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

Abstract

We combine the charge equilibration (QEq) method with the modified embedded atom model (MEAM) to describe a series of intermetallic Ga_xPd_y 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.

1. 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 [1,2,3,4,5], methanol steam reforming [6,7], methanol and dimethyl ether syntheses [8,9,10,11], and methanol oxidation [12]. Since the first publication of the Ga-Pd phase diagram [13,14], the diverse intermetallic phases of Ga_xPd_y, particularly GaPd₂, has intensified interest in understanding the diverse intermetallic phases of Ga_xPd_y 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 (DFT) 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 Ga_1-xSn_xPd₂ (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.

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.

2. Methods and Models

2.1. DFT Calculations

The DFT Calculations were performed using the Vienna Ab-initio 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 Ga_xPd_y 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 (11 × 7 × 11, 8 × 8 × 5, 6 × 6 × 6, 10 × 10 × 10, 9 × 5 × 12, 9 × 12 × 6, 9 × 12 × 3, and 13 × 13 × 13) 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 strain-stress relationship [28]. Bader atomic charge calculation was utilized to determine the partial charges of Ga and Pd atoms for all systems discussed [29,30,31,32].

2.2. EAM and Modified EAM Potentials

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

$$E=\sum _i{F}_i\left({\rho }_i\right)+\frac{1}{2}\sum _{i,j,i\ne j}{\varphi }_{ij}\left(r_{ij}\right),$$

(1)

$${\rho }_i=\sum _{j,j\ne i}{f}_j\left(r_{ij}\right),$$

(2)

where $F_i\left({\rho }_i\right)$ is the energy for embedding an atom i into a site associated with the background electron density ${\rho }_i$. This term was derived from the summation of the densities of surrounding neighbors $f_j\left(r_{ij}\right)$, Equation (2), therefore adding the many-body interaction to the two-body interaction described by ${\varphi }_{ij}\left(r_{ij}\right)$. Various formulae for the two terms in Equations (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 [34,35,36]. 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 $F_i\left({\rho }_i\right)$ is essentially the same as that in the EAM potential proposed by Baskes [36] and is written as

$$F_i\left({\rho }_i\right)=A{E}_c\frac{{\rho }_i}{{\rho }^0}\ln \left(\frac{{\rho }_i}{{\rho }^0}\right),$$

(3)

where A is an adjustable parameter (fitted to DFT), and $E_c$ and ${\rho }_0$ 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 ${\rho }_i$. In the EAM potential ${\rho }_i$ originates from the superposition of spherical partial electron densities centered on neighboring atoms j (Equation (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

$$\mathsf{\Gamma }=\sum _{l=1}^3{t}^{\left(l\right)}{\left({\rho }^l/{\rho }^0\right)}^2,$$

(4)

where $t^l$ (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 ${\beta }^{\left(k\right)}$ (k = 0–3) fitted to DFT calculations. In turn, ${\rho }_i$ is taken as

$${\rho }_i={\rho }^0\sqrt{1+\mathsf{\Gamma }}.$$

(5)

For the pair potential, the same formula as in the modified EAM potential developed by Baskes et al. [35,36,37], is employed here:

$${\varphi }_{ij}\left(r_{ij}\right)=\frac{2}{Z}\left\{E^u\left(R\right)-F\left({\rho }^0\left(R\right)\right)\right\},$$

(6)

where Z is the number of nearest neighbor atoms of the reference structure, $E^u\left(R\right)$ is the energy per atom of the reference structure as a function of nearest neighbor distance R, and $F\left({\rho }^0\left(R\right)\right)$ has the same definition as that in Equation (1), but refers to the reference structure. The $E^u\left(R\right)$ 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. [34,35,36], is written as

$$E^u\left(r_{ij}\right)=-E_c\left(1+a^{*}+\frac{r_e}{R}\delta a^{{*}^3}\right)e^{-a^{*}},$$

(7)

$$a^{*}=\alpha \left(\frac{R}{r_e}-1\right),$$

(8)

$${\alpha }^2=\frac{9\mathsf{\Omega }B}{E_c},$$

(9)

where $\mathsf{\Omega }$ and B are the equilibrium atomic volume and bulk modulus of the reference structure, respectively, and the $\delta$ parameter in Equation (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 ($S_{ik})$ as first proposed by Baskes [36] as follows

$$S_{ik}=\prod _{j\ne i,k}{S}_{ijk},$$

(10)

where $S_{ijk}$ describes how an atom k screens the interaction between atoms i and j. This is controlled by the C parameter in the ellipse equation:

$$x^2+\frac{y^2}{C}={\left(\frac{R_{ik}}{2}\right)}^2,$$

(11)

$$C=\frac{2\left(X_{ij}+X_{jk}\right)-{\left(X_{ij}-X_{jk}\right)}^2-1}{1-{\left(X_{ij}-X_{jk}\right)}^2},$$

(12)

$$X_{ij}={\left(\frac{R_{ij}}{R_{ik}}\right)}^2\mathrm{and}{X}_{jk}={\left(\frac{R_{jk}}{R_{ik}}\right)}^2.$$

(13)

By drawing the ellipse on three atoms and imposing the two limiting ellipses via $C_{min}and{C}_{max}$ 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 > $C_{max}$), $S_{ijk}=0$ and the ij interaction is completely unscreened. On the other hand, if atom j is inside the inner ellipse, $S_{ijk}=1$ and the ij interaction is completely active. In turn, when atom j lies between the inner and outer ellipses, smooth screening is provided via:

$$S_{ijk}={\left[1-{\left(\frac{C_{max}-C}{C_{max}-C_{min}}\right)}^4\right]}^2.$$

(14)

This screening factor also determines if the partial electron density has the contribution from the second-nearest neighbor; by slightly lowering the $C_{min}$ 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.

2.3. 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 Equation (1) is thus extended to:

$$E_{total}=\sum _i{F}_i\left({\rho }_i\right)+\frac{1}{2}\sum _{i,j,i\ne j}{\varphi }_{ij}\left(r_{ij}\right)+\sum _i{E}_i\left(Q_i\right)+\sum _{i\ne j}\frac{Q_i{Q}_j}{4\pi {\epsilon }_0{r}_{ij}}$$

(15)

where $E_i\left(Q_i\right)$ is the energy of atom i as a function of charge Q from the QEq method. Expanding $E_i\left(Q_i\right)$ as a Taylor series from the charge-neutral metal atom up to the second order leads to

$$E_i\left(Q_i\right)=E_i\left(0\right)+Q_i{\left.·\left(\frac{\partial E_i}{\partial Q_i}\right)\right|}_0+\frac{1}{2}{Q}_i^2{\left.·\left(\frac{{\partial }^2{E}_i}{{\partial Q_i}^2}\right)\right|}_0$$

(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 $E_i\left(-1\right)$ and $E_i\left(+1\right)$ we get:

$$\left(\frac{\partial E_i}{\partial Q_i}\right)=\frac{1}{2}\left(IP+EA\right)={\chi }_i^0$$

(17)

and

$$\left(\frac{{\partial }^2{E}_i}{\partial Q_i^2}\right)=IP-EA=J_{ii}^0$$

(18)

where ${\chi }_i^0$ and $J_{ii}^0$ are interpreted as the electronegativity and the “self-Coulomb potential” of the isolated atom i, respectively. The constant $J_{ii}^0$ equals to the atomic hardness ${\eta }_i^0$ 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

$$E\left(Q_1,\dots ,Q_N\right)=\sum _i\left[E_i\left(0\right)+Q_i{·\chi }_i^0+\frac{1}{2}{Q}_i^2·J_{ii}^0\right]+\sum _{i\ne j}\frac{Q_i{Q}_j}{4\pi {\epsilon }_0{r}_{ij}}$$

(19)

The derivative of $E\left(Q_1,\dots ,Q_N\right)$ with respect to $Q_i$ leads to the electronegativity ${\chi }_i\left(Q_i\right)$ of atom i in the system:

$${\chi }_i\left(Q_i\right)={\chi }_i^0+Q_i{·J}_{ii}^0+\sum _{i\ne j}\frac{Q_i{Q}_j}{4\pi {\epsilon }_0{r}_{ij}}$$

(20)

Based on Sanderson’s principle, the electronegativity of all atoms within the metal particle or bulk crystal will equalize:

$${\chi }_1\left(Q_1\right)={\chi }_2\left(Q_2\right)=\dots ={\chi }_N\left(Q_N\right),$$

(21)

with the restriction that the total charge of the system must be conserved.

In what follows, we use two variations of the QEq approach. This involves the most common assessment of ${\chi }_i^0$ and $J_{ii}^0$ via Equations (17) and (18), thus using the experimental IP and EA of single Ga and Pd atoms, respectively [40,41,42]. On this basis, our setup denoted as EAM/QEq_exp refers to ${\chi }_{Ga}^0$ = 3.1490 eV/${\chi }_{Pd}^0$ = 4.4481eV and ${\eta }_{Ga}^0$ = 2.84775 eV/${\eta }_{Ga}^0$ = 3.88600 eV, respectively. Moreover, in a parallel setup denoted as MEAM/QEq_exp we perform a joint fitting of the MEAM and the QEq parameters.

2.4. Molecular Mechanics Calculations

The molecular mechanics simulations of the intermetallic bulk crystals were formed with the General Utility Lattice Program (GULP) package [43]. Starting from the experimental crystal structures, full optimization of all atom positions and cell vectors were performed before assessing elastic constants and partial charges. GULP was also used for fitting the MEAM/QEq parameters to best reproduce the DFT reference data. In turn, our molecular mechanics calculations dedicated to intermetallic clusters and surface slab models were performed with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) program [44]. In all calculations, we describe the Coulomb interactions via the damped shifted force potential with a real space cut-off distance of 6 Å and a damping parameter of 0.05 Å⁻¹, respectively [45].

3. Results

The embedded atom models readily provide quite accurate interaction potentials for studying bulk metals and alloys. Since the employed three-body functions explicitly consider the coordination of atomic coordination when assigning binding strength, it is also quite common to use EAM and MEAM potentials for modelling metal clusters and crystal faces. To characterize local charges, it appears as an intuitive extension of the EAM/MEAM description to simply add interaction terms stemming from the QEq model to our molecular mechanics description. Upon charge transfer between the atomic species of an alloy such as the Ga_xPd_y compounds investigated here, we must however account for the Coulomb interactions—which are not part of the original fitting of EAM/MEAM parameters.

In what follows, we follow two strategies to account for charge transfer in Ga_xPd_y alloys. Our first setup, denoted as MEAM_mix/QEq_exp reflects the direct combination of MEAM and conventional QEq_exp based on second order polynomial (Equation (16)) and experimental inputs to reproduce ionization energy and electron affinity of isolated Ga and Pd atoms (Equations (17) and (18)). The Ga-Ga and Pd-Pd MEAM parameters were taken from refs. [37,39], respectively, whereas details of mixing the MEAM_mix potentials are provided in

Table 1. MEAM and QEq parameters for describing Ga-Pd alloys. The parameters were obtained from (a) conventional mixing rules for MEAM_mix and conventional QEq_exp parameterization based on single atom ionization energy and electron affinity from the experiment (refs. [40,41,42], respectively, see also Equations (17) and (18)). Moreover, (b) we present a newly created MEAM_new/QEq_new model as obtained from fitting all interaction parameters.

Ga-Pd i−j	Mixing MEAM and Qeq (MEAMmix/QEqexp)		Newly Fitted MEAMnew/QEqnew
	Parameters	Source of Parameter
Ec/eV	3.782	DFT calc.	4.328305
re/Å	2.693	DFT calc.	2.612755
α	5.424	$0.5\left({\mathsf{\alpha }}^{Ga}+{\mathsf{\alpha }}^{Pd}\right)$	5.472011
δ	0.0735	$0.5\left({\mathsf{\delta }}^{Ga}+{\mathsf{\delta }}^{Pd}\right)$	0.012891
Cmin(i − i − j)	1.545	$0.5\left[C_{min}(i -i -i)+C_{min}(j -j -j)\right]$	0.440
Cmin(j − j − i)	1.545		1.100
Cmin(i − j−i)	1.545		1.545
Cmin(j − i − j)	1.545		1.545
Cmax(i − i − j)	2.8	$0.5\left[C_{max}(i -i -i)+C_{max}(j -j -j)\right]$	2.7
Cmax(j − j − i)	2.8		2.8
Cmax(i − j − i)	2.8		2.8
Cmax(j − i − j)	2.8		2.8
${\rho }_0^{Ga}/{\rho }_0^{Pd}$	1.0	${\rho }_0$ of Ga and Pd	1.0
${\chi }_{Ga}$/eV	3.1490	EA, IE from exp.	2.382024
${\eta }_{Ga}$/eV	2.84775	EA, IE from exp.	0.555958
${\chi }_{Pd}$/eV	4.4481	EA, IE from exp.	4.055200
${\eta }_{Pd}$/eV	3.88600	EA, IE from exp.	2.637226

. For comparison, the mixed MEAM model is compared to our recently developed EAM model for Ga-Pd interactions as reported in ref. [22]. We pointed out that both of these EAM/MEAM models do not include explicit Coulomb interaction terms. Thus, simply adding QEq_exp interactions to the EAM/MEAM potentials alters the structural and mechanical properties of the alloy, possibly leading to lower accuracy as compared to the original (stand-alone EAM or MEAM) parameterization.

In turn, our second setup is based on newly parameterized MEAM_new and QEq_new interaction potentials. For this, all of the involved parameters (Table 1) were optimized in a joint fitting procedure. In full analogy to our earlier study reported in ref. [22] we use GaPd₂ crystals (space group Pnma, no. 62) as reference (in terms of structure, lattice energy and elastic properties). The resulting MEAM_new and QEq_new parameters are shown in Table 1, whereas a comparison of the structural and elastic properties used as benchmarks to fitting the new potentials is provided in

Table 2. Structural data and mechanical properties of GaPd₂ (space group Pnma, no. 62) as used for fitting the Ga-Pd interactions in our previous EAM potential [22] and the present MEAM_new/QEq_new potential. Bulk and shear moduli are calculated from the Hill approximation [46], respectively.

Experiment [47]		Ab-Initio Reference [22]	EAM [22]	MEAMnew/QEqnew
a/Å	5.4829	5.4816	5.4072	5.3485
b/Å	4.0560	4.0545	4.1437	4.0693
c/Å	7.7863	7.8080	7.8601	7.8585
Elastic constants/10 GPa
c11		27.62	22.562	26.244
c12		13.765	15.703	14.805
c13		13.906	14.248	13.844
c22		24.992	22.421	27.395
c23		14.849	16.815	16.656
c33		26.876	24.107	27.194
c44		6.551	3.641	6.563
c55		4.035	2.268	3.374
c66		7.447	3.924	5.290
B		17.869	18.040	19.012
G		5.931	3.354	5.255

.

Comparing the two types of tailor-made Ga-Pd potentials, namely the EAM model from our previous work and the newly created MEAM_new/QEq_new model we clearly find improved reproduction of the structural data and mechanical properties of GaPd₂ as used for parameterization. However, considering the additional set of tunable parameters, this finding is quite unsurprising. More rigorous benchmarking is obtained by contrasting the results for the series of Ga_xPd_y crystals presented in

Table 3. Comparison of structural parameters (Å) of Ga_xPd_y crystals as obtained from the experiment, DFT, EAM, EAM/QEq_exp, MEAM_mix, MEAM_mix/QEq_exp, and MEAM_new/QEq_new potentials, respectively. As error margins, the percentage of the deviation from the experimental values is shown in square brackets.

		Exp.	DFT [22]	EAM [22]	EAM/ QEqexp	MEAMmix	MEAMmix /QEqexp	MEAMnew/ QEqnew
Ga5Pd [13]	a	6.448	6.507	6.609	6.382	6.297	6.296	6.213
			[0.9%]	[2.5%]	[1.0%]	[2.3%]	[2.4%]	[3.6%]
	b	6.448	6.507	6.609	6.382	6.297	6.296	6.213
			[0.9%]	[2.5%]	[1.0%]	[2.3%]	[2.4%]	[3.6%]
	c	10.003	10.104	10.253	10.901	10.567	10.565	10.580
			[1.0%]	[2.5%]	[9.0%]	[5.6%]	[5.6%]	[5.8%]
Ga7Pd3 [16]	a	8.772	8.865	8.935	8.938	7.120	7.135	8.770
			[1.1%]	[1.9%]	[1.9%]	[18.8%]	[18.7%]	[0.0%]
GaPd [48]	a	4.8969	4.952	4.9451	4.9620	5.052	5.046	4.928
			[1.1%]	[1.0%]	[1.3%]	[3.2%]	[3.0%]	[0.6%]
Ga3Pd5 [13]	a	5.42	5.514	5.46	5.42	5.572	5.558	5.413
			[1.7%]	[0.7%]	[0.0%]	[2.8%]	[2.5%]	[0.1%]
	b	10.51	10.679	10.59	10.45	11.103	11.067	10.445
			[1.6%]	[0.8%]	[0.6%]	[5.6%]	[5.3%]	[0.6%]
	c	4.03	4.064	4.06	4.16	4.047	4.064	4.110
			[0.8%]	[0.7%]	[3.2%]	[0.4%]	[0.8%]	[2.0%]
GaPd2 [47]	a	5.4829	5.572	5.4072	5.4094	5.580	5.570	5.348
			[1.6%]	[1.4%]	[1.3%]	[1.8%]	[1.6%]	[2.5%]
	b	4.0560	4.092	4.1437	4.1411	4.010	4.020	4.069
			[0.9%]	[2.2%]	[2.1%]	[1.1%]	[0.9%]	[0.3%]
	c	7.7863	7.865	7.8601	7.8553	8.317	8.303	7.858
			[1.0%]	[0.9%]	[0.9%]	[6.8%]	[6.6%]	[0.9%]
Ga2Pd5 [14]	a	5.485	5.507	5.475	5.416	5.572	5.565	5.399
			[0.4%]	[0.2%]	[1.3%]	[1.6%]	[1.5%]	[1.6%]
	b	4.083	4.111	4.075	4.094	3.967	3.973	3.961
			[0.7%]	[0.2%]	[0.3%]	[2.8%]	[2.7%]	[3.0%]
	c	18.369	18.608	18.334	18.477	19.474	19.450	18.787
			[1.3%]	[0.2%]	[0.6%]	[6.0%]	[5.9%]	[2.3%]

and

Table 4. Comparison of lattice energies (eV/unit cell) of Ga_xPd_y crystals calculated from DFT, EAM, EAM/QEq_exp, MEAM_mix, MEAM_mix/QEq_exp, and MEAM_new/QEq_new methods. As error margins, the percentage of the deviation from the DFT values is shown in square brackets.

	DFT	EAM	EAM/QEqexp	MEAMmix	MEAMmix /QEqexp	MEAMnew/ QEqnew
Ga5Pd	−85.55	−81.03	−81.85	−77.88	−78.69	−85.87
		[4.5%]	[3.7%]	[7.7%]	[6.9%]	[0.3%]
Ga7Pd3	−162.75	−147.22	−149.61	−134.05	−136.36	−163.22
		[15.5%]	[13.1%]	[28.7%]	[26.4%]	[0.5%]
GaPd	−37.97	−34.10	−34.76	−29.65	−30.29	−38.21
		[3.9%]	[3.2%]	[8.3%]	[7.7%]	[0.2%]
Ga3Pd5	−80.08	−70.63	−71.98	−62.00	−63.22	−81.62
		[9.5%]	[8.1%]	[18.1%]	[16.9%]	[1.5%]
GaPd2	−61.26	−53.20	−54.16	−46.87	−47.76	−62.52
		[8.1%]	[7.1%]	[14.4%]	[13.5%]	[1.3%]
Ga2Pd5	−143.90	−123.29	−125.32	−110.13	−112.05	−146.15
		[20.6%]	[18.6%]	[33.8%]	[31.9%]	[2.3%]

. In Table 3, we listed the unit cell constants as observed from experiment, DFT calculations, conventional EAM or MEAM modelling in absence of charge polarization and with the account of charge transfer between the alloy components from QEq. Likewise, the same series of interaction models is benchmarked in terms of the lattice energy in Table 4.

Our previous EAM-based model appears quite robust in terms of the observed structures, indeed showing agreement within 3% to the experiment. While the new MEAM_new/QEq_new interaction potential is slightly less accurate in terms of unit cell dimensions, we find drastically better reproduction of the lattice energy and (to more moderate extend) the elastic properties as compared to the DFT reference calculations. Indeed, all of the investigated models show deviations in the ballpark of 10–30% whereas the MEAM_new/QEq_new offers the assessment of cohesive energies at <2.5% precision (Table 4). Likewise, we found the elastic moduli c_ij (i,j = 1–6) of GaPd₂ within up to 50% error margins for the EAM model, whereas the MEAM_new/QEq_new method leads to <10% deviation from the DFT results.

We point out that the molecular mechanics models based on EAM or MEAM without QEq do consider alloying effects in an implicit manner, and thus already offer reasonable accuracy for modelling processes like mechanical deformation etc. However, when investigating Ga_xPd_y compounds from the perspective of catalysis, it is of particular interest to monitor the local charge distribution in the alloys. For the bulk crystals, this can be accomplished by means of DFT calculations and Bader analysis for assigning local electron density to atomic sites. In

Table 5. Comparison of the partial charges of Ga and Pd atoms in Ga_xPd_y crystals as calculated with DFT, EAM/QEq_exp, MEAM_mix/QEq_exp, and MEAM_new/QEq_new methods. The numbers shown in parentheses refer to the total charge of all Ga and Pd species of the corresponding unit cell, respectively.

Crystal	DFT (Bader Charge)		EAM/QEqexp		MEAMmix/QEqexp		MEAMnew/QEqnew
	Ga	Pd	Ga	Pd	Ga	Pd	Ga	Pd
Ga5Pd	0.12 × 3, 0.15 × 10, 0.18 × 3, 0.23 × 4 (3.32)	−0.83 × 4 (−3.32)	0.0308 × 4, 0.0315 × 16 (0.63)	−0.157 × 4 (−0.63)	0.0313 × 4, 0.0314 × 16 (0.63)	−0.157 × 4 (−0.63)	0.063 × 4, 0.070 × 16, (1.37)	−0.343 × 4 (−1.37)
Ga7Pd3	0.29 × 6, 0.30 × 10, 0.44 × 12 (10.02)	−0.82 × 6, −0.85 × 6 (−10.02)	0.060 × 16, 0.059 × 12 (1.67)	−0.139 × 12 (−1.67)	0.0608 × 4, 0.0582 × 8, 0.0593 × 16 (1.66)	−0.137 × 4, −0.139 × 8 (−1.66)	0.1501 × 4, 0.1557 × 8, 0.1413 × 16 (4.11)	−0.341 × 4, −0.343 × 8 (−4.11)
GaPd	0.50 × 4 (2.00)	−0.50 × 4 (−2.00)	0.11 × 4 (0.44)	−0.11 × 4 (−0.44)	0.11 × 4 (0.44)	−0.11 × 4 (−0.44)	0.33 × 4 (1.32)	−0.33 × 4 (−1.32)
Ga3Pd5	0.58 × 2, 0.59 × 4 (3.52)	−0.33 × 4, −0.34 × 2, −0.38 × 4 (−3.52)	0.139 × 4, 0.138 × 2 (0.83)	−0.084 × 6, −0.028 × 4 (−0.83)	0.1383 × 4, 0.1356 × 2 (0.82)	−0.0829 × 2 −0.0831 × 4, −0.0815 × 4, (−0.82)	0.51 × 4, 0.52 × 2 (3.08)	−0.307 × 2, −0.312 × 2, −0.311 × 2, −0.304 × 4, (−3.08)
GaPd2	0.61 × 4 (2.44)	−0.28 × 4, −0.33 × 4 (−2.44)	0.15 × 4 (0.60)	−0.07 × 4, −0.08 × 4 (−0.60)	0.15 × 4 (0.60)	−0.0743 × 4, −0.0744 × 4 (−0.60)	0.60 × 4 (2.40)	−0.301 × 2, −0.302 × 2, −0.297 × 2, −0.298 × 2 (−2.40)
Ga2Pd5	0.63 × 4, 0.65 × 4 (5.12)	−0.22 × 4, −0.23 × 4, −0.26 × 4, −0.27 × 4, −0.30 × 4 (−5.12)	0.162 × 4, 0.165 × 4 (1.31)	−0.066 × 4, −0.063 × 4, −0.065 × 8, −0.068 × 4 (−1.31)	0.158 × 4, 0.165 × 4 (1.29)	−0.066 × 4, −0.062 × 4, −0.065 × 4, −0.067 × 4, −0.063 × 4 (−1.29)	0.658 × 4, 0.745 × 4 (5.61)	−0.286 × 4, −0.269 × 4, −0.278 × 4, −0.295 × 4, −0.275 × 4 (−5.61)

, we show the partial charges observed in a series of Ga_xPd_y crystals to compare the DFT-based reference with the QEq charges as obtained from the EAM/QEq_exp, MEAM_mix/QEq_exp and MEAM_new/QEq_new methods. We find that the local charges predicted by the MEAM_new/QEq_new 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.

To one side the improvement achieved by our MEAM_new/QEq_new model appears particularly useful for the investigation of physical vapor deposition processes. To the 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 MEAM_new/QEq_new 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 (Figure 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 (Figure 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, (Figure 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.

To further elaborate this field of application for the MEAM_new/QEq_new potential, we also investigated the (001), (010), and (100) faces of GaPd₂. In particular for semi-hydrogenation, 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/m², respectively. This indicates that the (010) face exhibits the highest stability, consistent with experimental findings [47]. Moreover, the surface energy of the (010) face was also explored from DFT calculations [49], leading to an ab-initio-based reference value of 1.31 J/m²—which we find nicely reproduced by the MEAM_new/QEq_new potential.

The underlying GaPd₂ slabs were chosen generously large to ensure bulk behavior in the center (see Table 5 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 GaPd₂. This suggests that the catalyst performance of the (010) face for semi-hydrogenation 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].

4. Conclusions

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 Ga_xPd_y 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 MEAM_new/QEq_new 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. Moreover, QEq offers an appealing perspective in the direction of electro-catalysis as the implementation of net surfaces as a function of voltage is readily available. Likewise, our models offer the investigation of substrate-induced polarization of nanoparticles and Ga-Pd based liquid alloy droplets—as recently demonstrated for SCALMS (Supported Catalytically Active Liquid Metal Solutions) systems [50].