Letter pubs.acs.org/JPCL
Cite This: J. Phys. Chem. Lett. 2018, 9, 1480−1484
Quantum-Chemical Approach to NMR Chemical Shifts in Paramagnetic Solids Applied to LiFePO4 and LiCoPO4 Arobendo Mondal and Martin Kaupp* Institut für Chemie, Theoretische Chemie/Quantenchemie, Technische Universität Berlin, Sekr. C7, Straße des 17. Juni 135, 10623 Berlin, Germany S Supporting Information *
ABSTRACT: A novel protocol to compute and analyze NMR chemical shifts for extended paramagnetic solids, accounting comprehensively for Fermi-contact (FC), pseudocontact (PC), and orbital shifts, is reported and applied to the important lithium ion battery cathode materials LiFePO4 and LiCoPO4. Using an EPR-parameter-based ansatz, the approach combines periodic (hybrid) DFT computation of hyperfine and orbital-shielding tensors with an incremental cluster model for g- and zero-field-splitting (ZFS) D-tensors. The cluster model allows the use of advanced multireference wave function methods (such as CASSCF or NEVPT2). Application of this protocol shows that the 7Li shifts in the high-voltage cathode material LiCoPO4 are dominated by spin−orbit-induced PC contributions, in contrast with previous assumptions, fundamentally changing interpretations of the shifts in terms of covalency. PC contributions are smaller for the 7Li shifts of the related LiFePO4, where FC and orbital shifts dominate. The 31P shifts of both materials finally are almost pure FC shifts. Nevertheless, large ZFS contributions can give rise to non-Curie temperature dependences for both 7Li and 31P shifts.
N
neighboring transition-metal sites onto the Li atoms. It is known for molecular systems, however, that magnetic anisotropy around certain metal sites, induced by spin−orbit (SO) coupling, may give rise to so-called pseudocontact (PC) shifts that relate to the HFC anisotropy and may be parametrized by the electronic g-tensor and the zero-fieldsplitting (ZFS) D-tensor.10−12 Long-range PC shifts are used widely, for example, in the structure refinement of metalloproteins,13 and we have recently demonstrated the successful ab initio simulation of long-range PC shifts for an entire (cobalt-substituted) metalloprotein domain.14 In the regime of extended paramagnetic solids, the treatment of such PC contributions is still in its infancy. The first attempts to incorporate them via the electronic g-tensor in a doubletstate formalism were restricted to semilocal DFT functionals, and the effects on the overall shifts were moderate.15,16 We have recently also added the relevant orbital shifts to such solidstate calculations, and we found that the periodic g-tensor calculations may be replaced by an incremental cluster approach, making use of the essential locality of the g-tensor in such materials.17 So far, however, pNMR shift calculations on extended solids have completely neglected the potentially important ZFS effects. Moreover, it is known that currently available DFT approaches may be inadequate for treating ZFS
uclear magnetic resonance spectroscopy of paramagnetic substances (pNMR) is currently gaining an enormous boost due to, on one hand, substantial instrumental developments such as fast magic-angle spinning (MAS) combined with high-field instruments and, on the other hand, an improved theoretical and computational machinery allowing interpretation and even prediction.1,2 While the theoretical progress has been most notable for molecular systems, the need for quantum-chemical support is equally apparent for extended periodic magnetic solids. Here we will focus on olivine-type lithium metal phosphates (LiMPO4; M = Fe, Co). The study of these and related paramagnetic solids by NMR methods has received large interest3,4 due to their high potential as cathode materials in lithium ion batteries (and also in view of their interesting magneto-electric properties at lower temperatures;5 LiCoPO4 is also closely related to cobalt phosphates of interest in water oxidation catalysis6). The role of NMR spectroscopy on lithium-ion batteries can hardly be overestimated; it allows even the in situ study under charging or decharging conditions.7 The 7Li NMR shifts of such paramagnetic electrode materials, including phosphates like those studied here, have been interpreted extensively in terms of hyperfine-coupling (HFC) pathways as well as M−O and O−Li covalency in attempts to aid in the analysis of electronic structure and properties.3,4,8,9 In general, discussions and computations of such pNMR shifts, for these and related extended solids, have so far centered on the so-called Fermicontact (FC) shifts, which are directly related to the isotropic HFCs and thus to spin-density delocalization from the various © XXXX American Chemical Society
Received: February 7, 2018 Accepted: March 7, 2018 Published: March 7, 2018 1480
DOI: 10.1021/acs.jpclett.8b00407 J. Phys. Chem. Lett. 2018, 9, 1480−1484
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The Journal of Physical Chemistry Letters and g-tensors for highly correlated transition-metal centers (such as quartet CoII), and multireference ab initio wave function approaches may be required.18,19 Here we show that (a) PC shifts, including those enabled by ZFS, can be included into solid-state pNMR shift calculations at suitable multireference wave function levels by a cluster approach, combined with periodic (hybrid) DFT computations of hyperfine and orbital-shielding contributions, and that (b) PC shifts can be so large, for example, for the important LiCoPO4 cathode material, as to completely change the interpretation of the (7Li) NMR shifts. Our computations are based on a modern quantum-chemical implementation12 of Kurland−McGarvey theory,10 extended to the Curie−Weiss regime of a paramagnetic spin-coupled solid (eq 1).17 This formalism derives the hyperfine part of the pNMR shift tensor from EPR spin Hamiltonian parameters (g-, HFC-, and D-tensors; cf. refs 20 and 21 for related approaches) and accounts for residual exchange couplings in the Curie− Weiss temperature range by introducing the Weiss constant Θ into the temperature denominator (prefactor of the hyperfine shielding terms), that is δI = δIorb +
⎛ 1 ⎞⎛ 1 ⎜ ⎟⎜ ⎜ I kBgNμ N ⎝ T − Θ ⎠⎝ n μB
n
⎞
i
⎠
∑ gi⟨SS⟩i AI ⎟⎟
Figure 1. 2 × 2 × 2 supercell of olivine-type LiMPO4 used in the periodic calculations.
diffraction (XRD) structures and structures optimized at DFT level (using the PBE functional;24 Table S1 and Figure S1). This supercell has been used to compute δorb I (at PBE level) and AI (using PBE-based hybrid functionals with variable exactexchange (EXX) admixture17), with periodic boundary conditions (PBC) in the Gaussian-augmented plane-wave CP2K code.25 In contrast, for the g- and ZFS D-tensor calculations, mononuclear units were cut from the solid, and the phosphates were saturated with hydrogen atoms while conserving the correct oxidation states of metal centers and phosphorus (Figure 2; see Supporting Information for details). The
(1)
where μB, μN kB, T, gIN, and AI are, respectively, the Bohr magneton, the nuclear magneton, the Boltzmann constant, the absolute temperature, the nuclear g-value and HFC tensor of nucleus I. The Weiss constant Θ (−7522 and −82.1 K23 for M = Co, Fe, respectively) has been taken from orientationaveraged experimental single-crystal susceptibility measurements (our computations refer to T = 320 K). gi is the gtensor of spin center i. The (electron) spin dyadic ⟨SS⟩i represents a thermal average of the two spin operators over the eigenstates of the site ZFS Hamiltonian12 ⟨SaSb⟩ =
Q pq
∑qp Q pq⟨q|Sa|p⟩⟨p|Sb|q⟩ ∑q exp( −Eq /kBT )
, a , b = {x , y , z} (2)
⎧ e−Eq / kBT , Eq = Ep ⎪ ⎪ =⎨ kT ⎪− B [e−Ep / kBT − e−Eq / kBT ], Eq ≠ Ep ⎪ Ep − Eq ⎩
Figure 2. Local cluster with six Li ions used for the incremental cluster-model computations of g- and ZFS D-tensors for both materials.
The off-diagonal elements of the symmetric matrix Qpq bring in magnetic couplings between the eigenstates of the ZFS Hamiltonian, important for the correct behavior when going to low temperatures.12,20,21 The sum on the right hand side of eq 1 (normalized by the number of spin centers n interacting with nucleus I) allows us to assemble the magnetic anisotropy in the solid from local g- and ZFS D-tensors of the individual spin centers within the above-mentioned incremental cluster model. This procedure is based on the assumption that singleion anisotropies are much larger than anisotropic exchange interactions, which is clearly expected to hold true for the two materials studied here, where the transition-metal sites are separated by at least four bonds. Otherwise, the approach would have to be extended to also include anisotropic exchange. A 2 × 2 × 2 supercell of the olivine-type structure of both LiFePO4 and LiCoPO4, used in the pNMR calculations, is shown in Figure 1. We employed both the experimental X-ray
multireference second-order n-electron valence perturbation theory level (NEVPT226) has been applied to the g- and Dtensor computations on the resulting complexes, using the ORCA quantum-chemical code27 (DFT methods underestimate both quantities for M = Co, whereas CASSCF overshoots; cf. Tables S3 and S4). The computed site tensors have been inserted back into the extended structure to assemble the supercell tensors, making use of point-group symmetry. This procedure has been shown to give excellent agreement with periodic calculations for less anisotropic g-tensors,17 and it does so also for LiFePO4 (Table S2). For the particularly sensitive, large g-tensor of LiCoPO4, it leads to an overestimate by ca. 9% for giso (compare PBE/PBC against PBE cluster-model results in Table S3), and we expect a similar overshooting also for D. Combination of all 1481
DOI: 10.1021/acs.jpclett.8b00407 J. Phys. Chem. Lett. 2018, 9, 1480−1484
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The Journal of Physical Chemistry Letters
2.728, g22 = 2.4334, g33 = 2.030, giso = 2.397, D = 60.9 cm−1, and E/D = 0.151 (at the optimized structure, cf. Tables S3 and S4). This may be compared to experimental estimates from neutron diffraction giso = 2.36, (g33 − g11)/giso ≈ 0.3.28,29 We can also extract an effective g-tensor for the lowest Kramers doublet of the system and compare it with EPR data (see Table S8).5 Together the data indicate that due to some inaccuracies in the cluster treatment (Table S3) and due to a general ca. 10% overestimate at NEVPT2 level for the D-tensors of distorted octahedral CoII sites,18,19 for this most critical and anisotropic case we may overshoot both g-shift- (deviations from ge) and Dtensors by up to ∼20%. Inserting our currently best D- and g-tensor results (NEVPT2 level) into eq 1 to compute 7Li NMR shifts for LiCoPO4 as a function of the EXX admixture in the HFC computations, we obtain the 7Li shifts shown in Figure 4
contributions in eq 1 is done using in-house scripts, providing the currently most complete computational evaluation of pNMR shifts for extended paramagnetic solids in the Curie− Weiss temperature regime. We start with LiFePO4, where reliable g- and D-tensor data are available from single-crystal measurements to judge independently the accuracy of our computation of these quantities:23 g11 = 2.22, g22 = 2.13, g33 = 2.02, giso = 2.12, D = 10.1 cm−1, and E/D = 0.178. Computationally, we obtain g11 = 2.177, g22 = 2.123, g33 = 2.002, giso = 2.101, D = 9.7 cm−1, and E/D = 0.202 (NEVPT2 level at the PBE-optimized structure, Tables S2 and S4), in excellent agreement with experiment. This suggests a high accuracy of the incremental NEVPT2based cluster modeling of g- and ZFS tensors for this system. The magnetic anisotropy is much smaller than in the cobalt case28 (see below) but still significant. Figure 3 shows the 7Li shifts obtained with the help of these g- and D-tensor data for LiFePO4 (see also Tables S5 and S6),
Figure 4. Isotropic 7Li shifts (relative to LiClaq) computed for LiCoPO4 as a function of EXX admixture in the PBE-based hybrid functional used for the HFC calculations. g- and D-tensors obtained at NEVPT2 level, orbital shifts at PBE level, at the optimized structure, Θ = −75 K. The shifts are broken down into individual contributions (see text and eq S410). See Table S9 for further details and Table S10 and Figure S3 for results at the XRD structure.
Figure 3. Isotropic 7Li shifts (relative to LiClaq) computed for LiFePO4 as a function of EXX admixture in the PBE-based hybrid functional used for the HFC calculations. g- and D-tensors obtained at NEVPT2 level, orbital shifts at PBE level, at the optimized structure, Θ = −82 K. The shifts are broken down into individual contributions (see text and eqs S4 and S510). See Table S5 for further details and Table S6 and Figure S2 for results at the XRD structure.
(Tables S9 and S10). The three PC terms (eq S4), except the one depending on g⟨̃ SS⟩AIdip(red), for which preliminary DFT attempts have been reported in a doublet-state formalism,15−17 had so far not been considered for extended solids. For LiCoPO4, the PC terms dominate the overall negative Li shifts. Even at 40% EXX admixture for the HFC calculations and taking into account a computed ca. +10 ppm orbital shift (Table S9), we would arrive at best at around −20 ppm for the overall shift when neglecting the PC shifts.30 This falls significantly short of the observed experimental range of −86 to −111 ppm.8,31,32 The PC shift contributions overall account for about −133 ppm and would thus let us overshoot to about −140 to −160 ppm at realistic EXX admixtures of 20−35% for the HFCs (Figure 4, Tables S9 and S10). Apart from possible remaining error sources regarding δorb, HFCs, Θ, and simulation versus measurement temperature, we regard the likely ∼20% too large g-shift- and D-tensors at the chosen computational level (see above and SI) as the main suspects for this discrepancy. It seems that the cluster model accounts for about half of the errors in the g- and D-tensors (see above), and the accurate but not perfect NEVPT2 computations account for the other half. Scaling those two quantities down by a corresponding amount provides shifts close to the experimental range (Figure S4). Even after scaling them down in this way, the previously neglected PC shift contributions are clearly decisive for the
dependent on the EXX admixture of the hybrid functionals used for the periodic HFC calculations (relative to 1 M aqueous LiCl, cf. SI). The sign of the FC shifts, and thus of the total shifts, even changes with increasing EXX admixture, as the (small) isotropic HFCs also change sign (Table S7). At 25− 30% EXX admixture, the total shifts are still somewhat too high compared with experiment; at 35−40% the values are within the experimental range (Figure 3). Figure 3 also provides a color-coded breakdown into different contributions to the isotropic shifts, as detailed in eqs S4 and S5 in the Supporting Information:10 the terms dominated by AIFC that make up the FC shift δFC (depending in the numerator of eq S4 on, respectively, ge⟨SS⟩AIFC (light blue), Δgiso⟨SS⟩AIFC(dark blue), Δg⟨̃ SS⟩AIFC (magenta)), those accounting for the PC shift δPC (Δg⟨̃ SS⟩AIdip(red), ge⟨SS⟩AIdip (yellow), Δgiso⟨SS⟩AIdip (cyan)), as well as δorb (green). In contrast with LiCoPO4 (see below), the PC shift contributions are relatively small, summing up to only ca. 2−4 ppm (depending on input structure). Indeed, at the chosen computational levels, the orbital shifts (PBE results) of ca. 6−8 ppm (Tables S5 and S6) are larger here than the overall PC contributions. Turning to LiCoPO4, the direct comparison with experimental g- and D-tensors is somewhat hampered by limited experimental data. Our best NEVPT2 calculations give g11 = 1482
DOI: 10.1021/acs.jpclett.8b00407 J. Phys. Chem. Lett. 2018, 9, 1480−1484
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The Journal of Physical Chemistry Letters overall much more negative 7Li shifts in LiCoPO4 compared with LiFePO4, consistent with the appreciable importance of single-ion anisotropy also for the low-temperature magnetoelectric properties of this material.22 The ca. 100 ppm negative, previously neglected PC shifts for LiCoPO4 significantly affect the interpretation of the 7Li shifts in terms of hyperfine couplings and covalency. If one would extract 7Li HFCs and thus lithium spin-density values from the overall shifts, after correcting for the temperature-independent contributions7 one would obtain a far too negative HFC for LiCoPO4 (Table S11) and a much more reasonable value for LiFePO4 (Table S7). Importantly, inclusion of the orbital shifts, which so far have not been considered in most pNMR shift calculations for extended solids,17 is also crucial when trying to quantitatively account for the lithium shifts. Application of the same computational levels to the 31P shifts shows a clear dominance of the FC shifts (Figures S2 and S3 and Tables S5, S6, S9, and S10) due to the much larger delocalization of spin density (Tables S7 and S11) onto the phosphorus atoms. For LiFePO4, the PC shift terms now account for less than +5 ppm out of ca. + 3400 ppm (at 25% EXX admixture, with an orbital shift of about +10 ppm). For LiCoPO4, the various 31P PC shift contributions have opposite signs and together account for less than −10 ppm (with orbital shifts of about −20 ppm), also negligible compared with the ca. + 2400 ppm FC shifts at the 25% EXX level. About +400 ppm of the FC shifts are contributed by the spin−orbit-induced deviation of the isotropic g-value from the free-electron one via the term depending on Δgiso⟨SS⟩AIFC compared with about +160 ppm for LiFePO4. Figure 5 shows the inverse temperature dependence of the computed 7Li shift contributions for LiCoPO4. Deviations from
influence of ZFS even for those terms we consider to be FC shifts. In summary, a new computational methodology that combines high-level ab initio multireference wave function calculations of g- and D-tensors on clusters with periodic solidstate DFT calculations of hyperfine couplings and orbital shieldings provides the first access to full NMR shift calculations for paramagnetic solids, including the major spin−orbit-related (“pseudocontact”) contributions arising for magnetically anisotropic metal centers. These contributions have been shown to be crucial for the quantitative and even for the qualitative computation and proper interpretation of the 7Li NMR shifts of the important LiCoPO4 material, somewhat less important for the 7Li shifts for the related LiFePO4, and comparatively unimportant for the 31P shifts in the same materials (but affecting the temperature dependence!). Magnetic anisotropy effects are expected to be similarly crucial for NMR shifts in many other relevant paramagnetic materials, for example, when containing Co or Ni and in cases where FC shifts are small due to various reasons, but also clearly for compounds that incorporate heavier d- or f-metal centers. Inclusion of pseudocontact as well as orbital shift contributions in computational studies should allow improved interpretations of NMR measurements for paramagnetic solids in various fields of research.
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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.jpclett.8b00407. Further theoretical aspects, computational details, Tables and Figures with structural data, EPR parameters, shift analyses, and temperature dependences of shift contributions (PDF)
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Martin Kaupp: 0000-0003-1582-2819 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work has been carried out within the framework of the pNMR initial training network (Marie Curie Actions, EU Seventh Framework Programme, FP7/2007-2013, REA grant no. 317127), and we thank the groups of Clare P. Grey (Cambridge), Andrew J. Pell (Stockholm), and Juha Vaara (Oulu), as well as Dr. Shadan Ghassemi Tabrizi (Berlin) for helpful discussions. Further support by the UniCat Berlin DFG excellence cluster and HPC resources from the North-German Supercomputing Alliance (HLRN) are gratefully acknowledged.
Figure 5. Inverse temperature dependence of the computed 7Li shifts (relative to LiClaq) and of individual shift contributions for LiCoPO4 (at optimized structure). HFC tensor obtained at PBE0 level (25% EXX); g-tensor and D-tensor obtained at NEVPT2 level. Curvature indicates deviations from a Curie 1/T behavior.
a linear Curie behavior arise from the dominant ZFS-derived contributions (in particular, the one depending on ge⟨SS⟩AIdip). Deviations from Curie behavior are also apparent from experimental 7Li shift plots.9 Interestingly, even the 7Li shifts in LiFePO4 (Figure S5) and the 31P shifts in both materials (Figures S6 and S7) exhibit some deviations from Curie behavior, despite being clearly dominated by FC terms. This is due to significant deviations of ⟨SS⟩i from the S(S + 1)/3 expected in a doublet-state formalism due to an appreciable
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REFERENCES
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