The stability orders of a number of alkaline earth oxide cluster isomers
In the last few years, considerable effort has been directed to the understanding of metallic and semiconductor clusters. Clusters are aggregates of atoms or molecules intermediate in size between individual atoms and bulk matter, and their studies provide an interesting way to develop materials with varying properties by changing size and shape. Hence, studies of cluster properties as a function of size have received prominence in recent years. While much progress has been made on clusters of metals and semiconductors, metal oxide particles are often considered to be bulk fragments. However, their structure and properties could be entirely different in small clusters [
In this work, we have performed a comparative study of the structures, stabilities, and properties of some alkaline earth metal oxides (
It is interesting to study a similar system like calcium oxide in order to assess whether those trends are a general feature of alkaline earth oxide clusters or not. From the theoretical point of view, Ca2+ is larger than Mg2+, so we can expect ionic size effects to play an important role in determining structural differences. Besides, Ca2+ is approximately six times as polarizable as Mg2+, and the polarizabilities of the oxide anions are also larger in CaO because the bonding is weaker than in MgO. Calcium oxide also crystallizes in the close-packed “rock-salt” structure and is primarily an ionic material, with some degree of covalency in its bonding. It is considered as a prototype oxide from the theoretical point of view, with a wide band gap (7.1 eV) [
Barium oxide is an oxide with interesting electronic and structural properties. It is also a precursor to the ferroelectric perovskite oxide BaTiO3 and a component of the earth’s mantle. Barium strontium oxide coated carbon nanotubes serve as field emitters [
Theoretical work on ionic materials has been centered mostly in the family of alkali metal halides, and studies of metal oxide clusters have been comparatively scarce, despite their importance in many branches of surface physics, such as heterogeneous catalysis or corrosion. The mass spectra and collision induced fragmentation data for stoichiometric
We aim to study the electronic properties of the clusters of these alkaline earth metal oxides using the density functional approach.
In the calculations reported in the paper, first-principles density functional (DF) calculations were performed using the DMol [
Complete geometry optimizations for all structures
Various structures, including the slab, hexagonal, octagonal, ladder, and other types, were studied for various numbers of formula units of the four alkaline earth metal oxides. Various theoretical studies at different levels of calculations have been reported in the literature [
For CaO, too, the LDA-PWC calculated binding energy (5.08 eV) is in better agreement with the experimental [
Optimized structures from LDA and GGA calculations, along with the initial geometries for the (MO)2 systems.
Figure
Calculated binding energies, HOMO-LUMO gaps, and Fermi energies (in eV) for (MO)3 clusters.
M | Structure | Binding energy | HOMO-LUMO gap | Fermi energy | |||
---|---|---|---|---|---|---|---|
LDA (PWC) | GGA (PW91) | LDA | GGA | LDA | GGA | ||
Mg | Hexagonal | −23.33 | −21.12 | 3.05 | 2.98 | −3.66 | −3.41 |
| |||||||
Ca | Ladder | −25.61 | −23.13 | 1.79 | 1.70 | −2.02 | −1.80 |
Hexagonal | −25.60 | −23.22 | 2.34 | 2.20 | −1.87 | −1.70 | |
| |||||||
Sr | Ladder | −25.63 | −23.29 | 1.87 | 1.69 | −1.92 | −1.80 |
Hexagonal | −25.47 | −23.23 | 2.36 | 2.10 | −1.81 | −1.70 | |
| |||||||
Ba | Ladder | −26.93 | −24.60 | 2.03 | 1.88 | −1.76 | −1.66 |
Hexagonal | −26.81 | −24.57 | 2.47 | 2.25 | −1.69 | −1.58 |
(a) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (MgO)3 system. (b) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (CaO)3 system. (c) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (SrO)3 system.
In this ring structure (Figure
In the ladder structure, there are two types of atoms—the central ones having a coordination number of 3, while the outer atoms having a coordination of 2 only and are more unsaturated. As a result, the Mulliken charge densities on the outer Ca atoms are 1.249, compared to 1.292 for the central Ca atom. Likewise, the terminal two-coordinate oxygens have a smaller negative charge (−1.240), while the central one has a partial charge of −1.311. The central bond length is also longer (2.294 Å) compared to the outer ones (1.952 Å). The increase in the central bond length can be understood in terms of the increased coordination of the central ions. The external field produced by the larger number of surrounding ions increases the ionic character of the central Ca–O bond, which resembles the Ca–O lattice limit (2.405 Å), while the terminal atoms are closer to the molecular limit (1.822 Å). The bond orders are 0.837 and 0.610, respectively. The Ca–O bond lengths and bond orders are 2.059 Å and 0.566, respectively, in the Ca–O–Ca face, and 2.108 Å and 0.662 in the O–Ca–O face. Again, this difference is due to the substantial ionic radius of O2−, and the O–Ca–O face has two of these ions, compared to only one in the opposite face.
Figure
Binding energies, HOMO-LUMO gaps, and Fermi energies (in eV) for (MO)4 clusters.
M | Structure | Binding energy | HOMO-LUMO gap | Fermi energy | ||||
---|---|---|---|---|---|---|---|---|
LDA (PWC) | GGA (PW91) | B3LYP/6-311G(d)a | LDA | GGA | LDA | GGA | ||
Mg | Slab | −33.87 | −30.38 | −27.04 | 2.70 | 2.65 | −3.45 | −3.22 |
Octagonal | −32.41 | −29.47 | −26.12 | 3.22 | 3.14 | −3.79 | −3.51 | |
Ladder | −32.18 | −29.03 | −25.63 | 2.09 | 2.08 | −3.60 | −3.35 | |
| ||||||||
Ca | Slab | −38.50 | −34.63 | −32.37 | 3.14 | 2.89 | −2.00 | −1.81 |
Octagonal | −34.91 | −31.70 | −29.56 | 1.15 | 1.35 | −1.86 | −1.75 | |
Ladder | −35.58 | −32.17 | −28.79 | 1.88 | 1.78 | −1.95 | −1.74 | |
| ||||||||
Sr | Slab | −37.70 | −34.67 | — | 3.32 | 2.96 | −2.02 | −1.82 |
Octagonal | −34.87 | −31.69 | — | 1.33 | 1.17 | −1.82 | −1.69 | |
Ladder | −35.30 | −33.69 | — | 1.96 | 1.92 | −1.87 | −1.69 | |
| ||||||||
Ba | Slab | −39.83 | −36.15 | — | 3.65 | 3.38 | −2.01 | −1.78 |
Octagonal | −36.48 | −33.31 | — | 1.67 | 1.48 | −1.72 | −1.61 | |
Ladder | −36.77 | −33.57 | — | 2.09 | 2.24 | −1.73 | −1.60 |
(a) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (MgO)4 system. (b) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (CaO)4 system. (c) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (SrO)4 system. The (BaO)4 system is similar.
An interesting result is that, although the slab structure is preferred in all cases, the next important structure is the ring for (MgO)4, but for the other metal oxides, it is the ladder structure. Unlike (MgO)4, the initial octagonal structure undergoes considerable distortion in all other cases.
Binding energies, Fermi energies, and HOMO-LUMO gaps (in eV) for (MO)5 clusters.
M | Structure | Binding energy | HOMO-LUMO gap | Fermi energy | |||
---|---|---|---|---|---|---|---|
LDA (PWC) | GGA (PW91) | LDA | GGA | LDA | GGA | ||
Mg | Ladder | −41.65 | −37.62 | 2.20 | 2.18 | −3.59 | −3.34 |
Hexagonal | −41.65 | −37.62 | 2.21 | 2.20 | −3.58 | −3.34 | |
MgO-I | −41.52 | −37.57 | 2.25 | 2.19 | −3.59 | −3.34 | |
Decagonal | −41.08 | −37.41 | 3.36 | 3.30 | −3.79 | −3.47 | |
Chair | −42.87 | −38.57 | 2.30 | 2.24 | −3.54 | −3.28 | |
MgO-II | −41.52 | −37.57 | 2.24 | 2.17 | −3.59 | −3.34 | |
| |||||||
Ca | Ladder | −45.53 | −41.19 | 1.91 | 1.82 | −1.92 | −1.70 |
Hexagonal | −45.53 | −41.19 | 1.92 | 1.84 | −1.93 | −1.71 | |
CaO-I | −45.11 | −40.88 | 1.87 | 1.75 | −1.91 | −1.70 | |
Decagonal | −43.19 | −39.37 | 1.99 | 1.92 | −1.52 | −1.35 | |
Chair | −47.74 | −43.01 | 2.00 | 1.90 | −2.16 | −1.90 | |
CaO-II | −45.05 | −40.88 | 1.41 | 1.76 | −1.86 | −1.70 | |
| |||||||
Sr | Ladder | −44.92 | −42.81 | 1.91 | 1.89 | −1.81 | −1.63 |
Hexagonal | −43.60 | −41.83 | 1.01 | 1.05 | −1.84 | −1.63 | |
SrO-I | −44.68 | −40.47 | 1.65 | 1.56 | −1.81 | −1.66 | |
Decagonal | −42.50 | −38.81 | 1.94 | 1.71 | −1.51 | −1.37 | |
SrO-II | −44.68 | −40.60 | 1.64 | 1.46 | −1.81 | −1.67 | |
| |||||||
Ba | Ladder | −46.54 | −47.44 | 1.97 | 2.14 | −1.64 | −1.52 |
Hexagonal | — | −46.95 | — | 1.77 | — | −1.53 | |
BaO-I | −46.42 | −42.37 | 1.98 | 1.82 | −1.69 | −1.57 | |
Decagonal | — | −43.07 | — | 1.86 | — | — | |
BaO-II | −46.41 | −42.35 | 1.95 | 1.80 | −1.68 | −1.56 |
(a) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (MgO)5 system. (b) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (CaO)5 system. (c) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (SrO)5 system. (d) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (BaO)5 system. Dashes represent structures that did not optimize.
Binding energies, HOMO-LUMO gaps, and Fermi energies (in eV) for (MO)6 clusters.
Structure | Binding energy | HOMO-LUMO gap | Fermi energy | ||||
---|---|---|---|---|---|---|---|
LDA (PWC) | GGA (PW91) | B3LYP/6-311G(d)a | LDA | GGA | LDA | GGA | |
Slab | −54.46 | −48.95 | −44.40 | 2.88 | 2.85 | −3.37 | −3.14 |
Hexagonal | −54.67 | −49.30 | −44.11 | 3.34 | 3.29 | −3.47 | −3.22 |
Ladder | −51.13 | −46.21 | — | 2.26 | 2.22 | −3.58 | −3.34 |
| |||||||
Slab | −60.34 | −54.31 | −51.01 | 2.95 | 2.78 | −1.95 | −1.74 |
Hexagonal | −59.90 | −54.05 | −50.70 | 2.98 | 2.86 | −1.96 | −1.74 |
Ladder | −55.48 | −50.21 | — | 1.93 | 1.86 | −1.90 | −1.69 |
| |||||||
Slab | −59.37 | −53.62 | — | 2.87 | 2.66 | −1.88 | −1.67 |
Hexagonal | −58.81 | −53.26 | — | 2.82 | 2.63 | −1.88 | −1.68 |
Ladder | −54.54 | −49.56 | — | 1.88 | 1.74 | −1.75 | −1.61 |
| |||||||
Slab | −60.89 | −55.14 | — | 3.02 | 2.89 | −1.79 | −1.58 |
Hexagonal | −60.53 | −55.00 | — | 3.06 | 2.89 | −1.83 | −1.60 |
Ladder | −56.29 | −51.32 | — | 1.87 | 1.82 | −1.56 | −1.45 |
Table
For CaO, we do not observe any Ca–O bond compression with increasing number of atoms in the terminal rings; that is, the bond lengths do not vary too much when going from the slab (2.108 Å) to the hexagonal ring (2.106 Å) structure. There is, however, an interior ring expansion (2.290 Å) in the slab structure, similar to the MgO system, due to increased polarization of the Ca–O bond under the influence of the terminal rings. The terminal Ca–O bond distance in the three-ring stack is observed to be the mean of the 2.405 Å lattice value and the molecular 1.822 Å distance [
This behavior continues down the series. For (SrO)6, the optimized interior Sr–O bond distance (2.422 Å) is similar to the lattice value (2.565 Å), while the terminal ring distance (2.245 Å) is smaller and closer to the gas phase value (1.920 Å). For (BaO)6, the inner and outer Ba–O bond distances are 2.564 Å and 2.379 Å, respectively, compared to the gas phase and bulk values of 1.940 Å and 2.762 Å, respectively. An interesting trend is also observed. As the atomic number of the metal ion increases, the interplanar distance in the stacked hexagonal ring approaches the M–O distance in the ring, suggesting that spherical clusters become more important for the heavier alkaline earth metal oxides. For example, for (MgO)6, the ring and interplanar distances are 1.891 and 1.980 Å, respectively. The corresponding distances are 2.118 Å and 2.159 Å (CaO), 2.272 Å and 2.290 Å (SrO), and 2.422 Å and 2.431 Å (BaO).
The inner ring bond distance is also only slightly larger than the bulk value for the (MgO)6 slab, and the deviation from the bulk value increases with increasing atomic number of the metal, suggesting a faster overall convergence to bulk properties for MgO clusters than for other alkaline metal oxide clusters.
Figure
(a) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (MgO)6 system. (b) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (CaO)6 system. (c) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (SrO)6 system. (d) Optimized structures from LDA and GGA calculations, along with the initial geometries for the (BaO)6 system.
For small MgO clusters, the experimental [
As noted previously, the preferred geometry for the (MgO)6 cluster is tubular, which becomes more stable than the rectangular bulk-like cluster due to stabilization of the occupied levels. The isosurfaces of the HOMO and LUMO reveal that these comprise mainly the 2p orbitals of oxygen and 3s and 3p orbitals of Mg, respectively, but the LUMO also has an oxygen 3s component (see Figure
Isosurfaces (isovalue = 0.03 Ha) of the HOMOs and LUMOs of the (MgO)6 hexagonal and cubic clusters.
The electronic density of states (DOS) near the Fermi level for the two structures is shown in Figure
Calculated partial densities of states for the (a) hexagonal and (b) slab (MgO)6.
The DOS plots for the nanotube and cube-like structure are qualitatively similar, but one important difference is noticeable. In the hexagonal structure, there is greater involvement of the Mg2+ ion 3d orbitals near the Fermi level, although the population analysis reveals that the Mg electron configurations in the two are similar (3s0.342p63p0.463d0.26 for the hexagonal structure and 3s0.362p63p0.453d0.26 for the terminal slab ions). The peak height for the slab structure is also smaller. The computed electron configuration of the magnesium ion differs from the expected 2p6. The 3s, 3p, and 3d orbitals are occupied, leading to a formal charge on Mg closer to one than the expected two.
The densities of states were also calculated for the slab structures of the other (MO)6 systems. These are shown in Figure
Calculated partial densities of states for the (a) (CaO)6, (b) (SrO)6, and (c) (BaO)6 slabs.
It can be seen from Figure
The bond lengths and total energy per molecule for the lowest energy structures are given in Table
Bond lengths (
MO | Number of units | Preferred structure | Binding energy |
|
---|---|---|---|---|
MgO | 1 | Linear | −3.69 | 1.743 |
2 | Rhombus | −6.75 | 1.858 | |
3 | Hexagonal | −7.78 | 1.819 | |
4 | Slab | 8.47 | 1.936 | |
5 | Chair | −8.57 | 1.788–1.967 | |
6 | Hexagonal | −9.08 | 1.891, 1.980 | |
| ||||
CaO | 1 | Linear | −5.08 | 1.818 |
2 | Rhombus | −8.01 | 2.005 | |
3 | Ladder | −8.53 | 1.952–2.294 | |
4 | Slab | −9.63 | 2.126 | |
5 | Chair | −9.55 | 1.940–2.204 | |
6 | Slab | −10.06 | 2.108–2.289 | |
| ||||
SrO | 1 | Linear | −5.59 | 1.928 |
2 | Rhombus | −8.19 | 2.140 | |
3 | Ladder | −8.54 | 2.074–2.389 | |
4 | Slab | −9.38 | 2.249, 2.256 | |
5 | Ladder | −9.07 | 2.204–2.341 | |
6 | Slab | −9.90 | 2.238–2.422 | |
| ||||
BaO | 1 | Linear | −6.55 | 2.031 |
2 | Rhombus | −8.75 | 2.277 | |
3 | Ladder | −8.98 | 2.189–2.512 | |
4 | Slab | −9.97 | 2.413–2.417 | |
5 | Ladder | −9.31 | 2.205–2.532 | |
6 | Slab | −10.15 | 2.374–2.566 |
The increased stability of slab structures (both
An important observation from the optimized three-dimensional isomers is the chair-type structures of
In the case of
The similar values of the calculated Mulliken charges, approximately +1 on the metal ions in the various metal oxide clusters, make it difficult to assign an electronic cause for the slight preference for hexagonal structures in the case of MgO and slab structures for the other metal oxides. This leads us to believe that it is a packing effect rather than an electronic one. As stated in the sections above, due to the small cation size in MgO, the Mg–O bond is short, and, consequently, the four-membered ring in the slab structure is too strained. In order to accommodate the small cation and the large anion in the four-membered ring, the Mg–O bond length increases, leading to a weakening of the bonding and consequent instability. On the other hand, the octagonal ring structure is too open and is hence not favored for any of the metal oxides. This leaves the six-membered ring as a compromise for MgO systems.
An alternate explanation for the preference for the six-membered ring structure in the case of MgO could be the existence of aromaticity. In order to quantify aromaticity, we used the nucleus-independent chemical shift (NICS) method proposed by Schleyer et al. [
An important finding of the present study is that hexagonal tube-like structures are preferred for
An outstanding result of the present study is the similar stabilities of the hexagonal-ring-based structures and the rock-salt-like slab-shaped isomers. While this observation is important as such and contradicts the exclusive nature of the latter structural shapes proposed previously [
It is difficult to find experimental verification for our results, as neutral clusters are difficult to study experimentally. Their structures are usually inferred indirectly from the mass spectra of ionized clusters, the more abundant species being interpreted as the more stable. However, the results from such studies on alkaline earth metal oxides are contradictory and depend on the process of formation of the clusters. Two conclusions, however, result from these studies. Firstly, for small clusters, hexagonal stacked rings are preferred for
These experiments also suggest that the hexagonal ring and rectangular slab structures are topologically equivalent. Deformation along one of the directions orthogonal to the rings stack transforms it into the slab structure. As noted earlier, this intense vibration mode for (MgO)6 occurs at a low wavenumber (691 cm−1). Thus, experimental knowledge of abundance of masses alone cannot distinguish between the two structures, and sophisticated calculations such as the present ones can only decide the relative stabilities. Our earlier studies [
The authors declare no conflict of interests with any financial organization regarding the material presented in the paper.
The authors thank the Council of Scientific and Industrial Research (CSIR) for financial support (Grant no. 01(2554)/12/EMR-1). Ritu Gaba and Upasana Issar thank the CSIR and the University Grants Commission (UGC), respectively, for Senior Research Fellowships.