Approaching Bulk from the Nanoscale: Extrapolation of Binding Energy from Rock-Salt Cuts of Alkaline Earth Metal Oxides

A systematic DFT study is performed on (MgO)n, (CaO)n, (SrO)n, and (BaO)n clusters with 6 < n < 50, and which display a cuboid 2 × 2 × 2 atomic motif seen in the bulk, rock-salt, configuration. The stability and energy progression of these clusters are used to predict the energies of infinitely long nanorods, or nanowires, slabs, and the bulk global minimum energy.

other hand, within the modelling community there is a high interest in finding series of representative nanostructures of computationally manageable size that can be exploited to predict physical properties and reactivity of bulk, either directly or by extrapolation. 7 Our first application of the proposed method is to alkaline earth metal oxides, the cuboid [1 0 0] cuts of which as shown by experiment resemble the shape of observed nanowires. 8 We predict properties of nanowires and slabs that can be constructed from the smallest cuboid metal oxide (MO) 4 nanocluster, which is in fact only metastable in the gas phase. Note, that properties of nanowires, or nanorods, and slabs have attracted interest 9 due to applications in field emission devices 10 and sensing. 11 Early studies of nanoclusters typically reported the tentative global energy minimum atomic configuration of a compound as a function of the number of formula units, n, it is composed of. The structural motif of these global minima for a compound will also typically change with size n; the smallest global minima may not resemble cuts that can be taken from the crystal structure of the bulk phase. Nanoclusters of zinc oxide is a good example of this change in non-bulklike structural motifs: the smallest global minima are rings, bubbles, double bubbles. 1,12 In contrast, many of the relatively more stable global minima for the smaller nanoclusters of alkaline earth oxides have cuboid atomic configurations and therefore resemble cuts from their bulk phase (the NaCl rock-salt structure).
Even if the stoichiometry is maintained, nanoclusters display a wide structural diversity. Take, for example, the relatively small (MgO) 5 cluster, which is predicted to have five unique metastable structures. 13 Given that the properties of materials are dependent on their atomic and electronic structure, nanoclusters can potentially display a diverse range of properties compared to their corresponding bulk.
Of the alkaline earth metal oxides (MO) n clusters, magnesium oxide clusters, which have applications in gas and photoactive 14 sensing devices, have been extensively investigated. 6a,15 Structural and electronic properties have been reported for (MgO) n clusters up to n = 40, 6a,15c,15d,15f,16 and for nanotubes up to n = 33. 15b,15d, 16 For the smallest (n ≤ 30) (MgO) n clusters, another structural motif was predicted to be that of the tentative global minima for atomic structures resembling hexagonal-barrels. 13a Conversely the global minima clusters of the Ca, Sr, and Ba analogues show a preference for rock-salt cuts at sizes as small as n = 4 and specifically a preference for cuboid rock-salt cuts persists at arbitrarily small sizes, even where barrel shapes, bubbles or non-cuboid rock-salt cuts are also local minima. 17 An extrapolation approach has been previously employed for magnesium and calcium oxides using the normalized clustering energy (NCE) of the global minima in the cluster size range of n = 2-40 16,18 to predict the energy at the bulk limit. In our early work on the stability of (MO) n clusters with n = 2-24, we found that while most of these clusters have rock-salt-like global minima, not all do, 13a,17a and, therefore, the stability trends might have an undue noise that should hinder accurate prediction. But this problem can be solved by a judicious construction of cluster series using one building unit, i.e. following the same structural motif will be demonstrated below.
In this paper, we will investigate ever-larger atomic structures that can be constructed from the n = 4, 2 × 2 × 2 cuboid building unit; first expanding in one, then in two, and finally in all three dimensions. These series of calculations will allow us to extrapolate the energies of a one-dimensional nanowire and a twodimensional slab. This method will be applied to the whole series of stable alkaline earth metal oxides: MgO, CaO, SrO, and BaO. Our hypothesis is that extrapolation using data from clusters that do not include the non-cuboid global minima should yield a more accurate value of the bulk energy. We also extrapolate directly from the energy of cubic cuts to obtain an estimate of the bulk energy. In the following section we outline the methods used. After that we present our results along with a discussion and conclude with a summary of our findings.

Cluster Geometry Optimization
The initial unrelaxed structures for most cuboid structures for (MO) n n ≤ 25 were taken from our previous data, 13a,17a while all other clusters were generated for this study with the metal-oxygen bond length set to that found in the bulk. 19 The geometry of all clusters was optimized in FHI-aims 20 with the PBEsol 21 GGA functional and tight settings for the density grid, Hartree potential, basis functions cut-offs and tolerances imposed on convergence. The geometry was considered converged when the maximum residual force component per atom was below 1 × 10 -3 eV Å -1 . FHI-aims tier 1 basis functions was used for all cations, which is analogous to DNVP(1d1f), i.e. double numerical valence with polarization, whereas FHI-aims tier 2 basis functions for anions, which is analogous to TNVP(2d1f1g), i.e. triple numerical valence with enhanced polarization.

Cluster Band Gap Calculations
Single point calculations were performed on the optimized geometries using the PBEsol0 functional and a tight basis set. Tier 1 basis functions were used for all cations, and tier 2 for oxygen anions.

Calculation of the Normalized Clustering Energy
For all the cuboid clusters the Normalized Clustering Energy (NCE) was calculated. The NCE 22 is a measure of how much more stable a cluster is compared to the n = 1 smallest stoichiometric global minimum configuration, i.e. the fully relaxed dimer molecule. For a total energy E(n) of a (MO) n cluster, and with the corresponding energy of the M-O dimer, E(1), the normalized clustering energy is The NCE can be extended to one-two-and three-dimensional structures by taking n to be the number of formula units in the unit cell for which the energy is given; these energies will be referred to as generalized NCEs (gNCE).

Direct Calculation of the NCE for Periodic Structures
The energies of the 1D, 2D and 3D oxides were directly calculated using FHI-aims and the same functionals as for the geometry relaxation and employed periodic boundary conditions. For the one-and two-dimensional materials vacuum gaps were utilized to ensure that the separation between the materials and their periodic images was large enough that they did not interact with each other.

Building Blocks
This study focuses on clusters of the group 2 oxides in sizes ranging from 2 × 2 × 3 atoms, n = 6, up to n = 50 (all possible cuboids except the 4 × 4 × 5 cuboid). Later a direct extrapolation of the bulk energy will be obtained from the energies of the series of perfect cube cuts where we will notice that a reasonable linear fit is hampered by the inclusion of the smallest cluster. Moreover, given that the composition of the 2 × 2 × 2 clusters contains only three-coordinated corners atoms (see Fig. 2a), i.e. all atoms severely undercoordinated, the energy of this smallest cluster is not used in any of the three-step extrapolation.  Energies for the finite systems, clusters composed of p × q × r blue blocks, are calculated; whereas energies for the extended systems (green blocks) are determined by extrapolation as follows: nanowires with cross-section p × q from data taken from the set of r p × q × r clusters; slabs with thickness of p blocks from data taken from the set of q nanowires with cross-section p × q; and bulk from data taken from the set of p slabs.
clusters are defined in terms of N x × N y × N z , where each number, N, is the number of atoms along the axis.

Clusters to Nanowires
Initially, the 2 × 2 × N z clusters were considered, with the geometry optimized in FHI-aims. The NCE was calculated for N z ranging from 3 to 25 (n = 6 to 50) and plotted as a function of the number of formula units, n, shown in Fig. 3.
From Fig. 3 we see the NCE is lowest for MgO clusters and highest for BaO clusters, with the NCE for CaO, and SrO clusters approximately midway between. The similarity of NCE energy for Ca, and Sr clusters is expected due to their more similar sizes and ionization energies -shown in Table 1.
As expected, stabilizing effects increase with cluster size as well as decreasing cation size. In Fig. 3 we also observe an inverse relationship between cluster size and NCE. When the NCE is plotted as a function of 1/n, shown in Fig. 4 a linear fit to the data can be applied, which can be extrapolated to infinitely long nanowires, as 1/n tends to 0.
The vertical-axis intercepts of the linear fits from Fig. 4 are the predicted generalized NCE (gNCE) for infinitely long nanowires with a 2 × 2 atom cross-section. Next we systematically increased the nanarods in transverse directions from N x = 2 and N y = 2, to N x = 4 and N y = 4, over a range of N z values and the NCE calculated. Linear fits of the data made to predict the energy of nanowires with various cross-sections. For example, in Fig. 5, the NCE is plotted as a function of 1/n for clusters with a 4 × 4 atom cross-section, predicts nanowire energies that are lower than the corresponding 2 × 2 atom cross-section analogues.
The 3 × 3 × N z data set are limited as we have decided to only investigate charge-neutral clusters, so for this set only even values of N z were considered. The 3 × 3 × N z data are also expected -and found -to predict higher gNCEs than even-atom diameter nanowires. This is because all the clusters used to predict the gNCE are polar -see Fig. 2d -and are less stable (higher in energy) than their non-polar counterparts. 17a Although barium is more polarizable than the smaller cations, this effect is most prominent in (BaO) n clusters and diminishes with a decrease in cation size. Table 2       for the full range of clusters investigated, and their associated uncertainty. From Table 2 we can see the predicted gNCE for an infinitely long SrO nanowire with a 2 × 3 atom cross-section is -4.679(7) eV, and that the uncertainty on the 2 × 2 nanowires, for which the greatest number of unique clusters were analyzed, is smaller than all others.

Nanowires to Slabs
The extrapolated nanowire data from Table 2 can then be used to extrapolate data further to 2D structures namely the two-, three-and four-atom thick slabs, from two-, three-and fouratom width nanowires, and is shown in Fig. 6.
As with the cluster-to-wire calculation, extrapolation of a linear fit to the data is used to predict the gNCE for two-, three-, and four-atom thick slabs, given in Table 3.

Extrapolation to the Bulk
The slab gNCEs can be used to predict the bulk energies; as shown in Fig. 7, the vertical axis intercepts now correspond to our predicted bulk gNCE for the group 2 oxides, which has been tabulated in Table 4 and compared to the energies derived using the same exchange and correlation density functionals for the 3D periodic structures.
We compared the extrapolated bulk gNCE using the wire to slab to bulk three-step method, to the predicted gNCE when all clusters are considered in one-step. The NCE for all cubic clusters produced in this work were plotted as a function of 1/n and fit in a one-step process, as shown in Fig. 8 with the extrapolated bulk gNCE also given in Table 4.
The three-and one-step methods predict energies which bracket the values resulting from DFT using periodic boundary conditions (PBC). Energies derived using the one-step approach   that includes/excludes the smallest cube clusters are significantly further/nearer to the target PBC values. The three-and one-step methods are consistently higher and lower than the target energy. Moreover, the three-and one-step methods provide energies that are closer to the target values for smaller and larger cations, respectively.

HOMO-LUMO Gaps for 2 × 2 × N z Clusters
Additional single-point calculations were performed on the optimized 2 × 2 × N z set of clusters using FHI-aims with the PBEsol0 functional. The overall HOMO-LUMO gap was calcu-lated and is shown in Fig. 9b. In contrast to the bulk, the size of the HOMO-LUMO gap is not clearly correlated with cation size. To explain the observed trends, we recall that the magnesium ion has the largest difference in ionic radius as compared to the oxygen anion, which results in strong ionic relaxations away from ideal cubic lattice positions. Thus, a significant stabilization of the electronic levels for states localized on magnesium should be expected (polarization). This explains the significant drop in the position of the LUMO energies for MgO compared to the other three compounds -see Fig. 9a. The trends we observe in the HOMO energies, shown in Fig. 9a, are in qualitative agreement with experimentally derived and computational values of the ionization potentials for the respective bulk compounds. 23 SPECIAL EDITION:   Table 4 Extrapolated bulk gNCE for MgO, CaO, SrO, and BaO using the three-step method, the one-step method (A = broken, B = solid line in Fig. 8

Figure 7
Extrapolated gNCE for two-, three-, and four-atom thick slabs shown as points on the graph, with a linear fit extrapolating to the predicted gNCE for an infinitely thick slab; the bulk.