Toward Routine Gauge-Including Projector Augmented-Wave

Article pubs.acs.org/JPCC

Toward Routine Gauge-Including Projector Augmented-Wave Calculations for Metallic Systems: The Case of ScT2Al (T = Ni, Pd, Pt, Cu, Ag, Au) Heusler Phases Ary R. Ferreira, Karsten Reuter, and Christoph Scheurer* Department of Chemistry, Technische Universität München, Lichtenbergstrasse 4, D-85748 Garching, Germany S Supporting Information *

ABSTRACT: We use the gauge-including projector augmented-waves (GIPAW) method to report, for the first time, theoretical 27Al Knight shifts in metallic systems other than metallic Al. We consider metallic Al and a set of six intermetallic compounds, for which experimental chemical shifts were recently made available in the literature. The orbital and spin components of the chemical shielding tensors are computed from the same ground-state spinpolarized electronic structure, converged under the influence of a uniform external magnetic field. A linear response formalism is used to compute the orbital part, while the spin part is approximated by the linear relationship between the external field and the Fermi contact contribution to the induced magnetic hyperfine field at the nuclear position. Core spin-polarization effects are taken into account by means of a perturbative approach. Our results show that the GIPAW approach yields reasonably acceptable chemical shifts with affordable k-meshes in the irreducible Brillouin zone, enabling separation between contributions to metallic shifts from the electron orbital and from the electron spin susceptibilities.

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INTRODUCTION The combination of solid-state nuclear magnetic resonance (SS-NMR) experiments with theoretical spectral parameters within the gauge-including projector augmented-waves (GIPAW) formalism,1,2 is common practice in structural characterization of materials.3,4 Within the frozen core approximation (FCA) for extended periodic systems and density functional perturbation theory (DFPT),5−7 the GIPAW method was originally developed to predict the orbital contribution, σ⃡o(r′), to the total chemical shielding tensor, σ⃡tot(r′). In this approach, only the charged nature of electrons is taken into account in formulating the effect of a uniform external magnetic field, Bext, on the electronic structure of a solid sample. This is enough for nonmagnetic insulating materials, which have neither unpaired electrons nor partially filled bands at the Fermi surface. However, in SS-NMR experiments on metallic systems, it is common to observe strong paramagnetic resonance shifts due to interaction between the magnetic moments of spin-polarized conduction electrons and the nuclear spin. Such contributions to the measured resonant frequencies are known as Knight shifts, and it is hard to find arguments to separate them from the competing diamagnetic shielding and paramagnetic deshielding effects.8 An extension of the GIPAW method for computation of this additional spin contribution in metals, σ⃡s(r′), was proposed by d’Avezac et al.9 In that work, as previously described for σ⃡o(r′) in insulators,1,2 a linear response formalism was used for the © 2016 American Chemical Society

calculation of σ⃡s(r′) upon a redefinition of the Green’s function operator, necessary for the computation of first-order quantities.10 Recently, Laskowski and Blaha11 showed that the σ⃡s(r′) component can be obtained from the ground-state (GS) spin-polarized electronic structure, converged under the constraint of a given total magnetization, ms, mimicking the induced spin magnetic moment of the system caused by Bext. In that work, the authors reported σ⃡o(r′) values computed with a gauge-invariant perturbation method in a full-potential augmented plane wave (APW) basis set,12,13 while the σ⃡s(r′) contributions were estimated from the linear relationship between Bext and the induced magnetic hyperfine field, Bhf(r′), instead of the first-order field B(1) s (r′), as proposed by d’Avezac et al.9 with the GIPAW method. Actually, this strategy has already been adopted for metallic carbon nanotubes in a previous report,14 where Gaussian-type basis sets with periodic boundary conditions were used within density functional theory (DFT).15,16 Additionally, we have to mention that in both plane waves (PWs) based approaches,9,11 it has been found for elemental crystals that the convergence of computed values for σtot ⃡ (r′) = σo⃡ (r′) + σs⃡ (r′) with respect to sampling in the Brillouin zone (BZ) requires up to tens of thousands of kpoints. Received: August 19, 2016 Revised: October 20, 2016 Published: October 21, 2016 25530

DOI: 10.1021/acs.jpcc.6b08418 J. Phys. Chem. C 2016, 120, 25530−25540

Article

The Journal of Physical Chemistry C

to vk,k′|ũ(0) n,k ⟩, where vk,k′ is the k-dependent velocity operator. The sum in eq 3 includes all the empty orbitals e, and the energies ϵe,k are the unperturbed pseudoeigenstates of |ũ(0) n,k ⟩. In (0) practical calculations, the closure relation ∑e |ũ(0) e,k ⟩⟨ũe,k | = 1 − (0) ∑o |ũ(0) o,k ⟩⟨ũo,k | allows the implementation of a summation over occupied states o instead. The aforementioned redefined Green’s function, used in ref 9, follows the extension of the DFPT approach to latticedynamical calculations proposed in ref 10 for metallic systems. In order to deal with the absence of a band gap when computing the |ũ(1) n,k ⟩ states, a smearing function is used to set the occupation of a given energy level i at k-point k as fFi , k = f [−(ϵF − ϵi , k )/T ], where ϵF is the ground-state Fermi energy and T is a fictitious electronic temperature. So the adapted k-dependent Green’s function is defined as

In the present work, we use the GIPAW method to compute Al SS-NMR chemical shifts for a set of metallic systems, following the approach reported in ref 11 for the computation of σ⃡s(r′). Besides metallic Al, we also report SS-NMR spectral parameters for a set of ternary ScT2Al (T = Ni, Pd, Pt, Cu, Ag, Au) Heusler intermetallic compounds, for which recent experimental data has been made available in the literature.8 We show that it is possible to obtain reasonably acceptable results with an affordable number of k-points, which drops significantly for the primitive cells of the intermetallics under study if compared to metallic Al. The paper is organized as follows. In the Theory section, we describe the theoretical methodology and the technical details of our calculations. Next, we present the results obtained for metallic Al with the GIPAW method, in light of previous reports with alternative approaches.9,11 Then, we discuss our results obtained for the ScT2Al intermetallics, with the focus on the performance of different atomic data (potentials, charge densities, and wave functions) made available for end users by code developers and expert users. The last section contains our conclusions. 27

.k(ϵ) =

j

THEORY Methods. All calculations were performed with the Quantum ESPRESSO17 (QE) open-source software suite version 5.3.0, which offers an implementation of DFT with PWs and pseudopotentials (PPs) or projector augmentedwaves (PAW)18 data sets. The GIPAW calculations of the orbital component σ⃡o(r′) for spin-polarized metallic systems were carried out with the QE-GIPAW module SVN revision 408, which required minor adaptations for the achievement of the results. For Bext magnitudes commonly used in SS-NMR experiments, the total first-order induced field can be approximated by linear combination of the orbital and spin contributions:9 =

B(1) o (r′)

+

B(1) s (r′)

BFc(R) = (1)

(2)

where the three terms on the right-hand side of eq 2 are the bare, paramagnetic correction, and diamagnetic correction terms, respectively. These terms result from the use of the GIPAW operators within DFPT for computation of j(1)(r′). For insulators, all three components depend on the ground-state ikr (0) pseudovalence Bloch states, |Ψ̃(0) n,k ⟩ = e |ũn,k ⟩, corresponding to occupied bands with index n at k-point k, where |ũ(0) n,k ⟩ is a cell (1) periodic function. In order to compute the j(1) bare(r′) and jΔp (r′) components, it is also necessary to obtain the perturbed eigenstates to first-order, |ũ(1) n,k ⟩, by applying the k-dependent Green’s function: .k(ϵ) =

∑ e

|ue(0) ̃ , k ⟩⟨ue(0) ̃ ,k | ϵ − ϵe , k

ϵj , k − ϵ

|uj(0) ̃ , k ⟩⟨uj(0) ̃ ,k | (4)

2 μ g μ sρ ̅ 3 0 e B s

(5)

Here μ0 is the permeability of vacuum (4π × 10−7 T2·m3·J−1), ge is the electron g-factor (2.002319), μB is the Bohr magneton (9.274 × 10−24 J·T−1), and s is the electron’s spin. The electronic spin density, ρ¯s = ρ¯↑ − ρ¯↓, is averaged inside a sphere centered at R = 0 with Thomson radius RT = Zre, where Z is the element atomic number and re is the classical electron radius. An alternative to assuming a finite size nucleus for the calculation of BFc(R) is to replace ρ¯s by the value of the innermost electronic spin density in the PAW data set radial grid, ρs. For comparison purposes, we also consider this strategy in the present work and we denote all quantities derived from ρ¯s and ρs, with and without an overbar, respectively. We also point out that, as reported in ref 11, the anisotropic (or dipolar) contribution to Bhf(r′) is negligible for the metals with high-symmetry structures considered in the present study. This is not true for other systems like the small molecules containing C and H atoms reported in ref 19, and we have performed some preliminary tests (not reported) with the current implementation of the QE-GIPAW module against those results. Even upon the all-electron (AE) wave functions reconstruction provided by the PAW method, the FCA may not allow the correct simulation of BFc(R). This is a consequence of the inherent nonrelaxation of the non-spin-polarized atomic partial waves, |ϕλ⟩, of core states, available in the PAW data sets. The

The respective chemical shielding tensors are obtained from the ratio between the external and induced fields, as for example, B(1) o (r′) = − σo⃡ (r′)Bext . In the first GIPAW approach developed for norm-conserving pseudopotentials (NCPPs),1 B(1) o (r′) is computed from the application of the Biot−Savart law by use of the first-order induced current: j(1)(r′) = j(1) (r′) + j(1) (r′) + j(1) (r′) bare Δp Δd

fFj , k − f [−(ϵF − ϵ/T )]

where the sum is over fully and partially occupied states j, and all the first-order expectation values are obtained with the assumption of no variations of the Fermi energy to first-order, that is, ϵ(1) F = 0. Since in the present work we follow ref 11 for the computation of σ⃡s(r′), it is necessary to induce a spinsplitting in the GS electronic structure of the metallic system by defining a value for ms, constraining its self-consistent calculation. So, when eq 4 is applied to nonperturbed eigenstates with different spin channels ξ (↑ or ↓), the respective Fermi energies, ϵ↑F and ϵ↓F, must be taken into account. According to ref 11, the spin component of the induced field in eq 1 can be approximated by the Fermi contact contribution to Bhf(r′) at the nuclear position R, for which we used the following expression in SI units:

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B(1) ind (r′)

∑

(3) 25531


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previous version 0.3.1, made available by Kücu̧ ̈kbenli et al. (PSLR);26 and a library released by Jollet et al. (JTH).27 The PSL and PSLR libraries were generated with the LD1 code, which is part of the QE package, while the JTH set was generated with the ATOMPAW code.28 In the case of the sets PSL and PSLR, there are different possible combinations of the respective PAW data sets. So, for the ScT2Al compounds, we define a label with the set name, followed by two letters indicating the type of PAW data set for Al and T elements, respectively, according to the nomenclature adopted in each distribution (n, nl, dn, spn, spfn). A third letter for Sc is suppressed because there is only one set of atomic data available for this element in both libraries. In order to compare theoretical and experimental chemical shifts, it is necessary to select a suitable zero-shift compound for which σ⃡s(r′) = 0 and, therefore, σref ⃡ (r′) = σo⃡ (r′). In GIPAW calculations of shielding parameters, the FCA introduces errors that can be canceled in the calculation of theoretical chemical shifts, δth, provided there is an electronic similarity between the target and reference zero-shift compounds.29 In that case, it is possible to simply compute

spin polarization of valence electrons induces a spin polarization of core electrons that can be relevant for an accurate computation of the Fermi contact term. In our results, this effect is accounted by means of the perturbative approach proposed in ref 20, where the spin-up and spin-down charge densities for spin channel ξ, at position r in the PAW data set radial grid, are given by the sum of valence and core contributions, ρξ(r) = ρvξ(r) + ρcξ(r). The spin density due to valence electrons is recovered for each spin channel according to the PAW approach: ρ vξ (r) = ρ ξ̃ (r) + ρ1ξ (r) − ρ1̃ ξ (r)

(6)

where ρ̃ξ(r) is the so-called bare contribution that depends on ξ ξ the variational functions |Ψ̃(0) n,k ⟩, while ρ1 (r) and ρ̃ 1 (r) are the PAW (or GIPAW) one-center reconstruction terms. Computation of the PAW-related densities in eq 6 requires the ̃ (0) quantities ⟨ϕλ|r⟩⟨r|ϕκ⟩, ⟨ϕ̃ λ|r⟩⟨r|ϕ̃ κ⟩, and ⟨Ψ̃(0) n |p̃λ⟩⟨p̃κ|Ψn ⟩, where |ϕ̃ λ⟩ and |p̃λ⟩ are the atomic pseudo partial waves and their dual projectors constructed so that ⟨p̃λ|ϕ̃ κ⟩ = δλκ, respectively. Since the isotropic Fermi contact interaction depends on the electronic spin density at R, only reconstruction terms for which λ and κ indexes correspond to s orbitals simultaneously are taken into account. The nonzero spin density due to core electrons, ρcξ(r), depends on spinpolarized atomic core partial waves, |ϕλξ⟩. These are constructed from a linear combination of the original nonspin-polarized counterparts |ϕλ⟩ (available in the PAW data sets) in a first-order perturbative approach, where the perturbing potential is a functional of charge and spin densities. In our work, as in ref 19, these spin densities are averaged around the nucleus up to a radius of 5 au, using the PAW data set radial grid. More details can be found in ref 20, and we point out that core-polarization corrections for hyperfine parameters is an experimental feature available in the QEGIPAW module. Technical Details. The Perdew−Burke−Ernzerhof (PBE)21 generalized gradient approximation was used to describe the exchange−correlation functional in all computations. Numerical accuracy was ensured at this level of theory by converging the simulated SS-NMR spectral parameters and total electronic energies with respect to the PW basis set and the k-point samplings in the irreducible BZ, which were determined by the Monkhorst−Pack procedure.22 A Fermi− Dirac probability distribution is used as the smearing function in eq 4 and also for convergence of the self-consistent calculation of the GS spin-polarized electronic structure. For its broadening parameter, kbT, we consider values in the range from 2 to 10 mRy for tests. In a PP approach like the GIPAW method, the choice of the correct set of atomic data able to yield reliable results is not straightforward in the case of properties that are essentially local, like the SS-NMR spectral parameters. Recently, there have been efforts toward the establishment of a precision-based indicator (the so-called Δ gauge), which can be used to benchmark different PAW data sets for a given solid-state DFT code.23 However, it is important to remark that such an approach is based on parameters of the equations of state of elemental crystalline solids. So its transferability to more complex systems/properties is not guaranteed.24 For the sake of comparison, three sources of PAW data sets compatible with the QE suite were selected: namely, the PSLibrary project (PSL) version 1.0.0, provided by Dal Corso;25 a revision of its

δth = σref − σtot

(7)

where σ = Tr[σ⃡(r′)]/3 is the isotropic value (the trace) of the chemical shielding tensor. Such an approach yielded good results for the 27Al nuclide in different phases of alumina, Al2O3, reported in a previous work,30 where the PSL and PSLR sets were used. However, in an opposite situation, that is, when the surrounding electronic environments of the target nuclei are dissimilar in the two compounds, the source of errors may be different. So, an alternative is to use an internal (computational) reference with a nonzero experimental chemical shift, 29 δexp ref , and calculate exp δthcomp = (σref + δref ) − σtot

(8)

Aluminum phosphate, AlPO4, was the reference compound used in refs 9 and 11 for metallic Al; therefore, we also report our δth and δcomp values relative to AlPO4. We also adopt the th same experimental shift, δexp ref = 45 ppm, with respect to aluminum chloride, AlCl3, in heavy water adopted in ref 9. The PAW data sets used for P and O from the PSL distribution were the respective n and nl, following its nomenclature. Regarding the structures used in the calculations, all the ternary ScT2Al Heusler intermetallic compounds were reported in ref 8 with a cubic MnCu2Al-type structure and the Fm3¯m space group, where the Al and Sc atoms form a NaCl-type substructure occupying the 4a and 4b Wyckoff positions, respectively, while T atoms are located at all cubic voids in 8c sites. For all these structures, a primitive cell with four atoms was used with the experimental lattice parameters reported in ref 8. For metallic Al, we used the primitive cell with a single atom derived from the face-centered cubic conventional structure with space group Fm3¯m.

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RESULTS AND DISCUSSION Metallic Al. Since our focus in the present work is on 27Al SS-NMR chemical shifts in different systems, we start by presenting some results obtained for metallic Al. As previously reported,9,11 up to 70 000 k-points can be necessary to represent the irreducible BZ in a discrete grid so that the values computed for σo are converged for this nuclide. We show in Figure 1 a broad view of this convergence test, which was 25532


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Figure 1. Convergence of the isotropic 27Al orbital component (σo) of chemical shielding with respect to the number of k-points in the irreducible BZ for metallic Al. The test was carried out with kbT = 8 mRy and Ecut = 60 Ry, by use of the PSL-nl PAW data set. An induced spin magnetic moment ms = 0.003μB was used to compute the GS wave functions.

Figure 2. Dependence of 27Al isotropic spin (σs) and orbital (σo) components of chemical shielding in metallic Al on the smearing parameter (kbT). The PSL-nl PAW data set for Al is used with Ecut = 70 Ry and an induced spin magnetic moment ms = 0.003μB.

carried out with a smearing broadening parameter of kbT = 8 mRy. It can be seen that the result is quite compatible with prior reports for the same system.9,11 For this first test, we selected arbitrarily the Al PAW data set PSL-nl, and a kinetic energy cutoff (Ecut) of 60 Ry was checked to be suitable. In order to have all properties at issue converged in the following calculations with the Al atomic data sets PSL-nl, PSL-n, PSLR, and JTH, the respective values used for Ecut were 70, 70, 80, and 80 Ry. We also provide in Table S1, some details of the calculations with k-point meshes preceding the convergence of σo to within 1 ppm, where it is possible to see that all the quantities related to σs are already very well converged. In view of the strong paramagnetic resonance shifts observed experimentally for this system, we believe that a 72 × 72 × 72 mesh (24 984 k-points) is enough for the tests that follow. As reported in ref 11, where equivalent simulations were performed with an APW basis set, there is a noticeable dependence of σo and σs components on the smearing parameter kbT. In order to estimate this effect, the authors reported the coefficient of a linear correlation for both components, calculated as [σ(kbT2) − σ(kbT1)]/(kbT2 − kbT1), with kbT1 and kbT2 equal to 4 and 8 mRy, respectively. As can be seen in Figure 2, in that same kbT range, their correlation plots are qualitatively quite similar. The authors in ref 11 found metallic Al coefficients equal to −4 and 11 ppm/ mRy for σo and σs, respectively. For the correlation with σo, we found a value of −13 ppm/mRy with all Al PAW data sets. The origin of that discrepancy between our GIPAW results and their APW results is more likely to be the use of FCA than the current implementation of the approach proposed by d’Avezac et al.9 To confirm that, we can mention that, in some preliminary tests (not reported), we computed σo also for metallic Li. From these tests, we have found a value of 82.91 ppm, which is very close to the 81 ppm reported in refs 9 and 11. The correlation coefficients computed by us for σs are available in Table 1. The small differences between our GIPAW results and the APW results from ref 11 also can be assigned to the use of FCA for the 27Al nuclide. Resorting again to our preliminary tests with metallic Li, our σs values around −161

ppm, computed for this system with a NCPP, are quite different from the −266 and −264 ppm values computed with an APW basis set11 and a linear response formalism within the GIPAW method,9 respectively. These results obtained for Li point to a high sensitivity of the approach used in the present work to the FCA. We have discussed in a previous publication31 that the Li nucleus is a challenge for the computation of properties that have a strong dependence on the correct description of its 1s state with the GIPAW method. At this point, we also mention another set of preliminary tests (not reported) made for the 45 Sc nuclide, where, when trying to reproduce the ρcore values for the isolated atom with electronic configuration 3d1{1↑, 0↓}4s2 reported in ref 20, we found that NCPPs had systematically yielded better results than ultrasoft pseudopotentials (USPPs). Beyond this issue, we also opted to not report results for 45Sc due to difficulty in finding a suitable zero-shift compound for it, recalling that, for some reason, metallic Sc is not among the systems studied in ref 11. In a last analysis of the plots in Figure 2, it must be noted that despite the dependence of σo and σs components on the smearing parameter kbT, there is a considerable cancelation when both are added up to compute σtot. From Table 1, we point out that the linear dependence of both σs and σ¯s on kbT is virtually the same, with a shift of only about 167 ppm in the curves. It proves that this dependence is not related to the approximation of a nucleus with a finite size. Nevertheless, it must be understood that there is a possibility to tune the computed chemical shifts by convenient selection of points in the PAW radial grid, in order to define the spin density in eq 5. That brings a certain arbitrariness to the approach and, in order to expose this, we decided to report results considering both ρ¯s and ρs. Additionally, we must mention the deviation from linear behavior observed in Figure 2 for both spin components with kbT = 2 mRy, which is probably related to the relatively small number of k-points in this test. Nevertheless, such small values for kbT were not considered either in ref 11 or in ref 9. The authors in ref 11 have shown that both BFc and ms depend linearly on Bext up to high external fields of 200 T. This is a prerequisite to assume that the spin component of magnetic 25533


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mind that these two quantities are influenced by the FCA in different ways. If the former depends only on the difference ϵF↑ − ϵF↓, the latter is quite sensitive to the PAW data set, since it depends on the reconstructed ρvξ(r) and also on the first-order perturbative ρcξ(r) (see ref 20). This is a plausible justification for the difference between our slope, σs = BFc/Bext = 1629.1 ppm, and the 1590.5 ppm value reported in ref 11. We also point out that our value for σs was computed from a linear regression of the respective correlation in Figure 3. In other computations for metallic Al with different PAW data sets, we assume a fixed value for ms = 0.003μB. With the PSL-nl atomic data, this is equivalent to an external field of 95.88 T, which induces a spin splitting of 0.82 mRy, and such extrapolation can also introduce small errors. On the basis of the tests discussed above, the same values of ms = 0.003μB and kbT = 8 mRy and the same 72 × 72 × 72 kpoint mesh were used to check the performance of all Al PAW data sets selected in our study. The results are summarized in Table 1, and we point out that the convergence with respect to the k-point mesh was rechecked for each case. First, one can note that the two contributions to σtot, as well as the σref values computed for AlPO4, are different among the four sets of atomic data. Nevertheless, it can be seen that our δ¯th values are closer to the experimental value of 1640 ppm reported in ref 9 than the δcomp = 1874 ppm value reported in that same work, in th which the σs contributions were calculated with a linear response formalism. Moreover, the δ¯th values simulated with PSL-n and PSLR PAW data sets are in very good agreement with that experimental value. However, it can be noticed that use of ρs in eq 5 instead of ρ¯s yields a systematic overestimation of the experimental values. Yet, in general, our δ¯th and δth values computed within the GIPAW approach are closer to the full-potential APW results reported in ref 11 than the GIPAW simulations of ref 9. Another outcome that deserves attention is that, for a given atomic data set, the values of σo and σref are quite close to each other. This is in agreement with the GIPAW results reported in ref 9 as well as the APW simulations from ref 11. This cancelation between the orbital components of pure Al and ionic Al (for which σs = 0) suggests that the surrounding

susceptibility is proportional to the induced spin moment, and we show in Figure 3 that this dependence is also linear in our

Figure 3. Dependence of the Fermi contact contribution to induced magnetic hyperfine field (BFc) and of the induced spin magnetic moment (ms) on the external magnetic field (Bext), for metallic Al. The respective slopes of a linear fit and standard deviations (Δ) are also provided. The PSL-nl PAW data set is used with Ecut = 70 Ry and a smearing parameter kbT = 8 mRy.

simulations up to about Bext = 300 T. It is important to emphasize that ms is the total magnetization that constrains the self-consistent calculation of the |Ψ̃(0) n,k ⟩ states, from which Bext is simply computed by use of the spin splitting (ϵF↑ − ϵF↓). We found 3.117 × 10−5 for the slope ms/Bext, virtually the same value reported in ref 11. This result shows that the pseudovalence states obtained within the FCA and with a smaller number of k-points in our simulations are quite close to the ones previously simulated with the APW method. So, the main source of discrepancy in the σs values computed within the FCA must come from the approximation of taking as linear the relationship between Bext and BFc. It is important to bear in

Table 1. Isotropic 27Al Chemical Shifts and Magnetic Susceptibility Related Properties Computed for Metallic Al and for Reference Compound AlPO4 with Different PAW Data Setsa σo σ¯s σs σref δ¯th δth δ¯comp th δcomp th H χmol s b χmol o molc χo b χmol th c χmol th

PSL-nl

PSL-n

PSLR

JTH

482.86 (−13) −1619.93 (8) −1785.80 (9) 515.12 1652.19 1818.06 1697.19 1863.06 7.63 17.47 −4.54 −3.08 12.93 14.40

482.91 (−13) −1607.09 (12) −1770.40 (13) 517.24 1641.42 1804.73 1686.42 1849.73 7.70 17.32 −4.28 −2.78 13.04 14.54

498.41 (−13) −1600.58 (8) −1774.20 (9) 539.40 1641.57 1815.19 1686.57 1860.19 7.70 17.32 −4.30 −2.13 13.01 15.19

327.38 (−13) −1633.61 (8) −1696.57 (9) 360.12 1666.35 1729.31 1711.35 1774.31 7.63 17.47 −4.51 −3.07 12.97 14.40

a Chemical shifts are given in parts per million (ppm). Coefficients of the dependence of computed shieldings on smearing parameter kbT (see text) are given in parentheses in ppm/mRy. All susceptibilities are in units of 10−6 cm3·mol−1, and the magnetic field strength values H are in 107 A·m−1 (amperes per meter). bComputed with the isotropic value of χ⃡bare (see text). cComputed with the isotropic value of χ⃡o (see text).

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complete test. The first encouraging outcome from this test is that the number of k-points necessary to converge σtot to within less than 1 ppm is significantly smaller than for metallic Al (see Tables S1 and S2). Obviously a decrease is expected, as the volumes of the irreducible BZ for the intermetallic primitive cells are smaller. Also, since all of them have the same symmetry, we concluded after additional tests that a 38 × 38 × 38 mesh (3900 k-points) is suitable for the other five intermetallics. The next step is check the linear dependence of BFc on Bext for all intermetallics. This procedure was applied with the combination of PAW data sets PSL-n-n to all ScT2Al compounds, and the result is presented in Figure 4. The SS-

electronic environments are similar in both compounds. This indicates that eq 7 may be more suitable for metallic Al than eq 8. Still regarding the agreement between experimental and theoretical chemical shifts, the authors in ref 11 reported a lower value of δexp = 1595 ppm, more compatible with their simulations. Even not being so dissimilar from our results, with a relative difference of about −5% for the worst δ¯th value in Table 1, we found a third experimental value in the literature32 of 1630 ppm, for which a similar reference compound [aqueous Al(NO3)3] was used. Withal, we remark that the main differences between our and the GIPAW chemical shifts reported in ref 9 are the σs contributions, pointing to a good performance of the approximation σs = BFc/Bext and of the perturbative approach proposed in ref 20 for metallic Al. We also present in Table 1 the spin and orbital components mol of the simulated molar magnetic susceptibilities, χmol + th = χs mol χo , computed for metallic Al. The correction to σo due to surface currents that appear in the sample, σmacro, requires computation of the macroscopic magnetic susceptibility. We assume a spherical sample in the present work and, in the current implementation of the QE-GIPAW module, as described in ref 1, only contributions from j(1) bare(r′) (see eq 2) are taken into account for the computation of the so-called bare contribution to macroscopic susceptibility (see also ref 9), χ⃡bare. As described in refs 1 and 9, it depends on the application of the k-dependent momentum operator to the states |ũ(1) n,k ⟩. For nonzero temperatures in metallic systems, the authors in ref 9 used a second version of the orbital contribution to the macroscopic magnetic susceptibility, χ⃡o, where the k-dependent velocity operator is applied to |ũ(1) n,k ⟩ instead. We provide in Table 1 the two versions of orbital components of the simulated molar magnetic susceptibilities, χmol o . The spin component of the simulated molar magnetic susceptibility, χmol s , was calculated as χsmol =

MAl M 4π H

Figure 4. Dependence of the Fermi contact contribution to induced magnetic hyperfine field (BFc) for the six ScT2Al Heusler compounds on the external magnetic field (Bext). The PSL-n-n (PSL-n-dn for ScCu2Al) combination of PAW data sets is used with Ecut = 80 Ry and a smearing parameter kbT = 8 mRy. Taken from a linear regression for each line ms vs Bext (not shown), the values of ms (in units of 10−3μB) that would be induced by an external field of 11.74 T are given in parentheses. The respective slopes and standard deviations (Δ) are available in Table S3.

(9)

where MAl = 10 cm3·mol−1 is the metallic Al molar mass, M = msμB/Vcell is the magnetic dipole moment per unit volume in A· m−1 (amperes per meter), and H = Bext/μ0 is the magnetic field strength, also in A·m−1. To compute H, we assume μ/μ0 ≈ 1.0, where μ is the permeability of metallic Al. All values in Table 1 were computed under the constraint M = 1.68 × 103 A·m−1. It is possible to see that our results underestimate the −6 experimental value χmol cm3·mol−1 reported in exp = 16.5 × 10 mol ref 11, as well as the theoretical value χth = 15.5 × 10−6 cm3· mol−1 reported in that same reference. We point out that the values calculated by us with χ⃡o, instead of χ⃡bare, are in χmol o better agreement with the respective values obtained from APW simulations in ref 11, with a highlight to the performance of the Al atomic data set PSLR from ref 26. Moreover, the dissimilarities between the simulated χmol values listed in Table s 1 and those reported in ref 11 are very small. Intermetallics ScT2Al. In order to assess the performance of our approach as well as of the different PAW data sets, we benchmark them against experimental 27Al SS-NMR chemical shifts available in the literature for a set of six ternary ScT2Al (T = Ni, Pd, Pt, Cu, Ag, Au) Heusler intermetallic compounds.8 Since the critical point in Knight shift simulations with DFT and PWs for metals is k-point sampling in the irreducible BZ, we arbitrarily selected the system ScPd2Al to perform a

NMR experiments reported in ref 8 were carried out with an external field of 11.74 T, and we found that a range for ms between 0.001μB and 0.003μB covers the resulting BFc fields for Bext values up to 30 T for all compounds. Study of the magnetic properties of the intermetallics in ref 8 revealed that the compounds with T = Ni, Pd, Cu, and Ag are Pauli-paramagnetic, while only ScAu2Al showed superconductivity. Recalling that samples obtained for ScPt2Al did not favor such studies for this compound. There are some features of the transition metals (TM) T that could be considered to explain the slopes σs, such as the similarity of valence configurations between the two group 10 elements and between the two coinage elements. However, the computed values for BFc at the Al nuclei depend on the spin polarization of all valence s electrons induced by Bext, combined with the resulting spin polarization of core s states due to the spin magnetic moments from valence electrons. The associated mechanisms will be discussed further, based on the analysis of the respective spin densities. We also show in Figure 4 for each intermetallic compound the values of ms (in units of 10−3μB) that would be induced by an external field of 11.74 T, according to the linear regression made for the respective correlations. We extend these values for testing all remaining possible combinations of 25535


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The Journal of Physical Chemistry C Table 2. Isotropic 27Al Chemical Shifts and Related Properties Computed for Six ScT2Al Heusler Compoundsa σ¯s

σs

σo

PSL-n-n PSLR-n-n JTH

−612.02 −612.02 −599.00

−677.13 −677.13 −625.04

395.77 421.38 257.50

PSL-nl-spn PSLR-n-spn JTH

−593.65 −594.66 −577.29

−659.10 −664.10 −599.00

443.62 462.22 303.69

PSL-nl-n PSLR-n-spn JTH

−1000.33 −1009.68 −928.88

−1107.84 −1121.87 −963.60

518.31 551.94 369.73

PSL-nl-spn PSLR-n-dn JTH

−98.16 −95.49 −91.15

−112.19 −108.51 −95.49

272.46 293.17 122.81


−108.51 −108.51 −91.15

−121.54 −121.54 −95.49

297.99 320.58 147.76

PSL-n-n PSLR-n-dn

−476.79 −481.47

−528.21 −537.56

400.18 419.69

PAW set

Bext ScNi2Al 12.09 12.09 12.09 ScPd2Al 11.23 12.09 12.09 ScPt2Al 11.23 11.23 12.09 ScCu2Al 11.23 12.09 12.09 ScAg2Al 12.09 12.09 12.09 ScAu2Alc 11.23 11.23

B¯Fc

BFc

δ¯comp th

δcomp th

δexpb

7.40 7.40 7.24

8.19 8.19 7.56

778.50 775.04 746.62

843.61 840.15 772.66

851

6.67 7.19 6.98

7.40 8.03 7.24

710.26 716.84 678.72

775.71 786.28 700.43

826

11.23 11.34 11.23

12.44 12.60 11.65

1042.25 1042.14 964.27

1149.76 1154.33 998.99

na

1.10 1.15 1.10

1.26 1.31 1.15

385.93 386.72 373.46

399.96 399.74 377.80

426

1.31 1.31 1.10

1.47 1.47 1.15

370.75 372.33 348.51

383.78 385.36 352.85

445

5.35 5.41

5.93 6.04

638.86 646.18

690.28 702.27

649

All chemical shieldings and shifts are in parts per million (ppm), Fermi contact contributions to induced magnetic hyperfine field (BFc) are in milliteslas (mT), and external magnetic field (Bext) values are in teslas (T). bFrom ref 8. cNo JTH PAW data set was available for Au. a

PAW data sets, within each distribution (PSL, PSLR, and JTH). The full list of isotropic 27Al chemical shift related properties is available in Tables S4 and S5. Among all the simulated chemical shifts available in Supporting Information, we list in Table 2 only the δcomp th values that are in better agreement with the SS-NMR experiments reported in ref 8. We also provide the respective BFc, as well as their dependent quantities, computed with both ρ¯s and ρs in eq 5. It can be seen that, with the exception of values underestimate the experimental ScAu2Al, all the δcomp th results. Nevertheless, the intermetallic ScAg2Al is the only case for which the absolute value of the error exceeds 6%, being 13.76%. Yet even among the other five compounds, we point out that the relative chemical shifts obtained by taking one of them as a zero-shift reference are not in really good agreement with experiment, with the exceptions of ScCu2Al and ScNi2Al. Such lack of accuracy with respect to experimental values could compromise theoretical approaches concerning, for example, the resonance shifts measured for other intermetallics with compositions ScT0.5T′0.52Al, also studied in ref 8. However, we stress that the calculations performed in the present work consist of a critical benchmark not only for the GIPAW method but also for transferability of the PAW data sets in use. On the basis of the complete set of results available in Supporting Information, it is possible to see that an arbitrary choice of atomic data without appropriate tests can be the main source of inaccuracy in this kind of simulation within the FCA. In order to provide a statistical assessment of the performance of PAW data sets, we show in Figure 5 the correlations between the theoretical and experimental chemical shifts listed in Table 2. A visual analysis of that figure confirms that use of ρs instead of ρ¯s in eq 5 in general yields better results, with the exception of ScAu2Al. The values of 0.919 ± 0.105 and 1.053 ± 0.106 computed for the slopes of δ¯comp and th

Figure 5. Correlations between theoretical and experimental chemical shifts listed in Table 2. The red straight line through the data represents a hypothetical perfect agreement, while the dashed and solid black lines were derived from linear fits considering only the best and δcomp values computed for δ¯comp th th , respectively. The respective standard deviations (Δ) calculated for the intercepts are 69.5 and 70.1 ppm, while for the slopes, the Δ values are 0.105 and 0.106.

δcomp th , respectively, indicate good performance of the GIPAW method, at least for 27Al in the ScT2Al compounds studied in the present work. Moreover, before a visual comparison of Figure 5 and the equivalent analysis in ref 11, it is necessary to consider the big difference between the experimental 27Al spectral window for these five intermetallic systems (from 426 to 851 ppm) and the wide range of chemical shifts for the simple metals studied in ref 11 (from 260 to 15 700 ppm). The computed values of −7.4 ± 69.5 and −52.0 ± 70.1 for the intercepts of δ¯comp and δcomp correlations, respectively, also th th indicate satisfactory performance of the GIPAW method. The 25536


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Table 3. Macroscopic Magnetic Susceptibilities and Related Properties Computed for Six ScT2Al Heusler Compoundsa M

χmol s

PSL-n-n PSLR-n-n JTH

4.73 4.73 4.73

133.20 133.20 133.20

−28.94 −38.31 −60.98


2.49 2.63 2.49

80.87 85.63 80.87

−62.07 −41.17 −48.86

PSL-nl-n PSLR-n-spn JTH

2.22 2.22 2.22

71.36 71.36 71.36

−71.86 −146.79 −67.03

PSL-nl-spn PSLR-n-dn JTH

2.03 2.03 2.03

61.84 61.84 61.84

18.62 4.65 −7.11


1.84 1.84 1.84

66.60 66.60 66.60

16.08 13.63 −19.73

PSL-n-n PSLR-n-dn

1.86 1.86

66.60 66.60

−8.31 −16.20

PAW set

b χmol o

c χmol o

b χmol th

c χmol th

d χmol exp

ScNi2Al −38.27 −44.95 −73.43

104.26 94.89 72.23

94.93 88.26 59.77

166.00

−54.44 −45.55 −65.98

18.80 44.46 32.01

26.43 40.08 14.89

26.70

−113.15 −96.72 −127.88

−0.50 −75.43 4.33

−41.79 −25.36 −56.52

n.a.

31.19 290.18 4.10

80.47 66.49 54.73

93.04 352.03 65.94

55.00

11.55 9.71 −21.05

82.68 80.23 46.87

78.15 76.31 45.55

81.60

23.65 20.63

58.29 50.40

90.25 87.23

na

ScPd2Al

ScPt2Al

ScCu2Al

ScAg2Al

ScAu2Ale

All susceptibilities are in units of 10−6 cm3·mol−1. Values of the magnetic dipole moment per unit volume M are given in units of 102 A·m−1 (amperes per meter). bComputed with the isotropic value of χ⃡bare (see text). cComputed with the isotropic value of χ⃡o (see text). dFrom ref 8. eNo JTH PAW data set was available for Au. a

Table 4. Details of All Computed Quantities That Contribute to Isotropic Values of 27Al Orbital and Spin Components in ScT2Al Compounds and in Metallic Ala ScT2Al PAW σmacro σbare σΔd σΔp σΔp,QR ρbare ρ¯GIPAW ρGIPAW ρ¯core ρcore

Ni

Pd

Pt

Cu

Ag

Au

Alb

n 9.71 −7.97 0.79 −347.18 −19.52 −3.0 142.0 144.0 2.0 15.0

spn 13.92 −12.87 0.69 −334.84 9.78 −3.0 139.0 141.0 2.0 15.0

spn 32.55 −16.25 0.70 −250.55 −0.04 −2.0 216.0 219.0 2.0 23.0

dn −1.09 −16.09 0.79 −487.68 11.70 −3.0 24.0 24.0 0.0 3.0

spn −2.68 −19.71 0.70 −451.24 7.97 −3.0 27.0 28.0 0.0 3.0

dn 3.23 −16.88 0.64 −365.93 13.08 −2.0 104.0 106.0 1.0 11.0

n 3.60 −11.24 0.59 −288.99 8.88 −2.0 355.0 361.0 4.0 38.0

Chemical shieldings are given in parts per million (ppm) and spin densities are in units of 10−6a0−3. The PSLR, which has only one PAW data set for Al and one for Sc, was used in all calculations. For metallic Al, values of ms = (0.4 × 10−3)μB (Bext = 12.96 T) and kbT = 8 mRy were used. bIn metallic Al.

a

of ScPt2Al. Moreover, it is possible to see that, for all six compounds, the absolute values of the spin contributions σ¯s and σs (slopes of the lines in Figure 4) are considerably lower than the values computed for metallic Al. Following the previous discussion about the macroscopic magnetic susceptibilities computed for metallic Al, we list in Table 3 the respective orbital and spin components calculated for the intermetallics. Values of the magnetic dipole moment per unit volume M are given in units of 102 A·m−1 (amperes per meter), and these values are the ones that would be induced by the external magnetic field of 11.74 T used in the experiment reported in ref 8. Comparing experiment versus theory, it can be seen that, as well as for chemical shifts, the choice of the

exceptional behavior of ScAu2Al in our simulations can be assigned to a bad description of the bonding characters of Al in that compound, due to FCA itself or even due to the choice of a standard XC functional. Even before this scenario, the level of precision in the results listed in Table 2 provides to experimentalists the valuable separation between Knight shifts and the resonance shifts from diamagnetic shielding and paramagnetic deshielding effects. As discussed previously in ref 11, we also found (see Table 1) that the orbital part σo of metallic Al and of the ionic reference AlPO4 cancel out to a large extent, and the chemical shift is virtually given by only the spin contribution σs. As can be seen in Table 2, this is not true for intermetallics, with the exception 25537


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the general bonding nature in all theses phases is similar. The σs values, in contrast, present a more pronounced variation among the six intermetallics. In the same period, a massive deshielding of about 500 ppm is observed from group 10 to the coinage elements, which parallels their redox chemistry. Given the Fermi contact nature of σs, we can resort to the spin densities in Table 4 and discuss the spin polarization mechanisms at the Al nuclei. The values computed for ρ¯GIPAW, ρGIPAW, ρ¯core, and ρcore are perfectly compatible with the lower oxidation states expected for the coinage metals, which transfer less electron density to Al. Moreover, upon analyzing the same values, it is possible to see that an equally massive shielding of about −400 ppm sets period 6 (Pt, Au) apart from periods 4 and 5. This is also in line with the plots in Figure 4 and explains why ScNi2Al/ScPd2Al and ScCu2Al/ScAg2Al form pairs with similar Knight behavior. The exceptional results observed for ScAu2Al and ScPt2Al are related to a cancelation of these deshielding/ shielding effects in the former compound, while in the latter these effects are added up. The higher values of spin densities at the Al nuclei in these compounds (see Table 4) are related to the lanthanide contraction of period 6, which is related to the additional 14 electrons in the 4f orbitals of Au and Pt.

correct combination of PAW data sets is critical. Nevertheless, it is possible to see that, with some combinations, very good results for ScPd2Al (PSL-nl-spn), ScCu2Al (JTH), and ScAg2Al (PSL-nl-spn) can be obtained. The intermetallic ScNi2Al was the only one for which we could not obtain acceptable relative errors. Moreover, the use of χ⃡bare or χ⃡o for the computation of χmol o does not guarantee the best agreement with experiments. Besides these two sources of discrepancies, we must mention that there is no information regarding the permeability of the intermetallics in ref 8, and our results in Table 3 were also computed with the assumption that μ/μ0 ≈ 1.0. We list in Table 4 five contributions to σo (isotropic chemical shieldings in parts per million, ppm) according to the GIPAW formulation for USPPs described in ref 2. For a consistent comparison, we present only the values computed with the PSLR collection of atomic data that are in better agreement with the experimental chemical shieldings (see Supporting Information). We point out that the PSLR offers only one PAW data set for Al and one for Sc, and the general trends for the data in Table 4 are the same as the results listed in Table 2. The paramagnetic correction term σΔp (see also eq 2) is the dominant contribution to σo, as can be seen in Table 4. This term is strongly dependent on the chemical environment1 and it justifies the small difference between the values of σo computed for metallic Al and ScPt2Al, suggesting a certain similarity between the general bonding nature in both compounds. Moreover, the order of magnitude of the computed σΔp values shows how important are the GIPAW correction terms for the prediction of relative chemical shifts. They also point to a certain equivalence among the valence electrons of metallic Al, ScPt2Al, and the reference compound AlPO4 (as a distribution of electronic charges), for which σΔp = −230.85 ppm. Moreover, it can be noted that the σΔp values for all ternary ScT2Al compounds, as well as σo (see Table 2), are proportional to the slopes in Figure 4, that is, to σs. This emphasizes the relevance of the computation of both σo and σs contributions in works in which a series of compounds of different chemical nature are studied with a common standard reference. It is not always possible to discuss the detailed atomic and electronic structure in solid samples by considering only the isotropic Knight shifts, as already reported.14 Regarding the spin densities in Table 4, the ρ¯GIPAW and ρGIPAW terms are the dominant contributions, reinforcing the relevance of the GIPAW reconstruction for σs as well. Furthermore, looking also at the ρ¯core and ρcore values, it is clear that, in contrast to the orbital contributions, the induced distributions of valence electronic spins are dissimilar in metallic Al and ScPt2Al. In other words, the mechanisms associated with the spin polarization of Al core s states, induced by the spin-polarized valence electrons in both compounds, are not exactly the same. We also point out that we found positive signs for all computed ρ¯core and ρcore values that can be associated with an accumulation of charge associated with Al core 1s and 2s electrons of spin majority close to the nuclear positions, due to a repulsive interaction induced by valence s states.20 It is possible to group the intermetallics according to the valence configurations of their respective T transition metals, and this is quite consistent with the plots in Figure 4. From the shielding values listed in Table 2, it is noticeable that σo is generally deshielding and does not vary too much, slightly increasing from top to bottom and from right to left on the periodic table, about 250 ppm from Cu to Pt. It suggests that

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SUMMARY AND CONCLUSIONS We have computed 27Al SS-NMR chemical shifts for a set of ternary ScT2Al (T = Ni, Pd, Pt, Cu, Ag, Au) Heusler intermetallic compounds and also for metallic Al. We followed the approach recently reported by Laskowski and Blaha11 for simulation of isotropic values of the chemical shielding tensor σtot = σs + σo. The orbital component was computed by the extension to the GIPAW method for metallic systems described by d’Avezac et al.9 The spin component was approximated by the Fermi contact contribution to the induced magnetic hyperfine field at the nuclear position, where the first-order perturbative approach to core-level spin polarization proposed by Bahramy et al.20 was employed. From our simulations, we showed that it is possible to obtain reasonably acceptable results with an affordable number of k-points, enough to converge the target quantities, which drop significantly for the primitive cells of the intermetallics under study if compared to metallic Al. To our knowledge, this is the first report of theoretical Knight shifts in metallic systems other than simple metals. First of all, regarding the simulations performed for the ScT2Al compounds and the respective experimental data recently reported by Benndorf et al.,8 it is important to emphasize that our aim is not to offer a support to that particular work. In the scope of SS-NMR spectroscopy, combined theoretical and experimental investigations must be carried out in a collaborative way, where additional experiments with suitable zero-shift compounds are crucial to achieve optimal agreement between the simulated and experimental chemical shifts. More specifically, we aimed at expose the pros and cons of computing chemical shielding tensors in metallic systems within the FCA and the GIPAW method. The comparatively small number of k-points in the case of the intermetallics, combined with the computational cost advantage, which is normally attributed to the use of PPs with a PW basis set, are two positive aspects of the approach adopted in the present work. Even so, the complete set of theoretical chemical shifts available in Supporting Information exposes the susceptibility of the theoretical results to the choice of the PAW data sets for such simulations. Nevertheless, our calculations 25538


The Journal of Physical Chemistry C can be seen as a critical benchmark not only for the GIPAW method but also for transferability of PAW data sets in use. Since all of them were made available for regular end users in a transparent way by code developers and experts, we think that our report is in line with recent collaborative efforts to establish more systematic test procedures for PPs.23 We also point out some aspects of the approach adopted here that should be considered before it is applied to other systems. The core polarization correction for hyperfine parameters is an experimental feature available in the QEGIPAW module and only a few works19,33 have been reported that used this implementation, all with positive outcome. We have mentioned some preliminary tests, not detailed, that were conducted by considering 7Li and 45Sc nuclides, in spinpolarized and/or metallic systems. We recall that, for metallic Li, the well-known troubles in dealing with the overlap between its 1s and 2s states within the FCA31,34 have compromised the computation of σs. In the case of an isolated atom of Sc, when trying to reproduce the ρcore values reported in ref 20, we found that NCPPs systematically yielded better results than USPPs. These two cases reinforce that, besides the common parameters considered in PAW data set generation, the schemes for construction of the |ϕ̃ λ⟩ states from the AE counterparts |ϕλ⟩ are still open,28 and there is room for further development in that field, aimed to reach transferable atomic data sets more suitable for such applications. Finally, a last point also related to future developments is the huge number of k-points required in such simulations. Since the convergence of GS electronic structure is considerably cheaper than the linear response formalism used for calculation of chemical shieldings, a preselection routine of the most relevant k-points would be a valuable feature. In general, with an acceptable accuracy with respect to experimental values, our results have shown that it is possible to obtain the valuable quantitative contribution of Knight shifts to the observed chemical shifts, through computational simulations with DFT and within the FCA, for all metallic systems under study. Such simulations are relevant for advanced in situ SS-NMR studies of new materials for industrial use, like in the field of lithium ion batteries, mainly when samples are submitted to lithium intercalation/deintercalation processes.35,36 Moreover, they pave the way toward an expansion of the applicability of SS-NMR to the characterization of more complex materials, such as metallic glasses or metallic supported nanoparticles in the field of heterogeneous catalysis.

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ACKNOWLEDGMENTS

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REFERENCES

A.R.F. acknowledges a fellowship from the Brazilian National Council for Scientific and Technological Development (CNPq; Grant 249212/2013-7) and thanks Technische Universität München for support. We acknowledge the use of highperformance computing facilities and associated support services of SuperMUC, provided by the Leibniz-Rechenzentrum in Garching. Additionally, we emphasize that the features already implemented in the QE-GIPAW module SVN revision 408 were essential for the results achieved.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b08418. Five tables listing results obtained with all possible combinations of PAW data sets (PSL, PSLR, and JTH), details of k-point convergence tests, and linear relationship between Fermi contact contributions to BFc and Bext for the six ScT2Al Heusler compounds (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; phone +49 89 289 13616; fax +49 89 289 13622. Notes

The authors declare no competing financial interest. 25539


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The Journal of Physical Chemistry C (29) Zurek, E.; Pickard, C. J.; Walczak, B.; Autschbach, J. J. Phys. Chem. A 2006, 110, 11995−12004. (30) Ferreira, A. R.; Kücu̧ ̈kbenli, E.; Leitao, A. A.; de Gironcoli, S. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, No. 235119. (31) Ferreira, A. R.; Reuter, K.; Scheurer, C. RSC Adv. 2016, 6, 41015−41024. (32) Yuan, C.; Shen, X.; Cui, J.; Gu, L.; Yu, R.; Xi, X. Appl. Phys. Lett. 2012, 101, No. 021902. (33) Giacomazzi, L.; Martin-Samos, L.; Boukenter, A.; Ouerdane, Y.; Girard, S.; Richard, N. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 90, No. 014108. (34) Kresse, G.; Joubert, D. Phys. Rev. B: Condens. Matter Mater. Phys. 1999, 59, 1758−1775. (35) Ménétrier, M.; Saadoune, I.; Levasseur, S.; Delmas, C. J. Mater. Chem. 1999, 9, 1135−1140. (36) Levasseur, S.; Ménétrier, M.; Delmas, C. J. Power Sources 2002, 112, 419−427.

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