Energetic Order of Nb2O5 and Ta2O5 Polymorphs

Nb2O5 and Ta2O5 are materials with important technical applications, e.g., in electrochemistry as anode materials for Li or Na storage, or as selective oxidation catalysts. The properties of both oxides largely depend on their various polymorphs with different crystal structures. So far, the literature does not agree on which polymorphs of Nb2O5 and Ta2O5 are most stable under standard conditions. Herein, the relative stability of all relevant polymorphs of Nb2O5 and Ta2O5 is calculated using density‐functional theory and the recently developed rev2‐POB basis sets. First, a benchmark of the atomization‐free enthalpies of Nb2O5 is performed with selected generalized gradient approximation and hybrid functionals. The effect of London dispersion on structural and energetic properties is explicitly investigated. A large scattering of atomization energies is observed for the various functionals. The range‐separated hybrid functional RSHXLDA without dispersion correction proves to be the method of choice for the systems under investigation. For this reason, the stability order obtained with RSHXLDA is consisdered as most reliable.


Introduction
In the past years, Nb 2 O 5 and Ta 2 O 5 have been considered materials with important technical applications, Nb 2 O 5 e.g., as anode material for Na-and Li-ion batteries [1] or as a catalyst for aldol condensation. [2] Ta 2 O 5 is e.g., used in electronics, [3] as a catalyst for methanol oxidation, [4] or as an electrocatalyst for the oxygen reduction reaction. [5] It has been frequently reported that the properties of both oxides largely depend on their various polymorphs with different crystal structures. However, the literature so far does neither agree on the most stable polymorphs of the two solids, [3] nor on the order of stability. Numerous phases of Nb 2 O 5 [6][7][8][9][10][11][12][13][14][15] and Ta 2 O 5 were found, but only a few studies on their stabilities are available.
Kato et al. obtained H-Nb 2 O 5 in a crystalgrowth experiment at room temperature. [6] Andersson produced N-Nb 2 O 5 at 900°C in a sealed Pt tube with 10-20 weight percent of water. [7] R-Nb 2 O 5 was reported by Gruehn after heating H 3 NbO 4 at 550-650°C. [8] Schäfer et al. obtained B-Nb 2 O 5 and F-Nb 2 O 5 from NbOCl 3 . [16] The crystal structures of B-Nb 2 O 5 and F-Nb 2 O 5 were analyzed by Ercit et al. [10] and Mertin et al. [9] A high-pressure form of Z-Nb 2 O 5 was obtained by Zibrov et al. by compressing H-Nb 2 O 5 at 1050°C. [12] B-Ta 2 O 5 was found by Izumi et al. after cooling TaOCl 3 from 620 to 570 K. [13] Holleweger et al. synthesized β-Ta 2 O 5 by deposition of tantalum oxide films at 500°C on silicon substrates. [14] A highpressure form of Z-Ta 2 O 5 was obtained by Zibrov et al. via compressing of B-Ta 2 O 5 at 1200°C. [17] Another high-pressure form of F-Ta 2 O 5 was formed from tantalum hydroxide at room temperature. [15] TT-Ta 2 O 5 is a low-temperature phase obtained via hydrolysis of 2H-TaS 2 . Its crystal structure was determined by Aleshina and Loginova. [18] Hummel et al. synthesized L- Ta 2 O 5 and T-Ta 2 O 5 [19] from TT-Ta 2 O 5 . The high-temperature phase HT-Ta 2 O 5 was obtained by Liu et al. using the conventional solid-state reaction method and the advanced laser irradiation technique. [20] Jacob et al. measured the formation enthalpy for H-Nb 2 O 5 depending on the temperature. [21] Khan et al. reported a theoretical and experimental study on the optoelectronic properties of Nb 3 O 7 (OH) and H-Nb 2 O 5. [22] They used plane waves and generalized gradient approximation (GGA) functionals for structure optimization, and the TB-mBJ approach was used for electronic structure calculations. They could reproduce their experimental value for the optical bandgap of 3.2 eV with a deviation of 0.2 eV, but did not study the relative stability of polymorphs. Pinto et al. calculated the structural, electronic, and thermodynamic properties of T-Nb 2 O 5 and B-Nb 2 O 5 at the GGA þ U level using plane waves. [23] B-Nb 2 O 5 is reported to be more stable than T-Nb 2 O 5 .
In this work, the relative stability of relevant Nb 2 O 5 and Ta 2 O 5 polymorphs was calculated at the density functional theory (DFT) level, to identify the thermodynamically most stable phases under standard conditions. Structures with more than two partially occupied atom positions (T-Nb 2 O 5 , HT-Ta 2 O 5 , T-Ta 2 O 5 , L-Ta 2 O 5 ) were not taken into account. Such partial occupancies require an elaborate configuration analysis in theoretical model calculations. Ignoring these structures is justified because all of these are high-temperature phases that are no candidates DOI: 10.1002/pssb.202200052 Nb 2 O 5 and Ta 2 O 5 are materials with important technical applications, e.g., in electrochemistry as anode materials for Li or Na storage, or as selective oxidation catalysts. The properties of both oxides largely depend on their various polymorphs with different crystal structures. So far, the literature does not agree on which polymorphs of Nb 2 O 5 and Ta 2 O 5 are most stable under standard conditions. Herein, the relative stability of all relevant polymorphs of Nb 2 O 5 and Ta 2 O 5 is calculated using density-functional theory and the recently developed rev2-POB basis sets. First, a benchmark of the atomization-free enthalpies of Nb 2 O 5 is performed with selected generalized gradient approximation and hybrid functionals. The effect of London dispersion on structural and energetic properties is explicitly investigated. A large scattering of atomization energies is observed for the various functionals. The range-separated hybrid functional RSHXLDA without dispersion correction proves to be the method of choice for the systems under investigation. For this reason, the stability order obtained with RSHXLDA is consisdered as most reliable.
for the most stable polymorph. The high-pressure phases Z-Nb 2 O 5 and Z-Ta 2 O 5 were ignored because preliminary calculations on hybrid DFT levels showed that they are more than 200 kJ mol À1 less stable than the other structures without external pressure. The structure of the low-temperature phase TT-Ta 2 O 5 is higher by more than 300 kJ mol À1 and is ignored as well.
Additional to the experimentally reported structures of Nb 2 O 5 and Ta 2 O 5 , the set of polymorphs was extended by some hypothetical polymorphs. These were constructed using experimental V 2 O 5 [24,25] and As 2 O 5 [26] structures and substituting V or As by Nb/Ta. Also, some Nb 2 O 5 polymorphs were added to the Ta 2 O 5 set by substituting Ta with Nb and vice versa.
As a preliminary step, an assessment of selected GGA and hybrid functionals was performed to identify the most suitable functional for the energetic properties of Nb 2 O 5 and Ta 2 O 5 . We also investigated the effect of London dispersion on the calculated thermodynamic properties. The atomization free enthalpy of H-Nb 2 O 5 was benchmarked against the study of Jacob et al., [21] since this is the only available experimental reference data on thermodynamical properties of the two oxides to our knowledge. Using the best functionality of this benchmark test, the relative energies and free enthalpies of all relevant Nb 2 O 5 and Ta 2 O 5 polymorphs were calculated.

Computational Details
All calculations were performed with CRYSTAL17 (version 1.0.2), [27] a quantum-chemical program that uses linear combinations of Gaussian-type atomic orbitals as basis functions of the crystal orbitals. This is advantageous for efficient hybrid DFT calculations compared to plane-wave methods. For all calculations, the optimized rev2-POB-TZVP basis sets [28] were used which reduce the basis set superposition error and provide improved structural properties of solids. The integration truncation parameters TOLINTEG were set to 7 7 7 7 14. Monkhorst-Pack grids for integration in reciprocal space were converged individually for every unit cell (see below).
First, a benchmark of selected density functionals for the calculation of the atomization free enthalpy of H-Nb 2 O 5 was performed. For this purpose, the standard GGA functional PBE [29] (with and without D3(BJ) correction [30] ), the global hybrid functionals B1WC, [31] B97H, [32,33] PBE0 [34] (with and without D3(BJ) correction), PW1PW [35] (with and without D3(BJ) correction), M06 [36] (with and without D3(BJ) correction), WC1LYP, [37] and the range-separated (RS) hybrid functionals HSE06, [38] HISS, [39,40] and RSHXLDA [41,42] were used. This selection was in most cases based on the performance of the functionals in the benchmark test of Tran et al. [43] for simple cubic solids. It has to be noted that the results of our benchmark test for Nb 2 O 5 and Ta 2 O 5 do not coincide with the previous study, [43] due to the higher electronic and structural complexity of the transition metal oxides. In the following, we use the shorthand notation D3 for the D3(BJ) correction.
The Monkhorst-Pack grids were converged for each unit cell of the individual polymorphs. For space groups 12 (N-Nb 2 O 5 ), 15 The thermodynamic correction Δ therm (see the following) is the difference between the polymorph DFT energy and the polymorph-free enthalpy.
H M 2 O 5 and S M 2 O 5 were calculated as the sum of the individual components for translation, rotation, vibration, and electronic excitation. Translation and rotation contributions are not present in solids, and electronic excitations were neglected because the electronic band gaps were larger than the thermal energy RT (all bandgaps are larger than 3.5 eV on the hybrid DFT level). The vibrational enthalpy and entropy were calculated from the Boltzmann distribution of the vibrational frequencies ν i , the temperature T, the pressure P, and the cell volume V (using the ideal gas constant R and the Planck constant h).
Exemplarily, the phonon densities of states of the most stable polymorph H-Nb 2 O 5 for Nb and Ta are shown in the Supporting Information.
The translational enthalpy for gas-phase Nb, Ta, and O was calculated from the ideal gas law the translational entropy is available from the NIST chemical webbook [44] (161.059 J (K À1 mol À1 ) for O, 185.22 J (K À1 mol À1 ) for Ta, and 186.29 J (K À1 mol À1 ) for Nb). The atomization free enthalpy of M 2 O 5 was calculated as The calculated G at values were compared with formation free enthalpies of Nb 2 O 5 from Jacob et al. [21] using The gas-phase free formation enthalpies in (677.5 kJ mol À1 for Nb, 726.8 kJ mol À1 for Ta, and 201.2 kJ mol À1 for O) were also taken from NIST chemical webbook. [44] www.advancedsciencenews.com www.pss-b.com

Structures of the Selected Polymorphs
Experimental structural data of the selected polymorphs are collected in Table 1. ICSD numbers are provided where available. In this case, one oxygen atom was removed from the primitive unit cell.

Atomization-Free Enthalpies of H-Nb 2 O 5
In Table 2, the calculated atomization enthalpies, entropies, and free enthalpies of H-Nb 2 O 5 are compared to available experimental data from Ref. [21].
The atomization energies, enthalpies, and free enthalpies of the different methods differ by up to 500 kJ mol À1 , which corresponds to 10% of the experimental values. For H at , the smallest Δ is obtained with RSHXLDA, WC1LYP, and PBE (2.5, À11.2, À22.7 kJ mol À1 ). Since the atomization entropy (only calculated with PBE-D3) closely agrees with the experiment, the trends are the same for G at . The global hybrid functionals (PW1PW, M06, PBE0, B1WC, and B97H), as well as the short-range and medium-range, corrected RS hybrids HSE06 and HISS strongly underestimate G at , ranging from À55.9 kJ mol À1 (B1WC) to À384.0 kJ mol À1 (HISS). This is not improved by D3 correction (Δ ¼ À199.4 kJ mol À1 for PBE0-D3 and À113.5 kJ mol À1 for PW1PW-D3). The long-range corrected RS hybrid functional RSHXLDA (Δ ¼ þ5.2 kJ mol À1 ), the global hybrid functional WC1LYP (À8.5 kJ mol À1 ), and the standard GGA functional PBE (À20.0 kJ mol À1 ) provide the most accurate values for ΔG at . As expected, the D3 correction significantly increases the atomization energy in all cases (except M06 since dispersion is already included in its parameterization), which leads to a strong overestimation of G at with PBE-D3, by þ103.5 kJ mol À1 .
The range-separated hybrid functional RSHXLDA without dispersion correction proves to be the method of choice for the systems and the properties under investigation. The energetic order of the polymorphs was accordingly calculated using RSHXLDA. But first, we compare the structural and electronic properties of B-Nb 2 O 5 , calculated with all selected functionals, with available experimental reference data.

Lattice Parameters and Atom Positions of B-Nb 2 O 5
B-Nb 2 O 5 (space group 15) is a simple polymorph with only 4 symmetry-independent atoms. That makes it a suitable candidate for benchmarking atom positions and lattice parameters. Table 3 shows a comparison of calculated and measured lattice parameters of B-Nb 2 O 5 . [10] Accurate results (Δ < 0.4%) were obtained with the RS hybrid functionals RSHXLDA (long-range corrected) and HSE06 (shortrange corrected), the global hybrids PW1PW, PBE0, M06-D3 and M06, and with the standard GGA-functional PBE-D3. Among these, M06-D3 is the most accurate (0.16%). PBE without D3 correction has a larger deviation Δ of 1.29%, which is, however, in accordance with the benchmark test of Tran et al. [43] London dispersion correction significantly improves the PBE structures (Δ ¼ 0.30%). Table 4 shows the comparison of calculated and measured atom positions of B-Nb 2 O 5.
[10]  The atomic positions can be calculated accurately with all functionals (Δ between 0.001 and 0.003). With the global hybrid functional B97H and the standard GGA functional PBE, the largest deviations are observed (0.0033 and 0.0032, respectively). The hybrid functional M06 (with and without D3) provides the smallest deviations (0.0008 and 0.0009, respectively). Table 5 shows the comparison of calculated and measured electronic bandgaps of H-Nb 2 O 5. [22] In the discussion of the deviation of the bandgaps, it must be taken into account that the literature value is an optical gap. Usually, fundamental band gaps are larger than optical gaps due to excitonic effects. The slight bandgap changes observed for the D3 corrected functionals are due to the different geometries. The hybrid functionals HSE06 and PW1PW provide the best agreement with experiment with Δ values of À0.08 and 0.13 eV, again in line with earlier studies. As expected, PBE strongly underestimates the experimental gap due to the wellknown self-interaction error. The large deviation observed for RSHXLDA (Δ ¼ þ4.84 eV) is striking. This means that although

Relative Energies of the Nb 2 O 5 and Ta 2 O 5 Polymorphs
In the following section, we present the relative energies of the selected Nb 2 O 5 and Ta 2 O 5 polymorphs, obtained with selected functionals RSHXLDA, M06-D3, PBE, and PBE-D3. RSHXLDA gives the most reliable results for free atomization enthalpies (see Table 2) and is therefore considered as most reliable. Both PBE and PBE-D3 were selected to investigate the effect of dispersion correction. M06-D3 was chosen since it gave good lattice parameters (Table 3) and was among the more accurate DFT functionals in previous benchmark tests. [43] It is important to note that PBE, PBE-D3, and M06-D3 are only used to show the influence of the functional choice on the energetic order of the polymorphs. Figure 1 shows the calculated relative energies per formula unit in kJ mol À1 . The polymorphs are sorted by the energetic order obtained with RSHXLDA, which gave the most accurate results for the atomization-free enthalpies (Section 3.2). According to RSHXLDA, R-Nb 2 O 5 (12_2) is the most stable polymorph. However, the energies of the polymorphs with space groups 3 (0.5 kJ mol À1 ), 12 (0.8 kJ mol À1 ), 15 (1.8 kJ mol À1 ), and 59 (2.0 kJ mol À1 ) are only slightly higher with differences that are within the usual DFT error (all relative energies listed in Table 6). In most cases, PBE parallels the energetic order obtained with RSHXLDA, although space group 3 is the most stable polymorph. The large difference between PBE and PBE-D3 shows that the D3 correction has a significant effect on the relative stability. In earlier studies, it was shown that the C 6 parameters of the D3 method are too large for solids (due to missing reference systems with high coordination numbers), and lead to inconsistent relative stabilities of titania polymorphs. [45] Since also the atomization energy is overestimated with PBE-D3 (see Table 2), we conclude that the PBE-D3 is not appropriate for the present investigation. M06-D3 shows a similar trend as PBE-D3, both predicting space group 15 as the most stable polymorph for Nb 2 O 5 . The reason for this is a drawback in the D3 correction for large coordination numbers. Since the D3-correction is originally made for molecules, the reference compounds all have low coordination numbers. For solids with higher coordination numbers, the D3-correction just uses the highest available reference coordination number, which is much lower than the real coordination number (2.922 for Nb, see Table 7). This leads to an overestimation of the D3-energy. [46] The polymorph with space group 15 has larger Nb coordination numbers (see Table 7), which increases this effect. The improved version D4 which diminishes this effect is not implemented in the present CRYSTAL code.   For Ta 2 O 5 , PBE and RSHXLDA predict H-Nb 2 O 5 as the most stable polymorph (both same energy). However, the energies of the polymorphs with space groups 12_2 (2.2 kJ mol À1 ), 59 (3.7 kJ mol À1 ), 15 (2.0 kJ mol À1 ), and 3 (2.2 kJ mol À1 ) are only slightly higher with differences that are within the usual DFT error. M06-D3 and PBE-D3 again differ significantly from the other functionals and predict B-Ta 2 O 5 as the most stable polymorph (for the same reason as for Nb 2 O 5 ). For Ta 2 O 5 no value for F-Ta 2 O 5 (SG 72) could be calculated because of an SCF convergence failure in geometry optimization with RSHXLDA.
However, one has to keep in mind that these predictions are based on electronic energies. The presence of a particular polymorph in equilibrium at a given temperature is determined by the Gibbs free enthalpy G(T ). For this reason, we calculated the vibrational contributions to the thermodynamic functions under standard conditions from phonon calculations with PBE-D3. Here PBE-D3 was chosen as it is frequently used for the calculation of thermodynamic corrections, and also gave an accurate atomization entropy, see Table 2. Test calculations showed that PBE-D3 and PBE provide similar phonon frequencies and thus thermal contributions. Wherever possible, large supercells with lattice parameters >10 Å were used for the frequency calculations to obtain convergence with respect to long-wavelength phonons (supercell in Table 8). For comparison, we also present results for the primitive unit cell (normal in Table 8). Gas-phase thermodynamic data were taken from experimental databases (see Computational Details section).

Thermodynamic Corrections
The corrections at 298 K range from 20.8 to 33.1 kJ mol À1 for Nb 2 O 5 , and from 14.6 to 31.3 kJ mol À1 for Ta 2 O 5 for the primitive unit cells. Due to computer time limitations, only for four systems with larger supercells could be used for frequency calculations. The effect was quite different: only 1-3 kJ mol À1 for polymorphs with SG 12_2, 15, and 62, but þ8 kJ mol À1 for SG 59 (Nb 2 O 5 ), and þ9 kJ mol À1 for SG 12_2 and 59 (Ta 2 O 5 ). In these cases, the primitive cell vectors were too small to allow the calculation of low-frequency modes in the Γ-point approximation. [47] The primitive cell vectors of the polymorphs with SG 12 and 139 are larger than 10 Å, thus we assume that the present values are close to convergence. For SG 3, the lattice parameters a and c is about 20 Å while b is 3.8 Å. However, the primitive cell already contains 98 atoms, therefore the computational effort of the supercell frequency calculation is too high. For SG 19,53, and 72, the supercell effects were ignored because the relative DFT energies are already higher than 40 kJ mol À1 for Nb 2 O 5 and Ta 2 O 5 on the RSHXLDA level. The relative stability order of the polymorphs is only slightly changed by the vibration contributions obtained at 298 K. This is shown in the next section. Figure 2 shows the relative free enthalpies of the polymorphs of Nb 2 O 5 and Ta 2 O 5 per formula unit in kJ/mol calculated from the energies and the thermal corrections as previously.

Relative Free Enthalpies of the Nb 2 O 5 and Ta 2 O 5 Polymorphs
The thermodynamic correction changes the order of polymorphs with SG 3, 12, 12 _ 2, 59, and 15 which are energetically close to each other (also see Figure 3). The strong stabilization of the polymorph with SG 15 with M06-D3 and PBE-D3 compared to the other functionals remains. In the case of Nb 2 O 5 , H-Nb 2 O 5 (SG 3) is the most stable polymorph. However, the free enthalpies of the polymorphs with SG 12 (0.9 kJ mol À1 ), 12 _ 2 (3.7 kJ mol À1 ), 59 (6.1 kJ mol À1 ), 15 (8.6 kJ mol À1 ), and 139 (11.1 kJ mol À1 ) are only slightly higher with differences that are within the usual DFT error. For Ta 2 O 5 the lowest free enthalpy is also obtained for H-Nb 2 O 5 (SG 3). The polymorphs with SG 12 _ 2 (3.6 kJ mol À1 ), 59 (4.9 kJ mol À1 ), 15 (7.2 kJ mol À1 ), and 12 (8.7 kJ mol À1 ) are only slightly higher and close to the DFT error. The energetic difference of the polymorphs remains very small (see Table 9), so the identification of the most stable polymorph is difficult at the DFT level.
It is therefore interesting to investigate if the order of stability changes with increasing temperature, Figure 4 shows the temperature dependence of the free enthalpy of the polymorphs.
For Nb 2 O 5 , H-Nb 2 O 5 (SG 3) is the most stable polymorph in the investigated temperature range. This is the only experimental structure that was prepared at room temperature. [6] So this is a clear indication that the most stable structure was correctly determined. For Ta 2 O 5 , H-Nb 2 O 5 (SG 3) is also the most stable polymorph in the investigated temperature range. This is interesting because this structure has not yet been synthesized. It is still important to mention that the TT-Ta 2 O 5 structure [18] was reported to be a low-temperature phase but was not studied here because of partial occupations. So it has to be taken into account that this structure might be more stable than the hypothetical H-Nb 2 O 5 structure. The hypothetical polymorphs with SG 19, 62, 72, and 53 are significantly less stable than the others over the entire temperature range, so they are not shown in the plot. In addition, the change in free enthalpy with temperature is very different for some of the polymorphs. The polymorph with SG 15 is energetically similar to the stable polymorphs with space groups 3, 12, 12 _ 2, and 59 at room temperature. As T increases, the SG 15 polymorph becomes increasingly unstable and is

Conclusion
We calculated the relative stability of selected Nb 2 O 5 and Ta  For Nb our calculations confirm H-Nb 2 O 5 as the most stable structure as found in the experiment. [6] A quite interesting result is the high stability of the V 2 O 5 structure for Ta 2 O 5 and Nb 2 O 5 at low temperatures, which was unanimously confirmed by all

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.