Bonding network and stability of clusters: the case study of Al13TM4 pseudo-tenfold surfaces

The physical significance of clusters in Al13TM4 compounds is invesitgated through ab initio methods based on density functional theory.


Introduction
A large variety of intermetallic crystal structures are based on polyhedral entities, often called 'clusters', as first introduced by F. A. Cotton in the early 1960s to describe compounds containing metal-metal bonds (Cotton & Walton, 1982). This approach is very useful to represent the structures of complex intermetallic phases, like intermetallic clathrates, but also quasicrystals, their crystalline approximants and related phases (Steinhardt & Jeong, 1996;Suck et al., 2002;Abe et al., 2004). Typical clusters found in quasicrystals include the Mackay (Sugiyama et al., 1998), Bergman (Bergman et al., 1957) and Tsai clusters (Takakura et al., 2001), and numerous polyhedral shapes are used to describe complex intermetallics.
There is not usually a unique description of crystal structures in terms of structural building blocks. For example, a packing of pentagonal bipyramids was initially used by Henley to describe the Al 13 TM 4 structure types (TM = Fe, Co, Ru, Rh, Fig. 1). Later, based on quantum chemistry calculations (Armbrü ster et al., 2011), these compounds were represented as columns of elongated clusters containing strong TM-Al-TM molecular groups, resembling the 3D 'cage-compound' structure of the intermetallic clathrates. Dong's cluster+glue model, describing the structure by the [icosahedron](glue) 0,1 formula containing all the key information on alloy chemistry (Dong et al., 2007), uses two types of icosahedral clusters for Al 13 Fe 4 : Al 9 Fe 4 and Fe 2 Al 11 (Chen et al., 2014). Finally, the 18 À n bonding scheme applied to Al 13 Os 4 describes the structure by a stacking of Al 5 square pyramids and fluoritetype columns (Miyazaki et al., 2017).
The question of cluster stability in complex intermetallic compounds was reviewed a few years ago in the case of quasicrystals (Steurer, 2006). Several studies mention the influence of the intrinsic cluster structure on the bulk physical properties, such as their mechanical properties (Feuerbacher et al., 2001;Messerschmidt et al., 1999) or their electronic conductivity (Janot & de Boissieu, 1994;Janot, 1996;, 2006Fujiwara, 1989;Fujiwara et al., 1993;Zijlstra & Bose, 2003;Trambly de Laissardiè re & Fujiwara, 1994;Dolinšek et al., 2000;Stadnik et al., 2001;Widmer et al., 2006;Zijlstra & Janssen, 2000;Trambly de Laissardiè re, 2003, 2009). Molecular dynamics based modelling of crack and dislocation propagation in simple quasiperiodic model structures (Mikulla et al., 1998;Rö sch et al., 2004Rö sch et al., , 2005Rudhart et al., 2004) also questioned the role of clusters. Surface studies were found to be a pertinent approach to gain some insight into the question of cluster stability in quasicrystals (Dubois & Belin-Ferré, 2011;McGrath et al., 2010, and references therein). When prepared by sputtering and annealing cycles, quasicrystalline surfaces usually produce large, atomically flat terraces. The patterns observed by scanning tunnelling microscopy (STM) are attributed to signatures of the dissected clusters at the surface (Krajčí & Hafner, 2005). This then suggests that the clusters in icosahedral quasicrystals are not stable sub-units that maintain their shapes at the surface. Do these conclusions present a general character among complex phases? Recently, several surfaces of a cage compound -namely the type-I clathrate BaAuGe -were investigated by a combination of experimental and theoretical techniques. It was found that these surfaces preserve the cluster substructure, which is stabilized by a charge transfer mechanism (Anand et al., 2018a,b).
It follows that different surface behaviours occur among complex phases. In the following, we focus on the Al 13 TM 4 complex intermetallics (TM = Fe, Co, Ru, Rh), usually considered as four-layer decagonal quasicrystalline approximants, because they present a layered structure and pentagonal atomic arrangements (Goldman & Kelton, 1993). Experimentally, using single-crystal surfaces prepared by cycles of sputtering and annealing, significant differences were found among the considered pseudo-tenfold Al 13 TM 4 (TM = Fe, Co, Ru) surface structures (Fourné e et al., 2012;Shin et al., 2011), although common features were also observed. For all considered systems, the surface composition measured from XPS (X-ray photoelectron spectroscopy) as a function of surface sensitivity indicated no sign of chemical segregation. A surface plane selection is observed for all Al 13 TM 4 pseudo-tenfold surfaces, highlighted by STM measurements which identified consecutive terraces separated by a unique step height equal to half the lattice parameter. However, the absence of surface reconstruction is not observed systematically: the LEED (low-energy electron diffraction) patterns of the Al 13 Co 4 (100) and Al 13 Fe 4 (010) surfaces show rectangular and oblique surface unit cells, respectively, with dimensions consistent with the bulk values, while the surface of Al 13 Ru 4 (010) exhibits an atypical surface reconstruction, attributed to the occurrence of stripes running about 10 off the [001] direction. High-resolution STM images also identified different motifs at the surface for the different compounds (Fig. 2). Bipentagonal features are resolved at the Al 13 Co 4 (100) surface while pairs of bright features are observed at the Al 13 Fe 4 (010) surface, and fivefold motifs combined with pentagonal vacancies are visible at the Al 13 Ru 4 (010) surface .
For Al 13 Fe 4 (010), the combination of theoretical calculations and experimental observations led to a model preserving the Henley-type clusters at the surface. The corresponding simulated STM images were in good agreement with the experimental images . Further investigations based on dynamic LEED confirmed this result (Matilainen et al., 2015). Several complementary techniques have been employed to reach a reliable model of the Al 13 Co 4 (100) surface. The combination of STM, calculations based on density functional theory (DFT) and LEED converged towards a surface terminating at puckered layers (hereafter P-layers, Fig. 1  Fourné e et al., 2012). The surface structure showed heterogeneities, identified by STM, related to the partial occupancy of a few surface sites (Al 'glue' atoms, located in between Al bipentagonal motifs). The combination of surface X-ray diffraction and DFT pointed towards the same surface model, with partial occupancies for surface Co sites slightly buried in the P-type plane (Gaudry et al., 2016).
Whether or not clusters are preserved at the surface must be linked to the strength of their intrinsic bonds and, as such, is expected to strongly depend on the atomic and electronic structures of the considered compounds. While the dimensionality of the chemical bonding network in o-Al 13 Co 4 and Al 13 Fe 4 has been investigated on the basis of electrical transport measurements (Dolinšek & Smontara, 2011), no theoretical extensive and systematic comparison of the compounds in the Al 13 TM 4 (TM = Co, Fe, Ru, Rh) series has been carried out so far. Our work is based on a wide range of ab initio tools, based on DFT, in order to investigate the influence of the chemical composition on the pertinence of a bulk structure description based on clusters. Electronic structure calculations, including band structure calculations, projected density of states and projected crystal orbital Hamilton populations, highlight the different bonding characters of Al-Al, Al-TM and TM-TM pairs and how they contribute to the bulk cohesion. Methods based on infinitesimal displacements and harmonic approximations emphasize the impact of the chemical composition on phonon properties. Anisotropic thermal displacements are analysed and the singular behaviour induced by the cage structure is shown. Altogether, our results are used to discuss the relevance of the cage-versus layer-based descriptions. Finally, the interplay between the 3D bulk atomic arrangements and the 2D surface is discussed, based on surface energy calculations, before we present the conclusion.
Total energy calculations were performed using a cut-off energy (E cut ) and a number of k-points within the Brillouin zone so as to achieve an energy accuracy better than 0.1 meV per atom (E cut = 450 eV, Monkhorst-Pack k-points grid = 9 Â 7 Â 5 or equivalent). The reciprocal-space sampling was increased for electronic structure calculations (17 Â 13 Â 9 Monkhorst-Pack k-points grid) and the tetrahedron method with Blö chl corrections was used for Brillouin zone integrations. For the phonon calculations with supercells (2 Â 1 Â 1 or equivalent), we used a smaller k-point grid (2 Â 2 Â 2), in agreement with the setup of Mihalkovič & Widom (2007).
The phonon frequencies and the thermal displacements are determined using force constants derived from the calculation of the dynamic matrix based on the finite displacement method implemented in the Phonopy software (Togo & Tanaka, 2015). We used small atomic displacements (AE0.01 Å ). No imaginary mode was detected. The ellipsoid software was used to convert the thermal displacement parameters from the Cartesian to the crystal coordinate system (Deringer et al., 2014;George et al., 2015).
We used the projected crystal orbital Hamilton population (pCOHP) approach, implemented in the LOBSTER code (Dronskowski & Bloechl, 1993;Deringer et al., 2011;Maintz et al., 2013Maintz et al., , 2016 to analyse the chemical bonding. This method re-extracts Hamilton-weighted populations from plane-wave electronic structure calculations to develop a tool analogous to the crystal orbital Hamilton population (COHP) method (Deringer et al., 2011). The electron wavefunctions are projected onto the atomic local basis used for the DFT calculations: 3s3p for Al, 4s3d for Co and Fe, 4p5s4d for Ru and Rh. The charge spilling, i.e. electrons which cannot be projected onto the local basis, is found to be between 1 and 3% (1.09% for Al 13 Rh 4 and 2.79% for Al 13 Fe 4 ).

Surface energy calculations
The surfaces have been modelled with seven-layer-thick symmetric slabs, separated by a void thickness ('12 Å ). Surface energies ( clean ) were computed as a function of the Al chemical potential: where E slab is the total energy of the slab, i and N i the chemical potential and number of i species in the slab. The surface is considered to be in equilibrium with the underlying bulk, which constrains the chemical potentials in a range, i.e. ð17=13ÞÁH f Al À bulk Al 0 for Al, where ÁH f is the formation energy of the complex phase.
The Al 13 TM 4 compounds have been identified as promising catalysts towards hydrogenation reactions (Armbrü ster et al., 2009(Armbrü ster et al., , 2012Piccolo, 2013;Piccolo & Kibis, 2015). Therefore, to investigate a possible modification of the surface structure during operating conditions (H 2 atmosphere), the surface energies of the hydrogenated surfaces ( cover ) were evaluated by the sum of the clean surface energies ( clean ) and those with adsorbed species ( ads ) (Reuter et al., 2002;Reuter & Scheffler, 2003;Posada-Pé rez et al., 2017): cover = clean + ads (P, T, N H 2 ) where P, T and 2 Â N H 2 are the pressure, temperature and number of hydrogen atoms adsorbed on the surface. A simple thermodynamic model was used to compute ads :  where E H 2 is the energy of H 2 in vacuum, and Á H 2 is the chemical potential of H 2 calculated as Á H 2 ¼ Àk B T ln Z where Z is the partition function of the gas-phase H 2 molecule and k B is the Boltzmann constant. In the latter partition function, we only consider the translational and rotational contributions.

Atomic structures
The Al 13 TM 4 structures belong to the C2/m space group (102 and 51 atoms per cell for the conventional and unit cells, respectively). We also considered the Al 13 Co 4 orthorhombic phase (o-Al 13 Co 4 ), which crystallizes in the Pmn2 1 space group, with 102 atoms per cell. These two structures share similarities. They are described as a stacking of flat (F) and puckered (P) layers along the pseudo-tenfold axis ([010] for monoclinic crystals, [100] for the orthorhombic one). As mentioned in Section 1, these structures are also described by a stacking of clusters. In the following, the term 'cluster' refers to the Henley-type cluster ( Fig. 1), i.e. the pentagonal bipyramid.
The cell parameters deduced from the structural optimizations are gathered in Table 1. They are in good agreement with the experimental data. Here, full occupancies were considered in the theoretical approach, while partial occupancies are experimentally observed (Grin et al., 1994a,b) and contribute to the stability of the compounds (Mihalkovič & Widom, 2007).

Phonon band structures
To evaluate the anisotropic displacement parameters, phonon calculations have been carried out. The resulting phonon band spectra are presented in Fig. 3 for o-Al 13 Co 4 and monoclinic Al 13 TM 4 (TM = Co, Fe, Ru, Rh). There are 153 and 306 branches in the phonon band structure, corresponding to the 3 Â N degrees of freedom in the primitive unit cell for the monoclinic and orthorhombic structures, respectively. No band gap is observed, the optic modes arising from around 1.1-1.5 THz, in the A-point (respectively, Z-point) of the Brillouin zone for monoclinic (respectively, orthorhombic) structures. No clear difference between the averaged group velocities calculated perpendicularly or within the pseudotenfold axis is observed.
The phonon densities of states show a Debye behaviour at low energies and a maximum located around 4-6 THz. The position of the maximum is shifted to lower energies when moving from Al 13 Co 4 and Al 13 Fe 4 to Al 13 Ru 4 and Al 13 Rh 4 , because 4d metals are heavier. For Al 13 Co 4 , our results for the vibrational density of states are in agreement with the experimentally measured ones (Mihalkovič et al., 2000), and compare quite well with those reported by Mihalkovič & Widom (2007). However, in the latter case, the consideration of a single Al vacancy leads to a slight excess in the density of states at low frequency, which is not observed here because full occupancies were considered.
The previous phonon calculations were used to calculate the U ij anisotropic thermal displacements. Larger anisotropic displacements are found for several Al atoms in the flat plane, the largest one being observed for the Al atom of the TM-Al-TM molecular group, with values for U pseudo-tenfold =ðU 2 ? þ U 2 ? 0 Þ 1=2 between 0.1 and 0.3 (Fig. 4). This is in agreement with the bonding picture, the Al atom located in the cluster centre being involved in strong bonds within the neighbouring Al 5 TM atomic arrangements located in the P-type plane above or below, while the covalent-like interactions with the surrounding atoms in the flat plane are found to be negligible (see Section 4).

Electronic band structures
At low energy, the electronic structures of the Al 13 TM 4 compounds (Fig. 5)  Top and side views of the anisotropic displacements for atoms located in the Henley-type cluster (Al 13 Co 4 compound). Table 1 Cell parameters resulting from structural optimization, for Al 13 TM 4 monoclinic structures (C2/m space group) with TM = Fe, Co, Ru, Rh.
The case of the orthorhombic structure for Al 13 Co 4 (Pmn2 1 space group) is considered as well. free electron behaviour of the sp-like states. The orange colour of the bands shows the predominance of Al-sp states over TM-sp states. At higher energy, a strong maximum appears in the density of states (DOS), caused by localized and weakly dispersive TM-d states. This is in agreement with previous calculations (Mihalkovič & Widom, 2007;Manh et al., 1995). The uniform colour of bands suggests a strong hybridization between Al-sp states and TM-sp states as well as between Al-sp states and  Table 2). The electron per atom ratio (e/a) using the atomic values recently computed by Mizutani et al. (Mizutani & Sato, 2017;Mizutani et al., 2013;Mizutani, 2010) (1.00 for Rh, 1.03 for Co, 1.04 for Ru, 1.05 for Fe and 3.01 for Al) is 2.5 for the Al 13 TM 4 compounds, i.e. slightly larger than the values usually observed for Hume-Rothery phases (Ferro & Saccone, 2008), which classifies them as polar intermetallics. The presence of the pseudo-gap may then be due to a combination of the Hume-Rothery stabilization mechanism with hybridization effects, as already highlighted for Al 9 Co 2 (Trambly de Laissardiè re et al., 2005).

Network of chemical bonds
The bonding network in the Al 13 TM 4 compounds is investigated in order to gain some insight into the various surface structures observed, the broken bond model being largely employed to account for surface energies (Ruvireta et al., 2017).

Chemical bonding analysis
The chemical bond analysis based on the COHP curves and their integrated values (ICOHP, Table 3) revealed that the strongest bonds are homonuclear Al-Al bonds, located within the F-type atomic plane and ensuring the connection between clusters (Fig. 6). The strength of these bonds is rather high (larger than 2 eV per bond), despite quite large Al-Al distances (larger than 2.5 Å ). This is consistent with the idea that the metallic bond is very closely related to the covalent  Table 2 Position of the d-band maximum and density of states at the Fermi energy for the considered compounds.  (Manh et al., 1995) Figure 5 Band structures of the o-Al 13 Co 4 and monoclinic Al 13 TM 4 compounds (TM = Fe, Ru, Rh).
(shared-electron-pair) bond, and that each atom in a metal may be considered as forming covalent bonds with neighbouring atoms, the covalent bonds resonating among the available interatomic positions (Pauling, 1947). The shortest Al-TM distances also lead to strong bonds. They are identified as those of the TM-Al-TM molecular group located inside the cluster, oriented parallel to the pseudo-tenfold axis, connecting the F-and P-type atomic planes, in agreement with the previous analysis by Grin et al. (Armbrü ster et al., 2011), and consistent with the NMR study by Jeglič et al. (2009Jeglič et al. ( , 2010. Very weak TM-TM bonds are found ('0.2 eV per bond), with bonding distances close to 3 Å .

Cage-versus layer-based description
To probe the stacked-layer structure of the Al 13 TM 4 periodic decagonal approximants, we evaluate the P-type in-plane (S in P-type ) and inter-plane (S out ) bonding capacities by The ratio S in =ðS in þ S out Þ is calculated to be 24.5%, 26.8%, 24.4% and 28.4% for TM = Fe, Co, Ru, Rh, respectively. These results are consistent with the conclusions of Dolinšek & Smontara (2011), based on anisotropic resistivity measurements, stating that the stacked-layer description in terms of 2D atomic planes should only be regarded as a convenient geometrical approach to describe structures of the Al 13 TM 4 quasicrystalline approximants, whereas their physical properties are those of true 3D solids. Cluster stability is evaluated through the bonding capacity of TM atoms located in the P-type planes (TM P-type ). Such atoms are bounded to three different types of Al neighbours: the Al atom within the TM-Al-TM molecular group, the surrounding Al atoms, located in the P-type plane (pentagonal arrangement), and the ones outside the cluster, within the flat plane (Fig. 1). Our results are presented in Table 4. For all compounds, the contributions to the bonding capabilities of the TM-Al-TM molecular group are around 16-17%. It is the largest for Al 13 TM 4 with TM = Fe, Ru, because the corresponding states show almost no anti-bonding for TM = Fe, Ru (Fig. 7), while they are slightly anti-bonding for TM = Co, Rh.  Table 3 Average ICOHP/bond values (eV per bond), along with the corresponding average distance (in parentheses, in Å ), for the strongest Al-Al, TM-TM and Al-TM bonds in Al 13 TM 4 compounds.

Al-Al
Al-TM TM-TM

Figure 6
Structure of the o-Al 13 Co 4 F-type plane highlighting the strongest Al-Al and Co-Co bonds (red circles), which are inter-cluster bonds, linking together bipentagonal atomic arrangements. Light blue = Al; dark blue = Co.

Figure 7
COHP curves for the TM-Al bonds of the TM-Al-TM molecular group, along with those showing the interactions of TM P-type atoms with the surrounding Al atoms in the P-type plane (Al pentagonal arrangements). Table 4 Average ICOHP/bond values (eV per bond) and percentage contributions (the contributions of the bonding relative to a given atom are evaluated by the sum of the ICOHP/bond values of the nearest neighbouring interactions weighted by the respective bond frequencies) of the respective interactions to the net bonding capabilities around the TM atoms located in the P-type atomic plane.
Three types of Al neighbours are considered: the Al atoms within the TM-Al-TM molecular group, the surrounding Al atoms in the P-type plane (pentagonal arrangement), and those outside the cluster, in the F-type plane (Fig. 1). contribute 69-70% to the bonding capabilities. More than 50% of these interactions are attributed to the closest Al pentagonal arrangement, even if slight anti-bonding TM-Al P-type interactions at the Fermi level are revealed by the COHP curves. Intra-cluster contributions to the bonding capabilities of TM P-type atoms are the strongest for Al 13 Fe 4 , and the lowest for o-Al 13 Co 4 and Al 13 Rh 4 . An intermediate case is that of Al 13 Ru 4 , with Al-TM P-type bonds within the TM P-type -Al-TM P-type group as strong as those in Al 13 Fe 4 , while the strengths of TM-Al F-type bonds are similar in Al 13 Ru 4 , o-Al 13 Co 4 and Al 13 Rh 4 .

Bonding strengths and bonding distances
Finally, when looking at the variation of bonding strength as a function of bonding distance (Fig. 8), an exponential behaviour is observed, in agreement with the exponential decrease in the bond number n with bonding distance D n (in Å ) initially proposed by Pauling (1947Pauling ( , 1960: D n ¼ D 1 À A Â log 10 n [equation (29) in Herman (1999)].

Bonding network and pseudo-tenfold surfaces
From the bonding analysis, the P-type in-plane bonding capacities are evaluated to be in the range 20-30%. The clusters are found to be rather stable entities, the intra-cluster Al-TM P-type interactions contributing 69-70% to the TM P-type bonding capabilities. In the following, we discuss the consistency of these results with the pseudo-tenfold surface structures identified so far.

Clean surfaces
Two surface models are considered in the following. The A-type model preserves the Henley-type clusters at the surface, while the B-type surface model terminates at P-layers where on average all Al atoms are present and protruding Co atoms are missing. The surface energies of these two models are plotted in Fig. 9. The A-type model, preserving the cluster structure at the surface, is found to be the most stable within a large domain of chemical potentials, for both Al 13 Co 4 (100) and Al 13 Fe 4 (010), in agreement with the bonding situation of the compounds. However, the surface structure observed for Al 13 Co 4 (100) is the B-type one, corresponding to the narrow range in the Al-rich region.
The surface structures of the Al 13 TM 4 compounds with TM = Co and Fe reflect the duality between the description of the structure based (i) on a stacking of atomic planes (F-and P-type) perpendicular to the pseudo-tenfold direction, mirroring the periodic stacking of atomic planes with quasiperiodic in-plane atomic order found in decagonal quasicrystals, and (ii) on a stacking of Henley-type clusters. A dense Al-rich surface was identified for o-Al 13 Co 4 (100) and a highly corrugated surface, based on the preservation of the cluster structure at the surface, for m-Al 13 Fe 4 (010). They are consistent with the stronger character of intra-cluster bonds of m-Al 13 Fe 4 compared with o-Al 13 Co 4 . They are also consistent with the slightly stronger in-plane bonding capacities of the Al 13 TM 4 (TM = Fe, Co) P-type planes: 24.5% and 26.8% for m-Al 13 Fe 4 and o-Al 13 Co 4 , respectively.
Compared with the related m-Al 13 Fe 4 (010) and o-Al 13 Co 4 (100) surfaces where superstructures are absent, the reconstruction observed for m-Al 13 Ru 4 (010) is thought to act as a strain relief mechanism. Here, the Ru atoms located in the P-type atomic plane are strongly bonded to the Al atom of the Ru-Al-Ru molecular group: 2.30 eV per bond, but a bonding capacity similar to that of m-Al 13 Fe 4 (010), i.e. 16.6%. However, the surface structures of m-Al 13 Fe 4 (010) and m-Al 13 Ru 4 (010) present quite large differences, since a reconstruction is observed in the case of m-Al 13 Ru 4 (010).

Surface under hydrogen atmosphere
Both Al 13 Co 4 and Al 13 Fe 4 compounds have been identified as promising catalysts for hydrogenation reactions (Armbrü ster et al., 2009(Armbrü ster et al., , 2012Piccolo, 2013;Piccolo & Kibis, 2015). The performances of the catalysts are attributed to the Al 5 TM atomic arrangements at the surface, in agreement with the site isolation concept. While such atomic arrangements have been experimentally observed for m-Al 13 Fe 4 (010), the surface structure determined so far under ultra-high vacuum for Bonding strength of Al-Al bonds in Al 13 TM 4 compounds as a function of the distance. The fit uses the function n 0 exp½ðr 0 À rÞ=c. o-Al 13 Co 4 (100) consists of a dense Al-rich plane, leading to higher barriers for hydrogenation reactions (Krajčí & Hafner, 2011;Kandaskalov et al., 2017).
Experimental conditions are known to have a strong effect on the surface structures. For example, slight deviations from an ordered alloy's ideal stoichiometry in the subsurface or bulk region can drastically affect the surface composition (Ruban, 2002;Blum et al., 2002). Such effects have been observed on o-Al 13 Co 4 (100) using two single crystals grown by two different techniques, which may present slightly different bulk compositions (within the stability range of the o-Al 13 Co 4 phase) (Fourné e et al., 2012). However, in both cases, the surface is Al rich and no Co atoms were found to protrude at the surface.
When used as catalysts, the surface structure may evolve under operating conditions. In particular, exothermic adsorption on solid surfaces is known to reduce their surface energy (Mathur et al., 2005). In the following, we evaluate the modification of the surface energy due to the adsorption of atomic hydrogen. We focus on o-Al 13 Co 4 (100) using the two previous surface models (A-and B-type models). We consider two different coverages (4 and 8 atomic H per surface cell, Fig. 10). Atomic hydrogen is adsorbed on the most favourable adsorption sites (Krajčí & Hafner, 2011;Kandaskalov et al., 2014). The adsorption leads to a decrease in the surface energy, for temperatures below 400 and 200 K for the A-and B-type models, respectively. The stabilization is higher for the A-type model, because atomic hydrogen is more strongly adsorbed on the A-type model. The relative stabilization of the A-type model compared with the B-type one is found to be 0.07 and 0.14 J m À2 for the two considered coverages, respectively. The surface energy difference calculated between the Aand B-type models is 0.14 J m À2 for Al = Al bulk , i.e. larger or equal to the relative stabilization of the A-type model compared with the B-type one, when Al = Al bulk , and for the considered adsorption configurations and coverages. For small atomic hydrogen coverages (four atoms per surface cell), our calculations, realized with the PBE approach, suggest that the considered surface structures are rather stable. However, larger atomic hydrogen coverages are likely to modify the surface structure.

Conclusion
We reported a systematic investigation of the electronic structure, phonon properties and chemical bonding network of bulk Al 13 TM 4 compounds (TM = Co, Fe, Ru, Rh). Electronic structure calculations highlight rather strong hybridization. The strong TM-Al-TM bond within the Henley-type cluster leads to a strong anisotropy in the thermal displacement of the central Al atom. From the bonding analysis, the clusters are found to be rather stable entities, the intra-cluster Al-TM P-type interactions contributing 69-70% to the TM P-type bonding capabilities.
Structural differences between the o-Al 13 Co 4 (100) and m-Al 13 Fe 4 (010)  Modification of the o-Al 13 Co 4 (100) surface energy ( ads ) with atomic hydrogen adsorbates, for two different coverages, as a function of temperature and pressure.

Figure 9
Surface energy of the A-(full line) and B-type (dashed line) models, calculated for o-Al 13 Co 4 (100). strengths of the Al-TM bonds within the TM-Al-TM molecular groups provide an understanding of the different surface structures observed. Since intra-cluster bonds are stronger for m-Al 13 Fe 4 compared with o-Al 13 Co 4 , and in-plane bonding capacities of the P-type planes are stronger for o-Al 13 Co 4 compared with m-Al 13 Fe 4 , Henley-type clusters are preserved at the m-Al 13 Fe 4 (010) surface while they are truncated at the o-Al 13 Co 4 (100) surface.
The possible interactions with adsorbates lead to a decrease in the calculated surface energies, at low temperatures (T < 400-500 K and T < 200-300 K for the A-and B-type models, respectively). The stabilization depends on the chemical potentials of the adsorbate, as well as on those of the Al and TM atoms. This highlights the importance of operando conditions when considering applications for these surfaces.
Nevertheless, the specific atomic arrangements at the surface induced by the intrinsic cluster substructure of complex intermetallic compounds affect the surface properties. Several adsorption studies highlight the role of specific sites resulting from the cut by the surface plane through the cluster units identified in the bulk solid (Unal et al., 2009;Fourné e et al., 2014;Ledieu et al., 2009;Krajčí & Hafner, 2008). On fivefold Al-based quasicrystalline surfaces, it leads to the two famous 'dark stars' and 'white flowers' sites (Unal et al., 2009), which are identified as favourable atomic and molecular adsorption sites (Cai et al., 2003;McGrath et al., 2002). On approximant and related phases, signatures of the cluster substructure at the surface also lead to specific chemically active sites. For example, a possible reaction path for the semi-hydrogenation of acetylene on o-Al 13 Co 4 (100), identified as a performant catalyst for this reaction (Armbrü ster et al., 2009(Armbrü ster et al., , 2012, is predicted to involve the protruding clusters (CoAl 5 ensemble) (Krajčí & Hafner, 2011). Given the structural variety of complex intermetallics, described by stackings of very diversified clusters naturally present in the bulk structure, we can hope to control the surface properties by the selection of clusters that emerge at the surface as active centres, giving rise to multiple applications at the nanoscale.

Funding information
This work was supported by the European Integrated Center for the Development of New Metallic Alloys and Compounds. High Performance Computing resources were partially provided by GENCI under the allocation 99642, as well as the EXPLOR centre hosted by the Université de Lorraine (allocation 2017M4XXX0108). This work was supported by the French PIA project Lorraine Université d'Excellence, reference ANR-15-IDEX-04-LUE.