DFT-GIPAW 27Al NMR Simulations for Intermetallics: Accuracy Issues

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

DFT-GIPAW 27Al NMR Simulations for Intermetallics: Accuracy Issues and Magnetic Screening Mechanisms Ary R. Ferreira* Department of Physics, Universidade Federal de São Carlos (UFSCar), 13565-905, São Carlos, SP, Brazil

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S Supporting Information *

ABSTRACT: This study reports first-principles simulations of 27Al solid-state nuclear magnetic resonance (ssNMR) shifts (δexp iso ) carried out for a set of seven Al-containing intermetallic compounds. With the aim of assessing accuracy issues in such calculations using the gaugeincluding projector augmented waves method, the referred set was sought to cover the wide range of experimental 27Al δexp iso shifts reported in the literature for this type of compound in a representative way (from about −200 to 1600 ppm). From a technical/computational perspective, the findings allowed the detection of critical sources of inaccuracy and they were addressed to different approximations inherent in the computational methods. Among those, it is possible to highlight the formulation of the exchange-correlation energy functional in density functional theory, which at least for the 27Al nuclide has proved to be more critical than the frozen-core approximation, especially in the case of weakly magnetic aluminides. From a practical point of view, particularly regarding a possible use of such simulations for a more efficient interpretation of experimental ssNMR spectra, key physical insights into the magnetic screening mechanisms underlying the 27Al δexp iso shifts are reported, in which theory supports that in addition to the contributions of closed orbitals, those coming from unpaired electronic spins arise from two main competing hyperfine interactions: the first one with local magnetic moments at specific sites, resulting in a shielding effect and even negative shifts, and the second mechanism in which large Knight shifts to downfield gradually dominate in those compounds having a rather metallic character.

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INTRODUCTION First-principles density functional theory1,2 (DFT)-based computations of solid-state nuclear magnetic resonance (ssNMR) shifts can be considered routine for diamagnetic systems. It is nothing new that their aid may be crucial for unambiguous peak assignments in cases complicated by structural intricacies like static disorder, mixture of phases, or chemical exchange processes, for which plane-wave (PW)based methods have been the common choice, with highlight to the gauge-including projector augmented waves (GIPAW)3,4 approach. Note that its use is normally based on the premise of saving computational time, as in the pseudopotential (PP) approach.5,6 In that context, however, the predictive power of DFT faces major challenges when it comes to systems in which, in addition to possible structural issues, another complicating factor makes the ssNMR spectra difficult to interpret. In particular, it refers to a combination of magnetic screening mechanisms (MSMs) resulting from different aspects of the electron charge and spin densities around target nuclei. So, in addition to the contributions of closed orbitals to the observed exp ), those coming from experimental isotropic shifts (δiso unpaired electronic spins should also be brought into the calculation. Examples of an important class of systems for which such combination of MSMs cannot be ruled out are © XXXX American Chemical Society

paramagnetic transition-metal (TM) oxides and phosphates, on which intensive research has been carried out in the last 15 years,7−14 making use of different methods and codes to simulate their paramagnetic ssNMR (pNMR) shifts for a series of nuclides. Regarding the nature of the isotropic chemical shifts as well as the level of complexity of current strategies13,14 to compute their theoretical counterparts (δtheo iso ) for extended solids, it can be said that weakly magnetic intermetallics lie between diamagnetic and paramagnetic, insulating materials. Since although weakly magnetic, it is well known that in the presence of an external magnetic field (Bext), the conduction electrons are susceptible to a breaking of spin degeneracy, which, in simple metals, can result in large isotropic shifts to downfield due to a deshielding effect usually termed Knight shifts.15 Intermetallics are crystalline chemical systems whose wide diversity in terms of atomic arrangement and composition is increasingly revealed by modern characterization techniques. Naturally, such scenario raises expectations about their use as new functional materials in basic research and industry.16 Received: January 9, 2019 Revised: March 5, 2019 Published: March 14, 2019 A

DOI: 10.1021/acs.jpcc.9b00259 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Among the diverse range of possibilities forming this class of compounds, Al-containing intermetallics are of great interest, not only for structural, as well as oxidation-, corrosion-, and sulfidation-resistant applications,17 but also for more advanced purposes when it comes to magnetic systems, due to properties like those related to magnetic shape memory and magnetocaloric effects.18 Moreover, given the widely assorted crystal chemistry of Al-containing intermetallics, any experimental procedure able to provide atomic-level information complementary to usual X-ray diffraction is of significant relevance for their development. Within that scope, the combination of 27Al ssNMR experiments with electronic structure simulations has been highlighted as a promising and useful approach for crystal structure validation.19 In addition, such broaching can deliver valuable source of information regarding the chemical environments of NMR target nuclei. Although the GIPAW method has been used for nearly a decade to predict experimental chemical shifts for diamagnetic insulating materials, its application to intermetallic compounds is recent, and in a pioneering work focused on ScT2Al Heusler phases,20 it has been shown that a DFT-GIPAW-based protocol can yield satisfactory theoretical 27Al chemical shifts. Motivated by a need to further assess the performance of that protocol, the present work reports its application to a set of seven Al-containing intermetallics selected from ref 19. The set was defined to ensure its representativeness not only in terms of the crystal chemistry of the compounds, but also in terms of their respective reported 27Al δexp iso shifts (ranging from about −200 to 1600 ppm). Five of the seven intermetallics are the aluminides ZrAl3, ScAl3, TiAl3, TiAl, and VAl3, and this collection was complemented with two other intermetallic compounds having a more differentiated crystal chemistry, namely, AlB2 and CuAl2. The theoretical DFT results were primarily compared to experiments reported in the literature, which has enabled the detection of some critical inaccuracies that were addressed to different approximations inherent to the computational methods. Finally, further analyses of the theoretical findings provided key insights into the respective MSMs.

are expanded as linear combinations of atomic-like functions, while different schemes ensure a smooth connection with the PW representation in the interstices. In a distinctive formulation, the alternative pseudopotential (PP) method5,6 was grounded in the assumption that, in most of cases, only valence electrons are chemically relevant so that core states are not taken variational in the solid-state calculation. These states are actually obtained from atomic DFT simulations and are supposed to be transferable among different crystal systems. Thus, in PP-based simulations, the nuclei assume the role of nonpolarizable ionic centers in the termed frozen-core approximation (FCA), and in this way, the effect of core states on the variational pseudovalence Bloch states |Ψ̃n,k⟩, expanded with a finite set of PWs, is recovered through an adaptation of the system’s Hamiltonian, with n the band index and k a point in the reciprocal space. The avoidance of the |Ψ̃n,k⟩’s nodal structure near the nuclei limits the range of materials properties that are quantifiable with the PP method, with the ssNMR spectral parameters counting among those nonaccessible observables. The remedy came with the projector augmented waves (PAW) formalism25 (a generalization of the PP and LAPW methods) and the later introduction of the GIPAW method.3,4 Emphasizing that both PAW and GIPAW approaches were developed within the FCA, it has recently been shown that it is possible to compute 27Al spectral parameters within that approximation for intermetallic systems.20 This includes the distinctive contributions of closed orbitals and of metallic unpaired spins to the δtheo iso shifts, while mentioning that pioneering analogous simulations with the FPLAPW method have been previously reported.26 The reconstruction of the all-electron (AE) wave functions provided by the PAW method requires a series of quantities from atomic DFT simulations, such as the KS orbitals |ϕi⟩ and their pseudocounterparts |ϕ̃ i⟩ with respective dual projectors | p̃i⟩ constructed so that ⟨p̃i|ϕ̃ j⟩ = δij, among others. There are different schemes to generate this set of atomic data for each chemical element, which are made available by code developers and expert users as formatted files known as PAW data sets. Here, it is important to point out that, in addition to the selected XC DFT functional and the adoption of the FCA, the choice of a given collection of PAW data sets is another potential source of inaccuracy that shall not be overlooked, especially in the case of properties that are essentially local, such as ssNMR spectral parameters. It should also be recalled that in an ssNMR experiment with a metallic weakly magnetic material, the break of spin degeneracy comes essentially from the conduction electrons interacting with Bext as spinning charges. In the PAW simulations performed for the present work, this backdrop is mimicked by constraining the ground-state spin-polarized electronic structure to converge with an equivalent total magnetization (ms), which is an input of the calculation. This is nothing but an approach to emulate the presence of Bext and not an explicit incorporation of it in the simulation using a periodic supercell, which has already been proposed in the literature27,28 and is a matter of current research. The point is that this is one more approximation that will be assessed later on in this report. From the spin-polarized ground-state electronic structure, the electron densities for each spin channel ξ (↑ or ↓) are computed as a sum of valence and core contributions

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THEORY AND COMPUTATIONAL DETAILS Methods and Review of Approximations. In DFT simulations, in addition to the selection of a specific formulation of the exchange-correlation (XC) energy functional, another critical decision is on how to expand the Kohn−Sham (KS) orbitals. In the case of electrons in a crystal, the essential idea of partitioning the volume of the unit cell into two types of regions was fundamental for the development of the most commonly used methods. These two portions of the real space are: the set of spheres with radii rcut centered at each atomic site and the remaining interstitial region. The nearly free-electron nature of the valence states in the latter allows the use of a plane-wave (PW) basis set, which brings significant numerical advantages to the simulations. This is a common aspect of the main approaches proposed so far, and the task of dealing with the atomic-like nature of the core and semicore states in the vicinity of the atoms (spheres with rcut) marks their differences. One of these approaches is the full-potential (FP) linear augmented PWs (LAPWs) method21,22 and its variants,23,24 in which all of the electrons are explicitly included in the calculations. The variational states inside the spheres with rcut B


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The Journal of Physical Chemistry C ρξ (R) = ρ ξ̃ (R) +

∑ [ρ1ξ (ri) − ρ1̃ ξ (ri) + ρcξ (ri)] i

A last source of inaccuracy to be taken into account concerns the computation of the theoretical isotropic metallic shifts as

(1)

where ρ̃ξ(R) is the so-called bare contribution, which is Fourier transformed from reciprocal space and depends only on the variational valence |Ψ̃n,k⟩ ground states. The three terms in the average sum, ρξ1(ri), ρ̃ξ1(ri), and ρcξ(ri), are the two PAW one-center reconstruction terms25 and the core contribution, respectively. The latter is calculated using a perturbative approach, well is described in ref 29, and these three terms are computed for each point ri in the PAW data set radial grid. Given the relativistic origin of the Fermi contact interaction30 and the fact that all of the PAW data sets used in the present work are scalar-relativistic, a generalization of the relativistic contact interaction can be made by averaging ρξ(R) within a sphere centered at R, whose radius is the Thomson radius RT = Zre,31 with Z the atomic number and re the classical electron radius (RT = 36.63 fm is used for Al). In practice, this is achieved by limiting the set of points ri in the average sum of eq 1.32 Hereafter, such spin densities, computed with that relativistic approximation, will be labeled ρ̅s. Nonetheless, the PAW data set radial grids are actually adjustable parameters, which may not always be compatible with that approach. Hence, an alternative strategy to deal with this is to replace ρ̅s by the value of the innermost point in the PAW data set radial grid,32 and it is going to be labeled ρs. For comparison purposes, both are taken into account in the present work, and along the text, all quantities derived from ρ̅s and ρs will be denoted with and without an overbar, respectively. The matter at issue is that the arbitrary choice between using ρ̅s and ρs must be regarded as another eventual source of inaccuracy in such simulations. Continuing with the review of approximations and the listing of associated sources of inaccuracy, it has recently been shown that for paramagnetic, insulating solids,14 spin−orbit coupling (SOC) has to be taken into account in certain cases, since it introduces effects such as the g-shift from the free-electron gfactor, the relativistic part of the hyperfine tensor, and a contribution to the zero-field splitting, which allows a reasonable quantification of a complete set of isotropic contributions to δtheo iso , which are grouped in the termed Fermi contact and pseudocontact shifts. Note that the theoretical formalism used for the computation of those pNMR shifts is also a matter of current research. The above-mentioned SOC effects are neglected in this work for simple reasons. First of all, it simplifies and makes the calculations feasible. The application of the approaches described in refs 13 and 14 to metallic systems is something to be further explored. Moreover, besides being diamagnetic in the absence of Bext, the induced local magnetic moments on transition-metal (TM) sites computed for the seven Alcontaining intermetallics studied in this work (see further below) are small, about 2 orders of magnitude lower than those already reported for insulating, paramagnetic olivine-type LiTMPO4 compounds with essentially ionic bonds.13 Thus, one has to bear in mind that the choice of neglecting SOC makes the Fermi contact term the only isotropic contribution to the computed δtheo iso shifts. In the present work, instead of trying to assign possible disagreements between experimental and theoretical shifts to SOC effects, the focus will be on more elementary approximations.

exp theo δiso = σref − (σo + σs) + δref

(2)

with σo the isotropic orbital contribution given by the trace of the orbital contribution to the total chemical shielding tensor, Tr[σo/3], and σs, as aforementioned, computed from the isotropic part of the Fermi contact tensor.20 Since their experimental counterparts are not absolute quantities, to draw comparisons, the usual procedure is to select a suitable zeroshift diamagnetic compound for which σs is null. Its nonzero σo contribution is labeled as σref in eq 2, with σs and σo the spin and orbital contributions computed for the system under study, respectively. The term δexp ref is the experimental chemical shift of that reference compound with respect to a second reference, equivalent to the one used to report δexp iso . For instance, it is possible to mention aluminum chloride, AlCl3, in heavy water or aqueous aluminum nitrate, Al(NO3)3. Technical Details. All of the PAW and GIPAW results were obtained from electronic structure simulations performed using the Quantum ESPRESSO33,34 (QE) open-source software suite version 6.2. As commented in ref 20, small adaptations to that code were necessary to achieve the results following eq 2. Recalling that σs is approximated by the Fermi contact contribution to the induced magnetic hyperfine field at the nuclear position R, its calculation requires the respective spin densities given by the difference ρs(R) = ρ↑(R) − ρ↓(R). A detailed description of how the values of σs components are computed from ρs(R) with the PAW method can also be found in ref 20. Regarding the mimicking of Bext, all of the spinpolarized calculations have been carried out with ms = 0.1 × 10−3 μB. To assess the consistency of PAW simulations, a set of results obtained with an LAPW basis plus local orbitals (LAPW + lo), as implemented in the Elk open-source code35 version 4.3.06, were used. These simulations were taken as a standard for the numerical accuracy of the computed values of σs and it is worth mentioning that the strategy to mimic the presence of Bext with the QE code, introduced in the last section, is different when using the Elk code. In its implementation, the KS equations are solved in a two-step process, in which the ground-state electronic structure is primarily achieved in a firstvariational step with only scalar potentials in the Hamiltonian. Then, Bext (9.5 T in this work) is added to it in a secondvariational step and a corresponding break of the spin degeneracy of the conduction electrons is expected to be achieved. Therefore, it is reasonable to say that the FP-LAPW + lo σs values computed with Elk can ensure the consistency of the QE computed counterparts from two perspectives: the efficiency of the PAW data sets used and the successful mimicking of Bext. All calculations were carried out with the experimental crystal structures of all compounds.36−40 The numerical convergence of target properties with respect to the PWs basis set was investigated, and a value of 90 Ry was chosen for the kinetic energy cutoff (Ecut) in all systems, with a value of 8 times Ecut set for the kinetic energy cutoff for charge density and potential. Regarding the set of k-points in the first Brillouin zone, they were determined by the Monkhorst−Pack procedure41 and all of the samplings required for each simulation cell are available in Table S1. C


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The Journal of Physical Chemistry C

of the arbitrary choice on whether using ρ̅s or ρs to compute the chemical shifts. Moreover, since some extra calculations have been performed with an alternative PBE PAW data set for Al (see Supporting Information), that third source of inaccuracy related to the FCA is also addressed in that first set of results. Before discussing in detail the associated accuracy issues, it will be assumed that the δexp iso values reported in ref 19 are accurate and precise. Moreover, the wide spectral window in their range can always be taken into account in qualitative analyses. Nevertheless, a first glance at Figure 1 suggests that at least the LDA XC functional yields a substantial overall quantitative agreement between experiment and theory. So, for some reason, its well-known tendency to overbinding is beneficial for the prediction of δexp iso in these systems, with the exception of aluminum diboride, AlB2, whose results will be discussed later on. Regarding the performance of the GGA XC functionals, the Ti-containing aluminides reveal themselves as the most challenging system. However, a relevant remark in this context is that LDA has actually systematically overestimated the δtheo iso values computed with the GGAs, to different extents, for all five aluminides, which simply does not occur in the case of the other two intermetallics, AlB2 and CuAl2. Here, it is necessary to note that all of these first results were obtained at experimental structures and the effect of structural relaxation on δtheo iso will be discussed in the next section. With respect to the other two arbitrary options related to approximations inherent in the calculations, the results in Figure 1 provide a first argument to discard any issue associated with the FCA or the quality of the PAW data sets used. It has already been mentioned that inside the FCA, there are different ways to create them and an alternative PBE PAW data set for Al (see the Supporting Information) was used in the extra simulations carried out for TiAl and AlB2, whose respective δtheo iso values are also shown in Figure 1. So, without further discussion regarding that issue, it is clear that its impact was not as significant as those found when choosing different XC functionals. With regard to the use of ρ̅s or ρs to compute the chemical shifts, one can see that this arbitrary choice results in distinct effects on δtheo iso at the ends of the spectrum. As can be seen in the upfield range from about −200 to 200 ppm, comprising the theo shifts computed for TiAl and VAl3, it is notable that δ̅theo iso > δiso and that this difference decreases gradually moving to downfield, with virtually no impact for TiAl3 and ZrAl3 theo (from 200 to 700 ppm) and the inversion δ̅theo iso < δiso for ScAl3, AlB2, and CuAl2 (above 700 ppm). Notwithstanding these remarks, these results suggest that δ̅theo iso systematically reproduces better δexp iso . As commented in the last section, the FP-LAPW + lo σs values computed with Elk can be used to ensure the consistency of the QE computed counterparts from the perspectives of two distinct approximations: the FCA, as provided by the PAW method, and the strategy to mimic the influence of Bext on the electronic structure of the system. Note that it is a common assumption that AE FP-LAPW + lo calculations can provide numerical accuracy within a given XC functional for many computable quantities.52 The correlation found for the intermetallic compounds studied in the present work is depicted in Figure 2 and one can see that there is a rather good agreement between the computed σ̅ PAW and σ̅ LAPW . s s

The collection of PAW data sets provided by Dal Corso42 as the PSLibrary project version 1.0.0 was taken as standard. From that, those generated from scalar-relativistic atomic DFT simulations were selected, within the Perdew−Burke− Ernzerhof (PBE)43 generalized gradient approximation (GGA) for the XC functional. That generation procedure was performed using the atomic code, which is part of the QE package, following the formal relationship between ultrasoft PPs and the PAW method described in ref 44. For the sake of comparison, alternative PAW data sets were created from the same generation configurations but with other three different XC functionals, namely, the GGAs PW9145 and PBEsol46 and the local density approximation (LDA).47,48 In addition, as will be further discussed, extra simulations were performed for titanium aluminide, TiAl, using the PBE(β,μ,λ) family of nonempirically parametrized functionals.49 The FP-LAPW + lo results were all obtained with the default configurations of core electrons and local orbitals defined in Elk version 4.3.06. All of the XC functionals mentioned above are also implemented in this code with exception of PW91, for which it was necessary to resort to an external implementation available in the LIBXC50 library version 2.2.3. Finally, to compute the δtheo iso values according to eq 2, following refs 20, 26, and 51, aluminum phosphate, AlPO4, was selected as the reference compound, by adopting its experimental shift with respect to AlCl3 in heavy water δexp ref = 45 ppm.

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RESULTS AND DISCUSSION Performance of Standard XC Functionals. A comparison between experimental and theoretical 27Al isotropic chemical shifts computed for the seven intermetallic compounds is summarized in Figure 1, and a detailed list of all related quantities is available in Tables S2−S4. The four most popular XC functionals for solids have been used to compose this set of data and, in addition to that approximation, the results in Figure 1 also reveal the effect

Figure 1. Comparison between GIPAW theoretical and experimental 27 Al isotropic magnetic shifts in the intermetallic compounds at the respective experimental volumes. The black dashed line through the data represents a hypothetical perfect agreement. The symbols ●, ■, ◆, and ▲ stand for simulations with PBE, PW91, LDA, and PBEsol XC functionals, respectively. The filled and shaded symbols stand for theo δtheo iso and δ̅iso values, respectively. Additional calculations with an alternative PBE PAW data set for Al (see the Supporting Information) are represented by an asterisk (*) and a plus symbol (+), which stand theo for δtheo iso and δ̅iso values, respectively. D


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Table 1. Local Magnetic Moments at Specific Transition Metals and Aluminum Sites (mTM and mAl, Respectively, in 10−2 μB) Induced by Bext in the Structures of the Five Aluminides Simulated with Different XC Functionals PBE

LDA

intermetallic

mTM

mAl

mTM

mAl

ScAl3 ZrAl3 TiAl3 VAl3 TiAl

0.47 0.08 1.39 2.74 1.56

0.04 0.02 0.01 −0.02 0.00

0.47 0.08 1.33 2.56 1.52

0.05 0.02 0.02 −0.01 0.01

correlation can be understood if one takes into account that the contributions of σo and σs to δtheo iso are quite balanced in those compounds for which the shifts are small or negative (see Tables S2 and S3). Additionally, from the respective projected density of states (PDOS) plots in Figures S1 and S2, the relatively low PDOS with s-character at the Fermi level (EF) creates room for a competition between the assumed MSMs related to the Fermi contact interaction (pNMR vs Knight shifts), recalling also that those computed mTM values are about 2 orders of magnitude lower than those already reported for insulating, paramagnetic compounds in another work reporting DFT simulated pNMR shifts.13 Regarding the delocalization errors mentioned above, there are two most usual approaches to tackle them in DFT PWbased calculations. One is based on the Hubbard U correction (DFT + U) for density localization, while the other is the use of the so-called hybrid functionals. Note that in this latter, a fraction of the exact Hartree−Fock exchange energy is mixed with the DFT counterpart so that localization and delocalized errors cancel out to some extent. So, driven by the abovementioned trends in recent papers reporting pNMR simulations,10,13,14 the results that follow represent a first attempt to verify whether the success of LDA in predicting the 27 Al δexp iso shifts is nothing but a fortuitous outcome. In that effort, the performances of both approaches DFT + U and hybrid functionals have been assessed on top of the PBE XC functional, with a focus on the simulation of the σ̅ s component for the most critical case among the aluminides, TiAl. With respect to the DFT + U calculations, its simplified version proposed by Cococcioni and de Gironcoli56 was used, within which a first value for the Hubbard parameter (UTi‑d = 2.78 eV) was found for Ti d states following the linear response procedure also described in ref 56. Further details are available in the Supporting Information (see Figure S3). That first strategy resulted in mTi = 1.91 × 10−2 μB and σ̅ s = 2340.87 ppm, which are very high values compared to those obtained with PBE (see Tables 1 and S2). Additionally, it is worth mentioning that the respective structural relaxation resulted in a relative error in volume of 3.83%, mTi = 1.93 × 10−2 μB, and σ̅ s = 2367.02 ppm. Moreover, as can be seen in Figure S4d and as expected, the effect of the Hubbard correction on the PDOS around EF is the opening of the existing pseudogap by localizing the Ti d states, with a very small side effect observed on Ti semicore states (see Figure S4c) compared to the PBE analogous PDOS plots in Figure S4a. The nonrecommended empirical procedure of tuning the Hubbard parameter, in such a way to have the desired properties, has also been assessed. As shown in Figure S5, a value of UTi‑d = 5.05 eV was selected to yield σ̅ s = 608.48 ppm and mTi = 1.53 × 10−2 μB. However, after structural relaxation

Figure 2. Comparison between σ̅ s computed with the PAW and FPLAPW + lo methods for the 27Al nuclei in the intermetallic compounds at the respective experimental volumes. The black dashed line through the data represents a hypothetical perfect agreement. The symbols ●, ■, ◆, and ▲ stand for simulations with PBE, PW91, LDA, and PBEsol XC functionals, respectively. Additional calculations with an alternative PBE PAW data set for Al (see the Supporting Information) are represented by a plus symbol (+). In the inset, the tick labels are shown only in the top and left-hand-side axes with the same units of the outset.

In this way, at least for that property, it is possible to rule out, in principle, the impact of these two approximations. Beyond Standard GGA XC Functionals. The discussion of results made in the last section leads to a more thorough analysis of the effect of density functional approximations on the computed δtheo iso shifts. Despite the well-known success of LDA for metallic solids with delocalized electrons,53 it is appropriate to be cautious about assumptions regarding the suitability of XC functionals; therefore, tests are always welcome, especially in the case of intermetallic compounds, whose crystal chemistry may differ considerably from that of elemental metals, as evidenced by their own 27Al δexp iso shifts reported in ref 19. Delocalization errors are inherent to the XC approximated functionals used in DFT calculations,54 and they are commonly taken into account in pNMR shifts simulations.10,13,14 Here, it is quite pertinent to mention the remark made by Middlemiss et al.10 about the term “paramagnetic NMR” (pNMR). Within the scope of the present study, it refers to the particular situation in which local magnetic moments at nearby lattice sites, resulting from unpaired electrons, interact with nuclear magnetic moment at NMR target nuclei sites. So, it should not be confused with the paramagnetic contribution to the nuclear shielding tensor, which is accounted in the σo component (see eq 2). Moreover, grounded on NMR experiments reported in the work by Lue et al.,55 it is reasonable to state that the small or negative 27Al δexp iso shifts observed for aluminides result from an MSM that is a combination of pNMR and Knight shifts, in addition to the orbital contribution accounted in σo. The induced local magnetic moments at TM sites (mTM) computed with standard XC functionals for the five aluminides studied in this work are listed in Table 1. Although they are not PAW reconstructed quantities, it is possible to see that, indeed, they corroborate the MSM referred above. Despite a lack of linearity, increased mTM values exist in those systems for which δtheo iso tend to be small or negative. Naturally, such rough E


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The Journal of Physical Chemistry C Table 2. Properties Computed for TiAl Using LDA and the PBE(β,μ,λ) Family of Nonempirical Reparametrized XC Functionalsa XC functional

volume (Å3)

error (%)

LDA PBE(Gc,Gx,EL) PBE(Js,Gx,EL) PBE(Jr,Gx,EL) PBE(Gc,Gx,LO) PBEsol PBE(Js,Jr,EL) PBE(Jr,Gx,LO) PBE(Js,Jr,LO) PBE(Gc,Jr,EL) PBE

30.71 31.37 31.42 31.48 31.62 31.67 31.66 31.73 31.99 32.02 32.58

−5.01 −3.00 −2.82 −2.65 −2.21 −2.04 −2.09 −1.88 −1.05 −0.97 0.75

σ̅ s 592.37 1018.01 989.04 977.90 1068.49 1034.66 1023.43 1022.66 1083.25 1128.67 1232.28

σo

δ̅theo iso

mTi

108.09 145.65 133.39 128.21 151.56 139.68 130.11 134.70 138.13 132.11 142.92

−155.47 −609.01 −568.71 −552.82 −662.81 −618.08 −599.27 −601.53 −663.89 −705.19 −814.99

1.52 1.53 1.53 1.54 1.53 1.53 1.54 1.54 1.54 1.55 1.56

The volumes and respective errors listed refer to the resulting structural relaxation. Local magnetic moments at Ti sites (mTi) are given in 10−2 μB, and all shifts and shieldings are in ppm and were calculated at the experimental volume equal to 32.33 Å3 reported in ref 39.

a

Figure 3. Properties computed for TiAl as listed in Tables 2 and S6. (a) Plot of errors resulting from structural relaxation of the TiAl experimental cell taken from ref 39 vs the mean relative errors (mre) reported for lattice constants of a test set of 60 solids in ref 49. To identify the XC functionals corresponding to each point plotted in (b), (c), and (d), the reader should refer to Table 2, in which data are sorted by the relaxed TiAl volume. The filled and shaded circles correspond to properties computed at experimental and relaxed volumes, respectively.

equivalent PBE + UTi‑d = 2.78 eV simulation: the opening of the pseudogap by localizing Ti d states below EF. This is an expected convergence of results; however, a critical difference between PBE0 and HSE PDOS is that a severe suppression of the PDOS at EF is notable for the former, whereas it is less serious for later. This is an immediate implication of the inclusion of a fraction of the Hartree−Fock exchange energy, and these findings are quite in line with a recent report59 that demonstrates how hybrid functionals may compromise metallic properties that are sensitive to (or determined by) the electronic structure near EF, such as σ̅ s. It is notable in Figure 1 that among the standard GGA functionals taken into account so far, PBEsol has systematically approximated to the LDA results, yielding better (or less bad) agreement with experiments in the case of the two Ticontaining aluminides, TiAl and TiAl3. Moreover, the PBEsol computed shifts lie in between the LDA and PBE counterparts for the five aluminides, a trend that has also already been reported for lattice constants of a test set of 60 solids.49 The PBEsol functional is a nonempirical reparametrized version of

with that UTi‑d value, it has been found a huge error in volume of 12.04% and σ̅ s = 1761.67 ppm. The PDOS of that simulation carried out at the experimental volume is also available in Figure S4c,d, in which one can verify that it follows the same trends found with UTi‑d = 2.78 eV, but to a higher extent. Turning to the attempts of using hybrids, they have been carried out with the bare mixing of PBE057 and the Heyd− Scuseria−Ernzerhof (HSE)58 screened version. The referred results listed in Table S5 confirm that there is a good convergence of the total energy and of the bare contribution to spin density, ρbare (see ref 20), with respect to k-points. However, the same is not true for ρ̅GIPAW and thereby neither for σ̅ s. Since these simulations with so many k-points are computationally very demanding, these converge tests have not been continued. In any event, it is quite likely that σ̅ s will not converge to a value close to the one obtained with LDA and therefore result in a better prediction of the experimental shifts. Additionally, it can be seen in Figure S4f that the main impact on TiAl PDOS around EF follows the one found for the F


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is notable that isolated changes in β and λ have improved δ̅theo iso in a similar way, whereas changing only the μ parameter, which also appears in the exchange functional, has yielded even better δ̅itheo so values, which makes real sense looking also at the σ̅ s values plotted in Figure 3c, since this is a property that strongly depends on exchange. However, a less intuitive and more objective mindset behind the efficacy of changing from PBE to PBE(Gc,Gx,LO) was pointed in refs 49 and 60. Those functionals, for which both parameters β and μ were obtained from the same source, i.e., from the gradient expansion for exchange/correlation (Gx/Gc) or from Jellium surface/ response properties (Js/Jr), possibly benefit from error cancellation, which may have occurred to a lesser extent in the case of PBE(Js,Jr,LO). Notwithstanding those remarks, the results listed in Table 2 suggest that the other parameter that appears in the exchange functional, λ, may also play an important role in the prediction of δ̅theo iso . Changing from PBE(Gc,Gx,LO) to PBE(Gc,Gx,EL) is nothing more than tightening the Lieb−Oxford bound by replacing its original value (λLO = 2.273) to the exact value in the low-density limit of the electron liquid (λEL = 1.9555), according to ref 49. And so finally, if the correlation part of that functional is changed by resetting the β parameter with the values determined by fitting to Jellium surface energies (Js) or to Jellium response function (Jr), then the respective computed theo becomes even closer to δexp δ̅iso iso . That set of results achieved for TiAl suggests that a better (or less bad) prediction of δ̅theo iso , with acceptable lattice parameters, requires not only that μ is set from a gradient expansion in such a way that this expansion is recovered for exchange, as in the PBEsol XC functional. It has been shown that the use of a tighter λEL = 1.9555 yields further improvement of the result, and it has already been reported by Haas et al.49 that such change can influence the description of semicore states in some systems. Regarding the difference between the results obtained with PBE and LDA XC functionals, it is notable in the respective projected DOS (PDOS) shown in Figure S4a,b that indeed the very localized Ti 3s and 3p semicore states are more affected than the delocalized states around EF. The PDOS computed in the present work within the PAW approach are particular chemical renderings of the PW-based band structure computed within the FCA, i.e., it brings information regarding only valence and semicore states. Nevertheless, the effect of the XC functionals on the chemistry of the |Ψ̃n,k⟩ states in TiAl can become evident by looking at the differences between the PDOS of each spin channel (↑ and ↓), as shown in Figure 4. A qualitative assessment of these plots suggests that the LDA overbinding is damping down the spin susceptibility of states in whole spectrum, with the underbinding of the other two PBE-type GGAs tending to promote it. Furthermore, one can note that, indeed, not only the states around EF (see Figure 4a) but also the quite localized Ti 3s and 3p semicore states are affected (see Figure 4b,c, respectively). Furthermore, in a qualitative sense, the shifts of these states in the PDOS are proportional to the computed mTi plotted in Figure 3d, which is in line with an MSM associated with unpaired spins localized at TM sites in aluminides argued by Lue et al.55 Finally, just for the sake of completeness, an image analogous to Figure 1 is provided in Figure S8, depicting a comparison between PBE and the XC functionals PBE(Gc,Gx,LO), PBE(Gc,Gx,EL), and PBE(Jr,Gx,EL) (a detailed

PBE, in which the parameter β, which appears in the correlation functional, has been set by fitting it to Jellium surface energies. Additionally, a second parameter, μ, which appears in the exchange functional, guarantees that its secondorder gradient expansion is recovered.60 Actually, both PBE and PBEsol are two particular members of a set of five nonempirical PBE-type GGAs studied in ref 60 and that set was then expanded to 10 in a further study,49 by taking into account a third parameter, λ. Note that, while this parameter was originally determined in PBE so that the exchange energy alone obeys the Lieb−Oxford (LO) lower bound (λLO = 2.273),61 the authors in ref 49 used an optional value (λEL = 1.9555), tightening the LO bound. They used the notation PBE(β,μ,λ) to identify each member of that extended family of PBE-based XC functionals. Remarking that although they are not implemented in the QE version used in the present work, this is a straightforward task that can be done by following the current PBEsol implementation. Although far from the performance of LDA, it seems intuitive that the aforementioned beneficial use of PBEsol with respect to PBE in predicting δexp iso of the aluminides could be further explored using the family of XC functionals PBE(β,μ,λ). It is expected that such systematic investigation could provide any relevant insight into whether and how some key aspect of the Al−TM chemical bonds in these compounds can be used to explain the disagreements between theory and experiments exposed in Figure 1. The ssNMR-related data as well as the mTi values computed with each XC functional for the experimental cell of TiAl are listed in Table 2. In addition to these simulations, that cell has been relaxed with each PBElike functional as well as with LDA, and the respective computed properties are available in Table S6 and also in Figure 3. To assure the consistency of the simulations with the PBE(β,μ,λ) family of XC functionals, Figure 3a brings a comparison between the relative errors found for TiAl volumes and the mean relative error (mre) values reported in ref 49 for the lattice constants of the complete set of 60 solids. From that figure, it is possible to see that there is a fairly good linear relationship between the two set of relative errors. Regarding the asymmetry in the relative errors found for TiAl volumes, between −5.01 and 0.75%, it is compatible with the results reported in ref 49 for intermetallics with presumed similar crystal chemistry, namely, FeAl, CoAl, and NiAl. Before going into the matter of the impact of the different PBE reparametrizations on the computed quantities, Figure 3b provides an overview of the effect of structural relaxation on the 27Al δ̅theo iso computed for TiAl. To interpret the results, the reader must be aware that the order of XC functionals whose computed properties are plotted in that figure follows the order of ascending volumes listed in Table 2. The point is that from that figure, it is possible to state that, at least for TiAl, structural relaxation does not necessarily improve the agreement with its experimental shift (δexp iso = −170 ppm) as much as the use of the LDA XC functional does. Moreover, the same trend exists for σ̅ s and mTi in Figure 3c,d, respectively. Now, regarding the effect of the different PBE reparametrizations, its discussion can follow the one made in ref 49, focusing primarily on the three XC functionals that differ from PBE by a change in only one of the three parameters: PBE(Js,Jr,LO), PBE(Gc,Gx,LO), and PBE(Gc,Jr,EL). Recalling the notation proposed by the authors, PBE is PBE(Gc,Jr,LO) and PBEsol is PBE(Js,Gx,LO). So, basically from Figure 3b,c, it G


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this work,20,64 which indicates a fair possibility of revisiting the peak assignments made in the literature for AlB2 spectra, proposing that the actual experimental shift of pure AlB2 contributes to the sharp and intense peak around 1680 ppm in the spectrum reported in ref 63 and not to the broad peak at 880 ppm. Naturally, this also further explains the huge disagreement between the experimental 27Al chemical shift of AlB2 and its theoretical counterpart, as depicted in Figure 1. Regarding the broad peak at 880 ppm reported in ref 63, it could be assigned to chemical environments caused by Al deficiency in the own AlB2 structure, or in other phases found to be present in the samples of the binary Al−B system, like aluminum dodecaboride, AlB12. It is entirely true that this peak reassignment should be further investigated in a more systematic way, which is definitely out of the scope of this publication. In any event, some additional and computationally expensive simulations have been carried out to clarify two more tangible and immediate issues. First, the result of an analysis made for AlB2 using the family of XC functionals PBE(β,μ,λ), as performed for TiAl in previous section, is shown in Figure S7 (see also Tables S10 and S11). From that figure, it is clear that the effect of structural relaxation does not improve the agreement with the experimental shift of 880 ppm. A second issue regards the reported presence of AlB12 in the sample. The crystallographic cell of its most stable phase, αAlB12,65 contains 16 Al atoms, and their respective 27Al δ̅theo iso shifts, simulated with the PBE XC functional, are listed in Table S12 and are divided into two groups around 1400 and 1000 ppm. That table also shows that the effect of using LDA on the ρ̅s values was found to be as small as that obtained for CuAl2 and AlB2. Since these calculations for α-AlB12 are quite computationally expensive, the effect of structural relaxation on these shifts will not be explored in this work. With respect to the specific case of ssNMR spectra acquired for the binary Al− B system, the main message here is that the referred broad peak at 880 ppm is likely to be due to a more complex structure in terms of Al chemical environments. In a more general context, it clearly shows the usefulness of firstprinciples simulations of ssNMR spectral parameters in the case of mixture of phases. Overview of Magnetic Screening Mechanisms. This section brings forward a short overview of shift directions, as depicted in Figure 5, taking into account only the results obtained with the LDA XC functional. First, it is notable that, with the exception of AlB2, the σo component leads to a deshielding effect in all intermetallics. Looking at the plotted exp orbital shifts, δtheo iso/o = σref − σo + δref (see eq 2), it can also be seen that this deshielding with respect to AlPO4 is more pronounced in the aluminides than in AlB2 and CuAl2. Regarding the deshielding effect to a large extent, commonly expected for metallic systems, one can see that it is considerably more pronounced in AlB2 and CuAl2 than in the other intermetallics, resulting in the large shifts to downfield similarly to elemental Al (see ref 20). Note that this effect in ScAl3 and ZrAl3 is comparable to that existing in the chemical environments of Al in the ScT2Al Heusler phases, previously studied with the same approach in ref 20. Moreover, it seems that going to upfield δexp iso values (shown in parenthesis in Figure 5), the deshielding effect of σs decreases following the sequence TiAl3, VAl3, and TiAl, becoming a shielding effect more intense in the latter. The aluminide TiAl3 is an exception regarding that trend, but it has already been discussed that the

Figure 4. Differences between the PDOS of each spin channel in TiAl computed with PBE (blue), PBEsol (green), and LDA (red) XC functionals at the experimental volume.

list of all related quantities is available in Tables S7−S9). It is possible to see that the same trends observed in TiAl also hold for the other aluminides, albeit in a less intense way. The exception is TiAl3, suggesting that the prediction of 27Al δ̅theo iso shifts in both Ti-containing aluminides suffers from the same accuracy issue related to the description of the role of Ti semicore states in the Al−Ti chemical bonds. The Case of AlB2. The intermetallic AlB2 should be regarded as a case apart from the other six compounds studied in this work. First, it is a fact that there is a substantial difference between its experimental shift and that of CuAl2 (see Figure 1), as reported in ref 19. With a huge disagreement between experiment and theory found for AlB2, it is in principle irretrievable from the point of view of the approximations in the DFT-GIPAW calculations taken into account so far. Notwithstanding, another remarkable and contrasting fact is that the theoretical results indicate a great similarity between the Al chemical environments in both compounds, at least with respect to the particular aspects of their electronic structure that dictates their respective simulated 27Al δ̅theo iso shifts. It is also true that AlB2 has a very distinctive crystal structure among the other six intermetallics, consisting of hexagonal layers of Al atoms and graphitelike honeycomb layers of B atoms.40 Hence, it is supposed to have the most differentiated crystal chemistry, which gives room for questioning whether AlPO4 is a suitable reference compound to compute σref in eq 2 in that specific case. The reader is referred to ref 62 for more details about that issue regarding the suitability of a computational reference compound in GIPAW simulations. However, after revisiting the peak assignments made in the literature for this intermetallic, it was possible to envisage a structure-related accuracy issue that goes beyond the effect of structural relaxation. In a recent study, Turner et al.63 brought up challenges in synthesizing pure AlB2 and hence in obtaining an ssNMR spectrum with the only broad peak at 880 ppm expected for this compound. In the referred spectrum, two additional peaks at 1640 and 0 ppm were assigned to elemental Al and aluminum oxide, Al2O3, respectively, impurities claimed to be inevitably present in the samples. At this point, it is important to recall that the experimental shifts of these two latter compounds have already been successfully predicted with the GIPAW method and with the same PBE PAW data sets used in H


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plane-wave pseudopotential calculations, the results reveal a systematic failure of DFT XC functionals based on the generalized gradient approximation (GGA), especially for weakly magnetic aluminides, which, at least in the case of the 27Al nuclide, seems to be more critical than the adopted frozen-core approximation intrinsic to the GIPAW/PAW approaches. The results point to an effect related to an overestimation of the local magnetic moments at transitionmetal sites in such compounds. And it has been found that not even the use of “tailored” XC GGA functionals, by combining exact exchange or by resorting to a Hubbard Hamiltonian, has led to satisfactory results. The solution came with the simplest local density approximation (LDA), most likely due to its wellknown tendency to overbinding. From a practical point of view, particularly regarding a possible use of such simulations for a more efficient interpretation of experimental ssNMR spectra, key physical insights into the MSMs underlying the 27Al δexp iso shifts were reported in this work, making clearer the role of hyperfine interactions with spins of itinerant electrons and that of those localized at specific transition-metal sites (mTM). So, in addition to the contributions of closed orbitals to δexp iso , it has been proposed, following experimental evidences already reported in the literature, that additional contributions coming from unpaired electronic spins arise from two main competing hyperfine interactions: the first one related to those referred mTM localized magnetic moments, resulting in a shielding effect and even negative shifts, and the second type of interaction in which large Knight shifts to downfield gradually dominate in those compounds having a rather metallic character. It is a fact that DFT-based simulations of ssNMR chemical shifts in paramagnetic molecules and solid insulators (the aforementioned so-called pNMR shifts) is a topical issue. Recent efforts to propose protocols for pNMR shifts calculations with increasing levels of sophistication are mainly motivated by research in the field of alternate energy storage solutions, like all-solid-state batteries. With increased attention from chemists, intermetallic compounds place the simulations of ssNMR shifts in weakly magnetic metallic systems like the intermetallics studied in this work in an equivalent situation. The use of the GIPAW method in this context is mainly motivated by the expected gain in computational efficiency, comparable to the pseudopotential approach, an essential requirement for current trends in theoretical materials science like Big Data analytics. Further research in the topic of the present work should focus on the specification of more efficient theoretical protocols as well as on ensuring their effectiveness for other nuclides.

exp Figure 5. Orbital (δtheo iso/o = σref − σo + δref , in blue) and spin (−σs, in red) contributions to the metallic shieldings computed with the LDA XC functional according to eq 2. The distinctive Al local environments in the seven intermetallics are labeled with their respective experimental isotropic metallic shifts shown in parenthesis.

Ti-containing aluminides have proven to be the most challenging cases in this work. Now, recalling that, in addition to the contributions of closed orbitals to δexp iso through σo, it has been discussed in previous sections that those coming from unpaired electronic spins through σs may arise from two main competing MSMs. One is the so-called Knight shift MSM which comes from hyperfine interactions with spins of itinerant electrons existing in those compounds having a rather metallic character. This mechanism is normally expected to deshield the 27Al nuclei,19 and given the high PDOS with s-character at EF in CuAl2 and AlB2 (see Figure S1), it is reasonable to state that it prevails in these two intermetallics as well as in elemental Al (see ref 20), which is in agreement with the respective computed shieldings depicted in Figure 5. In the second accounted MSM, a supposed concurrent shielding effect comes from hyperfine interactions with local magnetic moments at transition-metal sites (mTM). As mentioned above, such mechanism has already 55 been used to justify negative δexp iso extracted from VAl3 spectra, and the relatively low PDOS with s-character at EF computed for the aluminides (see Figure S1) creates room for the prevalence of that mechanism.

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SUMMARY AND CONCLUSIONS In this work, first-principles simulations of 27Al ssNMR chemical shifts were performed for a set of seven Al-containing intermetallic compounds. With the aim of exploring accuracy issues in such calculations using the GIPAW method, the referred set was sought to cover the wide range of experimental 27 Al δexp iso shifts in a representative way (from about −200 to 1600 ppm). It has been found that, in addition to that representativeness in terms of the computed shifts, the set also provided comparative key insights into the respective underlying magnetic screening mechanisms (MSMs). From a technical/computational perspective, the findings allowed the detection of some critical sources of inaccuracy, which were addressed to different approximations inherent in the computational methods. Despite the simplicity of the

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S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.9b00259. Details on theoretical methods; additional spectral parameters related to quadupolar coupling; and the fsum rule computed for each compound (PDF)

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Ary R. Ferreira: 0000-0002-0718-3060 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

A.R.F. acknowledges the São Paulo State Research Foundation (FAPESP; grant 2016/12319-0) for a fellowship and also acknowledges UFSCar and Prof. José P. Rino for support. The author acknowledges the use of Petaflop computing facilities and associated support services of SDumont, provided by the Brazilian National Laboratory for Scientific Computing (LNCC) in Petrópolis, RJ.

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