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2006, 110, 21403-21407 Published on Web 10/06/2006
First-Principles Calculations within Periodic Boundary Conditions of the NMR Shielding Tensor for a Transition Metal Nucleus in a Solid State System: The Example of 51V in AlVO4 L. Truflandier,* M. Paris, C. Payen, and F. Boucher Institut des Mate´ riaux Jean Rouxel, UMR 6502, UniVersite´ de Nantes - CNRS, 2, rue de la Houssinie` re BP 32229, 44322 Nantes Cedex, France ReceiVed: July 27, 2006; In Final Form: August 30, 2006
We present the first density functional theory based calculations of NMR shielding parameters for a transition metal nucleus using periodic boundary conditions. These calculations employ the gauge-including projected augmented-wave pseudopotential approach. The quality of this method is discussed by comparing experimental and calculated chemical shift tensor eigenvalues for the quadrupolar 51V nucleus in the diamagnetic solidstate compound AlVO4. Furthermore, the combination of shielding tensor with fast and accurate projector augmented-wave electric field gradient tensor calculations allows us to determine the relative orientation of these two tensors.
Introduction Nuclear magnetic resonance (NMR) spectroscopy is a powerful technique to investigate the atomic structure of a solid.1 Indeed, NMR spectra can provide useful information on the local chemical environment, such as coordination numbers, bond types, number of nonequivalent sites, and degree of polyhedra distortion. In the case of solid state compounds, however, the assignment and interpretation of the resonance lines often remain ambiguous. This problem can be partially overcome by performing first-principles calculations of shielding and/or electric field gradient (EFG) tensors. For periodic systems, EFG tensor calculation methods have been developed in the field of density functional theory (DFT) using linearized augmented-planewave2-4 (LAPW) and projector augmented-wave5,6 (PAW) approaches. Applications for quadrupolar nuclei of transition metals have already been investigated4,7-10 and fairly good agreements with experimental values have been observed. Moreover, thanks to the sensitivity of the EFG parameters, combined with accurate geometry optimizations, determinations of imprecise X-ray diffraction (XRD) structures appear to be emerging outlooks.6,11,12 Despite the continuous development of quantum chemical methods, calculations of shielding parameters for a transition metal nucleus in an extended system have not been presented so far. In fact, the commonly used approaches were based on molecular calculations using localized atomic orbitals (LAO)13-15 and, unfortunately, investigations of inorganic systems depicted by means of finite models yield to strong convergence difficulties.16-20 Recently, the “gaugeincluding projected augmented-wave” (GIPAW) pseudopotential approach,21 explicitly taking into account the periodicity of the system by using a plane wave (PW) basis set, has been developed within the framework of the density functional perturbation theory22 in the presence of a magnetic field. The asset of the GIPAW approach compared to other pseudopotential methods,23,24 is the ability to keep the nodal properties of the 10.1021/jp0648137 CCC: $33.50
wave functions in the neighborhood of the nucleus. Considering the rigid contribution of core electrons with respect to NMR parameters,25 a precision comparable to all electron calculations can be achieved.21 Nevertheless, the application to large systems was, so far, limited to the only elements of the first three rows of the periodic table.26-29 Due to the wide range of applications associated with the transition metal solid-state compounds, calculation of the shielding parameters for transition metal nuclei embedded in infinite compounds would be a very useful tool for NMR solid-state studies. In this letter, we present the first 51V lines assignments in a diamagnetic inorganic large system, based on DFT shielding tensor calculations within periodic boundary conditions. Numerous NMR measurements have been performed on 51V in order to probe the vanadium sites in solids that are of interest in the fields of, for instance, heterogeneous catalysis.30,31 Here we focus our attention on the apparently simple compound AlVO4, its crystal structure being debated for several years.32-34 In fact, the combination of XRD measurements, NMR experiments, and first principle calculations has been necessary to fully understand its structural properties. In a very recent work, Hansen et al.35 have proposed assignments of the inequivalent Al and V crystallographic sites based on LAPW calculations of EFG parameters. These authors have shown that a high level of DFT structural optimizations is necessary and greatly improve the agreement between experimental36 and calculated EFG parameters.35 Using the GIPAW calculations, the assignments of the three experimental 51V shifts are unambiguous considering the optimized structure. The present results support earlier work on the atomic structure of AlVO4. We also report on PAW-EFG tensor calculations consistent with the published LAPW calculations. Furthermore, we show that combining GIPAW-shielding and PAW-EFG calculations gives access to the relative orientation of these two tensors. This point is discussed in terms of Eulerian angles which are © 2006 American Chemical Society
21404 J. Phys. Chem. B, Vol. 110, No. 43, 2006
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calculated and compared to the experimental values. Finally, we discuss (i) the reliability of the GIPAW approach for 3d transition metals shielding constants calculations, (ii) the utmost cautions needed to compare experimental and theoretical shielding tensor components, and (iii) the relevance of firstprinciples calculations to fill in, for specific cases, experimental deficiencies. Computational Details and Conventions All NMR parameter calculations and geometry optimizations were carried out at the DFT level of theory using the PW9137 functional. NMR shielding and EFG tensors were computed using the GIPAW and PAW methods, respectively, both implemented in the CASTEP21,26,38 code. The so-called “ultrasoft” pseudopotentials39 (USPP) were used to describe the interaction of the valence electrons with the nuclei and core electrons. The selections of core states were the common ones: 1s2 for O and 1s22s22p6 for Al. We used two and three projectors in each s and p channel for O and Al, respectively. For the third row atoms, the local potential must be augmented,21 thus two projectors were affected to the d angular momentum of the Al atom. The core radii, beyond which the pseudo wave functions match the all-electron ones, were set to 1.3 and 2.0 Bohr for O and Al, respectively. Difficulties appear from the fourth row elements, especially for the 3d transition metals,40,41 where one must carefully examine the influence of the pseudopotential expansion on the shielding tensor components. We observed42 that the inclusion of the 3s and 3p atomic functions into the valence is required to obtain accurate and converged values for the 51V NMR parameters. Using a core radius of 2.0 Bohr, one projector for the 3s state and two projectors per remaining channels were necessary. The same USPP settings for the V and O atoms were used for the DFT geometry optimizations, whereas only two projectors per angular momentum were used for the Al atom. For the atomic relaxation only, a force tolerance parameter of 0.05 eV/Å was used, while for the whole cell optimization, a stress tolerance parameter of 0.1 GPa was imposed. All calculations were converged with respect to the kinetic energy cutoff used for the PW basis set expansion and to the k-point density needed for the Brillouin zone integration. A cutoff of 600 eV (800 eV in the case of the complete cell) and a 3×3×2 Monkhorst-Pack k-point grid43 (9 irreducible k-points) were sufficient to achieve convergence. The conventions used to calculate the chemical shift parameters {δiso, δaniso, ηδ}, from chemical shift tensor eigenvalues {δxx, δyy, δzz}, are defined as follows:
1 isotropic component: δiso ) (δxx + δyy + δzz) 3
(1)
anisotropy component: δaniso ) δiso - δzz
(2)
δxx - δyy δaniso
(3)
|δzz - δiso| g |δxx - δiso| g |δyy - δiso|
(4)
asymmetry component: ηδ ) with
The shielding parameters {σiso, σaniso, ησ} are deduced from the calculated eigenvalues using relations similar to 1, 2, and 3. One obtains σiso ) δiso and ησ ) ηδ while σaniso ) -δaniso according to the relation
δij ) -a × [σij - σref]
(5)
Figure 1. Representation of the AlVO4 structure. The three symmetryinequivalent vanadium sites are labeled. The AlOx polyhedrons are shown in blue.
where δij and σij are respectively the chemical shift and absolute shielding tensor components, a is a slope (equal to unity in experiments), and σref is the isotropic shielding of a reference compound in liquid phase. Unfortunately, first-principles calculations of σref involve the consideration of rovibrational and intermolecular effects. To circumvent such tricky calculations, σref was evaluated assuming a linear regression between computed σiso and experimental δiso values and putting the slope to unity. The quadrupolar constant Cq and the asymmetry parameters ηq are calculated from the EFG tensor eigenvalues {Vxx, Vyy, Vzz}, with the convention |Vzz| g |Vxx| g |Vyy|,
Cq )
eQVzz , h
ηq )
Vyy - Vxx Vzz
(6)
where Q is the nuclear quadrupolar moment (Q ) -0.052 barns44) and Vzz the principal component. The orientation of the shielding tensor in the EFG tensor principal axes system is described by the Euler angles {ψ,χ,ξ} within the Rose conventions.45 When doing NMR measurements on powder samples, there are sixteen sets of Euler angles that leads to identical spectra.46 As a result, we narrowed the definition intervals of Euler angles down to the interval 0 e ψ e π and 0 e (χ, ξ ) e π/2 to obtain a unique set of Euler angles, as done by Skibsted et al.47 Results and Discussion AlVO4 Structure Considerations. The 3D AlVO4 crystal structure is described32,33 in the centrosymmetric P1h space group (Z ) 6). The unit cell contains three different types of VO4 tetrahedra (Figure 1) connected through corners with edgesharing AlO5 polyhedron and AlO6 octahedra. As already noted,35 the AlVO4 structure has to be optimized before calculating the NMR parameters. We have performed two different optimizations: one keeping the experimental cell parameters given by Coelho32 (labeled as Coelho-Opt) and the other one obtained by relaxing also the cell shape (labeled as Fully-Opt). The cell parameters published by Coelho have been preferred to those reported by Arisi et al.33 as the X-ray wavelength used in the former case (Cu KR radiation) is known with a better accuracy than in the latter one (synchrotron radiation with λ ) 1.2 Å). Bra´zdova´ et al.34 have previously done a complete relaxation of the atomic structure using the VASP48,49 code. While using the same computational parameters, small discrepancies are observed for the cell parameters
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J. Phys. Chem. B, Vol. 110, No. 43, 2006 21405
TABLE 1: Experimental and Theoretical Lattice Parameters of AlVO4 Coelhoa
experimental theoretical VASPb theoretical CASTEPc a
a (Å)
b (Å)
c (Å)
R (deg)
β (deg)
γ (deg)
vol (Å3)
6.541 6.600 (1.3%)d 6.569 (0.4%)
7.760 7.831 (0.9%) 7.794 (0.4%)
9.136 9.248 (1.2%) 9.199 (0.7%)
96.18 96.20 96.22
107.24 107.24 107.24
101.40 101.55 101.48
427.26 440.03 (3.0%) 433.72 (1.5%)
Reference 32. b Reference 34. c This work. d Percentages of deviation with respect to the experimental values are given in parentheses.
TABLE 2: Experimental and Calculated 51V Shielding Parameters (δiso, δaniso, ηδ) and EFG Quadrupolar Parameters (Cq,ηq) in AlVO4 site
V(1)
V(2)
V(3)
method experimenta Coelho-Opt this work Wien2kd Fully-Opt this work Wien2ke experimenta Coelho-Opt this work Wien2kd Fully-Opt This work Wien2ke experimenta Coelho-Opt this work Wien2kd Fully-Opt this work Wien2ke
δiso (ppm)
δaniso (ppm)
ηδ
Cq (MHz)
ηq
-744 ( 1 -738b(-738c)
-120 ( 6 -96
0.72 ( 0.10 0.55
2.35 ( 0.03 2.17 2.29 2.31 2.27 4.05 ( 0.05 3.86 3.99 -4.12 -4.09 3.08 ( 0.02 -3.10 -3.20 -3.24 -3.22
0.93 ( 0.03 0.94 0.85 0.86 0.80 0.84 ( 0.02 0.98 0.89 0.85 0.73 0.62 ( 0.02 0.72 0.63 0.74 0.67
-736b
-96
0.58
-661 ( 1 -670b (-667c)
87 ( 8 -77
0.74 ( 0.17 0.86
-671b
-80
0.77
-776 ( 1 -773b (-775c)
-82 ( 7 -62
0.88 ( 0.11 0.50
-773b
-65
0.53
Reference 36. b Calculated 51V chemical shifts δiso are reported relative to the reference values of -2004 and -2014 ppm, for the Coelho-Opt and Fully-Opt structures, respectively (see computational details). c Calculated 51V chemical shifts have been corrected with respect to a slope equal to 1.04,42 i.e. δiso* ) -1.04 × [σiso - 1975]. d Reference 35, calculated LAPW-EFG parameters were obtained after atomic coordinates optimization employing the Wien2k52 code, using the XRD (Coelho32) lattice parameters. e Reference 35; calculated LAPW-EFG parameters were obtained after atomic coordinates optimization employing the Wien2k code, using previously DFT optimized34 lattice parameters presented in Table 1. a
between the VASP results and our CASTEP optimization (FullyOpt), the latter being closer to the experimental values (Table 1). This improvement is related to the quality of our generated pseudopotential.50 51V Shielding Parameters. Using the isotropic chemical shift obtained with eq 5 for the Coelho-Opt structure (Table 2), the direct assignments of the three experimental 51V resonances are straightforward. The discrepancies are of the order of the ppm. These accurate results can even be improved when a correction is used on the slope of eq 5, with a value a ) 1.04(4).51 This corrective factor has been obtained from a linear least-squares regression between calculated 51V isotropic shieldings and experimental chemical shifts, using a series of eighteen vanadium sites in various vanadate extended systems, where atomic coordinate relaxation has been performed.42 Concerning the anisotropy parameters, the agreement is reasonable for V(1) and V(3) sites, whereas for the V(2) a sign inversion is observed (Table 2). Oppositely, the asymmetry parameters are poorly reproduced for the V(1) and V(3) sites, in contrast to V(2), which exhibits a calculated value within the experimental error limit. To check the overall quality of the correlation between experimental and calculated shielding parameters the eigenvalues {δxx, δyy, δzz} of the second rank tensor have been considered. Experimental values have been obtained from chemical shift parameters {δiso, δaniso, ηδ} using eqs 1 to 3, and theoretical values have been deduced from the shielding eigenvalues {σxx, σyy, σzz}, using eq 5 where a has been set equal to 1 or 1.04 when a correction was considered. Our results are collected in Table 3, along with the classification of the eigenvalues according to the relation 4. As expected, weak discrepancies are observed between corrected and noncorrected eigenvalues. For V(1) and V(3) nuclei, the order imposed by relation 4 is respected for calculated parameters. Nevertheless, the observed differences between experimental and theoretical eigenvalues are highly varying from 35 to 1 ppm (absolute values). Such deviations are masked in the average isotropic values and
TABLE 3: Classifications of the Experimental and Calculateda 51V Chemical Shift Tensor Eigenvalues in AlVO4 site
ii
δii calcb
δii* calcc
δii expt
V(1)
zz xx yy zz xx yy zz xx yy
-642 -812 -760 -592 -742 -675 -711 -820 -789
-638 -815 -760 -586 -742 -672 -710 -823 -791
-624 -847 -761 -748 -585 -650 -694 -853 -781
V(2)
V(3)
a For the Coelho-Opt structure (see text for details) and according to the relation |δzz - δiso| g |δxx - δiso| g |δyy - δiso|. b Using the equation δii ) -[σii - 2004]. c Using the corrected equation δii* ) -1.04 × [σii - 1975].
magnified in the anisotropy and asymmetry parameters. On the V(2) site, we notice an inversion of the δxx and δzz theoretical eigenvalues. The very weak difference of 5 ppm between theoretical |δzz - δiso| and |δxx - δiso| leads to an erroneous assignment of the eigenvalues and then tensor orientation. Consequently, one obtains a sign inversion for the calculated V(2) anisotropy. The mean absolute deviation of 15 ppm calculated with the nine eigenvalues (the wrong assignment for the δxx and δzz principal components being corrected) may be an estimation of the theoretical accuracy one can reach for the 51V shielding parameters. Finally, the quality of the agreement between experimental and theoretical tensor eigenvalues can be easily observed in Figure 2. 51V Electric Field Gradient Parameters and Tensors Orientations. In Table 2 we compare our PAW results with previous LAPW calculations and experimental values of EFG parameters {Cq,ηq} for the Coelho-Opt and Fully-Opt structures. Considering the Cq parameters, the PAW and LAPW methods agree well within less than 4% and 2% discrepancy for the Coelho-Opt and Fully-Opt structures, respectively. Differences
21406 J. Phys. Chem. B, Vol. 110, No. 43, 2006
Letters Concluding Remarks
Figure 2. Experimental versus calculated (corrected values, see Table 3) 51V chemical shift tensor eigenvalues for the three V sites in the AlVO4 compound. The solid line represents perfect agreement between calculation and experiment.
TABLE 4: Experimental and Calculated Euler Angles between Shielding and EFG Tensors, for the V(1), V(2), and V(3) Sites in AlVO4 site V(1) V(2) V(3) a
method a
exp. calc.b exp.a calc.b exp.a calc.b
ψ (deg)
χ (deg)
ξ (deg)
137 ( 30 0 114 ( 30 1 130 ( 30 8
27 ( 7 72 29 ( 23 80 25 ( 12 58
58 ( 30 70 88 ( 31 71 15 ( 20 80
Reference 36. b From the Coelho-Opt structure.
originating from the electronic structure approaches and computational parameters have been already discussed.6 The weak discrepancies observed may also be attributed to the small differences between the two optimized geometries. Anyhow, as Hansen et al. already noted,35 the agreement between theoretical and experimental values is excellent. We note that a complete optimization of the structure does not worsen the simulation, in agreement with the very weak modification we obtain for the lattice parameters. Concerning the V(2) site, the sign inversion of δaniso is present for both optimizations (Table 2) and the sign inversion of Cq between the Coelho-Opt and Fully-Opt structures, also found by Hansen et al., can be explained by an ηq very close to 1. We finally want to outline that EFG is directly evaluated from the ground state density; therefore, time resources needed in PAW-EFG compared to LAPW-EFG computations are greatly reduced. As the three calculated parameters δiso, δaniso, and ηδ of the chemical shift interaction, as well the Cq and ηq parameters of the quadrupolar interaction, are quite well reproduced, we can trust upon the orientation of theses tensors in the crystal frame and, hence, to their relative orientations. Nevertheless, the calculated Euler angles differ significantly from the measured ones (Table 4). We believe that these differences derive from the difficulties to obtain reliable experimental {ψ,χ,ξ} values in AlVO4 due to its small chemical shift anisotropy. More reliable comparison between experimental and calculated Euler angles would be obtained by studying compounds exhibiting larger chemical shift anisotropy. This will be shown in a forthcoming publication dealing with pyro- and metavanadates.42
In summary, accurate calculations of 51V isotropic chemical shift in large systems have been performed thanks to a method which uses translational symmetry in order to reproduce bulk effects. Therefore, direct assignment of 51V solid state NMR resonances is allowed. We have demonstrated that principal components of the shielding tensors should be considered for the correlation between calculated and experimental results, to avoid erroneous conclusions on the quality of the theoretical model. While powder NMR spectra give only access to relative orientation between EFG and shielding tensors, first-principles calculations also provide the orientations of theses tensors in the crystal frame which can only be obtained experimentally by single-crystal NMR measurements. Moreover, efficiency and accuracy of the EFG-PAW method used with our generated V pseudopotential allow applications on large systems not tractable by LAPW methods. We are currently extending this work to other 3d transition metals, such as 47/49Ti, and promising results have already been obtained.42 Acknowledgment. The computational presented in this work have been carried out at the “Centre Re´gional de Calcul Intensif des Pays de la Loire” (CCIPL), financed by the French Research Ministry, “the Re´gion Pays de la Loire”, and Nantes University. We thank CCIPL for CASTEP licenses financial support. L.T. gratefully acknowledges useful discussions with C. Pickard. We also wish to thank J.-C. Ricquier for graphic arts and N. Dupre´ for carefully reading of the manuscript. Supporting Information Available: Tables giving fractional atomic coordinates for Coelho-Opt and Fully-Opt AlVO4 structures, experimental and calculated NMR parameters for 27Al sites. This material is available free of charge via Internet at http://pubs.acs.org. References and Notes (1) Harris, R. K. Solid State Sci. 2004, 6, 1025. (2) Wimmer, E.; Krakauer, H.; Weinert, M.; Freeman, A. J. Phys. ReV. B 1981, 24, 864. (3) Blaha, P.; Schwarz, K.; Herzig, A. Phys. ReV. Lett. 1985, 54, 1192. (4) Blaha, P.; Schwarz, K.; Dederichs, P. H. Phys. ReV. B 1988, 37, 2792. (5) Blo¨chl, P. E. Phys. ReV. B 1994, 50, 17953. (6) Petrilli, H. M.; Blo¨chl, P. E.; Blaha, P.; Schwarz, K. Phys. ReV. B 1998, 57, 14690. (7) Schwarz, K.; Ambrosch-Draxl, C.; Blaha, P. Phys. ReV. B 1990, 42, 2051. (8) Dufek, P.; Blaha, P.; Schwarz, K. Phys. ReV. Lett. 1995, 75, 3545. (9) Bastow, T. J.; Burgar, M. I.; Maunders, C. Solid State Commun. 2002, 122, 629. (10) Bredow, T.; Heitjans, P.; Wilkening, M. Phys. ReV. B 2004, 70, 115111. (11) Hansen, M. R.; Madsen, G. K. H.; Jakobsen, H. J.; Skibsted, J. J. Phys. Chem. A 2005, 109, 1989. (12) Zhou, B.; Giavani, T.; Bildsøe, H.; Skibsted, J.; Jakobsen, H. J. Chem. Phys. Lett. 2005, 402, 133. (13) Dios, A. C. d. Prog. Nucl. Magn. Reson. Spectrosc. 1996, 29, 229. (14) Helgaker, T.; Jaszunski, M.; Ruud, K. Chem. ReV. 1999, 99, 293. (15) Facelli, J. C. Concepts Magn. Reson. Part A 2003, 20A, 42. (16) Xue, X.; Kanzaki, M. J. Phys. Chem. B 1999, 103, 10816. (17) Valerio, G.; Goursot, A. J. Phys. Chem. B 1999, 103, 51. (18) Tossell, J. A. Chem. Phys. Lett. 1999, 303, 435. (19) Tossell, J. A.; Cohen, R. E. J. Non-Cryst. Solids 2001, 120, 13. (20) Tossell, J. A.; J., H. J. Phys. Chem. B 2005, 109, 1794. (21) Pickard, C. J.; Mauri, F. Phys. ReV. B 2001, 63, 245101. (22) Gonze, X. Phys. ReV. A 1995, 52, 1096. (23) Mauri, F.; Pfrommer, B. G.; Louie, S. G. Phys. ReV. Lett. 1996, 77, 5300. (24) Sebastiani, D.; Parrinello, M. J. Phys. Chem. A 2001, 105, 1951. (25) Gregor, T.; Mauri, F.; Car, R. J. Chem. Phys. 1999, 111, 1815. (26) Profeta, M.; Pickard, C. J.; Mauri, F. J. Am. Chem. Soc. 2003, 125, 541.
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J. Phys. Chem. B, Vol. 110, No. 43, 2006 21407 (42) Truflandier, L.; Boucher, F. et al., submitted to Phys. ReV. B. (43) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (44) Pyykko¨, P. Mol. Phys. 2001, 99, 1617. (45) Rose, M. E. Elementary Theory of Angular Momentum; Wiley: New York, 1957. (46) Fernandez, C.; Bodart, P.; Amoureux, J. P. Solid State Nucl. Magn. Reson. 1994, 3, 79. (47) Skibsted, J.; Nielsen, N. C.; Bildsøe, H.; Jakobsen, H. J. Chem. Phys. Lett. 1992, 188, 405. (48) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 48, 13115. (49) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (50) In the previous work, the authors have used the VASP V_pv pseudopotential which keep in valence only the 3p state. For comparison, we have generated a new pseudopotential using the same parameters. New results match our 3s3p valence states pseudopotential. Finally, we suppose that our choice of pseudization functions which governs at least in part the pseudopotential convergence improve the pseudization scheme implemented in VASP. (51) The value of a and its standard error (in parentheses) do not depend significantly on the number of reference compounds. In all cases, the fitted value of a is different from the expected value a ) 1. This is probably due to the DFT limitations. This point will be discussed in detail in a forthcoming publication. (52) Blaha, P.; Schwarz, K.; Madsen, G. K. H.; Kvaniscka, D.; Luitz, J. In An Augmented Plane WaVe + Local Orbitals Program for Calculating Crystal Properties; Schwarz, K., Ed.; Techn. Universita¨t: Austria, 2001.
