DFT-study of the energetics of perovskite-type oxides LaMO₃ (M = Sc–Cu)

Abstract

The generalized gradient approximation to density functional theory is benchmarked for the calculation of formation enthalpies of lanthanide perovskite-type oxides LaMO₃ (M = Sc–Cu). Three different reaction pathways (from elements, mono and sesquioxides) have been investigated and the systematic errors associated with electron correlation due to overbinding of the oxygen molecule, electron self-interaction within localized 3d states, and geometrical relaxations are analyzed by critical comparison with a large number of experimental data. Calculated formation enthalpies from elements and sesquioxides are in good agreement with experiment when the overbinding of O₂ is corrected for using the Wang ad hoc factor of 131 kJ mol⁻¹ O₂. By contrast, the calculated formation enthalpies from monoxides are systematically too low which are attributable to strong self-interactions due to localized 3d states in MO. The effects of relaxation and choice of magnetic structure on the enthalpies of formation are analyzed.

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Introduction

Perovskite-type oxides (AMO₃) with transition metals on the M-sublattice have attracted considerable attention since they are of potential use in energy related technologies and environmental processes.² Redox energetics are of particular interest especially in the field of power generation by chemical looping combustion.³ However many complex systems have not been studied experimentally since experimental approaches to the redox thermodynamics are both costly and time-consuming. Ab initio methods and in particular Density Functional Theory (DFT) have been implemented in a range of available computational packages providing a cost efficient and inexpensive tool to examine the energetics at the atomic level of complex systems. The most popular implementations remain the Local Density Approximations (LDA) and Generalized Gradient Approximation (GGA) and although these provide a prediction of material properties with remarkable accuracy they are still hampered by the presence of some artificial features associated with electron correlation effects in transition metal oxides.

Several attempts in order to benchmark the computational techniques against experiment are available in the literature.^1,4,5 Wang et al.¹ and Chevrier et al.⁶ have analyzed the GGA errors in the calculation of formation enthalpies (from elements) for a range of binary 3d Transition Metal Oxides (TMOs). By detailed comparison with experiment, two systematic GGA errors for the calculation of oxidation energetics of binary TMOs were addressed: 1) the overbinding of the O₂ molecule and, 2) the lack of cancellation of electron self-interaction errors from the filled localized 3d states. Whereas the GGA overbinding error of the O₂ molecule gives too negative atomization energies,^7–9 the lack of cancellation of errors due to self-interaction gives typically too positive oxidation enthalpies. For the binary TMO compounds reported by Wang et al.¹ the average errors in the GGA formation enthalpies are always too high with an average error of 127 kJ mol⁻¹ O₂. Although the GGA overbinding of the oxygen molecule can easily be corrected for by comparison with experiment for a selection of non-transition metal compounds giving an overall correction of E^corr(O₂) = 131 kJ mol⁻¹ O₂,¹ the different electronic environment for the 3d metal cations in different TMOs cannot easily be dealt with ad hoc. Indeed, comparison with experiment for a large number of binary oxides shows that the calculated formation enthalpies of those TMOs containing atoms with localized 3d electrons (V, Cr, Mn, Fe, Ni and Cu) have a much higher standard deviation of about 20 kJ mol⁻¹ O₂ compared to those of non-transition metal oxides (where the standard deviation is less than 5 kJ mol⁻¹ O₂ and often less than typical errors in experiment).⁶ Also, the discrepancies in the formation enthalpies between experiment and theory of the metal oxides in low valent state (M₂O and MO) are larger than the errors in the high valent state such as M₂O₃ and MO₂^1,6 which is not surprising bearing in mind that 3d electrons are more and more energetically penalized compared to GGA + U (where localized 3d states are projected out and replaced by a Hubbard model) when the 3d states are being filled. Some improvement over GGA can be achieved generally using hybrid functionals (by adding a fraction of Hartree–Fock exchange to the total energy), for example for the calculation of band gap. Although these hybrid functionals give a qualitatively correct description of the O–O bond in O₂,¹⁰ the formation enthalpies of binary TMOs are systematically too low (by about 40 kJ mol⁻¹ O₂) with a standard deviation of 15 kJ mol⁻¹ O₂.¹ In other words GGA with the ad hoc O₂ correction, E^corr(O₂), of Wang et al.,¹ is in overall better agreement with experiment although the scatter is slightly larger. Turning to the formation enthalpies of ternary compounds AMO₃ from elements, very small errors associated with over-delocalization of the 3d states are expected.

Calculated effective Bader charges on the M cations in LaMO₃ (M = Mn–Ni) using GGA are in very good agreement with results from GGA + U calculations.¹⁷ In other words the correlation error associated with self-interaction of the localized 3d orbitals is expected to be very small in these compounds.

In the present paper we consider to what extent GGA can be used as an aid to the energetics of perovskite-type oxides. The formation energetics of ternary compounds such as perovskite-type oxides (AMO₃), and perovskite lanthanides LaMO₃ (M = Sc–Cu) are chosen as a benchmark for this study because of the large amount of available experimental data. Unlike binary compounds, calculation of formation enthalpies for ternary compounds allows the study of the energetics through non-redox reactions too, such as 0.5A₂O₃(c) + 0.5M₂O₃(c) = AMO₃ which is particularly interesting for benchmarking GGA. We shall also investigate the role of magnetic structure and address geometrical (structural) effects on the formation enthalpy for the critical comparison with experimental data since the calculation of formation enthalpies ab initio involves full relaxations of atomic positions, lattice parameters and spin configurations. Results for structural parameters from GGA therefore provide another source of discrepancy since the GGA lattice parameters and bond lengths typically deviate from those of experiment by a few percent.

It is previously shown that the energy difference between different magnetic configurations is small, both for first row transition metal oxides,⁴ and lanthanide perovskites.¹⁸ Choice of spin-configurations and relaxation on the formation enthalpies is investigated by the comparison of results carried out using experimental and optimized geometries within different magnetic configurations.

Methodology

We study the reaction enthalpies corresponding to the formation of the ternary oxides from the elements, monoxides and sesquioxides:

La(c) + M(c) + 1.5O₂(g) = LaMO₃(c) (1)

0.5La₂O₂(c) + 0.5M₂O₃(c) = LaMO₃(c) (2)

0.5La₂O₃(c) + MO(c) + 0.25O₂(g) = LaMO₃(c) (3)

Eqn (1) is a redox reaction, in which the transition metal (TM) in the reactants is in the elemental metallic state. We do not expect large errors in the formation enthalpies associated with the over-destabilization of the delocalized states at the GGA level of theory since, as discussed previously, the calculated Bader charges on the M-site in lanthanide perovskites are in very good agreement with those from GGA + U.¹⁷ This suggests that the error in GGA due to overbinding of the O₂ molecule is the predominant source of error which is expected to give too positive formation enthalpies which can easily be corrected for using the Wang correction E^corr(O₂) = 131 kJ mol⁻¹ O₂.¹ Turning to reaction (2) which is not a redox reaction and involves no oxygen molecules, a direct comparison with experiment is meaningful since the errors due to the lack of cancellation of self-interaction associated with the filled 3d states on the M-site to a large extent are expected to cancel out, even though some charge transfer is expected due to the differences in the ionicity of the M–O bond in AMO₃ and M₂O₃. However, although challenges due to changes in the electronic environment are expected to be small, many TM sesquioxides are either unstable or unknown and a direct comparison with experiment is not available. Also, the choice of crystal structure for these sesquioxides involves a prediction with several possible candidates and the need to find the one with the lowest energy. Extra care is therefore warranted bearing in mind the possible influence of structure type on the formation enthalpies. By contrast, since most monoxides are known experimentally and a large number of formation enthalpies obtained experimentally are reported, the investigation of the performance of GGA using eqn (3) enables us to analyze errors due to lack of cancellation of the self-interaction error on the M cation in a low valent state.

Computational details

The total energies of all species are calculated within the generalized gradient approximation (GGA) to DFT with Perdew–Burke–Ernzehof (PBE) functionals.^19,20 Projector augmented wave (PAW) pseudopotentials^21,22 as implemented in Vienna ab initio Simulation Package (VASP 5.2.11)²³ was used. In order to reach converged energies to within 3 meV per formula unit, an energy cutoff of 550 eV was used. A single k-point (Γ-point) was chosen in the magnetic supercell (see below). Gaussian smearing was used, and different smearing parameters were checked for the metallic and non-metallic compounds. Atom coordinates, unit cell dimensions, and cell volume were fully relaxed for each structure to obtain ionic forces smaller than 0.01 eV. In all calculations experimentally reported structural data were used as the starting point for the structural relaxation.

For the ternary magnetic compounds (M = Ti, V, Cr, Mn, and Fe), calculations were performed in five different magnetic structures; nonmagnetic, ferromagnetic and antiferromagnetic type A, C, and G as described by Wollan et al.²⁴ In order to produce the three different antiferromagnetic structures, supercells were generated by constructing a 2 × 2 × 1 conventional 16-M unit cell.

The experimentally observed magnetic configurations and magnetic moments are given in Table 1.^11–16 The total energies of the binary oxides and the pure metals are calculated with the same crystal and magnetic structure as reported by Wang et al.¹ In all cases, the magnetic moments are allowed to relax.

Table 1 Magnetic structure and magnetic moment of LaMO₃^a

Comp.	Magnetic Configuration		Magnetic Moments (μB)
	GGA	Exp.	GGA	Exp.	Hund's prediction
a NM: nonmagnetic, F: ferromagnetic, A-AF: antiferromagnetic type A, C-AF: antiferromagnetic type C, G-AF: antiferromagnetic type G.
LaScO3	NM	NM
LaTiO3	C-AF	G-AF^11	0.36	0.45^11	1
LaVO3	C-AF	C-AF^12	1.67	1.3^12	2
LaCrO3	G-AF	G-AF^13,14	2.64	2.63^13	3
				2.8^14
LaMnO3	C-AF	A-AF^14,15	3.43	3.7^15	4
	A-AF			3.9^14
LaFeO3	G-AF	G-AF^14	3.65	4.6^14	5
				3.9^16
LaCoO3		NM^14
LaNiO3		PM^14
LaCuO3		PM^14

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The results for the binary compounds and the pure metals are not reported specifically here since the results are fully consistent with previous reports.^1,6,33 The total energy of the oxygen molecule is calculated for a fully relaxed triplet O₂ molecule in a cubic box side 1000 pm, and with the experimental O–O distance of 121 pm.³⁴ For a spin polarized O₂ molecule the calculated equilibrium bond length is 122 pm, the total energy per molecule is −10.03 eV/O₂, while the corresponding dissociation energy is 6.31 eV/O₂. Although these values are in good agreement with the previous GGA reports (total energy: −9.95 eV/O₂,³⁵ dissociation energy: 6.23³⁵ and 6.02 eV/O₂),¹ the dissociation energy is more than 1 eV/O₂ higher than the experimental value (5.12 eV/O₂).³⁴

In Table 1 we report the calculated magnetic structures and compare with those found experimentally. These results are consistent with previous experimental work.^11–16 In one case, LaTiO₃, the calculated magnetic structure with the lowest formation enthalpy (G-AF) is different from that found experimentally (C-AF). However for LaTiO₃ the formation enthalpy of G-AF is only marginally higher than C-AF by about 0.3 kJ mol⁻¹M. For LaMnO₃ the calculations of the formation enthalpies carried out using the fully optimized and experimental crystal structures result in different antiferromagnetic magnetic structures of C- and A-type, respectively (see Table 2). This finding however is studied previously and is known to be correlated to the problems GGA has in predicting the ground-state magnetic ordering of LaMnO₃.³⁶

Table 2 Formation enthalpies of LaMO₃ from La, M, and 1.5 mol O₂ (kJ mol⁻¹M)^a

Compound	Experimental	GGA
		NM	F	A-AF	C-AF	G-AF	Corrected for O2
a N.C.: no convergence reached. b Analysis of previous results. The bold values are in the ground state. Normal fonts: fully relaxed reactants and products. Italic: energies using experimental structures for both reactants and products.
LaScO3	−1888.85 ± 2.67^25	−1681.36					−1877.86
		−1678.90
LaTiO3		−1509.85	−1511.28	−1511.73	−1512.05	−1511.74	−1708.55
		−1483.15	−1484.79	−1485.15	−1485.36	−1483.63
LaVO3	−1563.4^b^26	−1363.24	−1379.91	−1384.59	−1389.19	−1376.67	−1586.69
		−1357.93	−1375.88	−1381.62	−1384.40	−1374.85
LaCrO3	−1536.21 ± 5.14^27	−1208.86	−1303.05	−1305.94	−1307.84	−1310.46	−1511.87
	−1534.4^b^28	−1206.17	−1292.14	N.C.	N.C.	−1305.81
LaMnO3	−1425.1^b^26,29	−1144.57	−1254.04	−1255.84	−1258.15	−1245.00	−1454.65
	−1437.99^30	−1135.06	−1243.41	−1242.64	−1241.90	−1240.24
LaFeO3	−1334.7^31	−1085.23	−1141.79	−1142.12	−1143.39	−1154.52	−1351.02
	−1373.48 ± 2.61^27	−1076.29	−1137.70	−1128.93	−1133.29	−1148.62
LaCoO3	−1241.34 ± 2.13^27	−1048.57					−1245.07
	−1258.2^b^32	−1044.35
LaNiO3	−1192.41 ± 2.75^27	−1015.60					−1212.10
		−1014.00
LaCuO3		−941.89					−1138.39
		−938.73

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Results and discussion

Calculated values of the enthalpies of formation for LaMO₃ using eqn (1) (i.e. from elements) are reported in Table 2 using both experimental and optimized geometries. Available experimental data are also included.^25–32 The most stable form of the binary oxides is used and the magnetic structure with the lowest energy is marked in bold. Comparison with experiment shows that the calculated enthalpies (without E^corr(O₂)) are too high. The dominant source of error, as discussed previously, is due to overbinding of the O₂ molecule at the GGA level of theory. In the last column in Table 2 results are reported where the correction term of Wang, 1.5 E^corr(O₂) = 196.5 kJ mol⁻¹M (i.e. for 1.5 mol O₂),¹ is added to the magnetic configuration with the lowest energy. The O₂ corrected results are in very good agreement with experiment; no systematic deviations are found i.e. the average error is only 1.4 kJ mol⁻¹M although the scatter is quite large (σ = 41.2 kJ mol⁻¹M). The overall good agreement with experiment is not surprising bearing in mind that the calculation of Bader charges of the M-cations in LaMO₃¹⁷ are in good agreement with results from GGA + U. The large scatter in the formation enthalpies arises in part from discrepancies between experimental values for the same compound (e.g. the formation enthalpy for LaFeO₃ reported in ref. 31 and ref. 27 is 40 kJ mol⁻¹M) and possibly in part from cancellation of the geometrical effect and correlation since the calculations (marked as bold) have been carried out using fully optimized geometries.

Comparison of the calculated values marked as italics (experimental geometries are used for both reactants and products) and normal fonts (both product and reactants are fully optimized) shows that the enthalpies calculated from fully optimized geometries undershoot those calculated using the experimental ones by about 25 kJ mol⁻¹M for LaTiO₃, 15.5 kJ mol⁻¹M for LaMnO₃. For the remaining compounds the effect of relaxation is negligible for a comparison with experiment (less than 5 kJ mol⁻¹M) and geometrical effects have very little influence on the scatter. The O₂ corrected results carried out using experimental rather than optimized geometries deviate from experiment on the average by −4.9 kJ mol⁻¹ O₂. This suggests that the remaining but small error may be due to both systematic errors in experimental data used in the statistical analysis and an incomplete description of correlation effects in LaMO₃.

In Table 3 the enthalpies of formation of LaMO₃ from La₂O₃ and M₂O₃ (eqn (2)) are listed as well as the available experimental data.^{25–28,30,37} The calculations marked as bold are the magnetic configurations with the lowest energy (consistent with those marked as bold in Table 2). As expected, the calculated values are in good agreement with the experimental reports. The mean error is −16 kJ mol⁻¹M and the scatter is 34 kJ mol⁻¹M. With the exception of LaCrO₃ where the calculated values overshoot those reported experimentally by about 35 kJ mol⁻¹M, all compounds deviate from experiment by less than 20 kJ mol⁻¹M. The small discrepancy for LaCrO₃ is likely to be associated with the marked self-interaction in LaCrO₃ rather than in Cr₂O₃ since the formation energy of Cr₂O₃ calculated using GGA and corrected for O₂ overbinding is in very good agreement with experiment.

Table 3 Formation enthalpies of LaMO₃ from La₂O₃, M₂O₃ (kJ mol⁻¹M)^a

Comp.	Experimental	GGA
		NM	F	A-AF	C-AF	G-AF
a The bold values are in the ground state. b Analysis of previous results.
LaScO3	−38.64 ± 2.03^25	−18.64
LaTiO3		−4.64	−6.07	−6.52	−6.84	−6.53
LaVO3	−57.6^b^26	−14.43	−31.10	−35.78	−40.38	−27.86
LaCrO3	−73.06 ± 2.03^27	64.10	−30.08	−32.98	−34.82	−37.49
	−67.7^b^28
	−76.8 ± 5.2^37
LaMnO3	−50.3^b^26	64.13	−45.35	−47.14	−49.45	−36.30
	−63.19^30
LaFeO3	−64.58 ± 2.32^27	−0.33	−56.89	−57.22	−58.49	−69.63
	−64.60 ± 2.55^27
LaCoO3		5.78
LaNiO3		−26.32
LaCuO3		−45.99

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Enthalpies of formation of LaMO₃ from La₂O₃, MO and 0.25 mol of O₂ are reported in Table 4. Experimental data are also included.^{26,27,30,31,38} Not surprisingly, in contrast to the calculated enthalpies reported in Table 2 and 3, the calculational values are too negative compared to the experiment and the inclusion of the ad hoc correction of the O₂ molecule of Wang et al.¹ by 0.25 E^corr(O₂) (only 0.25 mol O₂ is included) worsens the agreement with experiment i.e. the mean error is −51 kJ mol⁻¹M and the standard deviation is 59 kJ mol⁻¹M. The main source for the discrepancy is the strong self-interaction in the monoxides which can be confirmed by the comparison between the calculated formation enthalpy for the monoxides and experiment.⁶ For LaMnO₃, for example, the calculated enthalpy are about 80 kJ mol⁻¹M too low compared to experiment whereas the calculated formation enthalpy (including O₂ corrections) overshoots experiment by about 65 kJ mol⁻¹M. The remaining discrepancy for LaMnO₃ can be attributed to relaxations (see Table 2) since the calculated enthalpy using experimental geometry is about 15 kJ mol⁻¹M lower than that of using fully optimized geometries. For LaCuO₃ the calculated formation enthalpy for CuO is lower than that of experiment, and the discrepancy between experiment and GGA is either associated with self-interaction in LaCuO₃ or errors in the experiment.

Table 4 Formation enthalpies of LaMO₃ from La₂O₃, MO, and 0.25 mol O₂ (kJ mol⁻¹M)^a

Comp.	Exp.	GGA
		NM	F	A-AF	C-AF	G-AF	Corr. for O2
a The bold values are in the ground state. b Analysis of previous results.
LaScO3		−315.21					−347.96
LaTiO3		−223.68	−225.11	−225.56	−225.88	−225.57	−258.63
LaVO3	−236.60^b^26	−246.01	−262.68	−267.36	−271.96	−259.44	−304.71
LaCrO3		−159.62	−253.80	−256.70	−258.54	−261.21	−293.96
LaMnO3	−144.40^b^26	−80.88	−190.36	−192.15	−194.46	−181.32	−227.21
	−155.93^30
LaFeO3	−168.90^31	−104.13	−160.69	−161.02	−162.29	−173.43	−206.18
	−207.68^27
LaCoO3	−107.64 ± 1.77^27	−106.90					−139.65
LaNiO3	−57.31 ± 2.55^27	−88.60					−121.35
	−56.94 ± 2.65^27
LaCuO3	−18.2^38	−21.95					−54.7

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This big mean error in the calculation of the enthalpy of oxidation from eqn (3) is in agreement with the previous reports which suggest that this error is attributed to the change in the localization character of the electronic states on transition metals.^1,5

Results listed in Table 2, 3 and 4 also allow examination of choice of spin configuration and furthermore higher order effects such as the influence on relaxation on the stability of different magnetic configurations. Comparing the GGA-enthalpies for a given compound carried out using non-magnetic and different magnetic configurations shows that the discrepancy is highest at the center of the transition metal block. For LaCrO₃, LaMnO₃ and LaFeO₃ the difference between the nonmagnetic and G-AF configurations is significant (∼100 kJ mol⁻¹M), whereas for LaVO₃ and LaTiO₃ the discrepancy is about 25 kJ mol⁻¹M. By contrast, the difference between different magnetic configurations are less than 20 kJ mol⁻¹M indicating very little coupling between the spin-configuration and structural relaxations. In fact, all the magnetic configurations give values with a standard deviation which is smaller than the errors in the experimental values. The largest difference is for LaFeO₃, where the ferromagnetic configuration is only 13 kJ mol⁻¹M higher than the G-AF magnetic configuration. Although this highlights the importance of using spin-polarized calculations for magnetic compounds, even a simple magnetic structure gives reasonable results for a critical comparison with experiment. Thus a ferromagnetic configuration simply provides a reasonable compromise between cost and efficiency often.

A few general guidelines for the calculation of thermochemical properties of lanthanide perovskite type oxides with an accuracy comparable with those of experiment are summarized. For non-redox reactions, exemplified here by eqn (2), accurate results can be obtained at the GGA level of theory. For redox reactions which involve a high occupancy of localized 3d bands, illustrated by eqn (3), the GGA calculated values are typically too low because of the strong electron self-interaction error in the binary transition metal monoxide. For redox reactions involving transition metals (eqn (1)), GGA provides accurate results when the overbinding of O₂ has been dealt with by e.g. adding an ad hoc correction term such as that of Wang et al.¹ The comparison of eqn (1), (2) and (3) leads us to confirm the previous report by Jain et al.⁵ where they suggest that although standard GGA performs better than GGA + U for the calculation of transition metals (capturing the energetics of delocalized metallic states), it does not perform as well in the case of transition metal oxides.

Conclusion

We have benchmarked DFT at the GGA level for the calculation of formation enthalpies of LaMO₃ by detailed comparison with a large volume of experimental data. Three different routes have been investigated providing a suitable test for the performance of the DFT since they highlight different systematic errors in the exchange correlation term of the total energy. In particular we analyzed the following GGA errors: 1) the overbinding of the O₂ molecule, 2) the error due to correlation of the localized 3d electrons and, 3) geometrical effects.

In eqn (1) the calculated formation enthalpies using GGA are too positive compared to experiment which is largely due to overbinding of the O₂ molecule. Following Wang et al.¹ the inclusion of an ad hoc correction term of 131 kJ mol⁻¹ O₂ to the calculated reaction enthalpies for all compounds results in very good agreement with experiment. In eqn (2) the agreement with experiment is very good at the GGA level of theory, which is not surprising, bearing in mind that very little charge transfer on the TM site has taken place and therefore no correction for self-interaction is needed for the critical comparison with experiment. By contrast, the third reaction where LaMO₃ has been formed from the TM monoxide, the high occupancy of the localized 3d bands (correlation) provides a challenge to GGA. This is associated with the energetic penalty of adding extra electron to the 3d orbitals which increases when decreasing the valence of M. In general, for such oxidation reactions, GGA will overestimate the total energy of the TMOs with lower valence M much more than the TMOs with higher valence M and therefore the formation energy obtained is significantly lower than that of experiment.

We also addressed the role of spin configurations for the ab inito calculation of material properties. If treating the material as nonmagnetic the error in formation enthalpy can be as large as 100 kJ mol⁻¹M emphasizing the importance of using spin-polarized calculations. However, any presumably small magnetic unit cell gives reasonable total energies.

Acknowledgements

This study was financially supported by the research council of Norway as part of the “RENERGI programme”. We gratefully acknowledge Notur for providing the computational facilities.

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Footnote

† Present address: SINTEF Materials and Chemistry, P.O. Box 124 Blindern, NO-0314 Oslo, Norway. E-mail: mehdi.pishahang@sintef.no; Fax: +47 2206 7350; Tel: +47 4010 0923

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