Abstract
Several recent studies have shown that SCAN, a functional belonging to the
meta-generalized gradient approximation (MGGA) family, leads to significantly
overestimated magnetic moments in itinerant ferromagnetic metals. However, this
behavior is not inherent to the MGGA level of approximation since TPSS, for
instance, does not lead to such severe overestimations. In order to provide a
broader view of the accuracy of MGGA functionals for magnetism, we extend the
assessment to more functionals, but also to antiferromagnetic solids. The
results show that to describe magnetism there is overall no real advantage in
using a MGGA functional compared to GGAs. For both types of approximation, an
improvement in ferromagnetic metals is necessarily accompanied by a
deterioration (underestimation) in antiferromagnetic insulators, and
vice-versa. We also provide some analysis in order to understand in more detail
the relation between the mathematical form of the functionals and the results.
https://arxiv.org/pdf/2004.04543.pdf
I. INTRODUCTION
The local density approximation1(LDA) and general-ized gradient approximation2,3(GGA) of density func-tional theory1,4(DFT) usually provide a fair descrip-tion of the magnetism in itinerant ferromagnetic (FM)3dmetals, albeit a slight overestimation of the magneticmoment can be obtained (see, e.g., Refs. 5–7). On theother hand, the LDA and GGA are inaccurate for anti-ferromagnetic (AFM) insulators, where the 3delectronsare more localized and the self-interaction error (SIE)8present in LDA and GGA is more important. As a conse-quence, the atomic moment around the transition-metalatom in AFM systems is clearly underestimated.
The exchange-correlation (xc) functionals of the meta-GGA (MGGA) level of approximation10,11should in gen-eral be more accurate since they use an additional ingre-dient, the kinetic-energy density (KED), which makespossible to remove a portion of the SIE.12The stronglyconstrained and appropriately normed (SCAN) MGGAfunctional proposed recently by Sunet al.13was con-structed in such a way that it satisfies all the 17 knownmathematical constraints that can be imposed on aMGGA functional, and was appropriately normed, i.e.made accurate, for particular systems. The SCAN func-tional has been shown to be accurate for both moleculesand solids,13–16including systems bound by noncova-lent interactions provided that a dispersion term isadded.17–19On the other hand, it has been realized thatSCAN leads to magnetic moments in bulk FM Fe, Co,and Ni that are by far too large.16,20–24The overestima-tion of the magnetic moment with SCAN has also beenobserved in alloys25,26and surface systems.
Nevertheless, this overestimation of magnetic momentsis not inherent to the MGGA, since other MGGA func-tionals like TPSS,28revTPSS,29and TM30lead to val-ues similar to PBE.20,23,31Interestingly, Mej ́ıa-Rodr ́ıguezand Trickey24showed that SCAN-L, a deorbitalized ver-sion of SCAN they proposed in Refs. 32 and 33, leads toa magnetic moment which is similar to PBE, while theresults for the geometry and binding energy of moleculesand solids stay close to the original SCAN.
Regarding the general performance of MGGA func-tionals for magnetism in solids, a few questions remain.For instance, not that many results for the atomic mag-netic moment in AFM systems have been reported. Re-cent tests on various Mn, Fe, and Ce oxides have shownthat SCAN underestimates the moment in some caseslike MnO or Fe2O3, but overestimates it in MnO2.35 In Refs. 36 and 37, the FM and AFM phases of VO2 were studied with numerous functionals including TPSS,revTPSS, MGGAMS0,38 MGGAMS2, 39 and SCAN. Itwas shown that the latter three functionals lead to moments that are larger than those predicted by TPSSand revTPSS, especially for the AFM phase. Comparisons with reference Monte-Carlo results for the AFMphase of VO2 indicate that MGGAMS0, MGGAMS2,and SCAN should be more accurate.36,37In Ref. 40, the high-Tcsuperconductor parent compound La2CuO4werestudied with TPSS, revTPSS, and SCAN, the latter giv-ing a value of the moment of the Cu atom in good agreement with experiment, while a clear underestimation is obtained with TPSS and revTPSS. A recent study byZhanget al. has shown that SCAN underestimates the atomic magnetic moment in MnO, FeO, CoO, and NiO.41Finally, it has been reported that SCAN leads to a magnetic moment in AFMα-Mn that is much larger than with PBE
Despite these results for FM and AFM systems, whatis missing is a more systematic study of the relative per-formance of MGGA functionals for magnetism. In partic-ular, besides SCAN and (rev)TPSS, not much is knownabout the performance of other MGGA functionals. Itis also not fully clear to which extent an increase (e.g.,with respect to PBE) of the moment in FM solids witha given MGGA necessarily translates into an increase
for AFM solids. In the present work, a more systematiccomparison of MGGA functionals for magnetism is pre-sented. FM and AFM systems are considered, as well asnonmagnetic (NM) ones. The latter may be wrongly de-scribed as magnetic with DFT methods.22,43The searchfor a possible magnetic ground state for the supposedlyNM systems is restricted to FM.
for AFM solids. In the present work, a more systematiccomparison of MGGA functionals for magnetism is pre-sented. FM and AFM systems are considered, as well asnonmagnetic (NM) ones. The latter may be wrongly de-scribed as magnetic with DFT methods.22,43The searchfor a possible magnetic ground state for the supposedlyNM systems is restricted to FM.
II. METHODS
The MGGA functionals that are considered forthe present work are BR89,44 TPSS,28 revTPSS,29 MGGAMS2,39 MVS,45 SCAN,13TM,30HLE17,46SCAN-L,32,33and TASK.47 Here, we just mention thatHLE17 consists in a simple empirical rescaling of TPSS exchange and correlation, which are multiplied by 1.25and 0.5, respectively, in order to achieve better resultsfor the band gaps of solids and excitation energies of molecules.46The very recent TASK from Aschebrockand K ̈ummel,47 which is an exchange functional thatis combined with LDA correlation,48also provides accurate band gaps, but in contrast to HLE17 it was constructed in a nonempirical way without tuning pa-rameters. All MGGAs except SCAN-L and BR89 aret-MGGAs since they depend on the Kohn-Sham (KS)KEDtσ= (1/2)∑Nσi=1∇ψ∗iσ·∇ψiσ(σis the spin index).SCAN-L is a deorbitalized version of SCAN. At-MGGAis deorbitalized32,49,50by replacingtσby an orbital-free(and thus necessarily approximate) expression that de-pends onρσ,∇ρσ, and∇2ρσ, and is thereby turned intoa∇2ρ-MGGA, which is an explicit functional of the elec-tron density. The BR89 exchange functional of Becke andRoussel, which was proposed as an accurate approxima-tion to the Hartree-Fock exchange energy,44depends onbothtσand∇2ρσ. BR89, which is combined in this workwith LDA correlation,48is tested since it differs radicallyfrom the other MGGAs in terms of construction. There-fore, it may be interesting to see the results obtained withsuch a functional.
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