O3 and AlPO4 Surfaces. For the interaction between metal cluster/particle and surface, HOMO and LUMO shapes as well as their energies play important roles. In the slab model, the HO and LU bands of the distorted Al2O3 and AlPO4 surfaces are localized on the Rh2 adsorption site, as shown in Figure 3A, where the term “distorted” means the structure of the surface moiety is not in equilibrium but distorted in the Rh2/Al2O3 and Rh2/AlPO4 systems due to the interaction with Rh2. These HO and LU band shapes must be reproduced by the embedded cluster model. In the bare cluster models of the distorted Al2O3 and AlPO4 surfaces, however, they are found on the edge (Figure 3B),89 indicating that the frontier orbitals do not exist G


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the total DOS of the Rh2/Al2O3 or AlPO4. We calculated such p-DOS in our previous study and found that almost all states of the LU band of the AlPO4 surface are observed in the valence band of Rh2/AlPO4 but in the case of Rh2/Al2O3 some states are observed in the conduction band.47 On the basis of these results, we discussed that the CT significantly occurred from the Rh2 to the LU band of the AlPO4 surface but such CT was weaker in the Rh2/Al2O3 than in the Rh2/AlPO4. To check if these results can be reproduced by the embedded cluster model, we calculated the p-DOS of the LUMO of the surface using the embedded cluster models with the VLP charges and the PE potential, as shown in parts B and C, respectively, of Figure 5. In the embedded cluster model of the Rh2/AlPO4, almost all p-DOS of the AlPO4 LUMO was found in the occupied level (Figures 5B2,C2). On the other hand, the pDOS of the Al2O3 LUMO was found in both occupied and unoccupied levels in the Rh2/Al2O3 (Figures 5B1,C1). These results are essentially the same as those of the slab model (Figures 5A1,A2). Moreover, the shape of p-DOS (red line) of the surface LUMO is similar between the slab model and the embedded cluster model with either the VLP charges or the PE potential. It is concluded that the band structure around the Fermi level can be discussed well with the cluster model when either a large number of PCs or the PE potential was employed in calculations.

Figure 4. PBE-calculated spin distributions of the Rh2/Al2O3 and Rh2/ AlPO4 for (A) slab model, (B) bare cluster model, and (C) embedded cluster models. Point charges of VLP (1.452 × 106 for Al2O3 and 1.016 × 106 for Rh2/AlPO4) were considered in calculations. The PE means periodic electrostatic potential.

charges or the PE potential was considered in calculations, the unreasonable features disappeared and the spin density was mainly found on the Rh2 moiety (Figure 4C). In both Rh2/Al2O3 and Rh2/AlPO4, the slab model indicated the presence of small spin density on the neighbor O atoms to the Rh2. Such small spin density could be reproduced by the cluster model with either the LP/VLP charges or the PE potential (Figure 4C). It is concluded that these embedded cluster models are useful for discussing the spin density distribution. 3.4. Density of States (DOS) of Rh2/Al2O3 and Rh2/ AlPO4. Density of states (DOS) and projected density of sates (p-DOS) are important properties in discussing solid surfaces. It is necessary to reproduce these properties by the cluster model. In Figure 5, DOSs and p-DOSs of the Rh2/Al2O3 and Rh2/AlPO4 are compared between the slab and the cluster models, where a black line represents total DOS, a red line is pDOS of the LU band of the surface, and a blue line is p-DOS of the HO band of the surface. In the cluster model with the VLP charges, the DOS around HOMO and LUMO is similar to those of the slab model in both of α and β spin DOSs, whereas orbital energies shift to lower value compared to the slab model, as shown in Figures 5A1,B1. In the cluster model with the PE potential, the DOS around HOMO and LUMO are similar to those of the slab model in both of α and β spin DOSs (Figures 5A1,C1). Similar results are observed in the case of the Rh2/AlPO4 system (see Figures 5A2,B2,C2). It is concluded that the DOSs around HOMO and LUMO are reproduced well by the embedded cluster model by employing either the VLP charges or the PE potential. However, one problem is found in the conduction band above the Fermi level; the DOS is not found around −3.0 to −2.0 eV in Rh2/Al2O3 and around −4.2 to −3.8 eV in Rh2/ AlPO4 in both α and β-spin DOSs of the slab model (cyan stripe in Figure 5A1,A2), but several α and β spin DOSs are found there (cyan stripe in Figure 5B1,B2,C1,C2) in the cluster models with the VLP charges and the PE potential. These states arise from the edge of cluster model because some unoccupied orbitals exist around the edge. This problem must be solved in the near future. If the charge transfer (CT) occurs to the LU band of the surface from the Rh2, the p-DOS of the LU band of the Al2O3 or AlPO4 surface moiety must be found in the valence band of

4. CONCLUSIONS An embedded cluster model with either a large number of point charges or periodic electrostatic potential was proposed for Al2O3 and AlPO4 surfaces in this work, where the PC values employed in the calculation were taken to be the same as the Bader charges obtained from periodic DFT calculation of slab model. PCs were not placed on three-dimensional space but placed on several layers of surface. The PE potential was derived so as to reproduce the infinite three-dimensional PC distribution obtained by the slab calculation. The embedded cluster model with PCs or the PE potential was applied to theoretical study of the interaction between the Rh2 cluster and such metal oxide supports as Al2O3 and AlPO4. The bare cluster model with neither PC nor the PE potential presented very poor computational results about the interaction energy, band structure, HOMO and LUMO shapes, and spin density distribution. When either a large number of PCs or the PE potential was employed, however, the embedded cluster model successfully reproduced computational results by the slab model. For instance, the interaction energy of the Rh2 cluster with the Al2O3 and AlPO4 surfaces was calculated to be similar to that evaluated with the slab model. The DOS around HO and LU bands and the shape of frontier orbital as well as the spin density distribution of cluster model were essentially the same as those of the slab model. On the basis of these results, it is concluded that the embedded cluster model with either a large number of PCs or the PE potential is recommended as a simple but robust model of the solid surface. The cluster model with the PE is useful to reproduce the computational results by the slab model because infinite three-dimensional charge distribution is considered in the PE potential. In the cluster model with PCs, a large number of PCs are placed on several layers of surface model. It is noted here that the embedded cluster model with either a large number of PCs or the PE potential is useful for nontransition-metal oxide surface, but further examination is H


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Figure 5. PBE-calculated total DOSs of the Rh2/Al2O3 and Rh2/AlPO4, p-DOSs of HOMO and LUMO of distorted the Al2O3 and AlPO4 surfaces of (A) slab model, (B) embedded cluster model with the VLP charges, and (C) embedded cluster model with the PE. Black (i), red (ii), and blue lines (iii) represent total DOS, p-DOS of the LU band of the distorted Al2O3 or AlPO4 surface, and p-DOS of the HO band of the distorted Al2O3 or AlPO4 surface. Vertical pink line represents the Fermi level. The vacuum level is set to be zero. No level is found in the cyan stripe in the case of slab model. The negative density of states correspond to the density of β spin states.

needed for applying this model to transition-metal oxide, defect electronic level, related properties, and so on.

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ASSOCIATED CONTENT

S Supporting Information *

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The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b04126. Full representation of ref 81, the derivation of oneelectron integral of periodic electrostatic (PE) potential, effects of cutoff energy in the Ewald summation method (Figure S1), effects of supercell size on computational results (Table S1), effects of atomic charge on computa-

tional results (Table S2), basis set effects (Table S3), cluster size effects (Table S4 and Figure S2), and HOMO and HOMO−1 of the cluster model of Rh2/ AlPO4 (Figure S3). XYZ data for cluster models of Rh2/ Al2O3 and Rh2/AlPO4 (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +81-75711-7907. ORCID

Shigeyoshi Sakaki: 0000-0002-1783-3282 I


Article

The Journal of Physical Chemistry C Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was carried out under “Element Strategy Initiative for Catalysts and Batteries (ESICB)”, which is financially supported by Ministry of Education, Culture, Science, Sports, and Technology (MEXT), Japan. M.M. is thankful to MEXT for financial support by JSPS KAKENHI (No. JP17K05750). S.S. acknowledges financial support from MEXT through a Grant-in-Aid of Specially Promoted Science and Technology (No. 22000009) and JSPS KAKENHI (No. 15H03770) and from the Ministry of Economy, Trade and Industry, Japan through the NEDO project. We thank the computational center at the Institute of Molecular Science, Okazaki, Japan, for using computers.

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models because we could not find any orbital that is similar to that of the slab model.

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Embedded Cluster Model for Al2O3 and AlPO4 Surfaces Using Point

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