Developments of Interorbital–band Interaction Analysis andEmbedded Cluster Model Incorporating Periodic ElectrostaticPotential for Supported Metal Catalysts

Abstract

担体表面上に金属微粒子が高分散した担持金属触媒において，金属微粒子と担体表面との間の金属–表面相互作用は，金属微粒子を担体表面上に安定に分散担持するとともに，金属と表面間の電荷移動などの相互作用により電子状態を変化させ，触媒活性に重要な影響を及ぼす．このような金属微粒子と担体表面との界面における電子状態の関与する相互作用を，実験の測定のみから解明することは困難であり，その本質の理解・予測のためには理論計算による検討が有用である．しかし，従来のスラブモデルを用いた平面波DFT法には，軌道間相互作用に基づく解析手法が乏しい，また，高精度電子状態計算の実行がコスト的に困難，という問題がある．このような問題を解決するために我々は，「射影状態密度を用いた軌道–バンド間相互作用解析手法」と「周期的静電ポテンシャルへの埋め込みクラスターモデル」の開発を行ってきた．本総説では，これらの手法の概要を述べるとともに，Rh₂/AlPO₄，Rh₂/Al₂O₃への適用例を紹介する．

Figures

Figure 1.

 Structures of (A) Rh₂/AlPO₄ and (B) Rh₂/Al₂O₃ systems: Green, yellow, red, and black balls indicate Al, P, O, and Rh atoms, respectively. The second and lower layers are represented by pale colors. (Adapted with permission from M. Matsui, M. Machida, and S. Sakaki, J. Phys. Chem. C 2015, 119, 19752–19762. Copyright (2018) American Chemical Society.)

Figure 2.

 Orbital plots of the lowest unoccupied band (LUMO band) of equilibrium and deformed surfaces, and difference electron density map for AlPO₄ and Al₂O₃ systems. The red/blue colors indicate the different sign of the wave function. The cyan area represents an increase in electron density and the orange one does its decrease. (Adapted with permission from M. Matsui, M. Machida, and S. Sakaki, J. Phys. Chem. C 2015, 119, 19752–19762. Copyright (2018) American Chemical Society.)

Figure 3.

 (A-1) Density of states (DOS) of equilibrium and deformed surfaces of AlPO₄, and (A-2) DOS of Rh₂/AlPO₄ and p-DOS onto the LUMO band of deformed AlPO₄ surface. (B-1) DOS of equilibrium and deformed ones of Al₂O₃, and (B-2) DOS of Rh₂/Al₂O₃ and p-DOS onto the LUMO band of deformed Al₂O₃ one. Vertical pink line represents the Fermi level. The vacuum level is set to be zero. The negative density of states corresponds to the density of β spin states. (Adapted with permission from M. Matsui, M. Machida, and S. Sakaki, J. Phys. Chem. C 2015, 119, 19752–19762. Copyright (2018) American Chemical Society.)

Figure 4.

 Schematic representation of (A) slab model, (B) bare cluster model, and (C) embedded cluster model incorporating electrostatic potential. (Adapted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

Figure 5.

 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. (Reprinted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

Figure 6.

 Frontier orbitals of deformed 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 ų), 1.079 × 10⁵(250 × 250 × 15 ų), and 1.452 × 10⁶(900 × 900 × 15 ų) 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 ų), 7.551 × 10⁵(270 × 250 × 15 ų), and 1.016 × 10⁶(970 × 920 × 15 ų) for the AlPO₄'s with VSP, MP, and VLP, respectively. PE means periodic electrostatic potential. (Reprinted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

Figure 7.

 DOSs of the Rh₂/Al₂O₃ and Rh₂/AlPO₄, p-DOSs of HOMO and LUMO of the deformed Al₂O₃ and AlPO₄ surfaces of (A) slab model, (B) embedded cluster model with the periodic electrostatic potential (PE model). Black, red, and blue lines represent total DOS, p-DOS onto the LUMO band of the deformed Al₂O₃ or AlPO₄ surface, and p-DOS onto the HOMO band of the deformed 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 corresponds to the density of β spin states. (Adapted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

Tables

Table 1  Adsorption, deformation, and interaction energies together with important geometrical parameters of Rh₂/AlPO₄ and Rh₂/Al₂O₃ systems. (Adapted with permission from M. Matsui, M. Machida, and S. Sakaki, J. Phys. Chem. C 2015, 119, 19752–19762. Copyright (2018) American Chemical Society)

Table 2  Interaction energies (E_int; in eV) of Rh₂/Al₂O₃ and Rh₂/AlPO₄ slab and cluster models. (Reprinted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

^aNumbers of point charges are 1060 (24 × 28 × 15 ų), 11940 (83 × 83 × 15 ų), 1.079 × 10⁵(250 × 250 × 15 ų), 5.879×10⁵(590 × 590 × 15 ų), and 1.452 × 10⁶(920 × 920 × 15 ų) 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 ų), 8310 (84 × 84 × 15 ų), 7.551 × 10⁵(270 × 250 × 15 ų), 4.115 × 10⁵(620 × 580 × 15 ų), and 1.016 × 10⁶(970 × 920 × 15 ų) 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 E_int 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.

Table 3  Frontier orbital energies (ε_HOMO and ε_LUMO relative to vacuum level;^a eV) and band gaps of deformed Al₂O₃ and AlPO₄ surfaces. (Reprinted with permission from M. Matsui and S. Sakaki, J. Phys. Chem. C 2017, 121, 20242–20253. Copyright (2018) American Chemical Society.)

^aIn 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. ^cNumbers of point charges are 1060 (24 × 28 × 15 ų), 11940 (83 × 83 × 15 ų), 1.079 × 10⁵(250 × 250 × 15 ų), 5.879 × 10⁵(590 × 590 × 15 ų), and 1.452 × 10⁶(920 × 920 × 15 ų) 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 ų), 8310 (88 × 84 × 15 ų), 7.551 × 10⁵(270 × 250 × 15 ų), 4.115 × 10⁵(620 × 580 × 15 ų), and 1.016 × 10⁶(970 × 920 × 15 ų) for the AlPO₄'s with VSP, SP, MP, LP, and VLP, respectively. ^dPE indicates periodic electrostatic potential. ^eBand gap = ε_LUMO − ε_HOMO unless no caution is presented. ^fB3LYP calculation cannot be performed with the slab model in VASP. ^gFrontier orbitals close to HOMO or LUMO.^hSCF calculation cannot be converged in PBE calculations. ⁱHOMO−2. ^jε_LUMO − ε_HOMO–2. ^kLUMO+1. ^lHOMO−4.

^mε_LUMO+1 − ε_HOMO–4.

参考文献

• [1]   M. A. Newton, Chem. Soc. Rev., 37, 2644 (2008). doi:10.1039/b707746g19020678
• [2]   C. T. Campbell, Nat. Chem., 4, 597 (2012). doi:10.1038/nchem.141222824888
• [3]   M. Machida, K. Murakami, S. Hinokuma, K. Uemura, K. Ikeue, M. Matsuda, M. Chai, Y. Nakahara, T. Sato, Chem. Mater., 21, 1796 (2009). DOI:10.1021/cm9005844
• [4]   K. Ikeue, K. Murakami, S. Hinokuma, K. Uemura, D. Zhang, M. Machida, Bull. Chem. Soc. Jpn., 83, 291 (2010). doi:10.1246/bcsj.20090256
• [5]   M. Machida, S. Minami, K. Ikeue, S. Hinokuma, Y. Nagao, T. Sato, Y. Nakahara, Chem. Mater., 26, 5799 (2014). doi:10.1021/cm503061g
• [6]   M. Machida, S. Minami, S. Hinokuma, H. Yoshida, Y. Nagao, T. Sato, Y. Nakahara, J. Phys. Chem. C, 119, 373 (2015). doi:10.1021/jp509649r
• [7]   M. Machida, T. Eidome, S. Minami, H. P. Buwono, S. Hinokuma, Y. Nagao, Y. Nakahara, J. Phys. Chem. C, 119, 11653 (2015). doi:10.1021/acs.jpcc.5b01846
• [8]   J. H. Kwak, J. Hu, D. Mei, C. W. Yi, D. H. Kim, C. H. F. Peden, L. F. Allard, J. Szanyi, Science, 325, 1670 (2009). doi:10.1126/science.117674519779194
• [9]   A. Bruix, J. A. Rodriguez, P. J. Ramírez, S. D. Senanayake, J. Evans, J. B. Park, D. Stacchiola, P. Liu, J. Hrbek, F. Illas, J. Am. Chem. Soc., 134, 8968 (2012). doi:10.1021/ja302070k22563752
• [10]   R. S. Mulliken, J. Am. Chem. Soc., 74, 811 (1952). doi:10.1021/ja01123a067
• [11]   G. H. Booth, A. Grüneis, G. Kresse, A. Alavi, Nature, 493, 365 (2012). doi:10.1038/nature1177023254929
• [12]   C. Müller, B. Paulus, Phys. Chem. Chem. Phys., 14, 7605 (2012). doi:10.1039/c2cp24020c22373600
• [13]   L. Maschio, D. Usvyat, F. R. Manby, S. Casassa, C. Pisani, M. Schütz, Phys. Rev. B, 76, 075101 (2007). doi:10.1103/PhysRevB.76.075101
• [14]   S. Casassa, M. Halo, L. Maschio, C. Roetti, C. Pisani, Theor. Chem. Acc., 117, 781 (2007). doi:10.1007/s00214-006-0198-x
• [15]   C. Pisani, M. Schütz, S. Casassa, D. Usvyat, L. Maschio, M. Lorenz, A. Erba, Phys. Chem. Chem. Phys., 14, 7615 (2012). doi:10.1039/c2cp23927b22334044
• [16]   P. Y. Ayala, K. N. Kudin, G. E. Scuseria, J. Chem. Phys., 115, 9698 (2001). doi:10.1063/1.1414369
• [17]   D. Usvyat, B. Civalleri, L. Maschio, R. Dovesi, C. Pisani, and M. Schütz, J. Chem. Phys., 134, 214105 (2011). PMID:21663342 DOI:10.1063/1.3595514
• [18]   M. Marsman, A. Grüneis, J. Paier, G. Kresse, J. Chem. Phys., 130, 184103 (2009). doi:10.1063/1.312624919449904
• [19]   A. Grüneis, G. H. Booth, M. Marsman, J. Spencer, A. Alavi, G. Kresse, J. Chem. Theory Comput., 7, 2780 (2011). doi:10.1021/ct200263g26605469
• [20]   S. J. Nolan, M. J. Gillan, D. Alfè, N. L. Allan, F. R. Manby, Phys. Rev. B, 80, 165109 (2009). doi:10.1103/PhysRevB.80.165109
• [21]   S. J. Nolan, P. J. Bygrave, N. L. Allan, F. R. Manby, J. Phys. Condens. Matter, 22, 074201 (2010). doi:10.1088/0953-8984/22/7/07420121386379
• [22]   T. Shiozaki, S. Hirata, J. Chem. Phys., 132, 151101 (2010). doi:10.1063/1.339607920423161
• [23]   M. Matsui, J. Phys. Chem. C, 118, 19294 (2014). doi:10.1021/jp5047084
• [24]   M. Matsui, M. Machida, S. Sakaki, J. Phys. Chem. C, 119, 19752 (2015). doi:10.1021/acs.jpcc.5b02691
• [25]   M. Matsui, S. Sakaki, J. Phys. Chem. C, 121, 20242 (2017). doi:10.1021/acs.jpcc.7b04126
• [26]   R. F. W. Bader, Atoms in Molecules: A Quantum Theory (Oxford University Press), 1990.
• [27]   G. Henkelman, A. Arnaldsson, H. Jónsson, Comput. Mater. Sci., 36, 354 (2006). doi:10.1016/j.commatsci.2005.04.010
• [28]   E. Sanville, S. D. Kenny, R. Smith, G. Henkelman, J. Comput. Chem., 28, 899 (2007). doi:10.1002/jcc.2057517238168
• [29]   W. Tang, E. Sanville, G. Henkelman, J. Phys. Condens. Matter, 21, 084204 (2009). doi:10.1088/0953-8984/21/8/08420421817356
• [30]   M. Yu, D. R. Trinkle, J. Chem. Phys., 134, 064111 (2011). doi:10.1063/1.355371621322665
• [31]   R. S. Mulliken, J. Chem. Phys., 23, 1833 (1955). doi:10.1063/1.1740588
• [32]   T. Hughbanks, R. Hoffmann, J. Am. Chem. Soc., 105, 3528 (1983). doi:10.1021/ja00349a027
• [33]   R. Dronskowski, P. E. Blöchl, J. Phys. Chem., 97, 8617 (1993). doi:10.1021/j100135a014
• [34]   D. Sańchez-Portal, E. Artacho, J. M. Soler, Solid State Commun., 95, 685 (1995). doi:10.1016/0038-1098(95)00341-X
• [35]   D. Sánchez-Portal, E. Artacho, J. M. Soler, J. Phys. Condens. Matter, 8, 3859 (1996). doi:10.1088/0953-8984/8/21/012
• [36]   M. D. Segall, C. J. Pickard, R. Shah, M. C. Payne, Mol. Phys., 89, 571 (1996). doi:10.1080/002689796173912
• [37]   M. D. Segall, R. Shah, C. J. Pickard, M. C. Payne, Phys. Rev. B, 54, 16317 (1996). doi:10.1103/PhysRevB.54.16317
• [38]   V. L. Deringer, A. L. Tchougréeff, R. Dronskowski, J. Phys. Chem. A, 115, 5461 (2011). doi:10.1021/jp202489s21548594
• [39]   N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, D. Vanderbilt, Rev. Mod. Phys., 84, 1419 (2012). doi:10.1103/RevModPhys.84.1419
• [40]   S. Dapprich, G. Frenking, J. Phys. Chem., 99, 9352 (1995). doi:10.1021/j100023a009
• [41]   H. A. Graetsch, Acta Crystallogr. C, 57, 665 (2001). doi:10.1107/S010827010100513311408662
• [42]   M. Digne, P. Sautet, P. Raybaud, P. Euzen, H. Toulhoat, J. Catal., 226, 54 (2004). doi:10.1016/j.jcat.2004.04.020
• [43]   X. R. Shi, D. S. Sholl, J. Phys. Chem. C, 116, 10623 (2012). doi:10.1021/jp301114n
• [44]   G. Kresse, J. Hafner, Phys. Rev. B, 47, 558 (1993). doi:10.1103/PhysRevB.47.558
• [45]   G. Kresse, J. Furthmüller, Comput. Mater. Sci., 6, 15 (1996). DOI:10.1016/0927-0256(96)00008-0
• [46]   G. Kresse, J. Furthmüller, Phys. Rev. B, 54, 11169 (1996). doi:10.1103/PhysRevB.54.11169
• [47]   F. Libisch, C. Huang, E. A. Carter, Acc. Chem. Res., 47, 2768 (2014). doi:10.1021/ar500086h24873211
• [48]   N. Govind, Y. A. Wang, A. J. R. da Silva, E. A. Carter, Chem. Phys. Lett., 295, 129 (1998). doi:10.1016/S0009-2614(98)00939-7
• [49]   N. Govind, Y. A. Wang, E. A. Carter, J. Chem. Phys., 110, 7677 (1999). doi:10.1063/1.478679
• [50]   P. Huang, E. A. Carter, J. Chem. Phys., 125, 084102 (2006). doi:10.1063/1.233642816964996
• [51]   C. Huang, M. Pavone, E. A. Carter, J. Chem. Phys., 134, 154110 (2011). doi:10.1063/1.357751621513378
• [52]   F. Libisch, C. Huang, P. Liao, M. Pavone, E. A. Carter, Phys. Rev. Lett., 109, 198303 (2012). doi:10.1103/PhysRevLett.109.19830323215432
• [53]   S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V. Brown, J. Cheng, J. B. Lassiter, E. A. Carter, P. Nordlander, N. J. Halas, Nano Lett., 13, 240 (2013). doi:10.1021/nl303940z23194158
• [54]   K. Yu, F. Libisch, E. A. Carter, J. Chem. Phys., 143, 102806 (2015). doi:10.1063/1.492226026373999
• [55]   K. Yu, E. A. Carter, Proc. Natl. Acad. Sci. USA, 114, E10861 (2017). doi:10.1073/pnas.171261111429203675
• [56]   J. D. Goodpaster, T. A. Barnes, F. R. Manby, T. F. Miller, III, J. Chem. Phys., 137, 224113 (2012). doi:10.1063/1.477022623248993
• [57]   F. R. Manby, M. Stella, J. D. Goodpaster, T. F. Miller, III, J. Chem. Theory Comput., 8, 2564 (2012). doi:10.1021/ct300544e22904692
• [58]   G. Knizia, G. K. L. Chan, Phys. Rev. Lett., 109, 186404 (2012). doi:10.1103/PhysRevLett.109.18640423215304
• [59]   A. M. Burow, M. Sierka, J. Döbler, J. Sauer, J. Chem. Phys., 130, 174710 (2009). doi:10.1063/1.312352719425801
• [60]   N. A. Richter, S. Sicolo, S. V. Levchenko, J. Sauer, M. Scheffler, Phys. Rev. Lett., 111, 045502 (2013). doi:10.1103/PhysRevLett.111.04550223931382
• [61]   C. Tuma, J. Sauer, Chem. Phys. Lett., 387, 388 (2004). DOI:10.1016/j.cplett.2004.02.056
• [62]   C. Tuma, J. Sauer, Phys. Chem. Chem. Phys., 8, 3955 (2006). doi:10.1039/B608262A17028686
• [63]   S. Tosoni, J. Sauer, Phys. Chem. Chem. Phys., 12, 14330 (2010). doi:10.1039/c0cp01261k20886145
• [64]   A. D. Boese, J. Sauer, Phys. Chem. Chem. Phys., 15, 16481 (2013). doi:10.1039/c3cp52321g23949344
• [65]   A. Mazheika, S. V. Levchenko, J. Phys. Chem. C, 120, 26934 (2016). doi:10.1021/acs.jpcc.6b09505
• [66]   K. N. Kudin, G. E. Scuseria, Chem. Phys. Lett., 283, 61 (1998). doi:10.1016/S0009-2614(97)01329-8
• [67]   K. N. Kudin, G. E. Scuseria, J. Chem. Phys., 121, 2886 (2004). doi:10.1063/1.177163415291598
• [68]   M. W. Schmidt, K. K. Baldridge, J. A. Boatz, S. T. Elbert, M. S. Gordon, J. H. Jensen, S. Koseki, N. Matsunaga, K. A. Nguyen, S. Su, T. L. Windus, M. Dupuis, J. A. Montgomery, J. Comput. Chem., 14, 1347 (1993). doi:10.1002/jcc.540141112
• [69]   M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E. Scuseria, M. A. Robb, J. R. Cheeseman, G. Scalmani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li, M. Caricato, A. Marenich, J. Bloino, B. G. Janesko, R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Ortiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Goings, B. Peng, A. Petrone, T. Henderson, D. Ranasinghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng, W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, T. Vreven, K. Throssell, J. A. Montgomery, Jr., J. E. Peralta, F. Ogliaro, M. Bearpark, J. J. Heyd, E. Brothers, K. N. Kudin, V. N. Staroverov, T. Keith, R. Kobayashi, J. Normand, K. Raghavachari, A. Rendell, J. C. Burant, S. S. Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene, C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin, K. Morokuma, O. Farkas, J. B. Foresman, D. J. Fox, Gaussian, Inc., Wallingford CT, 2016.