Quantum-chemical study of stable , metastable and high-pressure alumina polymorphs and aluminum hydroxides †

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Introduction
Alumina (Al 2 O 3 ) is one of the most important ceramic materials for technological applications.Corundum (α-Al2O3) is used in spark plugs due to its high electrical resistance, in parts of acid and brine pumps due to its corrosion resistance, in melting pots and thermocouple tubes due to its high temperature resistance and in prostheses because of its biocompatibility.Also under high pressure conditions there are numerous technological applications for aluminum oxide.For example ruby (Al 2 O 3 doted with Cr 3 + ) is used as a pressure calibrator in "diamond-anvil-cells" 1 and sapphire (Al 2 O 3 doted with various cations) as a window material in shock wave experiments 2 .Besides the thermodynamically stable corundum there is a variety of metastable alumina phases of technological importance.
The κ-modification is used for surface coating of cutting tools because of its extreme hardness 3,4 .Since the γ-phase provides a large surface due to its porous structure 5 it is used in as support and structural promoter in catalysts in synthesis 6 , for the reduction of automotive pollutants, oil refining and in absorbents 7,8 .
But alumina is also of high interest in recent fundamental material chemistry research.The properties of iron and titanium defects and aggregates in sapphire were theoretically † Electronic Supplementary Information (ESI) available.
investigated by Walsh and coworkers 9 .Bai and others have observed an improvement in the growth and dieletric properties of carbon nanotubes by hybridization with ceramic microparticles 10 .Carreon et al. have reported the synthesis of continuous cobalt-adeninate metal-organic framework (MOF) membranes supported on porous alumina tubes 11 .Very recently the synthesis and detailed structural studies of mesoporous alumina as thin films and as powders 12 were reported by Rønning et.al.In energy research it was used as coating material for high voltage cathodes for enhanced electrochemical performance 13 .Alumina was even used in photochemistry to create and stabilize aqueous solutions of electrons 14 and the fabrication of free-standing Al 2 O 3 nanosheets promise high mobility flexible graphene field effect transistors 15 .
Despite its technological importance, there was no comprehensive experimental or theoretical work that covered all known alumina phases.With this manuscript we now present the first extensive quantum-chemical investigation of the geometric and electronic structure and the relative stability for all known Al 2 O 3 modifications.
Aluminium hydroxides, a family of the seven compounds akdalaite (tohdite) 16 , bayerite 17 , boehmite 18 , diaspore 19 , doyleite 20 , gibbsite 17 and nordstrandite 20 were extensively studied by Demichelis et al..They recently published a complete, systematic, and homogeneous review investigating the physico-chemical properties at hybrid density functional theory level 21 employing the B3LYP functional.To investigate the role of electron correlation in the stability of the hydroxides, Casassa and Demichelis reinvestigated their findings with periodic local Møller−Plesset second-order perturbative approach, aiming at providing a reliable trend of sta-bility on the basis of a proper description of both the longrange Coulomb interactions and the short-range correlation effects 22 .
Some phases are produced from other precursors, for example δ -Al 2 O 3 , which is also formed during thermal oxidation of aluminum 23 .The most common processing routes are shown in Fig. 1.
Boehmite represents the main component of many bauxite minerals and can be synthesized by precipitation of certain aluminum salts in aqueous solution or by hydrothermal synthesis just as gibbsite and bayerite. 24Gibbsite is also a part of bauxite minerals 25 whereas the structurally related bayerite is rarely found in nature 23 .In addition to the above-mentioned ways it can be produced with the Bayer process 23 .Akdalaite is most commonly processed from gibbsite by hydrothermal synthesis 26 .
In the metastable polymorphs, the oxygen anions form either face-centered cubic (fcc) or hexagonal close-packed (hcp) lattices 23 .The distribution of cations within these lattices results in a large number of different polymorphs.The structures based on the fcc packing of the oxygen atoms include γ, η (cubic), θ (monoclinic), δ (either tetragonal or orthorhombic) and γ (tetragonal).The structures based on the hcp packing include α (trigonal), κ (orthorhombic), κ (hexagonal) and ι (orthorhombic).Additional phases named θ , θ , λ (all monoclinic), U (orthorhombic) and χ (cubic 27 or hexagonal 28 ) have been mentioned in the literature but were discarded in the present study because the reported structure data are incomplete.U-Al 2 O 3 was discovered in nanocomposite Al 2 O 3 -SiC coatings using XRD 29 .1][32] χ-Al 2 O 3 is formed during calcination of gibbsite and represents the structural transition from gibbsite to κ-Al 2 O 3 23 .It is assumed that the χ phase has a complex layer structure with a random stacking order where the oxygen atoms are similarly packed as in gibbsite 28 .
Moreover, we investigate several existing structural models of high-pressure alumina phases.It is known that corundum is transformed into the so-called Rh 2 O 3 phase at about 80-100 GPa which turns into the CaIrO 3 phase at about 130 GPa.In order to overcome the kinetic barriers between these modifications, temperatures above 1000 • C are needed. 33 34As there are to our knowledge no recent extensive studies of the structure and properties of these high-pressure phases, the results of the present theoretical study may be useful for future studies, e.g. on phase transformations, surface and adsorption studies as well as reaction paths.
Details of our computational approach can be found in section Computational details.

Aluminum hydroxides
The calculated lattice constants are given in table 1, 2, 3 and 4. They show good agreement with experimental values with relative errors of less than 0.9% for boehmite, 2.8% for gibbsite, 0.8% for bayerite, and 0.6% for akdalaite, respectively.
It should be mentioned that the layer structures may be slightly improved with an additional dispersion correction to DFT 35 .In this study no dispersion correction was applied because the effects are expected to be small.
In order to include the hydroxides to the comparison of the relative stability of the alumina phases where the atomization enthalpy ∆ A H 0 was taken as measure, the atomization enthalpy ∆ * A H 0 of Al 2 O 3 was recalculated from the reaction enthalpy ∆ R H 0 of the hydroxides and H 2 O and normalized to one formula unit Al 2 O 3 .
The enthalpies H 0 have been calculated including zero point energies and vibration contributions at 298 K.The atomic energies were corrected from basis set superposition error with the counterpoise method (ATOMBSSE).

Boehmite
Gibbsite and Bayerite : Akdalaite : The aluminum and oxygen atoms form double layers of octahedra between which the hydrogen atoms are located in a zig-zag fashion (Fig. 2).The exact position of the hydrogens and therefore the space group is not fully known.In a previous study Digne et al. 40 suggest that at room temperature, the space group Cmcm is most probable boehmite.Therefore we take the orthorhombic model (space group Cmcm, no.63) proposed by Christensen et al. 36 from neutron powder diffraction as a starting structure for our geometry optimizations.2.1.2Gibbsiteγ-Al(OH) 3 Saalfeld and Wedde 37 investigated the gibbsite structure with XRD and suggested that it consists of layers of AlO 6 -octahedra that share one edge along the plane whereby each oxygen atom is bonded to one hydrogen atom.Half of the hydrogen atoms form hydrogen bridges within the layers whereas the other half forms interlayer bridges (Fig. 3).The suggested monoclinic primitive unit cell contains eight formula units (space group P2 1 /n).As P2 1 /n is a non-standard space group, the atomic positions and lattice constants were transformed to the standardized space group P2 1 /c (No. 14) for the CRYSTAL calculations.For the applied transformation matrices see Ref. 41.

Bayeriteα-Al(OH) 3
The structure of bayerite is very similar to the structure of gibbsite regarding the layers.The main difference is the arrangement of the hydrogen bonds between the layers (see Fig. 3 and Fig. 4) 40 .Zigan et al. 38 have suggested a primitive unit cell containing eight formula units with the space group P2 1 /n based on XRPD results.As discussed above for gibbsite, the atomic positions and lattice constants were transformed to the standard setting in space group P2 1 /c.atomic positions of the hydrogen and oxygen atoms and the space group (P6 3 mc, no.186) 39 .The oxygen atoms (positions 2b, 2a and 6c) form close-packed layers that are stacked in an ABAC fashion.Within the hexagonal primitive unit cell eight aluminum atoms build slightly distorted AlO 6 -octahedra (position 6c and 2b) whereas the remaining two (position 2b) form also slightly distorted AlO 4 -tetrahedra (Fig. 5).Digne et al. 40 obtained a structure where the hydrogen atoms occupy the position 2a (0,0,z).Therefore we put the hydrogen atoms in the same position with a z-value of 0.125 which results in an initial O-H distance of about one Angstrom.
2.2 Alumina phases 2.2.1 α-Al 2 O 3 α-Al 2 O 3 crystallizes in the trigonal crystal system (space group R3c, no.167) and can be described with either a rhombohedral or a hexagonal lattice system.The oxygen atoms form an hcp packing of spheres where 2/3 of the octahedral vacancies are occupied by aluminum atoms.The conventional unit cell (hexagonal axes, see Fig. 6) contains 30 atoms (Al 12 O 18 , primitive cell: Al 4 O 6 ) where the oxygen atoms occupy position 18e (x,0,0.25)and the aluminum atoms occupy position 12c (0,0,z) 23 .Different from the ideal hcp cell the values of the xand z-coordinate (x=0.307 and z=0.352) 42 differ from the value 1/3 because the aluminum atoms move towards the unoccupied octahedral interstices and hereby induce a repositioning of the oxygen atoms as well 23 .
The calculated bulk properties are shown in Table 5.After relaxation the atomic positions are almost identical to the experimental data and the results for the lattice constants are very satisfying as well with a relative error of 0.8%.The calculated heat of atomization ∆ A H 0 of 3005 kJ/mol (corrected by the basis set superposition error (BSSE)) is 78 kJ mol below the experimental value and thus shows an acceptable error of 2.5%.A study with electron energy loss spectroscopy (EELS) 44 yielded a fundamental band gap of 8.5 eV that is very close to the calculated 8.6 eV .The band structure yields a HOCO-LUCO-transition at the Γ-point (HOCO, highest occu-   which is consistent with a previous theoretical study 45 .From the projected density of states (PDOS) it can be concluded that the valence band consists mainly of oxygen orbitals and the conduction band mainly of aluminum orbitals as it is expected for an ionic compound in the form of Al 3+ 2 O 2− 3 .Band structures, atomic positions and density of states for all investigated modifications are listed in the supplementary information.

κ-Al 2 O 3
The κ modification has orthorhombic symmetry 46 and the space group Pna2 1 (no.33) 47 .Studies with HREM, XRPD, TEM (transmission electron microscopy) and NMR (nuclear magnetic resonance) led to structures where the aluminum atoms are octahedrally as well as tetrahedrally coordinated by oxygen atoms. 48,49There are 40 atoms within the primitive unit cell (Al 16 O 24 ) in which the oxygen atoms form ABAC layers where 3/4 of the aluminum atoms are situated in octahedral and 1/4 in tetrahedral interstices (see Fig. 7).Within the space group Pna2 1 there is only the position 4a (x,y,z) which is consequently occupied by all atoms.The structure found by Ollivier et al. 49 with XRPD is serving as the starting structure in this work.
With a relative error of less than 1% pob-DZVP/PW1PW provides good results for the lattice constants (Table 6).For the band gap a vertical transition of 7.4 eV was calculated in this work whereas a theoretical study on the electronic structure of four Al 2 O 3 modifications by Lee et al. 51 predicted a band gap of 5.49 eV .In the same study the band gap of corundum was underestimated by 2 eV which can be addressed to the use of the LDA functional and the well-known selfinteraction error.

θ -Al 2 O 3
The conventional monoclinic unit cell (space group C2/m, no.12) contains 20 atoms (Al 8 O 12 , prim.cell: Al 4 O 6 ) that all occupy position 4i (x,0,z) (Fig. 8).The aluminum atoms are evenly distributed over the octahedral and tetrahedral interstices of the oxide lattice 7 .For the initial structure we used the data from an XRPD study of Husson and Repelin 52 .
The calculated lattice constants are listed in Table 7 and    agree well with experimental data presenting a relative error of 1.4%.An indirect band gap of 6.9 eV was calculated and may be compared to the value of 5.04 eV from Lee et al. 51 with an assumed underestimation of about 2 eV as discussed above.

Journal of Materials Chemistry A Accepted Manuscript
2.2.4 γ-Al 2 O 3 Because of the low crystallinity of this phase the exact determination of the structure is very complicated 53 .Ching et al. 54 even suggested that there is no longrange order at all to be found in γ-Al 2 O 3 .In many studies, primarily older ones, the structure is described as a defective cubic spinel type (space group Fd3m, no.227) that is composed of a close-packed oxygen lattice (position 32e) whose tetrahedral (position 8a) and octahedral interstices (position 16d) are occupied by aluminum atoms.As the anion-cation ratio is 4:3 in an ideal spinel, cation vacancies have to occur to ensure the correct stoichiometry of aluminum oxide (2:3) 55 .Moreover a slight tetragonal distortion was reported in 1964 based on XRD results (0.983 < a/c < 0.987) 56 .There have been a large number of experimental and theoretical studies to determine the octahedral-tetrahedral ratio in which the vacancies appear and to illuminate whether hydrogen is a part of the structure.It was concluded that hydrogen is only a part of the surface structure and thus does not appear within the bulk. 57,58The vacancy ratio is a controversial issue as investigations with XRD, neutron powder diffraction and electron microscopy reveal that vacancies are located only in octahedral positions 59,60 , or only in tetrahedral positions 61,62 or in both octahedral and tetrahedral positions 63 .The latter result is also backed up by NMR and theoretical studies by Lee et al. 64 .However, a variety of theoretical studies result in solely octahedral vacancies 58,[65][66][67] .It has also been reported that several "non-spinel" positions are occupied in γ-Al 2 O 3 68,69 and that the unit cell was tetragonal with a cubic = √ 2a tetragonal 69 .This tetragonal structure can be regarded as a contraction of the cubic lattice along one direction with the space group I4 1 /amd (no.141) which is a maximum subgroup of Fd3m.Calculated diffraction patterns of cubic and tetragonal models where "non-spinel" positions are occupied have agreed very well with experimental data unlike diffraction patterns of defective spinel structure models 53,70 .Paglia et al. have created supercells (160 atoms) with the space group Fd3m as well as I4 1 /amd where aluminum atoms occupy "non-spinel" positions.This approach led to structures with the general space group P1 whose total energies were a little bit higher than the one of a defective spinel structure, but whose diffraction patterns correspond very well with experimental findings.Due to the poor crystallinity such a supercell can only be an approximation to the real structure of γ-Al 2 O 3 , but at the same time it contains all the representative characteristics of this phase.In general a random vacancy distribution is found in defective structures which is reflected only by a statistical mean value   53 was not included as it was developed by relaxing a defective spinel model without preserving the crystallographic system which we want to maintain for all structures.
But we want to mention that Ferreira et al. have compared this spinel-like structure proposed by Menendez-Proupin and Gutierrez 53,71 to a triclinic structure proposed by Pinto et al. 72 regarding their thermodynamic stability, lattice vibrational modes, and bulk electronic properties using DFT calculations 73 .They found the spinel-like model to be thermodynamically more stable by 4.55 kcal/mol per formula unit on average from 0 to 1000 K. Also the simulated infrared spectra of the spinel-like model showed better agreement with experimental data.
For the defective spinel model (γ-Spinel) we started with a tripled primitive unit cell (Al 6 O 8 − → Al 18 O 24 ) and removed two cations in octahedral positions in a way that the resulting vacancies are as far away from each other as possible.This stoichiometric cell contains 40 atoms (Al 16 O 24 ) and is relaxed under the restriction that the cubic crystal system is maintained at all times (Fig. 9).All defective structures in this work were relaxed while maintaining the specific crystal system.For model (γ-P) of Paglia et al. we chose the most stable structure that they obtained by optimizing a 2×1×3-I4 1 /amd supercell with frozen lattice parameters (Fig. 10).
Regarding the large number of atoms in the primitive cell (160) and the symmetry lowering due to the removal of individual atoms, the calculation of the γ-P-model is very expen-  sive.Therefore no frequency calculation could be performed in this case.The lattice constants (Tables 8 and 9) agree very well to the experimental data with relative errors of 0.2 % (γ-Spinel) and 0.9 % (γ-Paglia).The calculated band gaps differ by about 1 eV from the experimental reference, but in contrast to the spinel-model a direct transition was calculated for the Paglia-model.Energetically the spinel-model lies 12 kJ/mol (per formula unit) below the Paglia-model and is closer to the experimentally found heat of atomization.This is in agreement with the findings of Ferreira et al. mentioned above.However, in conclusion we favor the Paglia-model as representative bulk structure for γ-Al 2 O 3 because (a) Paglia et al. showed that the diffraction patterns agree very well with experimentally found patterns and (b) the periodic defective spinel model would lead to a structure with a highly ordered distribution of vacancies.This feature is rather known for the thermodynamically more stable δ phase and would not fit the picture of a structure with a minor long-range order.Moreover, due to the enormous number of possible modifications Paglia et al. could not investigate all potential cells that can be derived from their approach, so it is likely that there exist energetically more favorable structures than the model we used.Based on a Rietveld refinement Zhou and Snyder 68 suggested that the 8a-and 16c-positions are unoccupied and furthermore that about 10% of the aluminum atoms are situated on the "non-spinel" position 32e.But it is not yet clarified whether the latter model is only restricted to the surface region or not.It was reported by Lippens and de Boer 56 that there is a slight tetragonal distortion (0.985 < a/c < 0.993) and that the oxide lattice is less ordered than in γ-Al 2 O 3 .In this study we employ Zhou and Snyders model (η-ZS) and the one proposed by Ernst (η-E).For both models we constructed two 1×1×3-supercells leading to a composition of Al 72 O 24 for η-ZS and Al 66 O 24 for η-E assuming that all positions are fully occupied.Hence 56 and 50 atoms, respectively, had to be removed to achieve the correct stoichiometry.We removed the atoms in such a way that all the remaining interatomic distances are as large as possible.This procedure We could not find a model of η-E where less than three atomic pairs exhibit a distance of 1.8 Å (Fig. 11a).After geometry optimization however those atoms lie at least 2.8 Å apart (Fig. 11b).For the η-ZS models we could only reduce the amount of 1.8 Å bonds to six (Fig. 12a) but in the relaxed structure the bonds are all ≥ 2.8 Å (Fig. 12b).

η-Al
Figures 12c and 11c show the distortion of the oxide lattice of η-ZS-Al 2 O 3 compared to η-E-Al 2 O 3 .Regarding the relative errors of the calculated lattice constants (0.6% vs. 2.6%) and the heat of atomization (2930 kJ/mol vs. 2890 kJ/mol) we conclude that η-E-Al 2 O 3 is the preferable model.Moreover, as mentioned above, the Zhou and Snyder model may be restricted to the surface region.Both structures have indirect band gaps (4.4 eV and 5.5 eV ) that are rather small compared to the other modifications.
2.2.6 δ -Al 2 O 3 δ -Al 2 O 3 was described by Wilson 76 and Lippens and De Boer 56 as a tripled cell of γ-Al 2 O 3 with a highly ordered cation arrangement. 61In previous studies a tetragonal (a δ =b δ =a γ , c δ =3a γ ) 56,77,78 and an orthorhombic unit cell was suggested (a δ =a γ , b δ =1.5a γ , c δ =2a γ ) 30,32,61 .The tetragonal unit cell was found in those studies that produced δ -Al 2 O 3 from boehmite, whereas orthorhombic symmetry was observed in studies that used other precursors.As there is not sufficient information available about the atomic positions of the orthorhombic cell we restricted ourselves to tetragonal δ -Al 2 O 3 .Tsybulya and Kryukova 79 have obtained refined lattice constants through XRPD and electron microscopy and suggested the space group P4 1 2 1 2 (no.92).The cation vacancies are distributed over octahedral positions. 76,78,80oincidently γ-Fe 2 O 3 is also composed of a tripled (spinel) cell with cation vacancies exclusively on octahedral positions and has the same space group P4 1 2 1 2 81 , therefore this model has already been used for calculations of δ -Al 2 O 3 82 .Repelin and Husson 78 made a "least-squares fitting" of X-ray diffrac-      We simplified the δ -RH model (80 atoms) concerning the actual occupancy of the six 4j (x,0,z) and 4k (x,1/2,z) positions of the aluminum atoms (each 83.3 %) to be able to build a cell with less than 240 atoms: One 4j and 4k position is fully occupied while three of the remaining four positions are occupied.Consequently there are four cation vacancies in this model which were positioned as far away from each other as possible (Fig. 13a).The crystallographic data for the δ -FE model already include the vacancy positions, thus no atoms had to be removed in that case (Fig. 13b).
As for the γ-P-model we could not perform frequency calculations for δ -FE due to the large size of the unit cell (160 Atoms).δ -RH provides better results for the calculated lattice parameters (Tables 11 and 12) with a relative error of 1.0% compared to δ -FE-Al 2 O 3 (1.9%).The calculated band gaps of 6.6 eV and 7.2 eV , respectively, are direct transitions at the Γ-point.Regarding the energies per formula unit, δ -FE-Al 2 O 3 is 9 kJ/mol more stable than δ -RH-Al 2 O 3 .However, this re-  sult must be taken with some care, since we used a simplified model for δ -RH and only analyzed one of a variety of possible vacancy arrangements.Moreover, the δ -RH-model is -unlike the δ -FE-model -based on crystallographic data of δ -Al 2 O 3 and is considered as the more appropriate model in the end.

ι-Al 2 O 3
The ι modification was already discovered in 1959 by Foster 84 .He found that X-ray diffraction patterns of a rapidly quenched melt of cryolite and aluminum oxide are very similar to those of a certain mullite compound.Even though there are later investigations of ι-Al 2 O 3 , e.g. by Korenko et al. 85 , there are no detailed experimental structural information yet.However, a theoretical study by Aryal et al. 86 revealed a possible structure model.It is based on the structure of a mullite compound (Al 2+2x Si 2−2x O 10−x , x =oxygen interstices per unit cell, space group Pbam, no.55) with a very high Al:Si ratio (Al 2.826 Si 0.174 O 4.588 ).Starting from this compound the remaining silicon atoms were substituted by aluminum atoms and some oxygen atoms were removed to achieve the correct stoichiometry.One 2a position (0,0,0) is fully occupied and two 4h positions (x,y,1/2) are partially occupied by aluminum atoms (58.7% and 41.3%).The oxygen atoms occupy a 4g Wyckoff position (x,y,0) and a 4h position (x,y,1/2) and a second 4h position by 25%.The unit cell suggested by Aryal et al. contains 240 atoms and is not used in this study because of the large computational costs.Instead we constructed a smaller cell (45 atoms) from a 1×1×3-supercell of Al-substituted mullite (Al 31 O 36 ) and removed 13 aluminum and 9 oxygen atoms (Al 18 O 27 ).First priority was to remove the aluminum atoms in a fashion that all Al-Al-distances are ≥ 3.0 Å. Afterwards selected oxygen atoms were removed with the aim that the remaining oxygen atoms stay in the vicinity of as many aluminum atoms as possible.We created three such models and chose the energetically lowest as the starting structure for geometry optimizations.
As there are no experimental structure data available for ι-Al 2 O 3 , the theoretical model by Aryal et al. is used as a reference.With a relative deviation of the lattice constants of 9.3% Taking into account the wellknown self-interaction error of LDA which is reduced with hybrids, the PW1PW result is considered as more accurate.For a final clarification of the electronic structure it would be necessary to run a calculation of the Aryal-model with the PW1PW functional, which is, however, computationally demanding.For comparison with the Al 2 O 3 polymorphs found at normal pressure we also studied metastable phases that can only be obtained under high pressure.With CRYSTAL it is possible to optimize the structure parameters of bulk unit cells with external hydrostatic pressure (EXTPRESS).This feature has been used in the following to calculate the structure and stability of high-pressure phases. .All Al atoms occupy the Wyckoff position 8d (x,y,z) and are octahedrally surrounded by oxygen atoms located at positions 8d and 4c (0,y,1/4).By a Rietveld refinement at 113 GPa and 300 K Lin et al. 89 obtained structure data that were used as starting point for geometry optimization in this work (see Fig. 15).
The calculated lattice parameters (see Table 14) agree rather well with the experimental reference values.The deviations are less than 1.1 %.The calculated direct band gap of 11.7 eV is very high compared to the other polymorphs.In Table 15 the calculated and measured lattice parameters are compared.There is good agreement between theory and experiment with deviations of the cell parameters being smaller than 1.2 %.In this case the calculated band gap is indirect and has a value of 10.8 eV.16 show the relative stability of corundum and both high pressure modifications at various pressures.There is qualitative agremeent between calculated and experi-

Relative Stability
The relative stability of the Al 2 O 3 phases and hydroxides is shown in Fig. 18 and Table 17.The hydroxides are thermodynamically more stable than all Al 2 O 3 phases.Within the oxides gibbsite represents the most stable with an adjusted relative enthalpy of -189 kJ/mol followed by bayerite (-180 kJ/mol), boehmite (-85 kJ/mol) and akdalaite (-9 kJ/mol).
These findings are in agreement with previous studies 21,22 where gibbsite was found to be more stable by 7.7 kJ/mol than bayerite and boehmite is less stable than bayerite by 20.8 kJ/mol at B3LYP level.3 it is plausible that there is no significant energetical difference between these two phases.As well as for the other self constructed models for the defective structures it can be assumed that energetically more favourable atomic configurations exist for that model.

Computational details
All quantum-chemical calculations were performed with a development version of the crystalline orbital program CRYS-TAL 92,93 .Structure optimizations were performed employing the hybrid DFT functional PW1PW 94 which has been shown to provide good structural and thermochemical results for oxides and other compounds [94][95][96] .The correlation functional is PW91 97 while the exchange functional is a mixture of 20% Hartree-Fock (HF) and 80% PW91 exchange.The performance of twelve DFT functionals in the study of crystal systems with the focus on aluminum hydroxides was investigated by Demichelis et al. 98 .They found that recent GGA functionals reproduce the structure of orthosilicates quite well, but fail for the H-bonded layered Al hydroxides, where the inclusion of HF exchange in the hybrid functionals leads to a significant improvement.
The pob-DZVP bases recently developed by Peintinger et al. 99 were employed that have been parameterized for solid state systems.These basis sets were optimized in the same way as described here recently 100 and are available in the supporting information.The Monkhorst-Pack k-point-lattices have been converged for each system.The values of the shrinking factors are given in the supplementary information.

Summary and Conclusion
In this work we have investigated the structure, electronic structure and relative stability of alumina polymorphs.The PW1PW functional and the small pob-DZVP basis sets have delivered satisfactory results and have proven to be suitable for calculations of Al 2 O 3 bulk properties for those modifications where sufficient experimental information about the atomic positions was available.To our knowledge this was the first time that models for the defective structures η-and δ -Al 2 O 3 were constructed and quantum-chemically investigated.Exceptions are γ -Al 2 O 3 and κ -Al 2 O 3 .Both represent defective structures with large unit cells and several partially occupied positions which makes the construction of a convergent cell very complex and time consuming.Besides the alumina phases four aluminum hydroxides were successfully investigated and added to the relative stability comparison.We received reasonable results for the lattice constants.A dispersion correction 101 is expected to further improve the agreement with experiment for the layer structures.Due to the computational expense frequency calculations for the thermodynamic functions were not feasible for all systems, but it was concluded that the relative energies only slightly deviate from the relative enthalpies because of cancellation effects.The following energetic order was obtained: gibbsite < bayerite This agrees with earlier studies except for δ -Al 2 O 3 which was found to be more stable than κ-Al 2 O 3 .A possible cause for this discrepancy is the simplified model for δ -Al 2 O 3 besides inaccuracies of the functional and basis set.For a further improvement of the accuracy a gcp-correction 102 is desirable which takes basis set errors into account in structure optimizations.The implementation of these corrections into CRYSTAL is currently in progress.Both high-pressure phases were part of this study as well.The calculated results agree with the experiments that the Rh 2 O 3 modification is the most stable phase at 113 GPa and the CaIrO 3 modification is the ground state at 150 GPa.Moreover the transition pressures is well reproduced by the calculations.So α-Al 2 O 3 transforms to the Rh 2 O 3 phase at about 88 GPa (exp.80-100 GPa) which is transformed to the CaIrO 3 phase at about 132 GPa (exp.130 GPa).

Acknowledgments
This work was supported by the German Research Foundation "Deutsche Forschungsgemeinschaft" (DFG) within the Collaborative Research Area SFB 813 "Chemistry at Spin Centers -Concepts, Mechanisms, Functions" in the Project C5 "Spin centers in molecular solids -from paramagnetic salts to organic conductors".All structural images were generated with Jmol: an open-source Java viewer for chemical structures in

Fig. 1
Fig.1Common calcination routes of aluminum hydroxides and phase transitions of metastable aluminum polymorphs towards the formation of corundum23

a
: Ref. 68, b : Ref. 50 of many different defective cell models.As a consequence it is reasonable to choose the cell size as large as possible but as a consequence also the number of possible configurations and the computational costs increase along with the cell size.Based on comparison of the diffraction patterns, Paglia et al. 69 concluded that γ-Al 2 O 3 produced from boehmite can be better described with space group I4 1 /amd than with Fd3m.It was noted that the experimental diffraction patterns can vary if different precursors are used.For example γ-Al 2 O 3 produced from amorphous precursors via CVD could be better described by Fd 3m.In this study we compare the structure model of Paglia et al. (space group I4 1 /amd, no.141) and the defective spinel model.A recent model for γ-Al 2 O 3 by Menendez-Proupin and Gutierrez

2 O 3
In analogy to γ-, η-Al 2 O 3 is a defective spinel structure (space group Fd3m, no.227) where several aluminum atoms occupy the 48f-position.Shirasuka et al. 74 concluded from an XRPD study that 5/8 of the aluminum atoms are situated on the octahedral 16c-and 16d-position and the remaining 3/8 are distributed over the tetrahedral 8a-and 48f-positions.The results of a high-resolution transmission electron microscopy (HRTEM) study by Ernst et al. 75 differ from those of Shirasuka et al.only in the distribution of the cations over the 8a-(5.35%)and 48f-position (32.15%) while the 16c-and 16d-positions are occupied to the same amount.
(a) before relaxation (b) after relaxation (c) oxide lattice after relaxation

a
: Ref. 78, b : Ref. 50 tion patterns of δ -Al 2 O 3 which resulted in different lattice constants (a δ ≈ a γ / √ 2 = 5.599 Å and c δ = 23.657Å) that match well with a tripled cell of tetragonal γ-Al 2 O 3 69 .In this work we have used the γ-Fe 2 O 3 -(δ -FE) as well as the Repelin-Husson-model (δ -RH, space group P4m2, no.115) as starting structures.It should be mentioned that Pecharroman et al. 83 concluded that the IR-and NMR spectra of δ -Al 2 O 3 would rather fit to a mix of θ -and γ-Al 2 O 3 than to a tripled spinel cell based on a spectra comparison of γ-Fe 2 O 3 and δ -Al 2 O 3 .

Fig. 14
Fig. 14 Primitive unit cell of ι-Al 2 O 3 2.2.8 γ -Al 2 O 3 In 2004 Paglia et al. 5 observed during the calcination of boehmite that at 750 • C γ-Al 2 O 3 did not turn into the δ -but into another modification which they named γ .They determined its structure as a tripled unit cell of γ-Al 2 O 3 with space group P4m2 (no.115), similar to Repelin and Hussons 78 structural description of δ -Al 2 O 3 .Nevertheless the structure of γ -Al 2 O 3 is way more complex as there appear octahedral as well as tetrahedral interstices and several positions are only partially occupied by aluminum atoms.Further increasing the calcination temperature resulted in a more ordered cation distribution so that the structure approaches that of δ -Al 2 O 3 but still contains a few partially occupied positions.In order to construct a supercell model with correct stoichiometry, 16 aluminum atoms had to be removed from the P4m2cell with fully occupied positions (Al 48 O 48 ), resulting in a cell with 80 atoms (Al 32 O 48 ).Unfortunately, all geometry optimizations of this model led to extreme convergence problems with CRYSTAL-PW1PW.Therefore this structure was excluded from the energetic comparison.

2. 2 .
9 κ -Al 2 O 3 Yamaguchi and Okumiya 88 investigated the κ -Al 2 O 3 phase with XRPD and suggested a structural model under the assumption that its structure is very similar to akdalaite (space group P6 3 mc, no.186).The oxygen atoms form a close-packed layered structure with the stacking sequence ABAC while the aluminum atoms are spread over several octahedral and tetrahedral interstices with most positions only partially occupied.The hexagonal unit cell proposed by Yamaguchi and Okumiya includes 16 oxygen and 32/3 aluminum atoms.Starting with a tripled Yamaguchi-Okumiyacell with all positions fully occupied one gets a cell with the composition Al 84 O 48 from which 52 aluminum atoms have to be removed.All of the models obtained in this way gave rise to severe SCF convergence problems, therefore, as for γ -Al 2 O 3 , further investigation was not possible

3 -
Al 2 O 3 The CaIrO 3 modification crystallizes in the orthorhombic crystal system as well (space group Cmcm, No. 63) and has a primitive unit cell containing ten

a
: Ref. 34 atoms (Al 4 O 6 , conventional.cell: Al 8 O 12 ).Half of the aluminum atoms are coordinated octahedrally (position 4c) and the other half tetrahedrally (position 4a) by oxygen atoms (position 4c and 8f).Geometry optimization was performed taking atomic positions of CaIrO 6 from Sugahara et al. 90 and lattice constants for CaIrO 3 -Al 2 O 3 at room temperature and 150 GPa (XRPD) from Ono and Oganov 34 as starting points.The optimized structure is shown in Fig. 16.

3 O 4 by
κ-Al 2 O 3 has a relative enthalpy ∆H(α − κ) of 23 kJ/mol, close to the experimental value of 15 kJ/mol which was found in a calometric study by Yokokawa et al. 50where also ∆H(α − γ) and ∆H(α − δ ) were determined.Theoretical studies by Lee et al. 51 (plane-wave LDA) and Conesa et al. 82 (plane-wave GGA) obtained values of 20 kJ/mol and 11.5 kJ/mol, respectively, for ∆E(α − κ).The latter work examined the impact of a dispersion correction (DFT-D) on the relative stability of several alumina phases.The uncorrected DFT calculations yielded 8.2 kJ/mol for ∆E(α − κ) which means that the dispersion correcture led to a small change of 3.3 kJ/mol in this case.Thus we conclude that the neglect of dispersion corrections in the present study will not affect the main results.With PW1PW the θ modification is less stable than κ-Al 2 O 3 with a relative energy ∆E(α − θ ) of 23 kJ/mol which agrees well with the LDA results of Lee et al.where the difference is 4 kJ/mol.At variance Conesa et al. determined the θ phase to be about 1 kJ/mol more stable than κ-Al 2 O 3 .There is no experimental reference value for ∆E(α − θ ), but one can conclude from the calcination sequence of boehmite (Fig. 1) that the θ phase is probably even slightly more stable than δ -Al 2 O 3 and thus more stable than κ-Al 2 O 3 .The experimental values of E(α − γ), 22 kJ/mol, and ∆E(α − δ ), 11 kJ/mol, are overestimated by 25 kJ/mol and 20 kJ/mol.From a solution calorimetry study of MgAl 2 O 4 • Al 8 Navrotsky et al. 91 a value for ∆H(α − γ) of about 23 kJ/mol was derived which almost matches the value from Yokokawa et al.As already mentioned energetically more favorable γcells may be derived from the Paglia-model if more defect

Fig. 18
Fig. 18 Relative stability of Al 2 O 3 phases and hydroxides

Table 16 Stability
∆H (kJ/mol) of high pressure phases at various pressures