Improved Description of Perovskite Oxide Crystal Structure and Electronic Properties using Self-Consistent Hubbard U Corrections from ACBN0
IImproved Description of Perovskite Oxide Crystal Structure and ElectronicProperties using Self-Consistent Hubbard U Corrections from ACBN0
Kevin J. May ∗ and Alexie M. Kolpak † Department of Mechanical EngineeringMassachusetts Institute of Technology,Cambridge, MA 02139 USA (Dated: April 21, 2020)The wide variety of complex physical behavior exhibited in transition metal oxides, particularlythe perovskites A B O , makes them a material family of interest in many research areas, but thedrastically different electronic structures possible in these oxides raises challenges in describing themaccurately within density functional theory (DFT) and related methods. Here we evaluate the abilityof the ACBN0, a recently developed first-principles approach to computing the Hubbard U correctionself-consistently, to describe the structural and electronic properties of the first-row transition metalperovskites with ( B = V − Ni). ACBN0 performs competitively with hybrid functional approachessuch as the Heyd-Scuseria-Ernzerhof (HSE) functional even when they are optimized empirically,at a fraction of the computational cost. ACBN0 also describes both the structure and band gap ofthe oxides more accurately than a conventional Hubbard U correction performed by using U valuestaken from the literature. I. INTRODUCTION
Density functional theory (DFT) is one of the mostoften-used computational approaches for modeling theelectronic structure of complex molecules and solids.However, the approximate exchange-correlation (XC)term in the total energy functional, informed by earlywork on the homogeneous electron gas , leads to sig-nificant inaccuracies in DFT. Notable examples are theunderestimation of fundamental gaps in the electronicstructure, or the prediction of metallic ground states intransition metal oxides in cases where the true groundstate is insulating. Transition metal oxides are materi-als of interest in a wide variety of applications, includ-ing renewable energy and catalysis. In certain casesthe trends captured by DFT are sufficient, but whenquantitative predictions (e.g. location of a catalyst ona “volcano” plot) or band gaps are needed, “beyond-DFT” methods are required. This is especially impor-tant in perovskite oxides with formula unit ABO (where A is usually a lanthanide or alkaline earth metal and B is usually a transition metal), which include bandinsulators , Mott-Hubbard insulators , charge trans-fer insulators , and correlated metals . Perovskites andother related structures have found interest in a wide va-riety of applications, ranging from fundamental physicsphenomena (metal-to-insulator transitions , topologi-cal insulators , magnetism , superconductivity ,ferroelectricity ) to catalysts , battery materials ,and oxide electronics . Being materials where elec-tron correlations play an important role in determiningthe properties, they are challenging to describe univer-sally using current theoretical approaches.Approximate XC functionals such as the local densityapproximation (LDA) or the various flavors of the gen-eralized gradient approximation (GGA) do not cancelout the self-interaction energy in the Coulomb (Hartree)functional, leading to excessive delocalization . This is an important reason for qualitatively incorrect predic-tions in systems where charge is strongly localized, suchas in many transition metal oxides. In addition, the totalDFT energy for a given system as a function of electronoccupation is smooth for approximate XC functionals,whereas for the exact Kohn-Sham (KS) potential the en-ergy is piece-wise linear, with derivative discontinuities atinteger occupation numbers . This is one of the reasonsfor the significant underestimation of fundamental gapsby approximate XC functionals . It is therefore un-surprising that several beyond-DFT methods introducederivative discontinuities in the total energy vs. elec-tron occupation. Hybrid functionals, where a fractionof the exact Hartree-Fock (HF) exchange acting on theKS orbitals is used, intuitively reduce delocalization viathe cancellation of self-interaction in the Hartree energy,but also introduce discontinuity into the XC potential .While the most commonly used mixing fraction of 25 ex-act exchange (75 approximate DFT exchange) was jus-tified for atomization energies of molecules , in prac-tice the mixing fraction is often used as an empiricalparameter in order to optimize the description of a de-sired material property, as has been done with perovskiteoxides . A self-consistent hybrid functional (sc-hybrid)based on the PBE0 functional has also been reported ,which avoids empiricism in the mixing fraction by set-ting it equal to the inverse of the static dielectric con-stant of the material. This relationship can be justifiedfrom a comparison of the hybrid functional exchange-correlation functional to the expression for self-energyin the Coulomb hole plus screened exchange (COHSEX)approximation , a static version of the GW approxi-mation. In the original paper , the dielectric constantis calculated using the coupled perturbed KS equations(CPKS) within first-order perturbation theory , themixing fraction is updated, and this process is repeateduntil self-consistency is achieved. In plane-wave calcu-lations, the static dielectric constant may computed the a r X i v : . [ c ond - m a t . s t r- e l ] A p r Berry phase technique , as has been demonstrated inrecent work . This approach has improved upon fixedand empirical mixing fraction hybrid functionals for a va-riety of semiconductors and insulators , though themethod is computationally expensive owing to the mul-tiple iterations of hybrid functional calculations neededfor each material.DFT+ U , inspired by the Hubbard model , is anotherapproach to improving the description of correlated ma-terials. In DFT+ U , a corrective term is added to thetotal DFT energy functional that energetically favors or-bitals in the chosen Hubbard manifold (typically d or f electrons but not exclusively) being either completelyempty or full via screened HF-like Coulomb ( U ) andexchange ( J ) interactions that act only on this set of lo-calized orbitals, usually between orbitals on a single sitebut potentially between neighboring sites as well , andremoving a “double-counting” term from the DFT energyfunctional. Unfortunately, there is no unique choice forthe set of localized orbitals onto which to project the KSorbitals, nor for the double-counting term or the methodof calculating the values of U and J themselves. Atomic-like orbitals (e.g. from the pseudopotentials) are oftenused as a basis , as are Wannier functions . Thecorrection is often applied in a simplified scheme usinga single interaction parameter which is often assumedto take both U and J interactions into account, an “ef-fective” U eff = U − J . In this work, U refers to suchan effective value unless J is explictly mentioned. Thevalue of U , similarly to the fraction of exact exchange inhybrid functionals, is often used as an empirical param-eter that is varied to produce the desired results. First-principles approaches to calculating U do exist, however.The linear response (LR) method defines U in such away that the curvature of the total energy as a functionof electron occupation is canceled out for non-integer oc-cupations, giving rise to a derivative discontinuity in theenergy . A frequency-dependent, screened U can also becalculated via the constrained random phase approxima-tion (cRPA) . Self-consistent values of U , meaningthat U is calculated iteratively until a convergence cri-teria is met, have been shown to improve on empiricalchoices of U for structural properties and defect forma-tion energies . The downside of some of these ap-proaches (which will be briefly described further in thenext section) is that they can be computationally de-manding for large cells when there are many unique sitesthat warrant treatment with DFT+ U .Recently, a new approach to calculate the value of theHubbard U has been reported , inspired by previouswork computing U via unrestricted HF orbitals . TheACBN0 method defines U based on the bare Coulomband exchange interactions and a renormalized occupa-tion matrix, where KS orbital occupations are reducedbased on the Mulliken population of each KS orbital pro-jected on the Hubbard manifold. The main advantagesare flexibility with respect to unique Hubbard sites andextremely low computational cost, negligible compared to the main DFT calculation, making ACBN0 particu-larly well-suited for high-throughput applications . Cor-rections are applied using U values for both metal d ( U dd )and oxygen p ( U pp ) sites. In principle, these values canbe calculated for any Hubbard manifold on any givenatomic site. Due to the renormalization of the occupa-tion matrix, if few occupied KS states have strong char-acter of the chosen Hubbard manifold, the magnitude ofthe correction will be drastically reduced. This has theeffect of reducing computed values of U for more cova-lent materials. This method was originally tested on sev-eral benchmark materials (TiO , MnO, NiO and wurtziteZnO) and later on wide-gap semiconductors and severalother binary oxides , showing improved agreement withmore computationally-expensive beyond-DFT methodssuch as hybrid functionals and the GW approximation.It is worth mentioning that also recently, a new methodof computing U that is equivalent to the LR approach hasbeen demonstrated using density-functional perturbationtheory, which, similarly to ACBN0, allows for comput-ing self-consistent values of U on individual atomic siteswithout the use of large supercells .This work provides a further test of ACBN0 onthe theoretically-demanding transition metal perovskitesA B O , where B =Ti–Ni. Moreover, since there are fewstudies which look at DFT+ U on all of these materials(especially with first-principles calculations of U ), we of-fer comparison with fixed values of U chosen from valuesin the literature that were calculated using various meth-ods such as cluster configuration interaction (cluster-CI)calculations fit to experimental photoemission and X-ray absorption spectra, or the LR method mentioned pre-viously. This is referred to in the text as “PBE+ U Lit. dd ”.We chose these cluster-CI values since they often lie inthe median range of U values reported, and are all de-termined in the same way from experimental results. Itis important to mention that in general, values of U arenot transferable; factors such as the method of calculat-ing U , the XC functional, the pseudopotentials (also theXC functional used in generating them), the Hubbardmanifold to which U is being applied and the implemen-tation of the DFT+ U method all affect the value of afirst-principles U and the effect it has when applied. Weapply these values of U from the literature in an inten-tionally “na¨ıve” way and ignore the non-transferable na-ture of U to illustrate how using either empirically-chosenor first-principles values of U in this way can affect theresults of a DFT calculation. This is not intended as acomment on the “correctness” of those values of U or onthe results obtained using these corrections in their re-spective studies. We examine the prediction of magneticground state, lattice geometry, and electronic structurefor the 1st-row transition metal perovskites, and comparewith higher theory and experimental data when possi-ble, providing a necessary test of ACBN0 as well as aguide for treating these materials with computationally-inexpensive first-principles methods. II. THEORETICAL BACKGROUNDA. The Hubbard Correction in DFT: DFT+ U Here we briefly cover an introduction to several for-mulations of DFT+ U , omitting the derivations that arecovered in the original works. We refer to a thoroughreview of DFT+ U for more details and discussion on thedevelopment of the method. The basic idea of DFT+ U is that a correction to the DFT energy is applied by sep-arately treating electronic interactions between localized electrons, usually considered on a single atomic site. Itfrequently may be the only feasible choice for large sys-tems, owing to its negligible cost compared to the baseDFT calculation. Various implementations of the correc-tion are possible , but the basic form is: E DFT+ U [ n ( r )] = E DFT [ n ( r )]+ E Hub [ n Iσmm (cid:48) ] − E dc [ n Iσ ] (1)where E DFT is the energy of a DFT calculation (LDAor GGA), E Hub is the Hubbard correction term, E dc isa double-counting correction (to remove the energy cal-culated in E Hub from E DFT ) and n Iσmm (cid:48) is the occupationnumber of a localized orbital in the set ϕ m (the Hubbardbasis) on atomic site I , with spin index σ . This last termis often computed by projecting the KS orbitals onto aset of localized orbitals, such as pseudo-atomic orbitalsor Wannier functions, expressed as: n Iσmm (cid:48) = (cid:88) k,i f σki (cid:10) φ σki (cid:12)(cid:12) ϕ Im (cid:48) (cid:11) (cid:10) ϕ Im (cid:12)(cid:12) φ σki (cid:11) (2)where f σki is the occupation of φ σki , the i th KS orbitallabeled by k-point and spin index.While the early work on DFT+ U was done using aform for Eq. (1) that is more reminiscent of the Hub-bard model, it was not invariant upon rotation of thelocalized orbitals. A rotationally-invariant formulationwas developed by Liechtenstein et al. that is similar to aHF (HF) calculation: E Hub [ n Iσmm (cid:48) ] = 12 (cid:88) { m } ,σ, j (cid:8) (cid:104) m, m (cid:48)(cid:48) | V see | m (cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) n Iσmm (cid:48) n I − σm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) + ( (cid:104) m, m (cid:48)(cid:48) | V see | m (cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105)− (cid:104) m, m (cid:48)(cid:48) | V see | m (cid:48)(cid:48)(cid:48) , m (cid:48) (cid:105) ) (cid:0) n Iσmm (cid:48) n Iσm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:1)(cid:9) (3) E dc [ n Iσ ] = (cid:88) I (cid:26) U I n I ( n I − − J I (cid:2) n I ↑ ( n I ↑ −
1) + n I ↓ ( n I ↓ − (cid:3) (cid:27) (4)where V see are the screened Coulomb interactions be-tween electrons. The rotation invariance comes from the quadruplet integrals and the dependence on the traceof the occupation matrices in the Hubbard and double-counting terms, respectively. The Coulomb integrals (cid:104) m, m (cid:48)(cid:48) | V see | m (cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) are expressed as: (cid:90)(cid:90) ϕ † Im ( r ) ϕ Im (cid:48) ( r ) e | r − r (cid:48) | ϕ † Im (cid:48)(cid:48) ( r (cid:48) ) ϕ Im (cid:48)(cid:48)(cid:48) ( r (cid:48) ) d r (cid:48) d r (5)In the case of spherically symmetric Hubbard basis or-bitals (i.e. atomic orbitals), these integrals can be sep-arated into radial and angular parts. The parameters U and J can be expressed as atomic averages of theCoulomb integrals over the states with the same quantumnumber l : U = 1(2 l + 1) (cid:88) m,m (cid:48) (cid:104) m, m (cid:48) | V see | m, m (cid:48) (cid:105) = F (6) J = 12 l (2 l + 1) (cid:88) m (cid:54) = m (cid:48) ,m (cid:48) (cid:104) m (cid:48) , m | V see | m (cid:48) , m (cid:105) = F + F
14 (7)where l is the angular quantum number of the Hubbardorbitals (e.g. 2 for atomic-like d states), and F ν is aSlater integral from the radial part of Eq. (5). The aboveequations are strictly valid only for unscreened Coulombkernels and spherically-symmetric Hubbard basis sets,but they are often used to evaluate Slater integrals inscreened systems by working backwards: first computing U and J for the system then assuming the ratios be-tween the Slater integrals have the same values as thosefor symmetric atomic orbitals, and solving for F ν and V see . Further details are in the original paper and dis-cussed in later work .Eqs. (3) and (4) can be significantly simplified. Byretaining only the lowest order integrals in Eq. (5), only F remains and the functional can be rewritten as: E Hub (cid:2) n Iσmm (cid:48) (cid:3) − E dc (cid:2) n Iσ (cid:3) = (cid:88) I,σ U I eff (cid:2) n Iσ (1 − n Iσ ) (cid:3) (8)This results in the calculation of forces, stresses etc.being greatly simplified, and hence it is likely the mostwidely-implemented form of DFT+ U . This version givesextremely similar results to that of the full rotationally-invariant formulation, with the possible exception ofsome materials such as Fe-pnictides, heavy fermion ornon-collinear spin materials, and multi-band metals .This is due to the loss of the explicit higher-order inter-action J in materials where it is especially important,via the parameter U eff = U − J mentioned earlier. A cor-rection to the simplified version that takes an explicit J term into account without losing the simple form of theHubbard energy (DFT+ U + J ) has been derived from thesecond-quantized form of the total electronic interactionpotential, and verified on the CuO system . A similarly-derived expansion of DFT+ U , termed DFT+ U + V , thattakes into account inter-site Coulomb interactions, hasbeen shown to improve the treatment of materials thatexhibit a higher degree of covalency . These expandedschemes are not used in this work and therefore will notbe expounded upon.To interpret the effects of applying a U correction, wecan examine Eq. (8) and recognize that we can choosea representation of the Hubbard basis that diagonalizesthe occupation matrix : n Iσ v Iσj = λ Iσj v Iσj (9)with the constraint that 0 ≤ λ Iσj ≤ U cor-rection can be rewritten as E Hub [ n Iσmm (cid:48) ] − E dc [ n Iσ ] = (cid:88) I,σ (cid:88) j U I eff λ Iσj (1 − λ Iσj ) (10)where we see that the U parameter imposes an energeticpenalty for partial occupation of the localized orbitals,favoring fully occupied or fully empty orbitals. This in-troduces a difference in the potential seen by occupiedand unoccupied states and gives rise to a discontinuityin the potential as a function of occupation.When comparing DFT+ U to HF or hybrid functionals,there is some resemblance in the functional form, as seenin Eqs. (3) and (4). It could therefore be considered as asubstitution of a HF-like Hamiltonian for part of the den-sity that is normally treated by the Hartree and approxi-mate XC functionals. In this way it acts as a correction tothe self-interaction energy of localized states (where self-interaction error is the largest). DFT+ U differs fromhybrid functionals in that only a subset of states pro-jected onto localized orbitals are treated, as opposed tointeractions between all the KS states. The interaction isoften done in an orbital-averaged way for simplicity (i.e.the same U for all d states). In addition, the discontinu-ity in the energy as a function of the number of electronsonly arises from the subset of Hubbard orbitals. In otherwords, the potential is linearized with respect to occupa-tions of the localized states–not the number of electronsin the whole system–with discontinuities at integer occu-pations, considering the atomic states as isolated and incontact with the “bath” of the rest of the crystal. B. Self-Consistent Determination of U In order to improve confidence in the predictions madeby theory, it is desirable to minimize empirical parame-ters and perform fully ab initio calculations wheneverpossible. It is common to consider (incorrectly) thatvalues of U are transferable between systems where thechemical environment differs significantly, and there is aneed for methods of calculating the interaction parame-ters self-consistently for a given atom, crystal, magneticordering and localized basis set. There have been severalapproaches reported for calculating self-consistent valuesof U for use in DFT+ U , the two most common beingthe LR approach and the constrained random phase approximation (cRPA) . Here, we will briefly describethese and related methods. U from Linear Response The LR method developed by Cococcioni et al. in-terprets U as a correction that counteracts the curva-ture of the total energy as a function of the fractionalnumber of electrons present in the system, which is aresult of the unphysical self-interaction energy presentin approximate-XC DFT . This is a reformulation ofthe so-called ”constrained DFT” approach , where oneexplicitly calculates the change in energy as a functionof localized orbital occupation. This was typically notdone in plane wave DFT but via other methods (suchas muffin tin methods) where orbital occupation couldbe easily defined and fixed. A value of U could thenbe chosen that eliminates the unphysical curvature inthe exchange-correlation energy by calculating this cur-vature but then also calculating and subtracting the cur-vature in the total energy that arises from hybridization,from a non-interacting KS formulation of the same sys-tem. In LR, instead of varying the orbital occupation,a small perturbative potential is applied and varied, andthe resulting change in the total energy and localized or-bital occupation is used to determine the curvature in theenergy/occupation relationship. This method implicitlytakes screening into account via a self-consistent methodwhere the procedure is repeated until the calculated valueof U converges. The method also requires that a super-cell structure be used, which prevents interactions be-tween periodic images of Hubbard sites, which may becomputationally prohibitive, and may not work as wellfor closed-shell systems where the response of the systemto linear perturbation is very weak. U from the Constrained Random Phase Approximation The cRPA method finds use in both DFT andDFT+DMFT as it yields a frequency-dependent U . Theidea is that by separating the polarization of a systeminto localized (e.g. from d orbitals) and delocalizedstates, the inverse dielectric function can be factorizedand the effective interaction acting on the localized statescan be calculated while including the screening from theextended states. The basis for the localized states can beconstructed from just the manifold of interest for apply-ing U (e.g. a d – d model), or be expanded to include otherlocalized or itinerant states (e.g. a d – dp model). Calcu-lating interactions between multiple subsets of states isalso possible, such as when considering U dd , U pp and aninter-site term V dp (e.g. a dp – dp model). While the di-electric function should be calculated using both Hartreeand XC kernels, in cRPA only the Hartree term is in-cluded for simplicity (hence “constrained”). The cal-culated value of U is an expectation value of W r , theHartree kernel divided by the delocalized part of the di-electric function. The calculation of the localized anddelocalized parts of the polarization requires some carein materials where the bands are entangled .
3. Self-Consistent DFT+ U with ACBN0 Recently, Agapito et al. proposed a new schemefor self-consistently determining U , based in part onwork by Mosey and Carter that utilized unscreenedCoulomb and exchange interactions between Hubbard or-bitals (taken as unrestricted HF orbitals obtained for agiven system) to explicitly calculate U . While the au-thors have termed the approach a ”pseudohybrid Hub-bard density functional”, we wish to clarify that in itscurrent implementation it is not a variational scheme orapplied at each self-consistent step in the DFT calcula-tion, but rather consists a post-processing method on anexisting self-consistent DFT+ U calculation that is thenrepeated iteratively until the value of U is converged.Therefore it is not currently a functional. The generaloutline of this ACBN0 method (named for the authors)is as follows :i. The Hubbard orbitals are chosen as the pseudo-atomic orbitals (PAOs) present in the pseudopoten-tial files.ii. The PAOs are expressed as a combination of threeGaussian atomic orbitals, fit to the PAOs from astarting point of a Slater-type orbital (STO-3G) ba-sis. This greatly improves the computational effi-ciency of calculating HF-like interaction integrals.iii. The KS orbitals are projected onto the PAO basisvia a scheme developed by the same authors .iv. The occupation of the each KS orbital is “renormal-ized” by the Mulliken population of that orbital pro-jected on the Hubbard basis (including all atoms withstates that have the same quantum numbers in theunit cell).v. The Hartree ( U ) and exchange ( J ) terms are explic-itly calculated using the HF Coulomb and exchangekernels divided by the occupation numbers of theHubbard orbitals.The Coulomb and exchange energies are calculated asfollows. First, the KS orbital ( φ i ) occupations are renor-malized according to the Mulliken charge on the extendedbasis { ¯ m } (including all orbitals on all sites that havequantum numbers equal to the Hubbard orbitals in thesingle-site basis { m } ):¯ n k σφ i ≡ (cid:88) µ ∈{ ¯ m } (cid:88) ν c k σ † µi S k µν c k σνi (11) where k is an vector index for a specific k point in theBrillouin zone, c k σνi is the expansion coefficient of the lo-calized state ϕ ν as a component of KS state φ i , and S µν is the overlap matrix element for localized states ϕ µ and ϕ ν . A renormalized density matrix is then defined byadding the contributions of every KS state to the rele-vant Hubbard orbitals:¯ P σµν = 1 √ N k (cid:88) k ,i ¯ n k σφ i c k σ † µi c k σνi (12)Meanwhile, the occupations of the Hubbard orbitalsare: n σm = 1 √ N k (cid:88) k ,i,ν c k σ † mi S k mν c k σνi (13)where N k is the number of k points in the Brillouin zone.By using the HF expression for the energy, we then get E hub = 12 (cid:88) { m } ,σ (cid:104) ¯ P σmm (cid:48) ¯ P σ (cid:48) m (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:105) (cid:104) m, m (cid:48)(cid:48) | V ee | m (cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) + 12 (cid:88) { m } ,σ (cid:2) ¯ P σmm (cid:48) ¯ P σm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:3) (cid:104) m, m (cid:48) | V ee | m (cid:48)(cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) (14)which bears a clear resemblance to Eq. (3). By compar-ing this expression to another form of the energy givenby Eq. (8), E hub ≈ U (cid:88) { m } ,σ n σm n − σm (cid:48) + U − J (cid:88) m (cid:54) = m (cid:48) ,σ n σm n σm (cid:48) (15)we get as expressions for U and J : U = (cid:80) { m } ,σ (cid:104) ¯ P σmm (cid:48) ¯ P σ (cid:48) m (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:105) (cid:104) m, m (cid:48)(cid:48) | V ee | m (cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) (cid:80) m,σ (cid:54) = m (cid:48) ,σ (cid:48) n σm n σ (cid:48) m (cid:48) (16) J = (cid:80) { m } ,σ (cid:2) ¯ P σmm (cid:48) ¯ P σm (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (cid:3) (cid:104) m, m (cid:48) | V ee | m (cid:48)(cid:48) , m (cid:48)(cid:48)(cid:48) (cid:105) (cid:80) m (cid:54) = m (cid:48) ,σ n σm n σm (cid:48) (17)The strengths of ACBN0 are that the parameters canbe calculated for every atom in the unit cell, withoutneeding supercells or costly additional calculations. TheHF integrals are evaluated very quickly when the Hub-bard orbitals are projected onto a three-gaussian (3G)basis, in a fraction of the time needed for the DFT cal-culation. The renormalization should, in theory, reducethe calculated values of U and J for systems where theKS orbitals are not well-represented by a localized basis. III. COMPUTATIONAL METHODS
DFT calculations were performed using QuantumESPRESSO 6.1 , using optimized norm-conservingpseudopotentials from the SG15 library (La and Sr)and standard-accuracy (stringent for Cr) Pseudo-Dojo (transition metals and O), generated from the Op-timized Norm-Conserving Vanderbilt Pseudopotentialcode . Plane wave cutoff, k -point mesh, and self-consistency convergence threshold were converged withrespect to the total energy ( <
15 meV/atom), total force( < − Ry/a.u./atom), and unit cell pressure ( < × × ) and threshold of 10 − Ry. Convergence testresults and k-point paths for band diagrams are shownin the Supplemental Material . A plane wave cutoff of100 Ry was used (except for Cr, which used 120 Ry)with a Monkhorst-Pack grid of 4 × × − Ry were used for all calculations.Variable-cell relax calculations decreased the convergencethreshold to 10 − Ry for the final relaxation steps.DFT+ U was performed using U values calculated withACBN0, using Python scripts to both automate theself-consistent electronic structure calculations and de-termine the electron repulsion integrals. Detail on thecalculated U values, as well as first-principles values of U taken from the literature (since several first-principlesor empirical values of U have been reported, the medianvalue is used in this work), is shown in the SupplementalMaterial . The simplified rotationally-invariant imple-mentation of Dudarev et al. and Cococcioni et al. was used. Initial spin states and starting atomic mag-netizations were set according to the experimentally re-ported electronic configurations for each transition metalin the associated perovskite structure (such as high-spinFe , with t g : ↑ ↑ ↑ and e g : ↑ ↑ ) . Antiferromagnetic(AFM) AFM-A, AFM-C, AFM-G, ferromagnetic (FM)and nonmagnetic (NM) magnetic orderings were calcu-lated depending on what the primitive unit cell size andsymmetry allow. ACBN0 is not fundamentally limitedto a certain set of localized orbitals, but in the originalpaper and in this work, the atomic-like orbitals from thepseudopotentials are used for simplicity and for the con-venience of fitting a minimal three-Gaussian (3G) basisset for rapid evaluation of the electron repulsion integrals.The ACBN0 U correction was calculated and applied totransition metal 3 d states ( U dd ) and oxygen 2 p states( U pp ). Literature U values were only applied to the metal3 d states as is common practice (calculations labeled asPBE+ U Lit. dd ). We again mention that ACBN0 as imple-mented in this work is not a functional, but a method tocalculate U . Comparing energies between different calcu-lations using ACBN0 is not possible since presumably dif-ferent values of U would be used in each calculation. Todeal with this issue, we choose to present both an average U from the calculations of energetically-similar magneticstates, and the U from the experimentally-determinedground state. IV. RESULTS AND DISCUSSIONA. Computed Values of U Before discussing the main DFT+ U results, we firstpresent the values of U calculated by ACBN0 in this workand make some broad comparisons with other values of U presented in the literature. The results of ACBN0calculations of U are presented visually in Fig. 1. Themagnitude of U dd is between 1.5 and 3.5 eV, with a trendof increasing U dd with d shell filling. The increasing d – p hybridization as the d manifold is filled is apparent bythe relatively small increase in U dd for the metal cationsas well as the decrease in the U pp calculated for oxygen2 p states, which ranges from around 4.8 to 7.6 eV. Theselarger magnitudes of U pp on oxygen ions are in qualitativeagreement with the observation of experimental U pp val-ues from X-ray spectroscopy interpreted in the Zaanen,Sawatsky, and Allen (ZSA) scheme of oxide electronicstructure , though the values themselves correspondto an interpretation of U which is different to that used inthe present work and should not be quantitatively com-pared. The application of U pp to oxygen p states is animportant point of discussion and is covered in more de-tail in Sec. IV C 7.The values of U dd calculated in this work are typicallysmaller than those which have been used in the literatureto date for the same materials. We again caution that val-ues of U should not be considered transferable in general.Using the same DFT functional, code, and implementa-tion of U , even a change in pseudopotential (especiallyfrom untested pseudopotentials, or those generated witha different XC functional than the DFT calculation) canresult in changes on the order of eV. For example, theoriginal ACBN0 paper found values of U dd = 0 .
15 eVand U pp = 7 .
34 eV, using (untested) norm-conservingpseudopotentials from PSLibrary 1.0.0 . With the pseu-dopotentials used in our current work (see Computa-tional Methods, Sec. III), a test calculation on TiO yielded U dd = 0 .
12 eV and U pp = 8 . U correction–asignificant change in the pseudo-orbitals can affect thecalculated value of U by up to several eV . That be-ing said, the difference between ACBN0 and the varioussources from the literature varies between 0 eV (for U dd of LaCoO fitted to enthalpy of formation) and over 5eV (for U dd of LaMnO fitted to band gap, and LaCoO from LR). It is worth noting that for the two most com-mon forms of self-consistent U (constrained DFT/LR andcRPA) that typically, the lowest values of U (i.e. thosein closest agreement with ACBN0) were those calculatedfrom cRPA (introducing another difficulty in compari-son, namely the type of interaction model used in thecRPA) and the highest were for constrained DFT/LR.More detailed tables listing our calculated values of U (including the explicit U and J values), and values of U FIG. 1. Values of U obtained from ACBN0 for metal 3 d ( U dd )and oxygen 2 p ( U pp ) for the perovskite oxides studied in thiswork. Filled symbols refer to calculations using the experi-mental structure, while open symbols represent calculationsthat have been optimized with several iterations of relaxationand ACBN0 until the values of U change by less than 0.01 eV. U Lit. dd values from the literature used for comparing to conven-tional DFT+ U are included for comparison, with the rangeof literature U values found and cited in this paper boundedby grey rectangles. sourced from the literature are available in the Supple-mental Material . B. Crystal Structure
The structural parameters of perovskites LaMO (M= V–Ni) have been reported according to the definitionsshown in Fig. 2, similar to those reported by He and Fran-chini in their HSE hybrid functional study of first-rowtransition metal perovskites . HSE results mentionedrefer to this work unless otherwise noted. Optimizedstructures are analyzed only for the calculations usingthe experimentally-observed magnetic ordering (exceptfor paramagnetic LaNiO , where a non-magnetic state isused). These consist of the lattice parameters, unit cellvolume, various metal-oxygen bond lengths and metal-oxygen-metal bond angles. Crystallographic representa-tions have been chosen to be consistent among all per-ovskites (i.e., the space group unique axes are orientedin a such a way that allows a direct mapping of atomicsite positions between different materials). While the Heand Franchini also include the Jahn-Teller (JT) distor-tion modes Q2 and Q3 as parameters, their small mag-nitudes are not suitable for including in the mean abso-lute relative error (MARE) and will not be included inthis analysis for simplicity. They can be still calculatedfrom the information provided herein. One should ensurethat the same experimental reference structures are usedwhen comparing between different studies whenever pos-sible. In the following discussion this is the case unlessotherwise noted. θ θ θ M -O M -O M -O M -O M -O M -O V =( a × b )· c α a bc FIG. 2. Perovskite structure parameters used in the determi-nation of mean absolute relative error (MARE).
1. LaVO LaVO has monoclinic symmetry in the P /b spacegroup, with two unique V sites in the unit cell. Thestructural parameters, presented in Table I, reflect thisby including bond lengths and angles for both V sites.The MARE for ACBN0 is 0.69%, which compares fa-vorably to the PBE value of 0.88%, and especially tothe MARE of 2.6% obtained from PBE+ U Lit. dd . Thevalue of U dd used (3.0 eV) is both from cluster-CI modelfits to experimental spectra , and also from empiricalfits to band gap . This value is also close to cRPAresults . Despite this, PBE+ U Lit. dd describes the struc-ture of LaVO quite poorly. Hybrid functional calcula-tions, using both the commonly-used mixing fraction of0.25 (HSE-25) and an empirically-chosen value of mix-ing to improve the overall structural and electronic prop-erties (HSE-Opt), show improved structural agreementwith experiment at 0.48% and 0.35%, respectively . Itis interesting to note where the variation in MARE arisesfrom in the different methods. The largest error valuestypically arise from the bond angles; however PBE+ U Lit. dd also results in a significantly overestimated cell volume,and also incorrectly predicts some relative bond lengthsand angles such as M –O , > M –O , . He and Fran-chini’s PBE results, while the MARE similar to that re-ported here (0.98 vs. 0.88 %), differ significantly in someother parameters such as volume (0.2 vs. 1.08%). Thisillustrates some of the difficulty in comparing differentworks that utilize different codes, computation parame-ters and pseudopotentials, even though in general DFT isbecoming increasingly reproducible across various DFTimplementations . The main picture for LaVO is thatACBN0 marginally improves in all areas vs. PBE (which TABLE I. Structural parameters for AFM-C LaVO . Exper-imental data measured at 10 K is taken from Bordet et al. U Lit. dd = 3 . . Relative absolute error is shown in ital-ics (in %), with the mean absolute relative error (MARE)listed at the bottom of the table. ACBN0 calculations arePBE+ U dd + U pp . LaVO Expt. PBE PBE+ U Lit. dd ACBN0V (˚A ) 241.10 242.28 250.75 242.32 a (˚A) 5.5623 5.575 5.602 5.545 b (˚A) 5.5917 5.637 5.726 5.609 c (˚A) 7.7516 7.710 7.817 7.791 β ( ◦ ) 90.13 90.02 89.84 90.40 M –O (˚A) 1.978 1.961 2.014 1.963 M –O , (˚A) 1.989 2.025 2.101 1.990 M –O , (˚A) 2.042 2.023 2.021 2.057 M –O (˚A) 1.979 1.961 2.010 2.018 M –O , (˚A) 1.979 2.021 2.099 2.000 M –O , (˚A) 2.039 2.025 1.996 2.028 θ ( ◦ ) 156.74 158.80 152.51 156.15 θ , ( ◦ ) 156.12 156.64 152.53 154.08 θ , ( ◦ ) 157.83 156.84 152.97 156.53 MARE (%) still describes structure adequately with MARE <
2. LaCrO LaCrO has an orthorhombic structure with GdFeO (GFO) tilting distortions to the octahedra and spacegroup P nma (represented here in the
P bmn setting).As shown in Table II, ACBN0 (MARE 1.09%) performsslightly worse than PBE (MARE 0.94%), mostly due tothe poor description of bond lengths, despite slightly im-proved accuracy with regard to the lattice parametersand bond angles. PBE+ U Lit. dd results with a cluster-CIvalue of U dd = 4 . again result in a drasticallypoorer description of the structure. Other similar val-ues of U dd from fits to enthalpy of formation , band TABLE II. Structural parameters for AFM-G LaCrO . Ex-perimental data measured at 11 K is taken from Gilbu Tilset et al. U Lit. dd = 4 . . Relative absolute error is shownshaded in gray (in %), with the mean absolute relative error(MARE) listed at the bottom of the table. ACBN0 calcula-tions are PBE+ U dd + U pp . LaCrO Expt. PBE PBE+ U Lit. dd ACBN0V (˚A ) 233.60 237.54 244.14 237.26 a (˚A) 5.4718 5.521 5.588 5.522 b (˚A) 5.5093 5.519 5.557 5.519 c (˚A) 7.7491 7.796 7.863 7.785 M–O (˚A) 1.968 1.987 2.016 1.990 M–O , (˚A) 1.974 1.989 2.019 1.990 M–O , (˚A) 1.968 1.987 2.018 1.990 θ ( ◦ ) 159.59 157.67 154.33 156.07 θ ( ◦ ) 160.04 158.02 154.81 157.60 MARE (%) gap , and HSE calculations would likely provide thesame general result. The HSE results of He and Franchiniare referenced to a different (room temperature) exper-iment, but compared to the 11 K reference used here,PBE, HSE-25 and HSE-Opt (mixing 0.15) gave MAREvalues of 0.75%, 0.43% and 0.59%, respectively . HSE-Opt improves on the lattice parameters and bond lengthsbut worsens the error on the bond angles. HSE-25 showssimilar error on the bond lengths but drastically improvesall other structure descriptors considered here. Again,the difference between the previously reported PBE re-sults and the current work can likely again be explainedby computational differences such as choice of pseudopo-tential or DFT input parameters.
3. LaMnO LaMnO has the largest JT distortions among the 3 d perovskites studied here. The structural results are pre-sented in Table III. This has important consequences forthe calculated electronic structure, which is why a veryhigh structural accuracy is required in this material forpredicting electronic properties and ground states (dis-cussed in the next section). PBE and ACBN0 providealmost identical error, with MARE values of 0.93% and0.99%, respectively. This is in contrast to the work ofHe and Franchini, who report a large MARE for PBE(1.9%), caused by large inaccuracy in bond lengths thatdescribe the JT distortions, with the largest individualbond error being over 5% (the largest PBE bond lengtherror in this work is 1.62%) . Their calculations us-ing HSE-Opt and HSE-25 show some marginal improve-ment over the PBE and ACBN0 calculations here, butstill give a similar overall picture. The PBE+ U Lit. dd cal-culations once again show a significantly larger error at2.53%, with over 3% error on two of the three bondlengths, when using a value of U dd = 6 . . Awide range of U dd values have been used in the litera-ture, including 3.3 eV (from cRPA ) and 7.1 eV (fromconstrained DFT ). Hashimoto et al. reported thatPBE+ U Lit. dd = 2 . under full cell relaxation, butboth ACBN0 and PBE+ U Lit. dd fail to improve over thePBE case in this work. We performed a quick test with U dd = 2 . U Lit. dd describes theJT distortions in fully structurally-optimized LaMnO isunknown. One thing to note is that in these studies ,plane wave cutoffs between 30-40 Ry were used. We can-not claim that these results are unconverged with respectto calculating relaxed structures, but in this work, a cut-off of at least 100 Ry was found to be necessary to be con-verged with respect to cell pressure (within 0.5 kbar, seeSupplemental Material ). For energy differences lowercutoffs may be adequate, but quantitative comparisonof unit cell structure requires highly converged calcula-tion parameters to get accurate forces and stresses. Un-converged calculations may provide a fortuitous improve-ment in describing structure–additional test calculationswith both U dd = 0 . U dd = 2 . ,worsening accuracy for the plain PBE case but improvingaccuracy for the U dd = 2 . U correction, and thespecific implementation of DFT+ U used; the nature ofthe orbitals chosen and whether the J exchange termsare included explicitly vs. in a combined effective U canstrongly affect calculated values of U and the resultingmaterial properties. This will be discussed further in thenext section.
4. LaFeO Orthorhombic
P bnm
LaFeO has fully occupied e g and t g manifolds (high spin) that suppress JT dis-tortion. While PBE performs fairly well at describingthe structural parameters (MARE of 1.20%, Table IV),ACBN0 improves the accuracy of every unit cell param-eter (MARE 0.79%). The PBE+ U Lit. dd structure ( U dd =4 . ) again shows significantly worsened structuralaccuracy with a MARE of 2.90%. Hybrid functionals of- TABLE III. Structural parameters for AFM-A LaMnO . Ex-perimental data measured at 4.2 K is taken from Elemans et al. U Lit. dd = 6 . . Relative absolute error is shownshaded in gray (in %), with the mean absolute relative error(MARE) listed at the bottom of the table. ACBN0 calcula-tions are PBE+ U dd + U pp . LaMnO Expt. PBE PBE+ U Lit. dd ACBN0V (˚A ) 243.57 248.11 264.51 247.44 a (˚A) 5.532 5.563 5.631 5.549 b (˚A) 5.742 5.806 5.994 5.819 c (˚A) 7.668 7.681 7.837 7.663 M–O (˚A) 1.957 1.972 2.048 1.970 M–O , (˚A) 2.185 2.190 2.268 2.206 M–O , (˚A) 1.904 1.934 1.995 1.924 θ ( ◦ ) 156.69 153.63 146.21 153.04 θ ( ◦ ) 154.34 154.20 149.31 153.45 MARE (%) fer additional improvement vs. the ACBN0 results, withthe empirically-optimized HSE-Opt yielding a MARE of0.32% and HSE-25 yielding a MARE of 0.30% (note theseMARE values have been adjusted from the original publi-cation to correspond to the experimental data used here,which is very similar).
5. LaCoO Due to the smaller ionic radius of Co , LaCoO crys-tallizes in a rhombohedral structure with space group R ¯3 c , with slight GFO-type octahedral distortions. Struc-tural parameters and errors are listed in Table V. PBEand the smaller PBE+ U Lit. dd value of 4.2 eV (from cluster-CI calculations fit to experimental spectra ) performsimilarly, with MAREs of 1.20% and 1.28%, respectively.Increasing to a larger, LR U dd = 8 . , the error in-creases significantly to 3.60%. This provides yet anotherillustration of the pitfalls of choosing U dd uncritically,since the original work used LSDA+ U ; LSDA tends tooverbind and shorten bond lengths, and the addition of a U correction may increase the bond lengths closer to theexperimental value. ACBN0 provides the highest struc-tural accuracy for non-magnetic LaCoO , with a MAREof 0.12%, which compares very favorably to the HSE-25value of 0.42% and the HSE-Opt value of 0.44%. ACBN0and hybrid functionals are the only methods reportedhere that decrease the over-estimated unit cell volume0 TABLE IV. Structural parameters for AFM-G LaFeO . Ex-perimental room-temperature data is taken from Etter etal. U Lit. dd = 4 . . Relative absolute error is shownshaded in gray (in %), with the mean absolute relative error(MARE) listed at the bottom of the table. ACBN0 calcula-tions are PBE+ U dd + U pp . LaFeO Expt. PBE PBE+ U Lit. dd ACBN0V (˚A ) 242.88 247.51 252.38 245.13 a (˚A) 5.5549 5.558 5.595 5.547 b (˚A) 5.5663 5.653 5.679 5.617 c (˚A) 7.8549 7.877 7.944 7.867 M–O (˚A) 2.010 2.022 2.046 2.019 M–O , (˚A) 2.019 2.048 2.055 2.028 M–O , (˚A) 1.990 2.021 2.044 2.018 θ ( ◦ ) 155.26 153.70 152.17 154.00 θ ( ◦ ) 157.57 153.89 153.00 154.61 MARE (%)
TABLE V. Structural parameters for NM LaCoO . Exper-imental data measured at 4.2 K is taken from Thornton etal. U Lit. dd = 4 . and 8.5 eV . Relative absolute er-ror is shown shaded in gray (in %), with the mean absoluterelative error (MARE) listed at the bottom of the table. θ and θ describe O– ˆCo–O and Co– ˆO–Co angles, respectively.ACBN0 calculations are PBE+ U dd + U pp . LaCoO Expt. PBE PBE+ U Lit. dd ACBN04.2 eV 8.5 eVV (˚A ) 110.17 112.43 112.73 113.31 110.19 a (˚A) 5.3416 5.360 5.367 5.380 5.342 α ( ◦ ) 60.99 61.43 61.40 61.30 60.99 M–O (˚A) 1.924 1.947 1.949 1.952 1.926 θ ( ◦ ) 88.56 87.91 87.93 88.02 88.49 θ ( ◦ ) 163.10 159.58 159.51 159.64 162.25 MARE (%) of PBE–applying a U correction only to the d electronsresults in an increased cell volume. TABLE VI. Structural parameters for NM LaNiO . Experi-mental data measured at 1.5 K is taken from Garc´ıa-Mu˜noz et al. U Lit. dd = 5 . . Relative absolute error is shownshaded in gray (in %), with the mean absolute relative error(MARE) listed at the bottom of the table. θ and θ describeO– ˆNi–O and Ni– ˆO–Ni angles, respectively. ACBN0 calcula-tions are PBE+ U dd + U pp . LaNiO Expt. PBE PBE+ U Lit. dd ACBN0V (˚A ) 112.48 114.19 114.12 111.33 a (˚A) 5.3837 5.397 5.397 5.370 α ( ◦ ) 60.86 61.21 61.19 60.75 M–O (˚A) 1.933 1.950 1.949 1.925 θ ( ◦ ) 88.78 88.28 88.32 88.91 θ ( ◦ ) 164.82 161.97 162.20 165.47 MARE (%)
6. LaNiO LaNiO , similarly to LaCoO , has R ¯3 c symmetrywith GFO-type octahedral tilting. Structural param-eters and errors are listed in Table VI. PBE providesa fairly accurate picture of the structure but also sim-ilarly to LaCoO , overestimates the unit cell volume.PBE+ U Lit. dd = 5 . provides very marginal improve-ment in the structure, with a MARE value of 0.86%.LDA+ U results from Gou et al. optimized LaNiO withan estimated MARE of 0.3% and an empirical U of 6 eV,thought it should be noted that plain PBE resulted inthe best agreement with experimental Raman-active lat-tice modes and the large value of U destabilized the lat-tice by introducing imaginary phonon modes . ACBN0improves the picture without significantly introducinglarger errors to any of the structure parameters and yieldsa MARE of 0.40%. HSE-25 (HSE-Opt is zero mixingfraction, or plain PBE for this material) yields additionalimprovement with a MARE of 0.19%. While the geome-try improves with increasing mixing fraction (up to HSE-35 with MARE of 0.1%), the treatment of the electronicproperties worsens, as discussed in the next section.Figures 3 and 4 illustrate the results of this section,showing the MARE values for each material and the av-erage MARE for each method, along with detailed radarplots for each material that show each method’s relativeabsolute errors for cell volume, lattice parameters, bondlengths and bond angles. The PBE results agree fairlywell with the previously reported PBE calculations of Heand Franchini , and describe the structures of the 3 d LaBO perovskites fairly well with an average MAREof around 1%. Applying the values of U from the lit-erature usually results in a poorly described structure1(average MARE 2.3%), with the exceptions of LaCoO and LaNiO , where accuracy near the level of PBE is ob-tained. ACBN0 however, applying self-consistent valuesof U to both metal 3 d and oxygen 2 p states, significantlyimproves the predicted structures with an average MAREof less than 0.7%. There still are shortcomings when de-scribing some of the early transition metal compoundsthat have significant JT distortion and orbital ordering,since the structure and electronic properties are so closelyintertwined. While the previously-reported HSE-25 andHSE-Opt result in improved structural parameters vs.PBE (average MARE of 0.4% and 0.6% respectively),we will see in the next section that this does not neces-sarily translate to an improved overall picture includingelectronic properties, which is where the approach of ap-plying U pp along with U dd in a self-consistent approach,like with ACBN0, clearly has benefits. C. Electronic Structure
Bulk electronic structure properties for both experi-mental and optimized structures are presented in thefollowing sections for each perovskite. Band gaps aredetermined from the KS density of states (DOS) andcompared to experimental values where appropriate (i.e.not in the case of metallic LaNiO ). Magnetic moments(from unit cell absolute magnetization divided by thenumber of metal cations) are also compared with liter-ature values for magnetically ordered structures. Theenergies of AFM-A, AFM-C, AFM-G, FM and NM aretabulated relative to whichever magnetic structure is theexperimentally-observed ground state. We present pro-jected DOS (PDOS) for our calculations using PBE,ACBN0, and PBE+ U Lit. dd . For brevity, only band struc-tures for PBE and ACBN0 are compared; this is in orderto present the dispersion of the bands, which is not visiblefrom the PDOS.
1. LaVO The Mott insulator LaVO is not correctly describedby plain PBE DFT, which in this work predicts it asa AFM-A metal after geometry optimization. There isalso a type-G t g orbital ordering , which will notbe investigated here but may be included in future work.Table VII presents electronic structure parameters forLaVO , including band gap and magnetic moment com-pared with experimental values, as well as the relativeDFT-calculated energies of several possible magnetic or-derings compared to the experimentally-observed AFM-C order . Even with the correct AFM-C ordering, PBEpredicts a metallic ground state, as shown in Fig. 5a-b.ACBN0 predicts the correct AFM-C ground state andalso provides a very good estimate of the experimentally-observed band gap: a predicted 0.8 eV compared to theobserved 1.1 eV , introducing a gap between the oc- TABLE VII. Parameters obtained from the electronic struc-ture of LaVO , including band gap E g , magnetic moment perV cation µ and the energy difference ∆ E between various cal-culated magnetic ordering states for PBE, PBE+ U Lit. dd = 3 . , and ACBN0 (PBE+ U dd + U pp ). Experimental values forband gap and magnetic moment are also provided. LaVO AFM-C Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN0 E g (eV) 1.1 µ ( µ B /V) 1.3 U Lit. dd ACBN0∆ E (meV) AFM-A 42 -167 110AFM-G 313 -74 215FM 59 94 324NM 1547 5146 4264Optimized StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) AFM-A -66 97 94AFM-G 298 69 46FM 21 20 44NM 1326 5457 4096 cupied and unoccupied t g states. This can also easilybe seen in Fig. 5c-d and in the band structure of Fig. 6.The PBE+ U Lit. dd result and the HSE results of He andFranchini also result in a correct ground state, with thelatter giving a slightly larger estimate of the band gapfor HSE-Opt (1.46 eV). HSE-25 predicted a rather largevalue of 2.43 eV. Magnetic moments for ACBN0 andPBE+ U Lit. dd slightly overestimate the moment comparedto PBE, which is also larger than experiment. This isa common error in hybrid functionals as well. Anotherimportant feature to notice is the charge transfer (CT)gap, or the difference between the predominately oxygen-derived lower valence band and the unoccupied conduc-tion band of mostly d parentage. Experimentally thevalue is reported to be 4.0 eV , but PBE+ U Lit. dd pre-dicts a smaller gap of approximately 3 eV and a highermixing of O 2 p and V 3 d in the valence band vs ACBN0and the previously reported HSE results. ACBN0 pre-dicts a value near 4.2 eV, while HSE-Opt overestimatesthe experimental value, giving 4.9 eV. An additional em-pirical adjustment, HSE-10, can reduce this to 4.4 eVand gives a Mott-Hubbard (MH) gap of 0.89 eV. Whileboth PBE+ U Lit. dd and ACBN0 give similar band gaps, theadditional push of valence band oxygen 2 p states to lowerenergy from the U pp term in ACBN0 opens up the CTgap to a value agreeing closer to experiment and widensthe band dispersion of the valence band.2 FIG. 3. Mean absolute relative error (MARE) of perovskite structural parameters for PBE, PBE+ U Lit. dd , HSE (from He andFranchini ) and ACBN0 (PBE+ U dd + U pp ). HSE-25 refers to an exact exchange mixing fraction of 0.25, and HSE-Opt. is anempirically-optimized value to balance the description of both structural and electronic properties.FIG. 4. Relative absolute error of detailed subgroups of pervoskite structural parameters for PBE, PBE+ U Lit. dd , HSE (from Heand Franchini ) and ACBN0 (PBE+ U dd + U pp ). HSE-25 refers to an exact exchange mixing fraction of 0.25, and HSE-Opt. isan empirically-optimized value to balance the description of both structural and electronic properties. For lattice constants,bond lengths, and bond angles, and average error for each structure is used. Cell volume error for LaCoO goes beyond theaxis limits to improve clarity (12.83%).
2. LaCrO Table VIII presents electronic structure parametersof AFM-G LaCrO , an AFM insulator with an opti-cal band gap of 3.4 eV as reported by Arima et al .They note in this early work that the weaker MH tran-sition is completely indiscernible due to the stronger CTtransition, meaning the two gaps are nearly equal inwidth or correspond to the same gap, with significantCr 3 d –O 2 p hybridization in the valence band. Fromthe PBE+ U Lit. dd = 4 . calculations shown in Fig. 7f,this would seem to be a reasonable picture. Large valuesof exact exchange ( < , with a MHgap near 3.0 eV. However, this does not account for an important experimental observation–the green color ofLaCrO , which would require a gap in the optical range.He and Franchini also mention a study by Ong et al. that interprets the electronic structure in a different way.They applied empirical U dd corrections of 2.72, 5.44 and8.16 eV (the ACBN0-calculated value of U dd is 2.77 eV forCr, as shown in the Supplemental Material ) to comparethe simulated valence band PDOS to experimental X-rayphotoemission spectroscopy (XPS) spectra. They foundthat applying any U dd correction both worsened theircomparison with the experimental XPS spectra, and re-sulted in no features in the optical range near green lightin simulated reflectivity spectra. The implication is thatthe CT and MH gaps remain distinct, and two separatetransitions are present: the larger CT gap of 3.4 eV isresponsible for the previous experimental measurements,while the smaller MH gap near 2.2 eV explains the green3 FIG. 5. Projected density of states for AFM-C LaVO (on theO, V, and La states); a. experimental structure with PBE; b. optimized structure with PBE; c. experimental struc-ture with ACBN0 (PBE+ U dd + U pp ); d. optimized structurewith ACBN0 (PBE+ U dd + U pp ); e. experimental structurewith PBE+ U Lit. dd = 3 . ; f. optimized structure withPBE+ U Lit. dd = 3 . .FIG. 6. Band structure of AFM-C LaVO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure. FIG. 7. Projected density of states for AFM-G LaCrO (onthe O, Cr, and La states); a. experimental structure withPBE; b. optimized structure with PBE; c. experimentalstructure with ACBN0 (PBE+ U dd + U pp ); d. optimized struc-ture with ACBN0 (PBE+ U dd + U pp ); e. experimental struc-ture with PBE+ U Lit. dd = 4 . ; f. optimized structure withPBE+ U Lit. dd = 4 . .FIG. 8. Band structure of AFM-G LaCrO ; a. PBE op-timized structure; b. ACBN0 (PBE+ U dd + U pp ) optimizedstructure. TABLE VIII. Parameters obtained from the electronic struc-ture of LaCrO , including band gap E g , magnetic moment perCr cation µ and the energy difference ∆ E between various cal-culated magnetic ordering states for PBE, PBE+ U Lit. dd = 4 . , and ACBN0 (PBE+ U dd + U pp ). Experimental values forband gap and magnetic moment are also provided. The bandgap in brackets corresponds to a more recent interpretationof optical data . LaCrO AFM-G Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN0 E g (eV) 3.4 (2.4) µ ( µ B /Cr) 2.45-2.8 U Lit. dd ACBN0∆ E (meV) AFM-A 324 157 147AFM-C 146 76 71FM 519 250 233NM 4839 10979 10641Optimized StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) AFM-A 247 60 97AFM-C 128 30 45FM 388 90 155NM 4818 11179 10379 color of LaCrO and the corresponding peaks in reflectiv-ity measurements. Ong et al. conclude that there are nostrong electronic correlations in LaCrO , and that plainGGA is the most appropriate for describing this mate-rial. If one takes this interpretation as correct, it wouldappear at first glance that ACBN0, PBE+ U Lit. dd and HSEall do not describe this material correctly. ACBN0 andHSE result in CT gaps near 4.5 eV, while PBE results inan CT gap of near 3.4 eV. However, ACBN0 still resultsin a MH gap of near 2.6 eV, while HSE and PBE resultin MH gaps of 3 eV and 1.4 eV, respectively (see Figs. 7and 8). Ong et al. mentioned that previous experimen-tal spectra did not include features in the optical range,and suggested further experimental studies to find theirpredicted green light optical absorption.At the time there was no additional experimental evi-dence clarifying the electronic structure of LaCrO , butin 2013 Sushko et al. reported experimental measure-ments coupled with embedded cluster time-dependentDFT that discerned the multiple optical transitionspresent in this material. Spectroscopic ellipsometry re-vealed onset of absorption features near 2.3 eV and 3.2eV, occurring before a large 5 eV optical absorption on-set. They attributed the absorption features to familiesof t g – e g , t g – t g , and Cr 3 d –O 2 p transitions and con-clude that the true CT gap is near ∼ et al. suggested, while the green absorption feature(onset at ∼ ∼ t g – e g fundamental gap transitions and the previously-reported3.4 eV gap is due to inter-Cr t g – t g transitions. Theband structure in Fig. 8 illustrates these transitions with lines of appropriate energy superimposed over the bandstructure. This is more in line with trends in the CTgap from X-ray spectroscopy experiments , where thegaps are quite large since they are calculated from peakpositions rather than band edges ( ∼ ,and larger than the MH band gap) and generally decreasewith increasing d occupation. It is worth mentioning thatfor ACBN0, the spacing of the spin-down t g peak andO 2 p valence band peak is quite close to 7 eV, owing inpart to U pp forcing the oxygen 2 p states lower in energy.Further comparison of our calculations with experimen-tal spectra are available in the Supplemental Material .While ground state DFT strictly does not describe tran-sition energies, the ACBN0 results generally support thispicture in terms of the gaps and types of PDOS featurespresent, in contrast to those of PBE+ U Lit. dd , HSE-25 andHSE-Opt (HSE-10 provides a fairly similar picture toACBN0). This alternative picture significantly affectsthe band gap error, shown in Fig. 23, bringing it more inline with the rest of the perovskites.As shown in Table VIII, all the methods used in thisstudy, as well as the HSE results from He and Fran-chini, correctly predict the AFM-G magnetic orderingfor LaCrO . Magnetic moments are overestimatedslightly by PBE, and further overestimated by ACBN0and PBE+ U Lit. dd (although ACBN0 does to a lesser de-gree). Band structures for the PBE and ACBN0 opti-mized structures are shown in Fig. 8.
3. LaMnO LaMnO is an type-A AFM MH insulator with signif-icant JT distortions and e g orbital ordering . All themethods used in this work incorrectly predict a metallicFM ground state when the geometry and unit cell are op-timized; in addition, only PBE predicts the correct AFM-A ground state when the experimental structure is used(summarized in Table IX). This illustrates the particu-lar importance of the JT distortions in the existence of aband gap in this material. While there have been reportsof DFT+ U both improving and worsening the struc-tural and electronic properties of LaMnO , it is clear thatin an orbitally-ordered material and/or where the e g and t g bands exhibit markedly different localized or itiner-ant behavior, that the averaging used in both calculatingand applying U corrections in most commonly used im-plementations is likely inappropriate, and improvementsfrom such treatments are fortuitous. This is especiallytrue for the widely-used simplified rotationally-invariantimplementation of DFT+ U which considers the exchangeinteraction J as isotropic .Although in the DFT+ U implementation used in thiswork ACBN0 does not predict the correct ground state,for the AFM-A state it yields an accurate band gap of1.0 eV, with the e g bands being isolated from the otherbands (see Fig. 10), as reported in the HSE study ofHe and Franchini. Compare this with the ferromagnetic5 FIG. 9. Projected density of states for AFM-A LaMnO (onthe O, Mn, and La states); a. experimental structure withPBE; b. optimized structure with PBE; c. experimentalstructure with ACBN0 (PBE+ U dd + U pp ); d. optimized struc-ture with ACBN0 (PBE+ U dd + U pp ); e. experimental struc-ture with PBE+ U Lit. dd = 6 . ; f. optimized structure withPBE+ U Lit. dd = 6 . .FIG. 10. Band structure of AFM-A LaMnO ; a. PBE op-timized structure; b. ACBN0 (PBE+ U dd + U pp ) optimizedstructure. (FM) band structure presented in Fig. 12, where the moststriking difference is the change in these bands near theFermi level. For AFM-A LaMnO , HSE-25 grossly over-estimates the band gap (2.47 eV) and HSE-Opt gives areasonable value of 1.63 eV. PBE+ U Lit. dd highlights thepreviously mentioned failures of the simplified DFT+ U implementation for LaMnO by giving a band gap of only0.6 eV for U = 6 . . More notably, the spin-up t g states are pushed down below the oxygen valence band,in contrast to ACBN0 (see Fig. 9) and the hybrid func-tional results (for all mixing fractions). The use of U pp in ACBN0 allows the opening of the MH gap to experi-mental values without especially large values of U dd , andthe lowering of O 2 p energy preserves the general pictureof bonding given by PBE and HSE calculations. Thesetrends are also true in the FM case, shown in Fig. 11.The magnetic moment is again slightly overestimated byACBN0 and PBE+ U Lit. dd , while PBE gives a value atthe upper end of the experimental range. LaMnO isa widely-studied material and further discussion can befound in the literature .Owing to the difficulty in comparing not only self-consistent values of U but also the results of calculationsthat use different computational parameters (pseudopo-tential, XC functional, DFT+ U implementation, etc.)it is challenging to make definitive statements regard-ing the treatment of the ground state of LaMnO withinDFT and DFT+ U . It has been shown that explicit inclu-sion of orbital-dependent J corrections, as in the originalrotationally-invariant scheme by Liechtenstein et al. ,is necessary for stabilizing the experimentally observedAFM-A magnetic ordering and reproducing the e g or-bital ordering in LaMnO , at least when using LDA andPBEsol (GGA) functionals and computational parame-ters utilized in that work . Empirical values of U = 8eV and J = 2 eV in that work yielded a good overalldescription of correct ground state, band gap, and struc-tural properties. Other work using using PBE with ex-plicit U = 2 . J = 1 . U eff = 1 . . On theother hand, similar calculations using different DFT codewith a simplified U eff = 2 . .As an aside, we perform some simple test calculationswith explicit U and J on d and p states using the ACBN0-calculated values for both Mn and O (still calculated withthe Dudarev implementation: U = 3 .
62 eV, J = 1 .
18 eVfor Mn 3 d and U = 12 .
185 eV, J = 6 .
10 eV for O 2 p )yields an energy difference of 0.0002 eV, compared to thevalue of 0.030 eV in Table IX. If we keep the ACBN0correction on oxygen and increase U and J on Mn to6.0 eV and 2.0 eV, respectively, closer to the values ofMellan et al. , the AFM-A ordering is stabilized withthe DFT+ U correction, with only a subtle push of Mn d states deeper into the O 2 p deep valence band withthe larger values of U and J (shown in SupplementalMaterial ). Applying these same U values within the6 TABLE IX. Parameters obtained from the electronic struc-ture of LaMnO , including band gap E g , magnetic mo-ment per Mn cation µ and the energy difference ∆ E be-tween various calculated magnetic ordering states for PBE,PBE+ U Lit. dd = 6 . , and ACBN0 (PBE+ U dd + U pp ). Ex-perimental values for band gap and magnetic moment are alsoprovided. LaMnO AFM-A Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN0 E g (eV) 1.1-2.0 µ ( µ B /Mn) 3.4-3.9 U Lit. dd ACBN0∆ E (meV) AFM-C 277 415 309AFM-G 293 597 391FM 54 -169 -30NM 6400 14172 12285Optimized StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) AFM-C 213 279 241AFM-G 168 422 268FM -175 -369 -199NM 5079 18053 10614 simplified scheme does not stabilize the correct AFM-Aground state. Unfortunately, unit cell stress and pressureare not easily implemented in the generalized DFT+ U scheme, so only the calculations using the experimentalstructure has been performed for this additional compari-son. Therefore, the inability of ACBN0 to improve geom-etry and electronic structure in this work may potentiallybe determined by the implementation of DFT+ U ratherthan the ACBN0 approach itself, leaving room for futureimprovement.
4. LaFeO LaFeO is often considered an intermediate CT/MHinsulator , owing to the considerable O 2 p character inthe e g valence band. This material exhibits AFM-G mag-netic ordering with a band gap of 2.3 eV . Allmethods used in this work correctly predict the AFM-G ground state, which is much lower in energy than theother magnetic orderings listed in Table X. The projecteddensities of states for PBE, ACBN0 and PBE+ U Lit. dd =4 . in both experimental and optimized structuresare shown in Fig. 13, with band structures for the opti-mized structures of PBE and ACBN0 shown in Fig. 14.PBE gives a qualitatively correct picture of the elec-tronic structure but underestimates the band gap at 0.9eV. ACBN0 and PBE+ U Lit. dd both give band gaps muchcloser to experiment at 2.6 and 2.5 eV, respectively. Theband structures of PBE and ACBN0 are both qualita-tively similar, with the states above the valence band FIG. 11. Projected density of states for FM LaMnO (on theO, Mn, and La states); a. experimental structure with PBE; b. optimized structure with PBE; c. experimental struc-ture with ACBN0 (PBE+ U dd + U pp ); d. optimized structurewith ACBN0 (PBE+ U dd + U pp ); e. experimental structurewith PBE+ U Lit. dd = 6 . ; f. optimized structure withPBE+ U Lit. dd = 6 . .FIG. 12. Band structure of FM LaMnO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure.Lighter colors correspond to spin up, while darker colors cor-respond to spin down. TABLE X. Parameters obtained from the electronic structureof LaFeO , including band gap E g , magnetic moment perFe cation µ and the energy difference ∆ E between variouscalculated magnetic ordering states for PBE, PBE+ U Lit. dd =4 . , and ACBN0 (PBE+ U dd + U pp ). Experimental valuesfor band gap and magnetic moment are also provided. LaFeO AFM-G Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN0 E g (eV) 2.3 µ ( µ B /Fe) 3.9, 4.6 U Lit. dd ACBN0∆ E (meV) AFM-A 698 853 687AFM-C 419 363 300FM 838 1319 1064NM 4418 10352 10010Optimized StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) AFM-A 715 686 602AFM-C 385 303 284FM 921 1059 929NM 3866 10121 9439FIG. 13. Projected density of states for AFM-G LaFeO (onthe O, Fe, and La states); a. experimental structure withPBE; b. optimized structure with PBE; c. experimentalstructure with ACBN0 (PBE+ U dd + U pp ); d. optimized struc-ture with ACBN0 (PBE+ U dd + U pp ); e. experimental struc-ture with PBE+ U Lit. dd = 4 . ; f. optimized structure withPBE+ U Lit. dd = 4 . . FIG. 14. Band structure of AFM-G LaFeO ; a. PBE op-timized structure; b. ACBN0 (PBE+ U dd + U pp ) optimizedstructure. maximum being more or less simply shifted upwards inenergy. It is important to note the differences in thePDOS of ACBN0 and PBE+ U Lit. dd . ACBN0 produces apicture similar to that of PBE, except for the band gap;the valence band retains significant Fe–O hybridizationand remains separate from the deeper O 2 p valence band,also giving a similar picture to the HSE-Opt results ofHe and Franchini both quantitatively and qualitatively.This also puts it in good agreement with the photoe-mission data of Wadati et al. which was comparedwith the HSE results (see Supplemental Material ?? ).In contrast, despite the fairly accurate band gap, thePBE+ U Lit. dd calculation in this work results in an elec-tronic structure with significantly reduced hybridization,similar to the higher mixing fraction HSE calculations(HSE-35) by He and Franchini. The Fe e g parentage ofthe valence band is reduced, the valence band mergeswith the larger oxygen-derived valence band and occu-pied t g states are pushed outside the band width ofthe oxygen valence band, leading to a much more local-ized, ionic picture that does not agree with the aforemen-tioned experimental spectroscopic data. While the sametrend of increasing magnetic moment with U correction(also with hybrid functionals) continues with LaFeO ,the larger variation in the reported experimental valuesin Table X makes it difficult to make any claims abouttheir accuracy.8 TABLE XI. Parameters obtained from the electronic struc-ture of LaCoO , including band gap E g and the energy differ-ence ∆ E between various calculated magnetic ordering statesfor PBE, PBE+ U Lit. dd values of 4.2 eV and 8.5 eV , andACBN0 (PBE+ U dd + U pp ). Experimental values for band gapare also provided. Gap of “m” refers to a metallic system. LaCoO NM Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN04.2 eV 8.5 eV E g (eV) 0.3 m 0.8 1.2 0.8Relative Energy vs. NMExperimental StructurePBE PBE+ U Lit. dd ACBN04.2 eV 8.5 eV∆ E (meV) FM -102 -313 -1551 -331Optimized StructurePBE PBE+ U Lit. dd ACBN04.2 eV 8.5 eV∆ E (meV) FM -128 -618 -2334 -476
5. LaCoO A diamagnetic insulator (low-spin Co), LaCoO is notwell-described by plain DFT, which predicts a ferromag-netic metallic ground state. There still is no conclusiveunderstanding of the higher temperature magnetic be-havior of LaCoO , and a discussion of that topic is be-yond the scope of this work. It has been reported in theliterature that DFT+ U is at least capable of stabilizingthe correct low-spin insulating state, with U values ei-ther being varied empirically or calculated from firstprinciples . The values of U themselves includean empirical U eff = 6 . − .
65 = 5 .
85 eV , U = 3 . U = 4 . , LR U typically in the range of 7.8-8.5eV , U eff = 6 . , .
96 eV calculated from constrainedLDA , and a renormalized U = 4 . (a method from whichACBN0 takes inspiration). The ACBN0 U dd value on Coof ∼ U Lit. dd values of both 4.2 eV and 8.5eV are used for comparison.Both ACBN0 and PBE+ U dd = 4 . U dd = 8 . U gaps are fairlylarge compared to experiment (0.3 eV), as shown in Ta-ble XI. However, all the results fail to predict the correct FIG. 15. Projected density of states for NM LaCoO (onthe O, Co, and La states); a. experimental structure withPBE; b. optimized structure with PBE; c. experimen-tal structure with ACBN0 (PBE+ U dd + U pp ); d. optimizedstructure with ACBN0 (PBE+ U dd + U pp ); e. experimentalstructure with PBE+ U Lit. dd = 4 . ; f. optimized struc-ture with PBE+ U Lit. dd = 4 . ; g. experimental structurewith PBE+ U Lit. dd = 8 . ; h. optimized structure withPBE+ U Lit. dd = 8 . FIG. 16. Band structure of NM LaCoO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure. NM low- T ground state. This may be attributed to thefact that all the previous DFT+ U studies mentioned usedLDA+ U as opposed to the GGA+ U used here, and ourresults are consistent with those reported by Ritzmann et FIG. 17. Projected density of states for FM LaCoO (onthe O, Co, and La states); a. experimental structure withPBE; b. optimized structure with PBE; c. experimen-tal structure with ACBN0 (PBE+ U dd + U pp ) d. optimizedstructure with ACBN0 (PBE+ U dd + U pp ); e. experimentalstructure with PBE+ U Lit. dd = 4 . ; f. optimized struc-ture with PBE+ U Lit. dd = 4 . ; g. experimental structurewith PBE+ U Lit. dd = 8 . ; h. optimized structure withPBE+ U Lit. dd = 8 . al. . The tendency of larger U dd to extend bond lengthsin this material combined with the tendency of LDA tooverbind could explain why using the same value of U fora GGA+ U calculation fails to improve the description,and illustrates the non-transferability of U . The HSEresults of He and Franchini of course use PBE as thebase for mixing exact exchange, with HSE-25 giving avery large gap of 2.4 eV; the value of mixing for HSE-Opt is very small at 0.05, but yields a band gap of 0.1eV and also provides the best description of the struc-ture in that work. It should be mentioned that despitethe larger gap, ACBN0 provides a very similar pictureof hybridization to that of HSE-Opt, which also agreeswith the CT-like nature of the optical gap and makessense given the similarity of the former to the PBE result FIG. 18. Band structure of FM LaCoO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure.Lighter colors correspond to spin up, while darker colors cor-respond to spin down. and the very low exchange mixing fraction of the latter.As mentioned in the previous section, ACBN0 providesan excellent description of the structure of LaCoO . Thelarger, LR U of 8.5 eV significantly reduces the d charac-ter of the valence band and thus departs from the pictureprovided by PBE, ACBN0 and HSE. The band structuresof PBE and ACBN0 Fig. 16 and 18 further illustrate theelectronic structure as a simple shifting of the e g man-ifold higher in energy from the t g manifold, resultingin a band gap in the NM case (the FM state remainsmetallic).
6. LaNiO The last material to be studied is LaNiO , where thestrong covalent interaction between Ni and O screen re-sults in itinerant electrons that screen correlation to adegree and result in a paramagnetic (PM) metal, albeitstill one with important electron-electron interactions asrevealed by the T dependence of resistivity and heatcapacity . The electronic parameters of LaNiO are summarized in Table XII, with PDOS for both NMand FM states (for PBE, ACBN0 and PBE+ U Lit. dd ) shownin Fig. 19 and 21, respectively; and band structures forNM and FM states (for PBE and ACBN0) shown inFig. 20 and 22, respectively.PBE stabilizes a NM state with the experimental struc-ture. This is in contrast to Gou et al. but in agreementwith He and Franchini . The absolute energy differ-ence vs. the FM state is extremely small at 1 meV, about0 TABLE XII. Parameters obtained from the electronic struc-ture of LaNiO , including band gap E g and the energydifference ∆ E between various calculated magnetic order-ing states for PBE, PBE+ U Lit. dd = 5 . , and ACBN0(PBE+ U dd + U pp ). Gap of “m” refers to a metallic system. LaNiO NM Optimized StructureExpt. PBE PBE+ U Lit. dd ACBN0 E g (eV) m m m mRelative Energy vs. NMExperimental StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) FM 1 -839 -596Optimized StructurePBE PBE+ U Lit. dd ACBN0∆ E (meV) FM -33 -1000 -604 two orders of magnitude smaller than that reportedby the latter study. ACBN0 and PBE+ U Lit. dd = 5 . both stabilize FM ordering, similarly to previously-reported LSDA+ U and hybrid functional results .The PBE+ U Lit. dd calculation, similar to the aforemen-tioned LSDA+ U study, suppresses the contribution of Nistates near the Fermi level and pushes them to the bot-tom of the valence band, yielding a qualitatively incorrectpicture of the electronic structure. Aside from the factthat ACBN0 incorrectly stabilizes FM ordering in thebulk, the deviations from the PBE result are less extreme.Hybrid functionals of increasing mixing fraction have asimilar trend as when increasing U , but their descriptionof valence band spectra is significantly worse than LDA orDFT+ U . However, they also describe bound core statesmore accurately where DFT+ U does not (since the cor-rection is only applied to the valence states). Furtherstudy is needed to determine how ACBN0 performs incomparison with experimental spectra.It should be mentioned that all the reported DFT,DFT+ U and hybrid functional calculations are funda-mentally incapable of describing the electronic structureof LaNiO accurately. The delocalized states lead toscreened correlation effects that are not captured accu-rately with approximate XC functionals . Correctionssuch as hybrid functionals and DFT+ U are intendedto correct self-interaction error arising from inexact ex-change, and strictly speaking do not treat correlation.Many-body methods such as dynamical mean field theory(DMFT) are necessary to treat these correlation effectsin a meaningful way .The results of this section are summarized in Fig. 23.The absolute error is significantly improved usingACBN0 vs. PBE (the PBE error in LaCoO is dueto predicting a metallic state). The more stringent %MARE (since % errors for small gaps can be very high)demonstrates that ACBN0 is still improved vs. PBE,PBE+ U Lit. dd and HSE-25. Only HSE-Opt performs bet-ter on average, but as can be seen by the absolute er-rors, ACBN0 still outperforms HSE in several cases, most FIG. 19. Projected density of states for NM LaNiO (on theO, Ni, and La states); a. experimental structure with PBE; b. optimized structure with PBE; c. experimental struc-ture with ACBN0 (PBE+ U dd + U pp ); d. optimized structurewith ACBN0 (PBE+ U dd + U pp ); e. experimental structurewith PBE+ U Lit. dd = 5 . ; f. optimized structure withPBE+ U Lit. dd = 5 . .FIG. 20. Band structure of NM LaNiO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure. notably LaVO , LaCrO , and LaMnO . However, onemust be careful not to attribute too much meaning tojust the band gap, as simply applying an empirical U dd can result in a correct band gap but incorrect pictureof bonding, worsening structural description and otherproperties. The Supplemental Material has some com-parison to experimental X-ray photoemission and X-rayemission/absorption spectra that may be illustrative ofthis point . While ACBN0 can predict band gaps fairlywell, the overall failure to predict the correct ground state1 FIG. 21. Projected density of states for FM LaNiO (on theO, Ni, and La states); a. experimental structure with PBE; b. optimized structure with PBE; c. experimental struc-ture with ACBN0 (PBE+ U dd + U pp ); d. optimized structurewith ACBN0 (PBE+ U dd + U pp ); e. experimental structurewith PBE+ U Lit. dd = 5 . ; f. optimized structure withPBE+ U Lit. dd = 5 . FIG. 22. Band structure of FM LaNiO ; a. PBE optimizedstructure; b. ACBN0 (PBE+ U dd + U pp ) optimized structure.Lighter colors correspond to spin up, while darker colors cor-respond to spin down. of several of these oxides may be reflective of a need formore sophistication in applying U corrections. Many ma-terials may require separate corrections for t g , e g , etc.as has been noted in many early literature works. Hybridfunctionals do not have this problem, and as mentionedin Sec. I, can utilize first-principles self-consistent mixingfractions, but still suffer from high computational cost.Further developments in the implementation of DFT+ U may help bridge this gap.
7. Effect of U applied to oxygen 2 p states For most the perovskites studied in this work, the val-ues of U dd predicted by ACBN0 are significantly smallerin magnitude compared to the corresponding values of U referenced from in the literature (see SupplementalMaterial ). However, regardless of the transferability of U , these reference values of U dd were not calculated or fit-ted with a non-zero value of U pp applied to the oxygen 2 p states. This brings up the effect of U pp as an importantpoint for discussion.The use of non-zero U pp has been reported extensivelyon several transition metal oxides. In general, U pp canbe expected to be on the same order of magnitude as U dd , based on both cRPA calculations and on exten-sive experimental spectroscopy results . One notablecase is that of ZnO, where additional Hubbard correc-tions beyond U dd are necessary in order to open the bandgap to the experimental value. Even large values of U dd are not successful in this regard . Recent studies onbulk and nanowire ZnO have found that empiricalvalues of U dd ≈
10 eV and U pp ≈ U dd and U pp to match experimental XPSspectra, which resulted in values of 9.3 and 18.4 eV, re-spectively; based on the calculated dielectric tensor, theauthors concluded XPS spectra are not representativeof the ground state and are unsuitable for direct com-parison with calculated DOS . Interestingly, ACBN0results in U dd = ∼
13 eV and U pp = ∼ . . Severalother papers have investigated TiO , where the use ofboth U dd and U pp allows for the simultaneous improve-ment of both band gap and unit cell parameters. UsingLR for both Hubbard correction values yielded U dd = 3 . U pp = 10 .
65 eV for rutile TiO . However, anempirical U pp = 5 . U dd = 0 .
15 eV (owing to the nearlyempty d bands) and U pp = 7 . U pp used to give the correct band gap . Empiri-cal tuning of the Hubbard parameters on both titanium3 d and oxygen 2 p has also been investigated . Inthese studies, while U pp preserves the hybridized char-acter of the bonds, the lattice geometry is often worsecompared to LDA or GGA . Molecular organometallicNi magnets and cobalt oxyhydroxide catalysts have2 FIG. 23. The absolute band gap error for PBE and several corrective methods, and the MARE of the band gap predictionsfor each method. Asterisks denote data from He and Franchini . ACBN0 calculations are PBE+ U dd + U pp . Bars with grayhashed lines represent errors using an alternative interpretation of the electronic structure (see main text). also been studied using U pp . In many cases, justificationsbased on the experimental observation of large Coulom-bic interactions between O 2 p states, as well as preser-vation of strong d – p hybridization–strongly modified byapplying only U dd –have been put forward, but many ofthese studies still suffer from difficulty in applying the U correction self-consistently.The simultaneous, self-consistent determination ofboth U dd and U pp in this study has the effect of main-taining the metal-oxygen covalency present in the PBEcalculation (which is also the case for the HSE-Opt andHSE-25 cases when using hybrid functionals ) while alsoimproving the description of lattice geometry and theband gap. While U dd can sometimes be tuned to give amore accurate description of certain properties (usuallyband gap), by examining the pDOS in this work, we cansee that the bonding character is often significantly mod-ified, especially for larger values of U dd . Metal-oxygen co-valency is reduced, with the metal d states being pushedto lower energy with narrower DOS features. In con-trast, ACBN0 shifts both the O 2 p and metal 3 d bands;by not drastically changing the bonding character or in-troducing spurious charge transfer that can occur whenonly U dd is applied , the accuracy of several materialproperties can be improved simultaneously, which is en-couraging for researchers desiring to take a less empiricalapproach in their calculations. In addition, the nature ofthe renormalization procedure in ACBN0 will reduce themagnitude of U corrections when the KS states are notwell-represented by a basis of localized orbitals. Build-ing on the arguments put forward by other studies thatutilize a U pp correction, we believe the results presentedin this paper make a strong case for the incorporation ofHubbard correction terms on oxygen 2 p states in manytransition metal oxides, when calculated self-consistentlyfrom first principles. V. CONCLUSIONS
This work has demonstrated that ACBN0 improves thedescription of the first row transition metal perovskitescompared with PBE and a na¨ıve choice of U taken fromthe literature. ACBN0 also compares favorably with thehybrid functional HSE, offering improved descriptions ofband gaps vs. HSE-25 and performing competitivelyto an empirically optimized HSE-Opt for both structureand to a lesser degree, band gap, from completely first-principles values of U that directly depend on the chargedensity.Simply choosing a value of U from the literature isinsufficient when trying to obtain an overall picture ofmaterial properties. In addition, values of U can varyacross functionals, approaches to calculating U , and im-plementation of the DFT+ U method itself, leading toresults at odds with other published calculations in theliterature. We have also demonstrated the importance ofexplicit U and J values in some orbitally ordered mate-rials, which can also be easily performed with ACBN0.Overall, there still remains potential room for improve-ment in using and verifying ACBN0 that is mainly lim-ited by currently-available implementation in software.The use of unique values of U for specific subsets of or-bitals such as e g and t g may yet offer improved descrip-tions of materials such as LaMnO , in addition to thenecessity of using explicit U and J . ACBN0 should alsobe applicable to the DFT+U+V approach that offersimproved descriptions of covalent materials. If these de-velopments are fruitful, this method holds promise notonly in high-throughput applications but also in treatinga wide variety of complex materials with first-principlessite-specific U values at a reasonable computational cost. ACKNOWLEDGMENTS
We wish to thank Prof. Marco Buongiorno Nardelli(University of North Texas) for sharing an early ACBN0implementation, as well as Dr. Priya Gopal (University3of North Texas) and Dr. Andrew Supka (Central Michi-gan University) for helpful discussion. This work madeuse of computational resources from the National En-ergy Research Scientific Computing Center (NERSC), aDOE Office of Science User Facility supported by the Of-fice of Science of the U.S. Department of Energy underContract No. DE-AC02-05CH11231, as well as compu- tational resources from the Texas Advanced ComputingCenter (TACC) at The University of Texas at Austin.Financial support was received from the Skoltech-MITCenter for Electrochemical Energy Storage. 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TABLE S-I. ACBN0 calculations of U , J and U eff. = U − J forV 3 d and O 2 p in LaVO . Average values for energetically-competitive phases are bolded.LaVO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureV V O V V ONM U J U eff. U J U eff. U J U eff. U J U eff. U J U eff. AVG U eff. TABLE S-II. ACBN0 calculations of U , J and U eff. = U − J forCr 3 d and O 2 p in LaCrO . Average values for energetically-competitive phases are bolded.LaCrO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureCr Cr O Cr Cr ONM U J U eff. U J U eff. U J U eff. U J U eff. U J U eff. AVG U eff. TABLE S-III. ACBN0 calculations of U , J and U eff. = U − J for Mn 3 d and O 2 p in LaMnO . Average values forenergetically-competitive phases are bolded.LaMnO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureMn Mn O Mn Mn ONM U J U eff. U J U eff. U J U eff. U J U eff. U J U eff. AVG U eff. TABLE S-IV. ACBN0 calculations of U , J and U eff. = U − J for Fe 3 d and O 2 p in LaFeO . Average values forenergetically-competitive phases are bolded.LaFeO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureFe Fe O Fe Fe ONM U J U eff. U J U eff. U J U eff. U J U eff. U J U eff. AVG U eff. TABLE S-V. ACBN0 calculations of U , J and U eff. = U − J forCo 3 d and O 2 p in LaCoO . Average values for energetically-competitive phases are bolded.LaCoO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureCo O Co ONM U J U eff. U J U eff. AVG U eff. TABLE S-VI. ACBN0 calculations of U , J and U eff. = U − J for Ni 3 d and O 2 p in LaNiO . Average values forenergetically-competitive phases are bolded.LaNiO ACBN0 U Values (eV)MagneticState Quantity ExperimentalStructure OptimizedStructureNi O Ni ONM U J U eff. U J U eff. AVG U eff. TABLE S-VII. Values of U taken from the literature (”Lit. U ”). Cluster-CI refers to cluster configuration interaction calcula-tions, and those U values are an effective term given by U eff = U − J . Values used in this work are bolded. In LaCoO twovalues were used in separate calculations.B-site cation Literature UU eff (eV) Method Underlying XC Functional Code UsedV 2.49 cRPA ( t g - t g model) S90
PBE (GGA) VASP
Cluster-CI
S88
N/A N/A3.0 Fit to E gS89 LDA Not available3.16 cRPA ( t g - t g , p model) S90
PBE (GGA) VASP3.6 Fit to optical absorption spectra
S149
PBE (GGA) VASP3.85 Fit to E gS150 PBE (GGA) The ELK Code (FP-LAPW method)6.76 Constrained DFT
S88
LSDA Not available (LMTO-ASA method)Cr 1.97 cRPA ( t g - t g model) S90
PBE (GGA) VASP2.66 cRPA ( t g - t g , p model) S90
PBE (GGA) VASP3.5 Fit to Cr O enthalpy of formation S93
PBE (GGA) VASP3.8 Fit to HSE calculation
S94
PBEsol (GGA) VASP
Cluster-CI
S88
N/A N/A4.5 Fit to E g and µ S95
LSDA Not available (LMTO-ASA method)6.96 Constrained DFT
S88
LSDA Not available (LMTO-ASA method)Mn 3.3 cRPA ( d - dp model) S97
PBE (GGA) openMX4.0 Fit to enthalpy of formation
S151
PW91 (GGA) VASP5.0 Fit to E g and µ S95
LSDA Not available (LMTO-ASA method)
Cluster-CI
S88
N/A N/A7.1 Constrained DFT
S88
LSDA Not available (LMTO-ASA method)8.0 Fit to E gS152 LSDA Not availableFe 3.7 cRPA ( d - dp model) S153
PBE (GGA) WIEN2K (FP-LAPW method)4.0 Fit to enthalpy of formation
S151
PW91 (GGA) VASP4.0 Fit to E gS154 PBE (GGA) QUANTUM ESPRESSO
Cluster-CI
S88
N/A N/A5.1 Fit to HSE calculation
S94
PBEsol (GGA) VASP5.4 Fit to E g and µ S95
LSDA Not available (LMTO-ASA method)7.43 Constrained DFT
S88
LSDA Not available (LMTO-ASA method)Co 3.3 Fit to enthalpy of formation
S151
PW91 (GGA) VASP4.0 Unrestricted Hartree-Fock
S134
N/A GAMESS
Cluster-CI
S88
N/A N/A5.85 Fit to E g and µ S95
LDA Not available (LMTO-ASA method)6.9 Constrained DFT
S155
LDA WIEN2K, ELK (FP-LAPW method)6.96 Constrained DFT
S88
LDA Not available (LMTO-ASA method)
Linear response
S52
LDA QUANTUM ESPRESSONi 1.1 cRPA ( e g - e g model) S156
PBE (GGA) VASP5.64 Linear response S9 LSDA QUANTUMESPRESSO, VASP
Cluster-CI
S88
N/A N/A6.35 Fit to E g and µ S95
LSDA Not available (LMTO-ASA method)6.4 Fit to enthalpy of formation
S151
PW91 (GGA) VASP7.57 Constrained DFT
S88
LSDA Not available (LMTO-ASA method)
FIG. S-1. Pseudopotential convergence tests; total energy per atom vs. (a) plane wave cutoff, (b) k-point mesh, (c) scfconvergence threshold; total force per atom vs. (d) plane wave cutoff, (e) k-point mesh, (f) scf convergence threshold; totalcell pressure vs. (g) plane wave cutoff, (h) k-point mesh, (i) scf convergence threshold. The quantity ”k mesh” n refers to thedimensions of the k-point mesh, which is approximately n × n × . n for orthorhombic cells and n × n × n for rhombohedralcells. All quantities are referenced to a well-converged calculation with two of the three paramters fixed at 250 Ry plane wavecutoff, 9 × × − Ry.
TZ SRUX �� YXA Z DMHL B P FZP BQ � FIG. S-2. Paths through k-space for generating the bandstructures presented in the main text; (a)
P bnm ; (b) P /b ;(c) R ¯3 c . Images generated using XCrySDen S157 . P DO S ( a r b . un it s ) ACBN0 - Effective U - J P DO S ( a r b . un it s ) ACBN0 - Explicit U & J
2 4 6 8 E (eV)012345 P DO S ( a r b . un it s ) E AFM-A - E FM = 30 meV E AFM-A - E FM = 0.2 meVExplicit U & J - U = 6.0 eV, J = 2.0 eV E AFM-A - E FM = -6.3 meV FIG. S-3. Projected density of states for LaMnO ; (a) ACBN0with U eff = U − J from a simplified DFT+ U scheme (see tableS-III); (b) the same values of U and J but applied explicitly ina generalized rotationally-invariant DFT+ U implementation;(c) the same calculation as panel (b) but with U and J onMn increased to 6.0 and 2.0 eV, respectively. FIG. S-4. Total density of states for LaVO (dotted lines),and with Gaussian broadening applied (solid lines, 0.6 eV),comparing PBE, PBE+ U dd , and ACBN0 (PBE+ U dd + U pp ).For comparison is experimental spectra (solid lines, top offigure) from Chen et al. S158