Translational dependence of the geometry of metallic mono- and bilayers optimized on semi-ionic supports: the cases of Pd on γ-Al₂O₃(110), monoclinic ZrO₂(001), and rutile TiO₂(001)

Abstract

A simplified computational scheme for “deposition” of a metallic (Pd) monolayer on semi-ionic supports is considered. An arbitrary selection of the initial position of the deposited layer relative to the surface of the support cannot provide the best solution for an optimal atom–atom binding. The energy and/or geometry changes were checked relative to a slide of the mono- and bilayers in parallel to the support layer and re-optimization. Two types of possible consequences are shown at the slide length of 1–3 Å for γ-Al₂O₃(110), monoclinic ZrO₂(001) (or m-ZrO₂), and rutile TiO₂(001) (or TiO₂ below). First consequence option: either the Pd monolayer completely collapses after the shift/optimization, or it yields a new geometry (at γ-Al₂O₃(110) and m-ZrO₂) at different steps. Second consequence option: the shift leads to a loss of the Pd stabilization while the ordered monolayer's geometry is retained (at TiO₂ and m-ZrO₂). The extent of such destabilization varies depending on the strength of the interaction with the support. It is thus recommended to verify Pd stabilization energy by sliding/optimizing the monolayer along one or two monolayer directions in order to avoid its energy underestimation. The activity of such monolayers for modelling catalysis is demonstrated using H₂O dissociation in the Pd monolayer/rutile system.

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1. Introduction

The formation of clusters, monolayers or/and multilayer slabs is conventionally defined by a combination of three mechanisms, i.e., Stranski–Krastanov, Volmer–Weber, and Frank–van der Merwe. Two of them (with the exception of the Volmer–Weber type) pass through the monolayer formation but the details of its formation remain poorly understood. Modern research effectively completes this domain. The explosion of interest in 2D systems covers different types of materials from semiconductors^1,2 and dielectrics^3,4 to semimetals⁵ and metals.^6–13 The studies of atomically thin layers composed of under-coordinated metal atoms and named as “metallenes” are directed by their new possibilities in electrocatalysis¹⁴ and catalysis for oxygen reduction reactions.¹⁵ The main interest in “metallenes” is connected with multilayer slabs (3–5 layers¹⁵ and more¹⁶) because several methods to produce an atomic monolayer seem to be under study for different metals (Rh,¹² Pt,¹⁷ Au¹⁸) to the best of our knowledge. The monolayer metallenes can acquire new properties when deposited onto semi-ionic supports. The most detailed studies of mono- and bilayer formation were realized on TiO₂(110) (ref. 19–24) (please, see Discussion). A large fraction of small 2D Au clusters over Au/TiO₂(110) showed a turnover-of-frequency (TOF) maximum for the CO oxidation reaction¹⁹ in parallel revealing a non-metallic type of electric conductivity.²⁰ Such type of conductivity was also shown for small flat Pd clusters at the TiO₂(110) plane.²⁴ Their properties on other semi-ionic supports (such as carbides and nitrides) can be evaluated from numerous experimental and theoretical studies for flat square Me_x clusters, Me = Pd, Pt,¹¹ Cu,^10,11 Ag,¹¹ Au,^10,11 Ni,^9,10x = 4, 9, and 16. Continuous Pt monolayers, which can be considered as metallenes, were theoretically studied as catalysts for the water gas shift (WGS) reaction when deposited onto TiC(100) carbide.^6–8 Recently, the catalytic capabilities of a Pd monolayer deposited on a γ-Al₂O₃(100) plane were theoretically shown for catalysis of CH₄, H₂O, and O₂ dissociation.¹³ So, the problem of an accurate search for the most stable metallic monolayers on different supports becomes an important part of modeling.

Earlier, the passivation of a defective Si(110) plane by a thin Al₂O₃ layer was realized as alternate O/Al/O deposition or a Si(110) reaction with the existing Al₂O₃ fragment.^3,4 A reaction was admitted between the plane of regularly ordered Al or O atoms and the Si(110) support.^3,4 Such a method looks to be better suited for deposition of metallic Me slabs because the monolayer precursor can be cut from a selected direction in the Me bulk^13,25–27 being a good approximation for the final monolayer.¹³ It is possibly a consequence of the close Pd–Pd distances between the bulk and the deposited species, while the geometry of the latter can be different, for example, alternation of one parallel row of squares and four rows of triangles in the Pd monolayer deposited on γ-Al₂O₃(100).¹³ The layer was thus obtained with better Pd stabilization than single Pd atoms at favorable sites of the γ-Al₂O₃(100) plane.¹³ Closer Au–Au or Pt–Pt distances compared to respective values for bulk metals were illustrated in a new hybrid form of a flat Au₉Pt₉ monolayer without deposition on any support.²⁸

However, such joining/optimization of a metallic monolayer precursor and support depends on a large number of parameters and cannot “immediately” guarantee an optimal atom–atom binding and hence the highest metal stabilization. Within periodic boundary conditions, some limited freedom remains for monolayer sliding in parallel to the support. This aspect also applies to other systems, for example, to atomic H coverage at the W(110) surface when a quantitative agreement with experimental data requires a model with translational 1D motion of H atoms on the W-surface.²⁹ An additional problem arises due to the unclear density of the deposited Me atoms per selected unit cell (UC) to complete the monolayer. Some deviations from a totally flat shape of the Pd monolayers are partly a consequence of the non-flat surface of the support. This complicates an accurate assessment. A series of shifts of Pd monolayers over fixed γ-Al₂O₃(100), rutile TiO₂(001), monoclinic ZrO₂(001) supports are proposed to find their new optimal position. Below we characterized the qualitative changes of the geometry thus optimized and the order of values for possible errors. After the presentation of computational details (part 2), the structures of the Pd monolayers on γ-Al₂O₃(110) (part 3.1), rutile TiO₂(001) (part 3.2), and monoclinic ZrO₂(001) (part 3.3) are discussed. The example of H₂O dissociation in the Pd monolayer/rutile system is presented to illustrate the catalytic perspectives of the Pd monolayers (part 3.2.2), while the case of Pt and Pd monolayers is analyzed on the TiC support where the Pt monolayer formation was confirmed experimentally (part 4).

2. Computational details

Plane wave computations with periodic boundary conditions using the PBE³⁰ functional were performed within the projector augmented wave (PAW) method^31,32 implemented in VASP.^33,34 The GPU version of the VASP code^35,36 was used for geometry optimization. Dispersive energy corrections were considered zero damping D3 (ref. 37 and 38) levels. The PBE-D3 accuracy was verified via testing for the Au₈ cluster³⁷ and later for periodic systems³⁹ (molecular crystals and adsorbed molecules). The atomic charge density distribution was analyzed using Bader analysis.⁴⁰ The scripts provided by the transition state tools for VASP were used to build the initial images for the climbing image nudged elastic band (cNEB) calculations.^41,42 The energy cut-off was set to 500 eV. The Brillouin zone k-sampling was restricted to the Γ-point for the geometry optimization and transition state (TS) search via cNEB calculation. Spin polarized solution was considered for testing the basic Pd/γ-Al₂O₃(110) model. Vibrational frequencies were calculated using the finite difference method (0.015 Å atomic displacements) as implemented in VASP. For the selected reaction, the TS showed a single imaginary frequency corresponding to the reaction path. The figures of the 3D structure of the different models were drawn with MOLDRAW 2.0.⁴³ Steps of the reaction simulations (see the ESI†) were realized using wxMacMolPlt.⁴⁴

3. Results and discussion

3.1 Pd on γ-Al₂O₃(110)

One has to compare the respective Pd stabilization in each case (single Pd atom, cluster, slab) in order to roughly estimate the respective probabilities of such growth. The relative stabilization energy (eV per Pd) per Me atom is

U_Me = (U_tot − U_ox)/N (1)

where U_tot and U_ox are the total energy of the system (Me = Pd or Pt and oxide support) and of the support, respectively. A short list of U_Pd values at the PBE level are found in Table 1, while more complete data together with total energies (U_tot, U_ox) and U_Pd values at the PBE-D3 level are found in Table S2 of the ESI.† Going forward, one should note that no qualitative difference was observed between PBE and PBE-D3 levels.

Table 1 Relative stabilization energy (eV per Pd) per Pd atom U_Pd (1) for single atoms, clusters (N = 24), (111) monolayers with vacancies (N = 24–28), full monolayers (N = 28 for γ-Al₂O₃(110), 30 for TiO₂(001), 16 for m-ZrO₂), bilayers (N = 32 for ZrO₂, 48 for TiO₂(001) and γ-Al₂O₃(110)), and 4-layer Pd(110) (N = 72) deposited over the Al₅₆O₈₀ model of γ-Al₂O₃(110), the Ti₃₆O₇₂ model of the TiO₂(001) surface, the Zr₃₂O₆₄ model of the m-ZrO₂(001) surface, and the Ti₆₄C₆₄ model of the TiC(001) surface optimized at the PBE level. More details including total energies (for the PBE-D3 case as well) are found in Table S2 of the ESI†

	System	N	−UPd	Fig.
a For a smaller UC such as Al14O20 (see the text and ESI†). b After shift by 1 Å along the OY direction. c After shift by 2 Å along the OY direction. d After shift by 3 Å along the OY direction. e Shift by 1 Å along both the OX and OY directions. f Shift by 1 Å along the OX direction. g (111) model. h “band” type structure. i 1/1 model. j (100) model. k 4R–4R model. l “mix” model. m Thin Zr16O32 support model. n For Pt with UPt = −6.218 eV for bulk at the PBE level.
Al2O3	Atom	1	5.314	S1g†
			3.787	—
			3.755	—
			5.159^a	1b
	1-Layer	24	4.852	2a and b
			4.886^b	2c and d
			4.749^c	2e and f
			4.896^d	2g and h
		28	4.847	3f
			4.865^e	—
			4.862^f	3h
		30	4.865	3i
	2-Layer(100)	48	4.936^g	1c
			4.941^e^,^g	—
TiO2	Atom	1	3.318	S1a†
			3.111	S1b†
			2.726	S1c†
	Cluster	24	4.278	1e
	1-Layer (111)	24	4.094	S1d†
		24^f	4.071	S1e†
		24^h	4.190	4f
		27	4.263	4d and e
		28	4.305	6a
		30	4.710	4g and h
		30^f	4.362	4i and j
		32	4.353	S3b†
	2-Layer (111)	48	4.618	S1f†
	4-Layer (110)	72	4.714	1f
m-ZrO2	Atom	1	4.267	S1h†
	1-Layer	16	4.294^i	5a
			4.243^j	—
			4.325^g	—
			4.336^k	5b and c
			4.336^l	5d
	2-Layer (111)	32^m	4.920^g	S2b†
			4.836^g	S2a†
TiC	1-Layer	16	5.086	7c
		18	5.153	7b
			5.895^n	7a

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3.1.1 Monolayer preparation. Four steps were made to construct the monolayer model on the γ-Al₂O₃(110) support: 1) the initial Pd₂₄(100) monolayers were cut from the bulk in the (100) direction, joined to the oxide support model and optimized at fixed UC parameters defined by the oxide; 2) unidirectional shifts of a whole Pd layer with a variable step were considered relative to the γ-Al₂O₃(110) surface with subsequent optimization. It resulted in either disjoined parallel rows of squares and triangles (Fig. 2c, d, g and h), or in distorted Pd(111) monolayers (Fig. 2a, b, e and f); 3) an increase of the quantity of deposited Pd atoms (+4Pd) with the formation of new hybrid Pd₂₈ structures possessing the characteristic “chain” motif (a shift step was also used for this Pd₂₈ layer, Fig. 3h); 4) subsequent chemisorption of NH₃ resulted in modified hybrid Pd structures with another “wide chain” motif (Fig. 3f). In our work we shall consider the results of all the four steps.

3.1.2 Pd₂₄ monolayers with vacancies. The larger value of the first two cell vectors (11.174, 16.163 Å) of the γ-Al₂O₃(110) supercell with the Al₅₆O₈₀ content was first selected because it leads to a minimal mismatch with the appropriate Pd₂₄(100) fragment prepared from the bulk using the experimental Pd–Pd value of 2.757 Å.⁴⁵ The obtained after optimization model with one Pd layer (Fig. 2a and b) resembles a distorted Pd(111) plane. The transformation from the Pd(100) to Pd(111) plane could be explained by the lower (111) surface energy for fcc metals.^46,47 It is easy to see that the plane of symmetry is held for the Pd atoms in one UC. Three translational steps (by +1 Å, +2 Å, or +3 Å) along the OY axis result in total energy variations of −0.821, 3.297, or −3.530 eV (Table 2) after the optimizations at the new positions of the monolayers.

Table 2 Bader type Pd charges (|e|) for the initial and shifted/relaxed Pd₂₄ monolayers (by three steps by 1, 2, or 3 Å along the OY axis) at the surface of γ-Al₂O₃(110) and for the Pd₁₈ layer at γ-Al₂O₃(100). Atomic numbers are given in “No.” columns and Fig. 3 and S3† herein for γ-Al₂O₃(110) and Fig. 9b from ref. 13 for γ-Al₂O₃(100). More data (as POSCAR files as well) are supplied for γ-Al₂O₃(100) from ref. 13. Total charge Q (|e|), cohesion bulk energy U_Pd, total energy U_tot, and energy U_ox (all in eV) of the non-relaxed oxide support without Pd slabs are shown in the lowest row

γ-Al2O3(110)					γ-Al2O3(100)
No.	0	+1 Å	+2 Å	+3 Å	No.	0
a Optimized 2-layer Pd48 slab possesses −5.148 e at the same position. b PBE-D3. c Calculated bulk cohesion energy of −5.834 eV per Pd instead of −5.250 eV per Pd at the PBE level.
137	−0.592	−0.597	−0.285	0.028	117	0.113
138	0.108	−0.06	−0.198	−0.047	118	−0.196
139	−0.004	−0.027	0.076	0.011	119	0.333
140	0.116	0.014	−0.406	−0.592	120	0.114
141	−0.613	−0.597	−0.285	0.028	121	−0.248
142	0.050	−0.060	−0.198	−0.047	122	0.104
143	0.038	−0.027	0.076	0.011	123	−0.155
144	0.008	0.014	−0.406	−0.592	124	−0.258
145	−0.658	−0.622	−0.546	−0.622	125	0.109
146	0.143	0.019	0.039	0.018	126	0.090
147	−0.533	−0.304	−0.294	−0.314	127	−0.260
148	0.158	0.003	−0.475	−0.591	128	0.325
149	−0.691	−0.622	−0.546	−0.622	129	0.119
150	0.098	0.019	0.039	0.018	130	−0.259
151	−0.478	−0.304	−0.294	−0.314	131	0.107
152	0.041	0.003	−0.475	−0.591	132	−0.146
153	−0.627	−0.609	0.144	−0.052	133	−0.171
154	−0.627	−0.622	−0.163	−0.307	134	0.085
155	−0.049	−0.014	−0.337	0.001
156	−0.358	−0.297	−0.572	−0.626
157	−0.602	−0.609	0.144	−0.052
158	−0.671	−0.622	−0.163	−0.307
159	−0.036	−0.014	−0.337	0.001
160	−0.347	−0.297	−0.572	−0.626
Q	−6.126^a	−6.232	−6.034	−6.186	—	−0.194
−Uox	953.797	952.894	953.652	952.844	—	782.358
ΔUox	5.430	6.243	5.485	6.156	—	—
ΔUtot	0.000	−0.821	3.297	−3.530	—	—
−Utot	1075.617	1076.438	1073.141	1076.671	—	858.330
−UPd	4.852	4.886	4.749	4.896	—	4.221
−Utot^b	1099.454	1101.163	1097.848	1101.153	—	—
−UPd^b^,^c	5.293	5.364	5.226	5.363	—	—

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3.1.2.1 Decomposition of the total energy while shifting the Pd₂₄ slab. Dividing the total energy variations (from Table 2) per number of Pd atoms (24) at these three steps gives −0.034 (0–1), 0.137 (1–2), and −0.147 (2–3) eV per Pd per every shift. Despite the disconnected parts, the layer after the “+1 Å” shift is slightly more stable than the initial distorted (111) model (Fig. 2a and b) or the one after the shift by 2 Å (Fig. 2e and f) (Tables 1 and 2). Perturbed naked (without the Pd layer) and frozen (without optimization) supports at the new Pd positions possess energies of −953.797, −952.894, −953.652, and −952.844 eV, respectively, instead of the initial −959.137 eV. The destabilization of the oxide support due to the interaction with Pd₂₄ (5.430, 6.243, 5.485, and 6.156 eV, respectively) is rather large while changing in the opposite manner relative to the total energy at every 0–1, 1–2, 2–3 step. The higher the deformation, the higher the interaction energy in absolute value. It shows that respective changes of the total energy have important contribution from the chemical interaction between the monolayer and support.
3.1.2.2 Visualization of the shift of the Pd layer relative to the support. Due to the presence of two 4-atom Al chains (shown in magenta from top to bottom in Fig. 3a–d) at the surface, the respective chains of Pd atoms coordinated to the Al-chain are held in the “disordered” (Fig. 3a and c) and “ordered” (Fig. 3b and d) Pd₂₄ forms. The Pd atoms of these chains can relax after every shift, finally keeping or losing their favored bi-coordinated sites (Fig. S1g and 1b). In the second case, another Pd atom occupies the free bi-coordinated site. In order to demonstrate the slide of the Pd slab relative to the support with every +X Å step, the numbers of some O atoms of the upper oxide layers are shown together with the Pd numbers (using red and black notations for O and Pd, respectively, in color version of Fig. 3). The most convenient landmark to trace the monolayer motion relative to the oxide (from left to right) is the position of the Pd158 atom relative to that of O64. It can be easily seen that different atoms of the monolayer make different steps upon the imposed total shift. Only two atoms (Pd145 and Pd149) remain at their sites in all four models.

Fig. 1 (a) Al₅₆O₈₀ and (b) Al₁₄O₂₀ model of γ-Al₂O₃(110) with one Pd atom or (c) Al₅₆O₈₀ with a Pd₄₈ bilayer, (d) Ti₃₆O₇₂ model of the rutile TiO₂(001) surface with (e) the Pd₂₄ cluster and (f) 4-layer Pd₇₂(110) slab all optimized at the PBE-D3 level. The U_Pd energies (in eV per Pd) correspond to Table 1 when Pd atoms are involved. The atomic colors are given in red, magenta, green, and gray for O, Al, Ti, and Pd, respectively.

We selected a limited set of Pd atoms in order to show: 1) the changes of the geometries for the atoms with a drastic charge variation along with the 0–3 steps (Table 2), i.e., from −0.592 |e| to 0.028 |e| for Pd137‡ and inverse change from 0.116 |e| to −0.592 |e| for Pd140. These atoms perform the shifts in the opposite directions, i.e., Pd137 loses its position at a favorable site near Al atoms and Pd140 vice versa; 2) the neighboring Pd atoms with opposite charges, like the Pd140(−0.591 |e|)⋯Pd139(0.011 |e|)‡ pair, in the final model after the shift +3 Å (Table 2). This Pd140⋯Pd139 pair could be attractive sites for dissociation of small heteronuclear molecules (part 3.2.2). The charge selection was based on the Bader type charges because projected DOS (pDOS) analysis for Pd atoms also shows essential correlations with the Bader charges of the same atoms (for Pd/γ-Al₂O₃(100), will be shown elsewhere).

The most important result of the sliding and optimization is a highly ordered Pd₂₄ slab after the steps +1 Å (side and upper views in Fig. 2c and d) and +3 Å (Fig. 2g and h) which stabilize the system. No plane of symmetry is preserved for the Pd atoms in one UC after the +2 Å step as it was in the initial model (Fig. 2a and b). Disjoined “bands” for these two configurations include the fragments (squares, triangles) of the (100) and (111) planes. One band of squares (band no. 1 in Fig. 2d) is combined with one band of triangles (band no. 3 in Fig. 2d). The dashed lines therein show the missing band 2 of triangles due to rather long Pd⋯Pd distances of 3.6–3.7 Å (Fig. 2d). The variation of the total Bader charge of the Pd slab between the steps varies rather slightly between −6.034 and −6.232 |e| even if some atomic charges can change their sign in the course of an optimization after a single translation (Pd137, Pd141 etc. in Table 2). The difference in the absolute Pd monolayer charges (−0.169 versus −6.034 |e| at γ-Al₂O₃(100) and γ-Al₂O₃(110), respectively) can be explained by different types of the favored Pd sites, i.e., O–Pd–O at γ-Al₂O₃(100) and Al–Pd–Al at γ-Al₂O₃(110). A similar Al–Pd–Al site (the shortest Pd–Al are 2.522 and 2.529 Å in γ-Al₂O₃(100)) is absent at the γ-Al₂O₃(100) surface because it allows two additional shorter bonds with two O atoms (2.251 and 2.344 Å) and an essentially worse U_Pd value of −2.751 eV per Pd. As soon as the favored site for O–Pd–O at the γ-Al₂O₃(100) surface possesses U_Pd = −4.375 eV per Pd for a single atom which is slightly smaller (in absolute value) than U_Pd = −4.392 eV per Pd for the Pd₂₀ monolayer,¹³ the probability of Pd monolayer formation becomes higher at γ-Al₂O₃(100). Due to this deviation between γ-Al₂O₃(100) and γ-Al₂O₃(110), the Pd stabilization energies of the favorable site of a single atom and the Pd_n monolayer obey the |U_Pd(Pd₁/γ-Al₂O₃(100))| < |U_Pd(Pd₂₀/γ-Al₂O₃(100))| relation for (100),¹³ which is inverse |U_Pd(Pd₁/γ-Al₂O₃(110))| > |U_Pd(Pd_n/γ-Al₂O₃(110))|, n = 24–28, for γ-Al₂O₃(110). This illustrates better conditions for monolayer growth at the γ-Al₂O₃(100) plane.

Fig. 2 Side (a, c, e and g) and upper (b, d, f and h) views of the fragments containing 2 × 2 unit cells (UCs) of the Pd₂₄ monolayer at Al₅₆O₈₀ at (a) the initial position and after its slide/optimization by 1 (c and d), 2 (e and f) to 3 Å (g and h) along OY axis.

3.1.3 Continuous Pd₂₈ monolayers. Regarding the disconnected Pd atoms of band 2 (absent Pd–Pd bonds are shown by dashed lines in Fig. 2d) and the lower density of the Pd atoms per (110) surface (11.174 × 16.163/24 = 7.53 Ų per Pd atom) than what was achieved in the continuous Pd₂₀ monolayer at the (100) one (11.217²/20 = 6.29 Ų per Pd (ref. 13)), we evaluated that at least four additional Pd atoms could be added to build a continuous monolayer (11.174 × 16.163/6.29 = 28.67 Pd). We thus prepared a new Pd₂₈ fragment and allowed it to relax together with the support (it is shown together with adsorbed NH₃ in Fig. 3f and g, see explanations below) with the final density of the Pd atoms being 6.45 Ų per Pd. Its configuration was modified at the 4th step after the next chemisorption of NH₃ which resulted in modified hybrid Pd structures with another “wide chain” motif (Fig. 3f, where “wide chain” motif is shown by light cyan and blue Pd atoms in comparison with the “narrow chain” motif shown by magenta Pd atoms in the monolayer before NH₃ adsorption in Fig. 3g). One should note that NH₃ dissociation restored the initial “chain” (Fig. 3g) motif in the absence of NH₃ (not shown but is similar to the one in Fig. 3g). Comparable Pd densities at (100) and (110) planes can be indirectly confirmed by our attempt to add the monolayer up to 30 Pd atoms. It led to an “ejection” of two excessive Pd atoms (Fig. 3i) to the second layer without an U_Pd gain (−4.865 eV per Pd in Table 1) despite the higher Pd quantity.

Fig. 3 Upper (a–c and f–h) and side (e and i) views of the fragments containing (a–e) Pd₂₄ and (f, g and h) Pd₂₈, (i) Pd₃₀ monolayers and (e) whole UC, (a–c) upper layer and (f–h) without any part of the Al₅₆O₈₀ support at (a and e–g) the initial position and after its slide/optimization by 1 (b and h), 2 (c) to 3 Å (d) along OY. The characteristic motif for 2 × 2 Pd₂₈ UCs is shown (f and g) by Pd atoms given in light blue, magenta, and dark blue colors (H and N atoms of adsorbed NH₃ are given in gray (small spheres) and dark blue (small spheres), see text). In other cases (a–e) the atomic colors are given in red, magenta, green, and gray for O, Al, Ti, and Pd, respectively.

The importance of checking the shifts has also been shown for Pd₂₈ stability. Even if a modest energy gain was found to be present (from −4.847 to −4.865 or −4.862 eV per Pd, Table 1), both structures obtained by the shifts (of +1 Å along OX or along OX and OY together) clearly demonstrate a distorted (111) geometry (Fig. 3h). Surprisingly, this does not enhance the respective Pd₂₈ stability compared to the one for the most stable Pd₂₄ models with vacancies (or “empty space” between the bands, Fig. 2d) despite the larger number of Pd atoms. One could propose that the absence of essential energy variations relative to the sliding can be one of the signs of a complete Pd monolayer.

3.1.3.1 Difference between mono- and multilayers relative to the shifts. An important observation about the weak influence of the shift on the Pd stabilization of bilayers is noted for γ-Al₂O₃(110). The first example demonstrated that the Pd₄₈(100) bilayer can hardly be effectively stabilized upon the shift possibly due to a higher “rigidity” of the slab. The initial bilayer Pd₄₈(100) slab at γ-Al₂O₃(110) was tested with respect to the shift by +1 Å but the optimized (100) geometry is retained with a small increase (0.1%) of the relative stability expressed via the energy U_Pd (1). The shift of the Pd₄₈ bilayer at γ-Al₂O₃(110) (Table 1) was accompanied by switching from some bonds to others without a significant improvement of the stabilization (by only 0.005 eV per Pd) compared to the drastic change of the Pd₂₄ monolayer of 2.9% (Table 1). An even higher energy of 7.4% was gained by shifting a complete Pd₂₄ monolayer at the TiO₂(001) surface (part 3.2.1) below. An opposite case of relatively stronger stabilization of the Pd bilayer upon a shift (1.7%) was found for Pd₃₂(111) at m-ZrO₂(001) (Table 1, see part 3.3).

3.2 Pd/rutile TiO₂(001)

3.2.1 Geometry and stability. The example of step-by-step transformation from the initial Pd₃₂ precursor to the stable Pd₂₈ monolayer at TiO₂(001) (with the sizes of 13.77 × 13.77 Å) is shown in Fig. S3.† The initial grid Pd₃₂ (Fig. S3b†) was optimized from a Pd₃₂(100) fragment (Fig. S3a†) to the non-flat form with 4 Pd atoms above the others (Fig. S3b†). The deletion of these 4 atoms led to the Pd₂₈(111) monolayer with one di-vacancy (Fig. S3c† and 6a, the latter with an adsorbed water molecule), which was then completed up to the full Pd₃₀(111) layer by filling two obvious vacancies. The Pd₂₈(111) monolayer was applied for testing the reactivity below (part 3.2.2, Fig. 6). One should note that the surface density for Pd₃₀ at TiO₂(001) nearly coincides with the one for a complete hybrid Pd₂₀(1−4) monolayer at γ-Al₂O₃(100), i.e., 6.32 and 6.29 Ų per Pd despite the different Pd packing. This Pd₃₀ monolayer was also obtained starting from a Pd₂₄ monolayer precursor (Fig. 4a–c).

Fig. 4 Pd₃₀ monolayer with isolated (a–c) six, (d–f) three mono-vacancies, and (g–j) without vacancies including views of 2 × 2 cells (c, e and f). Atomic notations (a and b) are given before (a) and after (b) shift/optimization by 1 Å along OX. The case of “band” type Pd₂₄ monolayer is in (f) with the inverted OX and OY axes relative to the common case. In (a and c–e) ellipses show the Pd vacancies. The lost bonds upon the slide are shown by ellipse in (j). The color agreement corresponds to Fig. 1.

Two situations with the slide/optimization of the Pd₂₄ and Pd₃₀ monolayers at rutile TiO₂(001) can illustrate the second variant of the influence of support heterogeneity on their final structure, i.e., a conservation of the geometry with a destabilization upon the monolayer's shift. For a Pd₂₄ monolayer, both the initial (Fig. 4a) and shifted (Fig. 4b) models correspond to the same geometry with 6 monovacancies per UC (shown by 6 ellipses for one of the four UCs in Fig. 4c). These numerous mono-vacancies result in the lower stability of these Pd₂₄ monolayers (−4.094 and −4.071 eV per Pd) relative to a Pd₂₄ cluster (−4.278 eV per Pd, Fig. 1e). A reconstruction of the monolayer after its slide/optimization can be controlled regarding the Pd atomic numbers (Fig. 4a and b) and distances between the Pd atoms in (a) and (b). A saturation of three of them already stabilizes a Pd₂₇ monolayer (−4.263 eV per Pd, Fig. 4d and e).

One can note the possible energy gain/loss upon the shift of a Pd₂₄ monolayer (2.9%) and a Pd₄₈ bilayer (0.1%) parallel to γ-Al₂O₃(110). The minor change for the bilayer was assigned to its rigidity. The shift of a Pd₂₄ monolayer with 6 monovacancies relative to the TiO₂(001) surface mentioned above (previous passage) is also accompanied by a small U_Pd destabilization of 0.5%. This weak influence can be explained by a poor interaction which is not strongly perturbed (Fig. S1d and e†). Opposite partial loss of the bonds during the shift of the Pd₃₀ monolayer is evident from a comparison of the respective geometries before (Fig. 4g) and after the shift (Fig. 4i). The absent bonds are shown by an ellipse in the second case (side view in Fig. 4j). This lost binding led to the largest decrease of the Pd stabilization of 7.4% throughout all models on all three supports.

A similarity between the Pd₂₄ monolayers at both supports is manifested, revealing a higher stabilization of the denser models in the form of bands. Such a band at the TiO₂(001) surface possesses a Pd(111) geometry (Fig. 4i) while it forms a kind of hybrid structure including both squares and triangles (Fig. 2c, d, g and h) for Pd₂₄/γ-Al₂O₃(110). On both supports the U_Pd decrease has one order of value, i.e., 2.3% and 0.9% (the difference between the maximal and minimal values between four Pd₂₄ models at γ-Al₂O₃(110)), respectively. The importance of these values can be properly evaluated regarding analogous values for concurrent models of metallic species. In the case of TiO₂(001) the best stabilization of −4.278 eV per Pd for 24 Pd atoms is achieved for a bilayer cluster (Fig. S1f†). However, a drastic U_Pd increase takes place for a Pd₃₀ monolayer (−4.710 eV per Pd) so that it exceeds the stability of the Pd₄₈ bilayer slab (−4.618 eV per Pd) and the Pd₂₄ bilayer cluster. These intermediate U_Pd values for the Pd₂₄ monolayer show a possible concurrence between the monolayer and the small clusters at intermediate coverage of TiO₂(001).

3.2.2 Catalytic activity of the Pd₂₈ monolayer at TiO₂(001). The H₂O dissociation reaction is often considered as a limiting step for the WGS reaction which is one of the key directions of future H₂ production. We selected the case of the Pd₂₈ monolayer with one di-vacancy (shown by the ellipse in Fig. 6a) in order to illustrate H₂O dissociation in a Pd monolayer/rutile system. Earlier, no large difference was estimated for H₂O dissociation at a Pd monolayer with or without vacancies deposited on γ-Al₂O₃(100).¹³ Knowing that the higher H₂O adsorption energy corresponds to the most positive Pd atoms (Fig. 4 from ref. 13) we selected Pd124 with the largest Bader charge, 0.248 |e| (Table 3).

Table 3 Bader type Pd charges (|e|) for the reaction steps (REA, TS, PRO) of Pd₂₈ monolayers (by three steps by 1, 2, or 3 Å along the OY axis) at the surface of TiO₂(001). Atomic numbers are given in the “Pd” column and Fig. 6. The total charge Q (|e|) of the Pd₂₈ monolayer is shown in the lowest row. The Pd numbers are given in bold if the atom contacts with H₂O atoms

Pd	No H2O	H2O
		REA	TS	PRO
112	0.014	0.028	0.032	0.025
113	0.074	0.023	0.031	0.099
114	−0.004	−0.017	−0.004	0.002
115	−0.034	−0.086	−0.092	−0.053
116	−0.022	−0.062	0.007	0.171
117	−0.006	−0.048	−0.037	−0.002
118	−0.175	−0.186	−0.184	−0.209
119	0.080	0.089	0.086	0.075
120	0.243	0.222	0.232	0.230
121	−0.149	−0.138	−0.142	−0.158
122	0.233	0.255	0.253	0.256
123	−0.174	−0.153	−0.111	−0.111
124	0.248	0.366	0.425	0.433
125	−0.180	−0.236	−0.174	−0.147
126	0.005	0.072	0.089	0.027
127	−0.081	−0.012	−0.002	−0.066
128	−0.178	−0.192	−0.201	−0.185
129	0.070	0.071	0.080	0.087
130	−0.186	−0.172	−0.155	−0.194
131	0.159	0.126	0.119	0.161
132	0.055	0.080	−0.016	0.036
133	−0.045	−0.04	−0.008	−0.021
134	0.170	0.127	0.125	0.153
135	−0.213	−0.245	−0.239	−0.218
136	0.183	0.205	0.209	0.183
137	−0.170	−0.179	−0.175	−0.183
138	0.204	0.194	0.215	0.204
139	−0.182	−0.156	−0.179	−0.179
Q	−0.061	−0.064	0.184	0.416

---

The steps (Fig. 5) are shown for the reaction with a moderate barrier E^# of 0.69 eV and exothermic heat ΔU of −0.08 eV to be compared with the E^#/ΔU pairs of 1.05/0.46,⁴⁸ 1.09/0.59,⁴⁹ and 1.12/0.01 (ref. 50) at Pd(111) using PW91/USPPs, PW91/PAW, and PBE/Troullier–Martines norm-conserving PPs, respectively. Regarding the different approaches applied in ref. 48–50 one could also mention the 1.06/0.36 pair at a 5-layer Pd₉₀(100) but using the same PBE-D3/PAW level¹³ as considered herein.

Fig. 5 Pd₁₆ monolayers of (a) 1/1, (b and c) 4R–4R, and (d) “mix” types include (a, c and d) the views of 2 × 2 supercells. Atoms of the support are shown in (b). Different rhomb elements between (d) “mix” and (c) 4R–4R geometries are given via shaded tetragons. The color agreement corresponds to Fig. 1.

Fig. 6 (a) Reagents, (b) transition state, (c) products, and (d) energy profile of H₂O dissociation at the Pd₂₈ monolayer with one di-vacancy (shown by ellipse in (a)) deposited over rutile-TiO₂(001) optimized at the PBE-D3 level. (a) The activation energy E^# and total energies are shown in eV. The atomic colors are given in gray (small spheres), red, green, and gray (large spheres) for H, O, Ti, and Pd, respectively. Respective (d) reaction profile is unified with the one for H₂O dissociation at the 5-layer Pd₉₀(100) slab.¹³

Similar or smaller barriers E^# of 0.37 and 0.67 eV were calculated for H₂O dissociation earlier at Pd₁₈/γ-Al₂O₃(100) near two separate mono-vacancies and 0.40 eV at Pd₂₀/γ-Al₂O₃(100) without vacancies in the deposited Pd₂₀ monolayer.¹³ Exothermic heats ΔU were obtained in all cases (−0.31, −0.40, and −0.18 eV, respectively).¹³ It points to a comparable effect of the TiO₂(001) support on the dissociation reaction.¹³ From Table 3 one can evaluate the charging of the Pd₂₈ layer in the course of the reaction. While the reagents change the total charge of the monolayer (from −0.061 to −0.064 |e|) after adsorption very slightly, the dissociation raises it up to the final value of 0.416 |e| (Table 3). The maximum and minimum Pd charges are approximately the same, i.e., 0.248 to −0.213 |e| without H₂O and 0.256 to −0.218 |e| after the reaction. The numbers of Pd atom-participants (116, 124 with the OH group, 125, 132, 133 with H) are shown in bold in Table 3. It illustrates the response of the Pd monolayer to the reaction.

3.3 Pd/ZrO₂(001)

Both reconstruction and conservation were observed upon Pd sliding along the m-ZrO₂ surface. Starting from an initial Pd(100) monolayer joined with a m-ZrO₂ supercell (10.338 Å, 10.464 Å) the hybrid 1/1 model was optimized with a Pd density of 6.89 Ų per Pd (Fig. 5a) being the largest throughout those studied above. After a shift of this 1/1 model by 1 Å in both OX and OY directions, the 4R–4R (4-atom rhomb – 4-atom rhomb) geometry (Fig. 5b and c) was obtained. The shift of the 1/1 model by 0.73 Å in the OX direction only resulted in the new (111) less stable geometry with an energy of −4.325 eV (Table 1). After a shift of this 4R–4R model by 1 Å in both OX and OY directions, nearly the same “mix” geometry (Fig. 5c) was conserved with an energy of −4.336 eV. The minor difference between the latter and the 4R–4R one is illustrated for the “mix” model showing a rhomb fragment (Fig. 5c and d) which transforms to square from the rhombohedral geometry in the 4R–4R (no short diagonal in the shaded rhomb in Fig. 5d) model. The shift of the 4R–4R model by 0.73 Å in the OX direction also resulted in the (111) geometry with an energy of −971.278 eV (Table 1). So, the iso-energetic conservation upon the shift is related to the close 4R–4R and “mix” models, while other cases correspond to a monolayer's reconstruction. Compared to Al₂O₃(110) all these slabs are more stable than respective single Pd atoms (Table 1) as well as at rutile (part 3.2) which increases the possibility of slab formation from separate atoms.

For ZrO₂ we obtained a series of different monolayer geometries for the same Pd coverage including conventional low index (111) and (100) planes. It is instructive to compare the respective surface energies γ for these analogues. γ can be expressed via U_Pd and Pd density (A):

γ = ((U_Pd(slab) − U_Pd(bulk)) × N)/S = (U_Pd(slab) − U_Pd(bulk))/A (2)

where N is the Pd number per UC, S is the surface of the UC, A = S/N, U_Pd(slab), and U_Pd(bulk) are the respective stabilization energies. Using the U_Pd(slab) data of Table 1, one has 133, 134, and 146 meV Å⁻² for (4R–4R), (111), and (100) models at the PBE level. Comparing ZrO₂ with other systems, one should note a lower surface energy γ of γ-Al₂O₃(110) (60 meV Å⁻² for Pd₂₈ layer) or TiO₂ (78 meV Å⁻² for Pd₃₀) and a comparable value of 136 meV Å⁻² for Pd₂₀ at γ-Al₂O₃(110).

Our result in favor of a weak influence of shifts on the Pd(100) bilayer at γ-Al₂O₃(110) (part 3.1.3.1) contradicts the opposite example of the (111) bilayer at ZrO₂. Two similar Pd(111) models at a thinner ZrO₂ oxide slab were constructed independently by us. But its upper view (Fig. S2†) clearly shows relative displacement between the Pd₃₂(111) slabs relative the same support. An approximate estimation shows the shifts of −0.12 and 0.50 Å along the OX and OY axes, respectively. Regarding the Pd₄₈(100) bilayer at γ-Al₂O₃(001) (part 3.1.3.1, Table 1), a relatively strong U_Pd stabilization of the Pd₃₂(111) bilayer at the m-ZrO₂(001) Pd bilayer upon a shift was found (Table 1) of 1.7% despite the close Pd–O distances in both the (111) slabs (Table S1†). It is important to note that this shift was realized at the smaller Zr₁₆O₃₂ support model. Moreover, the more rigid model was obtained by fixing 20 lowest Zr/O atoms in the lower part of the Zr₁₆O₃₂ support instead of 8 lowest O atoms only in the larger Zr₃₂O₆₄ model. Nevertheless, the more rigid support led to 1.7% variation of the U_Pd value compared to a 0.01% variation for the Pd₄₈(111) bilayer at γ-Al₂O₃(001) (part 3.1.3.1). One may suggest that the additional “free” atoms can lead to the excess of this U_Pd growth of 1.7% (in absolute value). This example illustrates that the relaxation of a bilayer upon the shift depends on the type of support, thus achieving the value being comparable with the one for a non-complete monolayer at alumina (2.9% for Pd₂₄). It confirms the necessity to verify possible stabilization of mono- or bilayers relative to an optimization in a new position after a shift parallel to the support.

4. Discussion

The Pd₂₄ geometry changes drastically at different slide steps along the γ-Al₂O₃(110) support. It is important that a new ordered geometry of the monolayers appears. It signifies that the selection of the “best” support for Pd monolayer stabilization based on the best geometric coherence between the a priori known monolayer geometry (like Pd(111) or Pd(100)) and the geometry of the surface atoms of a support (Al, Ti, O) cannot predict that the combined model will have minimum energy. This is because a new Pd monolayer geometry can be formed instead of Pd(111) or Pd(100) after shifting and subsequent optimization. One example of new hybrid geometry of a Pt₉–Au₉ monolayer was theoretically proposed for the atoms with similar sizes.²⁸

The hybrid Pd₂₄ examples at γ-Al₂O₃(110) (Fig. 2c, d, g and h) are related to the case of the monolayers with vacancies. It can create an impression that a proper choice of a larger number of atoms and a better slide direction will result in the well-known Pd(111) layers irrespective of a slide like for Pd₃₀(111)/TiO₂(001). New Pd₂₈/γ-Al₂O₃(110) obtained by +4Pd addition and shift/optimization indeed resembles the (111) packing; however, it remains less stable than Pd₂₄ hybrids obtained due to the shifts of +1 and +3 Å. The advantage of the hybrid was also demonstrated for the case of Me = Pd, Pt, Rh at γ-Al₂O₃(100).¹³ It was shown that their monolayers can adopt a stable hybrid Pd(1/4) form named so because of alternated bands of squares (1 band) and triangles (4 bands). Our late evaluations confirmed that their U_Me stability exceeds the one of a single Me atom at γ-Al₂O₃(100) for Me = Pt and Rh also (it was demonstrated for Pd only¹³). This hybrid form is partly similar to the one obtained herein for Pd₂₄ at γ-Al₂O₃(110) (Fig. 2c, d, g and h), i.e., alternation of the bands of triangles and squares. The Me–Me distances in the hybrids have reasonable values relative to those in the bulk. The search for such new models can be achieved via slide testing. Our examples above for slides by +1 or +3 Å confirm this because they led to very close monolayer geometries.

Similar problems of the vacancies arise in 2D vdW hetero-structures. Removing some atoms from a borophene 2D monolayer has recently been proposed as a way for stabilization of a nodal-line semimetal.⁵ The stabilization of the respective 2D monolayer with vacancies could be enforced by a search for a suitable support with the required geometry of the surface atoms. The search for the slide direction along the appropriate support could be a solution for a stable siting of a semi-metal with ordered defects.

γ-Al₂O₃(110) (Fig. 1a), on the one hand, and rutile TiO₂(001) (Fig. 1d) or monoclinic ZrO₂(001) surfaces, on the other hand, reveal different trends for Pd monolayer formation. The dehydroxylated γ-Al₂O₃(110) surface possesses an attractive site for Pd which is bi-coordinated to a pair of Al atoms (Fig. S1g and 1b) where U_Pd = −5.314 eV per Pd exceeds the respective bulk value, i.e., −5.250 eV per Pd at the same PBE level or smaller (−5.637 eV per Pd) than the one of the bulk of −5.821 eV per Pd at the PBE-D3 level (Table S2†). This large energy was verified by using the Al₁₄O₂₀ model with a smaller UC for the γ-Al₂O₃(110) surface (Fig. 1b). The latter is sufficient to model the trapping of one Pd atom but is not convenient to simulate a slab. This model results in the smaller estimate of −5.159 eV per Pd (Table 1) but is in reasonable agreement with −5.314 eV per Pd obtained for the larger UC at the same PBE theory level. The other sites at the γ-Al₂O₃(110) surface result in a moderate stabilization (−3.787 and −3.755 eV per Pd). These sites together with the 8 favorable Pd positions (bi-coordinated to Al atoms) per UC form the upper and lower boundaries for the Pd₂₄ monolayer stabilization at γ-Al₂O₃(110), i.e., −4.896 and −4.886 eV per Pd (Table 1). These intermediate U_Pd values between the most stable and the others for the Pd₂₄ monolayer show a possible concurrence with the small clusters at intermediate coverage. As shown above (part 3.2), Pd stabilization in the Pd₃₀(111) monolayer at the rutile TiO₂(001) (Fig. 1d) surface exceeds the stabilization in any isolated atomic positions as we have observed for the hybrid Pd₂₀(1/4) monolayer at γ-Al₂O₃(100).

Experimental evidence for monolayer formation at the oxides under study is known for TiO₂(110).^19–24 Long discussion of the strong metal–support interaction (SMSI) concept included the behavior of metallic particles (Pt, Pd, Os, Ir, Rh, Ru) on various oxide supports (also at a mixture of anatase and rutile TiO₂ polymorphs) where the nearly equivalent CO or H₂ adsorption relative to deposited Me metal (CO/Me ∼ 1 or H₂/Me ∼ 1) in the reduced systems at 200 °C was not compatible with different hypotheses (collapse, poisoning) of the tentative nanoparticle/oxide structure.²¹ Electron micrographs of 2% Pd/TiO₂ and 2% Ir/TiO₂ gave no evidence for Pd or Ir crystallites greater than 1 nm.²¹ Later the authors²² proposed a flat structure of the Pt nanoparticle (up to monoatomic thickness) which correlated with the results of Baker et al. for Pt/TiO₂.²³ The last study demonstrated similar temperature intervals for the existence of metallic forms at oxides with SMSI, i.e., H₂ reduction at 152 °C was sufficient for the SMSI type structure. The authors²¹ also admitted Pt interaction with both cations and anions of the oxide.²² Mono- and bilayers of Au (ref. 19) and Pd monolayers²⁴ at TiO₂(110) were characterized as non-metallic species with a small gap. Possibly, due to the dominant (110) plane at the γ-Al₂O₃ surface^51,52 compared to the (100) plane, where the formation of Pd monolayers is more favorable, such observations were not fixed for γ-Al₂O₃. The nature of metal interactions with support atoms could be extended in the future using the QTAIM approach as realized for example in ref. 53 and 54.

A coherence between the experimental findings and stabilities predicted from theory can be achieved for Pt and Pd monolayers on the TiC(001) support where the formation of the Pt monolayer was electrochemically characterized.¹⁷ A stable Pt/Ti X-ray photoelectron spectroscopy signal ratio excludes the possibility of Pt agglomeration or detachment in the course of the hydrogen evolution reaction.¹⁷ Regarding the similarity between Pt and Pd monolayers shown earlier¹³ at γ-Al₂O₃(100), we also studied Pd monolayers at the Ti₆₄C₆₄(001) slab which is relatively easy to model. A measure of the stabilization of both Pt_N and Pd_N monolayers at TiC (12.242 × 12.242 Å), N = 16, 18, was done in the same way (Fig. 7). The Bader Pd charges vary between −0.106 and −0.318 |e| in agreement with the calculated Pt charge of −0.33 |e|.⁶ The high theoretical stabilization of the Pt monolayer is in agreement with experimental evidence of its formation.¹⁷ But the stabilization of the Pd monolayer is even higher than that of the Pt one which seems to be an argument in favor of similar stable Pd monolayers at TiC.

Fig. 7 A stabilization U_Me (1) of both (a) Pt_N and (b and c) Pd_N monolayers at TiC(001), N = (c) 16, (a and b) 18. The second value in the brackets (%) is the stabilization relative to the bulk value at the same PBE level or U_Me(Me_N)/U_Me(bulk) × 100%. 2 × 2 unit cells are considered without a support at (b and c). The atomic colors are given in olive, cyan, and gray for C, Ti, and Pt/Pd, respectively.

5. Conclusions

We showed that testing the shift of a Pd monolayer parallel to a support is a highly desirable subject after its initial deposition and relaxation. The number of possible variants for such sliding is limited by periodic boundary conditions so that the testing is not too time costly. Both the charge and energy depend on a proper choice of the monolayer position that can be important for both the reagent adsorption and reactivity of the monolayer. The greatest loss of the Pd stabilization of 7.4% was shown at rutile due to the corrupted Pd interaction with the support atoms. The evaluation of the required step could be given on the basis of the distance that provides the maximal number of the favored Al–Pd–Al contacts at γ-Al₂O₃(110). The drastic reconstruction upon a shift can be a sign of a non-complete monolayer. This possibility seems to be important for monolayers and bilayers. Due to the different O–Pd–O and Al–Pd–Al favorable sites at the γ-Al₂O₃(100) and γ-Al₂O₃(110) forms, respectively, a higher probability of Pd monolayer formation is predicted at γ-Al₂O₃(100) from our calculations with the smaller charge of the Pd monolayer (in absolute value). The surface Pd density spans a narrow interval from 6.45 (γ-Al₂O₃(110)) > 6.32 (rutile TiO₂(001)) > 6.29 Ų per Pd (γ-Al₂O₃(100)¹³) with slightly higher 6.89 Ų per Pd for m-ZrO₂(001) despite three different types of Pd packing in complete monolayers (without vacancies), i.e., (4R–4R), (111), (1/4) for ZrO₂, γ-Al₂O₃(110) and TiO₂, γ-Al₂O₃(100), respectively (the Pd(111) types at γ-Al₂O₃(110) and TiO₂ rather coincide). Respective surface energies γ (2) obey the sequence 60 < 78 < 133 < 136 meV Å⁻² for γ-Al₂O₃(110), TiO₂, ZrO₂, and γ-Al₂O₃(100). The activation barrier of 0.69 eV for water dissociation at the Pd₂₈ monolayer/rutile is pretty close to the moderate 0.67 eV at Pd₁₈/γ-Al₂O₃(100),¹³ which points to the catalytic capability of Pd monolayers on various supports. The high theoretical stabilization energies of Pt and Pd monolayers at TiC are in agreement with experimental evidence of monolayer formation (Pt).

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors are grateful for financial support from the Bulgarian Science Fund through contracts KΠ-06-RUSSIA-22 and the Russian Foundation of Basic Research within the grant 20-53-18001-Bolg_a. The research has been carried out using the equipment of the shared research facilities of HPC computing resources at Lomonosov Moscow State University.⁵⁵ This research used resources of the “Plateforme Technologique de Calcul Intensif (PTCI)” (http://www.ptci.unamur.be) located at the University of Namur, Belgium, which is supported by the FNRS-FRFC, the Walloon Region, and the University of Namur (Conventions No. 117, GEQ U.G006.15, 1610468 et RW/GEQ2016). The PTCI is member of the “Consortium des Équipements de Calcul Intensif (CÉCI)” (http://www.ceci-hpc.be).

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Footnotes

† Electronic supplementary information (ESI) available: Pd–O bond lengths in the bilayer Pd₃₂(111) slabs at m-ZrO₂(001) (Table S1, Fig. S2), complete data about with total energies (U_tot, U_ox) and U_Pd values at the PBE and PBE-D3 levels (Table S2), the geometries of single Pd atoms (Fig. S1a–c, g and h), monolayers (Fig. S1d and e and S3a–c), and bilayers (Fig. S1f) at γ-Al₂O₃(110) (Fig. S1g), TiO₂(001) (Fig. S1a–c and S3a–c), and m-ZrO₂(001) (Fig. S1h). Simulation steps of water dissociation at rutile can be visualized using the “H2O_dissociation_rutile_TiO2.xyz” file and wxMacMolPlt software. See DOI: 10.1039/d1ce01365c

‡ The atom changes its position (from top to bottom or from left to right) due to periodic boundary conditions (Fig. 3a–d).

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