Local Charge Distribution in GaxPdy Intermetallics: Characterizing Catalyst Surfaces from Large-Scale Molecular Mechanics Simulations

1 Local charge distribution in GaxPdy intermetallics: characterizing catalyst surfaces from large-scale molecular mechanics simulations Tanakorn Wonglakhon,1 Sven Maisel,2 Andreas Görling,2 Dirk Zahn1,* 1 Lehrstuhl für Theoretische Chemie / Computer Chemie Centrum Friedrich-Alexander Universität Erlangen-Nürnberg Nägelsbachstraße 25, 91052 Erlangen, Germany 2 Lehrstuhl für Theoretische Chemie, Friedrich-Alexander Universität Erlangen-Nürnberg Egerlandstraße 3, D-91058 Erlangen, Germany *Email: [email protected] 2 Abstract We combine the charge equilibration (QEq) method with the modified embedded atom model (MEAM) to describe a series of intermetallic GaxPdy compounds at near DFT accuracy. Apart from structure, energetics and elastic properties, a particular focus is dedicated to the partial charges on Ga and Pd sites in the bulk and on flat/terraced surfaces. By the example of GaPd2, we suggest a computationally very efficient approach to assessing the crystal faces and steps of interesting prospect for catalytic activity. To this end, we suggest enhanced catalytic activity of (010) faces by our simulation models that demonstrate particularly large charge transfer between surface Ga and Pd species, namely +0.8 and -0.4, whereas for the (100) and (001) faces local polarization is less than +0.6 and -0.3, respectively. Moreover, the study of rough surfaces is demonstrated from a small series of 10 nm sized simulation models featuring terraces. Local polarization of the atoms at the steps ranges from +0.5 to +1.1 and -0.5 to -0.3 for the Ga and Pd species, respectively. Keywords: Partial charges, Ga-Pd Intermetallic Compounds, MEAM Potentials, QEq method, catalytic activity 3 Introduction Intermetallic compounds of gallium (Ga) and palladium (Pd) have attracted considerable attention in catalysis research due to their potential in promoting various chemical reactions, including the partial hydrogenation of acetylene [1–5], methanol steam reforming [6,7], methanol and dimethyl ether syntheses [8–11], and methanol oxidation [12]. Since the first publication of the Ga-Pd phase diagram [13,14], the diverse intermetallic phases of GaxPdy, particularly GaPd2, has intensified interest in understanding the diverse intermetallic phases of GaxPdy and their catalytic properties [15]. To this end, recent studies by Armbrüster et al. [2,16] highlighted the exceptional selectivity of Ga7Pd3, GaPd, and GaPd2 in semi-hydrogenation reactions, surpassing conventional commercial catalysts. From the viewpoint of atomic simulation, density functional theory or quantum chemical calculations may provide valuable insights into the electronic and structural influences on catalytic activity. Notably, investigations into the electronic and structural influences on the catalytic activity of GaPd2 compounds, with Sn substitution on Ga atom as Ga1-xSnxPd2 (0 ≤ x ≤ 1), have revealed correlations between pronounced charge transfer from p elements to palladium and catalytic activity variation [17,18]. While informative, these theoretical studies are limited by computational cost and scalability, restricting the size of the models to hundreds of atoms. Thus, characterization of surfaces is limited to nm scale slab models and investigations of surface roughness are seriously hindered by the drastic computational effort needed. As a cost-effective alternative to DFT, molecular mechanics models featuring the charge equilibration method (QEq) offer near-DFT accuracy in the frame-work of interatomic potentials [19]. For intermetallic Ga-Pd systems, so far embedded atom model (EAM) potentials [20,21] were developed, however not yet incorporating the QEq approach [22]. To clearly discriminate the charge transfer between the two metal species and the metallic bonding in more general terms, we argue that a joint fitting process is needed to obtain a comprehensive combination of QEq and EAM potentials. 4 This is the aim of the present study from a modelling viewpoint. In turn, from the perspective of applications, we shall take use of the reduced computational costs to outline the study of complex catalyst surfaces and thus pave the road to millions of atoms models. Methods and Models DFT calculations The density functional theory (DFT) calculations were performed using the Vienna Abinitio Simulation Package (VASP) [23,24] with the projector augmented wave (PAW) approach [25] employed to represent the atomic cores of Ga and Pd atoms. The Perdew-Burke-Ernzerhof (PBE) exchange correlation functional [26] was utilized, along with a plane wave basis set with a kinetic energy cutoff of 700 eV. Convergence criteria included a tight threshold of 10-8 eV for wave functions and a first-order Methfessel-Paxton smearing with a width of 0.2 eV [27]. The structural characterization of solid GaxPdy intermetallic compounds relies on unit cell models and 3D periodic boundary conditions. Structural relaxation was performed until all forces were below 0.001 eV/Å. Γ-including k-point meshes of varying dimensions (11x7x11, 8x8x5, 6x6x6, 10x10x10, 9x5x12, 9x12x6, 9x12x3, and 13x13x13) were used in geometry optimization for Ga, Ga5Pd, Ga7Pd3, GaPd, Ga3Pd5, GaPd2, Ga2Pd5, and Pd, respectively. Additionally, for the calculation of mechanical properties, the mesh size was doubled. In VASP, the elastic tensor was computed through strain-stress analysis of the equilibrium structure. Six finite distortions were applied, and elastic constants were derived from the resulting strainstress relationship [28]. Bader atomic charge calculation was utilized to determine the partial charges of Ga and Pd atoms for all systems discussed [29–32]. EAM and modified EAM potentials 5 The EAM potential is widely used to model metals and alloys due to its ability to account for coordination-dependent binding strength. For Ga-Pd systems, we recently presented an EAM potential for modelling Ga-Pd intermetallic crystals and liquid alloys [22]. Essentially, the EAM is expressed as 𝐸=𝐹(𝜌) +1 2 𝜙𝑟, ,, (1) 𝜌= 𝑓𝑟 , ,(2) where 𝐹(𝜌) is the energy for embedding an atom i into a site associated with the background electron density 𝜌. This term was derived from the summation of the densities of surrounding neighbors 𝑓(𝑟), eq. (2), therefore adding the many-body interaction to the two-body interaction described by 𝜙(𝑟). Various formulae for the two terms in eqs. 1 and 2 have been proposed; our previous EAM potential for the Ga-Pd system is based on the formalism proposed by Mei et al. [33]. Although the EAM potential is sufficient for modelling various metals and alloys, for some materials with covalent bonds, such as silicon and gallium, modification of the EAM potential is needed. For this, Baskes modified the EAM to include angular terms, leading to the modified EAM potential (MEAM). Full details of the MEAM potentials are not repeated here and we instead refer to the references [34–36]. In what follows, only few aspects of the MEAM are discussed in detail to describe the background of the underlying parameters. For the MEAM potential, the embedding energy 𝐹(𝜌) is essentially the same as that in the EAM potential proposed by Baskes [36] and is written as 𝐹(𝜌)=𝐴𝐸𝜌 𝜌ln𝜌 𝜌, (3) where A is an adjustable parameter (fitted to DFT), and 𝐸 and 𝜌 are the cohesive energy and the electron density of the reference structure, respectively. The main change to the EAM potential lies in the treatment of the background electron density 6 𝜌. In the EAM potential 𝜌 originates from the superposition of spherical partial electron densities centered on neighboring atoms j (eq. 2), whereas in the MEAM, this is modified to take an angular term into account by weights originating to the projections of the distances between atoms i and j in x, y, and z directions - see reference [37] for details. The combination of those angular dependent terms reads Γ=𝑡()(𝜌/𝜌),   (4) where 𝑡 (l=1,3) are constants fitted to data from DFT calculations. The atomic electron densities are used to calculate the angular-dependent partial electron densities and are represented as exponential functions with decay constants 𝛽() (k = 0-3) fitted to DFT calculations. In turn, 𝜌 is taken as 𝜌=𝜌√1 + Γ . (5) For the pair potential, the same formula as in the EAM potential developed by Baskes is employed here: 𝜙𝑟=2 𝑍𝐸(𝑅)−𝐹𝜌(𝑅),(6) where Z is the number of nearest neighbor atoms of the reference structure, 𝐸(𝑅) is the energy per atom of the reference structure as a function of nearest neighbor distance R, and 𝐹𝜌(𝑅) has the same definition as that in eq. 1, but refers to the reference structure. The 𝐸(𝑅) can be calculated either from first-principles calculations or from the universal equation of state by Rose et al. [38] This universal equation of state, which is also used by Baskes et al, is written as 𝐸𝑟=−𝐸󰇡1+𝑎∗+𝑟 𝑅𝛿𝑎∗󰇢𝑒∗,(7) 𝑎∗=𝛼𝑅 𝑟−1, (8) 7 𝛼=9Ω𝐵 𝐸,(9) where Ω and B are the equilibrium atomic volume and bulk modulus of the reference structure, respectively, and the 𝛿 parameter in eq. 7 is the adjustable parameter used to make the potential more flexible and thus better represent the bulk modulus of the reference material. In the MEAM potential, the pair potential and the atomic electron density are scaled by the angular screening factor (𝑆) as first proposed by Baskes et al. as follows 𝑆 = 𝑆 , ,(10) where 𝑆 describes how an atom k screens the interaction between atoms i and j. This is controlled by the C parameter in the ellipse equation: 𝑥+𝑦 𝐶=𝑅 2,(11) 𝐶=2𝑋 +𝑋−𝑋 −𝑋−1 1−𝑋 −𝑋,(12) 𝑋 =𝑅 𝑅and 𝑋 =𝑅 𝑅.(13) By drawing the ellipse on three atoms and imposing the two limiting ellipses via 𝐶 𝑎𝑛𝑑 𝐶 parameters, the magnitude of screening of the ij interaction can be determined by the positions of the j atoms. If atom j is outside the outer ellipse (C > 𝐶), 𝑆 =0 and the ij interaction is completely unscreened. On the other hand, if atom j is inside the inner ellipse, 𝑆 =1 and the ij interaction is completely active. In 8 turn, when atom j lies between the inner and outer ellipses, smooth screening is provided via: 𝑆 =󰇩1− 𝐶 −𝐶 𝐶 −𝐶󰇪.(14) This screening factor also determines if the partial electron density has the contribution from the second-nearest neighbor; by slightly lowering the 𝐶 value (which is usually set to a default value of 2), the second-nearest neighbor is considered. The MEAM potential in this study also considers the contribution from the second nearest neighbor. Putting all parts together, in this study, we created the MEAM for the Ga-Pd intermetallic compounds based on the elementary MEAM potentials for Ga developed by Baskes et al. [37] and Pd by Lee et al. [39], but create newly fitted parameters stemming from integrating energy/forces derived from the MEAM and the QEq methods. Charge equilibration method (QEq) To allow the modelling of charge transfer and polarization effects in Ga-Pd systems, the charge equilibration method (QEq) is coupled with the EAM and MEAM potentials, respectively. The formulation of total energy according to eq. 1 is thus extended to: 𝐸 =𝐹(𝜌) +1 2 𝜙𝑟 ,, +𝐸(𝑄) + 𝑄𝑄 4𝜋𝜀𝑟  (15) where 𝐸(𝑄) is the energy of atom i as a function of charge Q from the QEq method. Expanding 𝐸(𝑄) as a Taylor series from the charge-neutral metal atom up to the second order leads to 9 𝐸(𝑄)=𝐸(0)+𝑄∙𝜕𝐸 𝜕𝑄+1 2𝑄∙󰇧𝜕𝐸 𝜕𝑄󰇨󰈅(16) Therein, the two derivatives are interpreted as QEq-specific force-field parameters that control the polarizability of the individual atoms i. The most common approach to assessing this data from the ionization potential IP and the electron affinity EA of the single atom, indeed by adding / subtracting 𝐸(−1) and 𝐸(+1) we get: 𝜕𝐸 𝜕𝑄=1 2(𝐼𝑃+𝐸𝐴)=𝜒(17) and 󰇧𝜕𝐸 𝜕𝑄󰇨=𝐼𝑃−𝐸𝐴=𝐽  (18) where 𝜒 and 𝐽  are interpreted as the electronegativity and the “self-Coulomb potential” of the isolated atom i, respectively. The constant 𝐽  equals to the atomic hardness 𝜂 multiplied by 2. By adding the Coulomb interactions to the summation of energies of all atoms, we then obtain the total electrostatic energy of all N atoms as 𝐸(𝑄,…,𝑄)=𝐸(0)+𝑄∙𝜒+1 2𝑄∙𝐽 + 𝑄𝑄 4𝜋𝜀𝑟  (19) The derivative of 𝐸(𝑄,…,𝑄) with respect to 𝑄 leads to the electronegativity 𝜒(𝑄) of atom i in the system: 𝜒(𝑄)=𝜒+𝑄∙𝐽 + 𝑄𝑄 4𝜋𝜀𝑟  (20) Based on Sanderson’s principle, the electronegativity of all atoms within the metal particle or bulk crystal will equalize: 𝜒(𝑄)=𝜒(𝑄)=⋯=𝜒(𝑄) , (21) with the restriction that the total charge of the system must be conserved. 16 bulk crystals, this can be accomplished by means of DFT calculations and Bader analysis for assigning local electron density to atomic sites. In table 4, we show the partial charges observed in a series of GaxPdy crystals to compare the DFT-based reference with the QEq charges as obtained from the EAM/QEqexp, MEAMmix/QEqexp and MEAMnew/QEqnew methods. We find that the local charges predicted by the MEAMnew/QEqnew approach best reproduce the Bader charges as obtained from the DFT reference. This particularly holds for the GaPd2 structure which is investigated for surface effects in the following. crystal DFT (Bader charge) EAM/QEqexp MEAMmix/QEqexp MEAMnew/QEqnew Ga Pd Ga Pd Ga Pd Ga Pd Ga5Pd 0.12 x 3, 0.15 x 10, 0.18 x 3, 0.23 x 4 (3.32) -0.83 x 4 (-3.32) 0.0308 x 4, 0.0315 x 16 (0.63) -0.157 x 4 (-0.63) 0.0313 x 4, 0.0314 x 16 (0.63) -0.157 x 4 (-0.63) 0.063 x 4, 0.070 x 16, (1.37) -0.343 x 4 (-1.37) Ga7Pd3 0.29 x 6, 0.30 x 10, 0.44 x 12 (10.02) -0.82 x 6, -0.85 x 6 (-10.02) 0.060 x 16, 0.059 x 12 (1.67) -0.139 x 12 (-1.67) 0.0608 x 4, 0.0582 x 8, 0.0593 x 16 (1.66) -0.137 x 4, -0.139 x 8 (-1.66) 0.1501 x 4, 0.1557 x 8, 0.1413 x 16 (4.11) -0.341 x 4, -0.343 x 8 (-4.11) GaPd 0.50 x 4 (2.00) -0.50 x 4 (-2.00) 0.11 x 4 (0.44) -0.11 x 4 (-0.44) 0.11 x 4 (0.44) -0.11 x 4 (-0.44) 0.33 x 4 (1.32) -0.33 x 4 (-1.32) Ga3Pd5 0.58 x 2, 0.59 x 4 (3.52) -0.33 x 4, -0.34 x 2, -0.38 x 4 (-3.52) 0.139 x 4, 0.138 x 2 (0.83) -0.084 x 6, -0.028 x 4 (-0.83) 0.1383 x 4, 0.1356 x 2 (0.82) -0.0829 x 2 -0.0831 x 4, -0.0815 x 4, (-0.82) 0.51 x 4, 0.52 x 2 (3.08) -0.307 x 2, -0.312 x 2, -0.311 x 2, -0.304 x 4, (-3.08) GaPd2 0.61 x 4 (2.44) -0.28 x 4, -0.33 x 4 (-2.44) 0.15 x 4 (0.60) -0.07 x 4, -0.08 x 4 (-0.60) 0.15 x 4 (0.60) -0.0743 x 4, -0.0744 x 4 (-0.60) 0.60 x 4 (2.40) -0.301 x 2, -0.302 x 2, -0.297 x 2, -0.298 x 2 (-2.40) Ga2Pd5 0.63 x 4, 0.65 x 4 (5.12) -0.22 x 4, -0.23 x 4, -0.26 x 4, -0.27 x 4, -0.30 x 4 (-5.12) 0.162 x 4, 0.165 x 4 (1.31) -0.066 x 4, -0.063 x 4, -0.065 x 8, -0.068 x 4 (-1.31) 0.158 x 4, 0.165 x 4 (1.29) -0.066 x 4, -0.062 x 4, -0.065 x 4, -0.067 x 4, -0.063 x 4 (-1.29) 0.658 x 4, 0.745 x 4 (5.61) -0.286 x 4, -0.269 x 4, -0.278 x 4, -0.295 x 4, -0.275 x 4 (-5.61) Table 4. Comparison of the partial charges of Ga and Pd atoms in GaxPdy crystals as calculated with DFT, EAM/QEqexp, MEAMmix/QEqexp, and MEAMnew/QEqnew methods. The numbers shown in parentheses refer to the total charge of all Ga and Pd species of the corresponding unit cell, respectively. To one side the improvement achieved by our MEAMnew/QEqnew model appears particularly useful for the investigation of physical vapor deposition processes. To the 17 other, a somewhat related topic is the characterization of rough surfaces, featuring island, plateaus or kinks. Surface reorganization upon such imperfections clearly benefits from interaction models that provide reliable cohesive energy – in terms of both structure and energy. However, with the help of the QEq method, we can now also characterize the distribution of local charges. The MEAMnew/QEqnew not only reasonably reproduces the DFT-Bader charges of the bulk crystal, but also offers the required computational efficiency to assess large scale slab models featuring rough surface landscapes – thus offering more sophisticated structure models to help accounting for “real catalyst” surfaces. This is illustrated in figure 1 for a small series of “rough surface” models, namely the (010) face of GaPd2 featuring a) steps, b) single ad-atom islands and c) holes resulting from removing single atoms from the ideal surface. The main driving force for charge transfer in the alloy is the difference in electronegativity of Ga and Pd. However, at surface sites we find individual electrostatic environment according to the local coordination by nearby atoms. Indeed, on the terrace model (fig. 1a), we find that Pd atoms on the plateau surface show -0.32 to -0.37 charge, whereas the steps feature -0.35 to -0.38 charge at the linear edge and -0.33 to -0.42 near kinks (k1,k2,k3). Likewise, the exposed Pd ad-atoms (fig.1b) shows different charging subject to local coordination, namely -0.40 for GaPd3 cluster-type arrangements as compared to -0.31 for Ga2Pd2 type motifs, respectively. Regarding holes in the (010) surface, (fig. 1c), Pd atoms next to a Ga vacancy (h1) and Pd vacancy (h3) show particularly negative charge up to -0.42. We argue that the charges of Ga and Pd atoms on the different GaPd2 surfaces are of paramount importance for adsorption and activation processes, such as the partial hydrogenation of acetylene. To this end, the extend of charge transfer stemming from the combination of alloying and surface effects at the individual catalyst sites is suggested as indicators for electron donating/withdrawing effects. 18 Figure 1. Local charge distributions of Ga and Pd atoms in different regions of rough GaPd 2 (010) surface models. Panels (a), (b), and (c) refer to a terrace model, adatoms and holes, respectively. The left side of each panel shows surface meshes outlining these areas, while the right side highlights atomic charge via a color code. The charge transfer stems from the difference in electronegativity of the Ga/Pd atomics and the local Coulomb interactions. On this basis, the negative (blue color) charge of Pd atoms show significant variation, namely from -0.29 to -0.44, upon local coordination at edges, island or holes, respectively. 19 To further elaborate this field of application for the MEAM new /QEq new potential, we also investigated the (001), (010), and (100) faces of GaPd 2 . In particular for semihydrogenation, this compound was demonstrated to provide excellent catalytic activity from the experiments of Armbrüster and co-workers [1,16]. Based on the MEAM new /QEq new potential, we calculated specific surface energies for the (001), (010), and (100) faces as 5.09, 1.39, and 3.73 J/m 2 , respectively. This indicates that the (010) face exhibits the highest stability, consistent with experimental findings [46]. Moreover, the surface energy of the (010) face was also explored from DFT calculations [51], leading to an ab-initio-based reference value of 1.31 J/m 2 – which we find nicely reproduced by the MEAM new /QEq new potential. The underlying GaPd 2 slabs were chosen generously large to ensure bulk behavior in the center (see table 3 for atomic charges), whilst the surface effects are monitored in terms of the charge transfer between surface atoms as illustrated in figure 2. In average, we find electron transfer to surface Pd atoms strongest for the (010) face of GaPd 2 . This suggests that the catalyst performance of the (010) face for semihydrogenation reaction may outperform the (100) and (001) faces. This inference is supported by the observed enhancement in electron transfer, which correlates with improved catalytic activity [17,18]. Figure 2. Comparison of GaPd 2 slabs featuring a) (001), b) (010), and c) (100) faces. Atomic charges as calculated from the MEAM new /QEq new model are illustrated by a color code. Atom notations include top1, top2, top3, and top4 for the topmost, second top, third top, and fourth top layers, respectively, while top1a and top1b denote atoms at the same topmost layer but different lateral positions. 20 Conclusion Our study underlines the importance of a systematic approach in developing accurate models for atomic charges in alloys. The original QEq approach based on parameters fitted to reproduce single-atom electron affinity and ionization energy is surely suited for metal and metal alloy clusters, but needs careful confirmation when investigating bulk crystals or surfaces thereof. Indeed, the reasonable performance of common EAM and MEAM potentials in absence of QEq stems from the implicit modelling of charge transfers within the hetero-terms of the atom-atom potentials. To that end, we recently fitted a new EAM potential for Ga-Pd interactions to better describe the mechanical properties of intermetallic GaxPdy compounds. In turn, when employing QEq as explicit term, we suggest to fully re-parameterize both, the QEq and the EAM/MEAM potentials. On this basis, we demonstrated the characterization of the catalyst GaPd2 at near DFT accuracy. The computational efficiency of our MEAMnew/QEqnew model outperforms DFT calculations by several orders of magnitude (subject to the choice of the basis set). Our approach therefore offers a new perspective into exploring complex alloy systems, such as extended crystallites, slab models or rough surfaces – as demonstrated by the case studies presented in this work. 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