376
W. Piskorz and F. Zasada
of the Hubbard U value has been systematically investigated [319–322], and it has
been shown that computationally expensive hybrid functionals exhibit little advantage over the properly selected GGA+U [323, 324]. It was shown, however, that
the U parameter is not universal for description of different properties of calculated
TMO systems. In particular case of Co 3 O 4 , it was shown that U = 3.0 eV provides
a better overall description of the electronic structure and surface reactivity while
U = 5.9 eV is better suited for description of the magnetic properties [320]. Generally, LDA calculations underestimate the spinel oxide lattice constants by 1–2%
relative to the experimental values, whereas GGA functionals overestimate lattice
parameters [325]. LDA describes the total energy of bulk oxides rather properly, however, fails severely when employed for bond energy calculations leading to errors of
several tens of per cent [326]. The GGA functionals generally yield much better bond
energies, surface energies, and adsorption energies. It has been reported [327] that
LSDA overestimates exchange interactions, possibly due to the overestimating of
p-d hybridisation [317, 328] whereas GGA functionals generally give better values
[329]. Since the value of U is usually fitted to other observables such as band gap
or the atomic magnetic moments, the computed exchange interactions can diverge
significantly from exact values [317].
Contrary to pure DFT, hybrid functionals have shown excellent results [330, 331].
While in pure HF calculations, the band gaps are overestimated, a hybrid approach
yields reasonable band gaps. More involved methods for an improved treatment
of electron correlation, e.g. the Møller–Plesset (MP2) expansion, quantum Monte
Carlo (QMC) approaches, or dynamical mean-field theory (DMFT) are currently too
computationally demanding to be generally applied to complex-structured oxides
and are found in practice only in cluster calculations [332].
The mere calculations of total energy of the unit cell systems can be used to distinguish between the normal ([A]
ted
[B 2 ]
oct O 4 , λ = 0), inverse ([B]
ted
[AB]
oct O 4 , λ = 1),
or mixed ([A 1−λ B λ ]
ted
[A λ B 2−λ ]
oct O 4 , 0 < λ < 1) spinel structure. The degree of
inversion, λ, is governed by several factors, e.g. the cation radius ratio, Coulomb
interactions between the cations, and crystal field effects of the octahedral site preference energy of cations [333]. The total energy calculations allow for assessing the
magnetic ordering of the paramagnetic centres, e.g. antiferromagnetic ordering in
cobalt spinel [334].
It was shown that DFT-based methods are able to reproduce the partially inverse
MgAl 2 O 4 spinel Raman spectra with high accuracy, and to solve some of the uncertainties in peak attributions [335]. In case of cubic LiMn 2 O 4 , the calculations helped
to understand transformation between cubic and orthorhombic structures [336]. The
first-principles calculations may also provide insight into influence of pressure on
spinel oxide structural. In such way, the prediction of structural and phase transitions
upon pressure was studied in case of ZnAl 2 O 4 and ZnGa 2 O 4 with the excellent reproduction of experimental trends [337]. As reported by Price et al., the sequence of
four transitions with increasing pressure was predicted for the cubic spinel phase of
calcium ferrite [338]. The more involved characteristic of bulk spinel oxides such as
detailed electronic properties, optical properties, or superconductivity, which involve
excited-state properties or high accuracy of the description of electronic correlation
W. Piskorz and F. Zasada
of the Hubbard U value has been systematically investigated [319–322], and it has
been shown that computationally expensive hybrid functionals exhibit little advantage over the properly selected GGA+U [323, 324]. It was shown, however, that
the U parameter is not universal for description of different properties of calculated
TMO systems. In particular case of Co 3 O 4 , it was shown that U = 3.0 eV provides
a better overall description of the electronic structure and surface reactivity while
U = 5.9 eV is better suited for description of the magnetic properties [320]. Generally, LDA calculations underestimate the spinel oxide lattice constants by 1–2%
relative to the experimental values, whereas GGA functionals overestimate lattice
parameters [325]. LDA describes the total energy of bulk oxides rather properly, however, fails severely when employed for bond energy calculations leading to errors of
several tens of per cent [326]. The GGA functionals generally yield much better bond
energies, surface energies, and adsorption energies. It has been reported [327] that
LSDA overestimates exchange interactions, possibly due to the overestimating of
p-d hybridisation [317, 328] whereas GGA functionals generally give better values
[329]. Since the value of U is usually fitted to other observables such as band gap
or the atomic magnetic moments, the computed exchange interactions can diverge
significantly from exact values [317].
Contrary to pure DFT, hybrid functionals have shown excellent results [330, 331].
While in pure HF calculations, the band gaps are overestimated, a hybrid approach
yields reasonable band gaps. More involved methods for an improved treatment
of electron correlation, e.g. the Møller–Plesset (MP2) expansion, quantum Monte
Carlo (QMC) approaches, or dynamical mean-field theory (DMFT) are currently too
computationally demanding to be generally applied to complex-structured oxides
and are found in practice only in cluster calculations [332].
The mere calculations of total energy of the unit cell systems can be used to distinguish between the normal ([A]
ted
[B 2 ]
oct O 4 , λ = 0), inverse ([B]
ted
[AB]
oct O 4 , λ = 1),
or mixed ([A 1−λ B λ ]
ted
[A λ B 2−λ ]
oct O 4 , 0 < λ < 1) spinel structure. The degree of
inversion, λ, is governed by several factors, e.g. the cation radius ratio, Coulomb
interactions between the cations, and crystal field effects of the octahedral site preference energy of cations [333]. The total energy calculations allow for assessing the
magnetic ordering of the paramagnetic centres, e.g. antiferromagnetic ordering in
cobalt spinel [334].
It was shown that DFT-based methods are able to reproduce the partially inverse
MgAl 2 O 4 spinel Raman spectra with high accuracy, and to solve some of the uncertainties in peak attributions [335]. In case of cubic LiMn 2 O 4 , the calculations helped
to understand transformation between cubic and orthorhombic structures [336]. The
first-principles calculations may also provide insight into influence of pressure on
spinel oxide structural. In such way, the prediction of structural and phase transitions
upon pressure was studied in case of ZnAl 2 O 4 and ZnGa 2 O 4 with the excellent reproduction of experimental trends [337]. As reported by Price et al., the sequence of
four transitions with increasing pressure was predicted for the cubic spinel phase of
calcium ferrite [338]. The more involved characteristic of bulk spinel oxides such as
detailed electronic properties, optical properties, or superconductivity, which involve
excited-state properties or high accuracy of the description of electronic correlation