ZORA Study of Cadmium Magnetic Shielding Tensors: Analysis

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Quantum Electronic Structure

A DFT/ZORA Study of Cadmium Magnetic Shielding Tensors: Analysis of Relativistic Effects and Electronic-State Approximations Sean T. Holmes, and Robert W. Schurko J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b01296 • Publication Date (Web): 05 Feb 2019 Downloaded from http://pubs.acs.org on February 6, 2019

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Journal of Chemical Theory and Computation

A DFT/ZORA Study of Cadmium Magnetic Shielding Tensors: Analysis of Relativistic Effects and Electronic-State Approximations

Sean T. Holmes and Robert W. Schurko* Department of Chemistry and Biochemistry, University of Windsor, Windsor, ON, Canada N9B 3P4 *Author to whom correspondence should be addressed. E-mail: [email protected] Tel: (519) 253-3000 x3548 - Fax: (519) 973-7098

1 ACS Paragon Plus Environment

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Abstract Theoretical considerations are discussed for the accurate prediction of cadmium magnetic shielding tensors using relativistic density functional theory (DFT). Comparison is made between calculations that model the extended lattice of the cadmium-containing solids using periodic boundary conditions and pseudopotentials with calculations that use clusters of atoms. The allelectron cluster-based calculations afford an opportunity to examine the importance of (i) relativistic effects on cadmium magnetic shielding tensors, as introduced through the ZORA Hamiltonian at either the scalar (SC) or spin-orbit (SO) levels and (ii) variation in the class of the DFT approximation. Twenty-three combinations of pseudopotentials or all-electron methods, DFT functionals, and relativistic treatments are assessed for the prediction of the principal components of the magnetic shielding tensors of thirty cadmium sites. We find that the inclusion of SO coupling can increase the cadmium magnetic shielding by as much as ca. 1100 ppm for a certain principal values; these effects are most pronounced for cadmium sites featuring bonds to other heavy atoms such as cadmium, iodine, or selenium. The best agreement with experimental values is found at the ZORA SO level in combination with a hybrid DFT method featuring a large admixture of Hartree-Fock exchange such as BH&HLYP. Finally, a theoretical examination is presented of the magnetic shielding tensor of the Cd(I) site in Cd2(AlCl4)2.


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1. Introduction Solid-state NMR (SSNMR) spectroscopy lends powerful insight into the chemistry of cadmium-containing compounds because of the sensitivity of cadmium magnetic shielding tensors to differences in molecular-level structure.1-7 The principal components of the magnetic shielding tensors, as well as their orientations within the molecular frame, vary appreciably between coordination environments, reflecting differences in chemical bonding, conformation, and dynamics.8-9 The NMR-active isotopes of cadmium, 111Cd and 113Cd, are both spin-½ nuclides having favorable NMR properties, including reasonably high gyromagnetic ratios (-5.693 × 107 rad T-1 s-1 and -5.955 × 107 rad T-1 s-1, respectively) and natural abundances (12.8% and 12.2%, respectively). Together, these aspects make cadmium SSNMR spectroscopy an important technique for structural elucidation of inorganic and bioinorganic solids. Cadmium currently finds wide use in general areas like energy technology (batteries, photovoltaics, photoresistors, quantum dots), anticorrosives (coatings, platings), dyes and pigments, plastic stabilizers, etc. Consequently, 111/113Cd NMR spectroscopy has been applied to the study of a diverse array of systems, including nanoparticles and quantum dots;10-16 batteries;17 luminescent materials;18 coordination polymers and metal-organic frameworks;19-21 zeolites,22 and metalloproteins and bioinorganic complexes.23-25 111/113Cd NMR spectroscopy is also used to analyze cadmium surrogate atoms that have been introduced into systems containing divalent metal atoms with unreceptive native nuclides;26 these investigations are especially important for studying the biologically-relevant ions Ca2+ and Zn2+ (43Ca and 67Zn are unreceptive nuclides that are often challenging to observe using routine SSNMR experiments and/or without isotopic enrichment), which are unsuitable for investigation by UV-Vis or electron paramagnetic resonance (EPR) spectroscopic techniques.25,27-28



Quantum chemical computations provide the most powerful means of exploring relationships between NMR parameters and structure.29-32 There are few detailed studies in the literature dealing with calculations of cadmium magnetic shielding tensors. To date, most reported calculations have used non-relativistic (NR) Hartree-Fock (HF) theory33-36 or density functional theory (DFT).37-41 Although relativistic effects,42-47 including spin-orbit (SO) coupling, are essential to obtain satisfactory agreement with experimental parameters for sixthperiod atoms such as platinum, mercury, and lead (for both the isotropic magnetic shielding constants and the principal components of the magnetic shielding tensors),48-51 the importance of relativistic effects for the prediction of the magnetic shielding of fifth-period atoms, such as cadmium, remains ambiguous. NR and/or scalar (SC) relativistic DFT approximations have been used with some success to calculate the magnetic shielding tensors of NMR-active isotopes elements such as yttrium, zirconium, niobium, molybdenum, silver, and tin.52-59 However, two recent studies by Alkan et al. have demonstrated the significance of relativistic effects, at the SO level, for the prediction of the principal components of tin and tellurium magnetic shielding tensors.60-61 Thus, a detailed examination of the importance of relativistic effects on the calculated principal components of the magnetic shielding tensors of fifth-period elements such as cadmium is needed. The earliest reports of first-principles calculations of cadmium magnetic shielding tensors in small molecules were provided by Nakatsuji et al.33-34 The results of these calculations are in reasonable agreement with experiment; however, a subsequent study by Ellis et al. suggested that the apparent agreement results from error cancellation due to (i) the use of insufficiently large basis sets, (ii) the neglect of electron correlation effects in HF calculations, and (iii) the lack of relativistic corrections.35 The significance of finite basis-set effects on cadmium magnetic


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shielding was later examined by Higashioji et al.36 DFT calculations of cadmium magnetic shielding tensors, which include electronic correlation effects, have been also explored.37-41 Several studies have discussed the importance of relativistic effects on cadmium magnetic shielding, although the conclusions of these reports disagree.62-64 Li et al. have used a combination of molecular dynamics simulations and relativistic DFT calculations to explore the time-averaged magnetic shielding of the Cd2+ ion in aqueous solution.62 The authors concluded that the relativistic contributions to the shielding do not vary substantially from system to system; because SO contributions did not vary, they mutually cancelled upon conversion to the chemical shift scale. Roukala et al. have discussed the calculation of cadmium magnetic shielding in Cd(CH3)2 and Cd(H2O)62+ using various relativistic treatments, including DiracHartree-Fock (DHF) theory and Breit-Pauli perturbation theory (BPPT).63 This analysis revealed that calculated cadmium shielding tensors vary among the different types of relativistic treatments considered in that work. Casella et al. used the zeroth-order regular approximation (ZORA) Hamiltonian to compute cadmium magnetic shielding constants. They concluded that relativistic corrections, at either the SC or SO level, do not lead to any significant improvement over NR calculations.64 Jokisaari et al. have illustrated small relativistic contributions to calculated carbon–cadmium scalar coupling tensors.65 Furthermore, all of these studies have focused on cadmium sites in the +2 oxidation state, although solids featuring the +1 oxidation state are also known.66 In the current study, we present the results of relativistic DFT calculations of the principal components of cadmium magnetic shielding tensors in a variety of coordination environments. Several computational protocols are employed, including various relativistic and electronic-state approximations. We compare periodic plane-wave pseudopotential and cluster-



based all-electron calculations as methods of modelling the lattice structures of the solids. The cluster-based calculations afford an opportunity to study the role of the ZORA SO Hamiltonian, as well as to explore the use of exchange-correlation functionals containing an admixture of Hartree-Fock exchange (HFX). We investigate a total of twenty-three computational protocols and assess the results through comparison with thirty experimental cadmium chemical shift tensors. Finally, we discuss the potential usefulness of these results for researchers using cadmium as a structural probe for NMR crystallographic investigations, as well as for those interested in relativistic effects and density functional approximations for the magnetic shielding tensors of heavy atoms in general.

2. Computational Methods 2.1. Geometry Optimizations. Calculations were performed on model structures obtained from previously reported X-ray diffraction (XRD) studies. The models were subjected to preliminary geometry optimizations using plane-wave DFT, as implemented in the CASTEP module of BIOVIA MATERIALS STUDIO 2018.67-68 In these structural refinements, the positions of hydrogen atoms were relaxed using the quasi-Newton BFGS energy-minimization scheme,69 whereas the positions of all other atoms, lattice constants, and space group symmetries were constrained to their experimental values. Calculations employed the PBE functional with a plane-wave cutoff energy of 700 eV and core-valence interactions modeled with ultrasoft pseudopotentials (USPPs) generated on the fly. Relativistic effects were incorporated using the ZORA SC pseudopotential formalism.70 Dispersion was included through the TkatchenkoScheffler two-body force field approximation.71 Integrals over the Brillouin zone were sampled using a Monkhorst-Pack grid with a k-point spacing of 0.07 Å−1.72 Thresholds for assessing


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structural convergence included a maximum change in energy of 5  10-6 eV atom-1, a maximum displacement of 5  10-4 Å atom-1, and a maximum Cartesian force of 0.01 eV Å-1. 2.2. Calculations of Magnetic Shielding. Cadmium magnetic shielding tensors were calculated in both plane-wave and cluster-based frameworks. The former used the gaugeincluding projector-augmented wave (GIPAW) formalism as implemented in CASTEP,73-74 whereas the latter used the gauge-including atomic orbital (GIAO) formalism,75-76 as implemented in Amsterdam Density Functional (ADF 2017).77-79 GIPAW calculations of magnetic shielding were performed using either USPPs with a cutoff energy of 700 eV or norm-conserving pseudopotentials (NCPPs) with a cutoff energy of 1000 eV.80 The two sets of GIPAW calculations are explored in more detail by examining the cadmium magnetic shielding in solid CdF2 with respect to the extension of the plane-wave basis set (Figure S1, Supporting Information). For calculations on isolated molecules, the isolated molecular state is approximated by employing large cubic unit cells with lattice constants of a = 20 Å. In the ADF calculations, clusters of atoms were used to represent the local lattice structures around the central cadmium atoms (Figure 1; a complete listing of the atomic coordinates for the clusters is available in the SI). Valence modification of terminal atoms using bond valence theory81-84 (VMTA/BV)50-51,60,85-86 was employed to terminate the clusters, assuring SCF convergence. In this model, the terminal atoms in the cluster are modified to compensate for missing bond strength (S) by replacing the terminal atoms with pseudoatoms with modified non-integer nuclear charges (Zmod): 𝑍mod = 𝑍nuc + ∆𝑆.


(1)


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In Eq. 1, Znuc is the formal charge of the terminal atoms, and S is calculated by the following relation: ∆𝑆 = 𝑉 ―

∑

exp

𝑘

(

𝑅𝑘0 ― 𝑅𝑘 𝑏𝑘

)

.

(2)

In Eq. 2, V is the unaltered valence of the terminal atom, and the exponential term contains the distance between atom pairs containing the terminal atom (Rk) and fitted bond-valence parameters (Rk0 and bk). These calculations employed the all-electron (AE) TZ2P basis set with a stringent linear dependence threshold parameter of 5 × 10-3 to avoid numerical problems associated with linear dependence of the basis functions.60 The convergence of the magnetic shielding in CdF2 is assessed using the basis sets DZ, DZP, TZP, TZ2P, and QZ4P (Figure S2). The correction to the xc response kernel, as introduced by Autschbach,87 was implemented in calculations employing the ZORA SO Hamiltonian. 2.3. Statistical Analysis. The correlation between calculated principal components of cadmium magnetic shielding tensors (m,ii) and experimental cadmium chemical shift tensors (m,ii) is modeled using linear regression: σ𝑚,𝑖𝑖 = 𝐴δ𝑚,𝑖𝑖 + 𝐵.

(3)

In Eq. 3, the indices m and i denote the cadmium site (m = 1, 2, …, M) and the principal component (i = 1, 2, 3), respectively. The slope of the correlation line (A) and the interpolated shielding of the reference system 0.1 M aqueous Cd(ClO4)2 (B) are adjusted for each model chemistry by considering the ninety principal components of the magnetic shielding tensors. The strengths of the regression models are described through coefficients of determination (R2). Root-mean-square errors are evaluated assuming an ideal slope of -1 (rms), or by accounting for systematic errors described by the slope of the best-fit line (rmsʹ).


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3. Results and Discussion 3.1. Overview. This study examines quantum chemical calculations of cadmium magnetic shielding tensors using both DFT and HF theory. The twenty-seven materials considered in this work have been studied previously by 111/113Cd SSNMR spectroscopy25,38,88-101 and diffraction methods,25,92,94,96,99-100,102-121 and feature thirty cadmium sites with coordination numbers ranging between four and eight, and bonding to atoms including carbon, nitrogen, oxygen, fluorine, sulfur, chlorine, selenium, and iodine (Table 1). Several theoretical aspects are considered for computing the principal components of the cadmium magnetic shielding tensors, which span a range of ca. 1500 ppm for this set of systems. First, we compare calculations performed using plane-wave DFT with those that model the extended structures of the cadmiumcontaining materials using clusters of atoms. Second, we explore the importance of relativistic effects on calculated magnetic shielding tensors using the ZORA Hamiltonian at the SC and SO levels, combined with cluster-based models. Third, we examine the effect of introducing an admixture of HFX into the calculation via hybrid functionals used in conjunction with SO coupling. Finally, we present calculations of the cadmium magnetic shielding tensor for a material containing Cd(I), which has not been studied experimentally with 111/113Cd SSNMR spectroscopy. The calculations presented in the following sections are grouped into four classes: GIPAW calculations employing USPPs and relativistic effects at the SC level (Class A); GIPAW calculations employing NCPPs and relativistic effects at the SC level (Class B); AE cluster-based calculations employing the ZORA SC Hamiltonian (Class C); and AE clusterbased calculations employing the ZORA SO Hamiltonian (Class D). Periodic calculations



(Classes A and B) employ the functionals PBE, PW91, BLYP, and RPBE; Class C calculations employ BP86 and HF, in addition to the previously listed functionals; Class D calculations employ these methods, as well as the hybrid functionals PBE0, B3LYP, and BH&HLYP. The discussions in sections 3.2 and 3.3 focus on results obtained using the BLYP functional, as these results are generally representative of those obtained using the other GGA functionals (the results of all calculations are summarized in Figures S3 – S7 and Tables S1 – S5). 3.2. Comparison of periodic and cluster-based calculations. GIPAW calculations of magnetic shielding tensors have been successfully used for a wide range of applications, since they employ periodic boundary conditions to account for the effects of long-range order in crystalline solids.74 There also exist well-established point-charge-based, fragment-based, or cluster-based protocols for modelling lattice effects of magnetic shielding tensors.122-125 In particular, cluster-based protocols have been used effectively for the prediction of magnetic shielding in a variety of materials.85,126-127 One advantage that cluster-based approaches maintain currently over periodic calculations is the ability to implement more advanced computational methods such as hybrid DFT at a reasonable computational cost (vide infra). This section discusses the results of calculations of cadmium magnetic shielding tensors obtained for Classes A – C (Figure 2, Table 2). There are substantial differences between the results obtained through the three classes, as indicated by the variation in the slopes of the linear regression lines (A), the interpolated values for the shielding of the reference compound (B), and the scatter around the best-fit lines (rms and rmsʹ, see Table 2 for definitions). For all computational classes, the agreement with experimental values is poor. Calculations using Classes A – C yield slopes of A = -0.38 ± 0.03, -1.07 ± 0.03, and -1.17 ± 0.03, respectively. Interpolated values for the shielding of the reference compound also vary, with Classes A – C


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yielding B = 4490 ± 8 ppm, 4106 ± 11 ppm, and 3670 ± 10 ppm, respectively. The rms errors are highest for Class A calculations (199 ppm) and lowest for Class B calculations (95 ppm). The rmsʹ errors are substantially lower for all classes because these quantities account for systematic discrepancies in the slope of the correlation lines: Class A calculations yield a value of 168 ppm, whereas Class B and C have rmsʹ values of 86 ppm and 71 ppm, respectively. The largest errors in any of the predicted principal components are observed for cadmium sites bound to heavy atoms such as iodine and selenium; these errors likely result from a combination of an insufficient treatment of relativistic effects and the choice of density functional method (vide infra). The differences in calculated magnetic shieldings between the two GIPAW methods and the AE method are not the result of inadequate modelling of lattice structure in the latter method. Calculations of the magnetic shielding for isolated cadmium-containing molecules of the form Cd(XHy)2 (X = C, N, O, F; y = 3, 2, 1, 0) illustrate that the differences in magnetic shielding are also present in the isolated molecules (Table 3). Calculations on the isolated molecules also afford insight into the sources of error in the calculations on the cluster models for solid materials. Differences in calculated magnetic shielding tensors in Classes A – C depend on the type of atom involved in the Cd–ligand bond. For example, the difference in the cadmium isotropic magnetic shielding in Cd(CH3)2 between Class A and C calculations is 1764 ppm, whereas this difference is only 800 ppm for CdF2. Class B and C calculations predict that the trend between cadmium isotropic magnetic shielding and the type of bonded atom follows the order CdF2 > Cd(OH)2 > Cd(NH2)2 > Cd(CH3)2, and that the chemical shift span follows the order Cd(CH3)2 > Cd(NH2)2 > Cd(OH)2 > CdF2. In contrast, Class A calculations do not predict any trend for either quantity. We intend to revisit the effects



of pseudopotential-generation protocols on the calculation of the magnetic shielding of heavy atoms. 3.3. Relativistic effects on cadmium magnetic shielding. There is some disagreement in the literature regarding the importance of relativistic effects for the calculation of cadmium magnetic shielding tensors (vide supra). Although there are suggestions that the SO contribution (defined here as ZORA/SO – ZORA/SC) to shielding tensors vary between systems,63 another report on a wider selection of materials suggests that the inclusion of relativistic corrections (at either the SO or SC levels) is not statistically significant.64 Calculations of cadmium magnetic shielding of the Cd(CH3)2 molecule at the ZORA SC and SO levels lend insight into the importance of SO coupling in these systems (Figure 3). At the SC and SO levels, the isotropic shieldings are 3034 ppm and 3500 ppm, respectively. The latter value is closer to the 4093 ppm calculated at the DHF level. (N.B. DHF calculations of cadmium magnetic shielding for small molecules have been examined by Roukala et al., and are used here as a point of reference).63 This result demonstrates that the inclusion of SO coupling in the DFT calculations brings the cadmium magnetic shielding into closer agreement with results obtained with the four-component DHF method. The deviation from DHF calculations is further minimized when the shielding is calculated using a hybrid functional or HF theory (vide infra). Next, we discuss the results of calculations of cadmium magnetic shielding tensors in cluster models of the solids obtained for Classes C and D (Figure 4, Table 2). Inclusion of SO coupling has a large impact on the interpolated values of the chemical shift reference and the slope of the correlation line. The shielding of the reference system is predicted to be higher (relative to the SC level) by ca. 540 ppm when SO coupling is included in the calculations. Inclusion of SO coupling also brings the magnitude of the slope of the correlation line closer to


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unity (SC: A = -1.17 ± 0.03; SO: A = -1.09 ± 0.03). The importance of SO coupling is less obvious when examining the rms and rmsʹ errors. The rms error decreases from 99 ppm (SC) to 87 ppm (SO), whereas the rmsʹ error increases from 71 ppm (SC) to 76 ppm (SO). For SC calculations, large discrepancies with experiment are observed for each principal value for cadmium sites featuring Cd–I or Cd–Se bonds. For SO calculations, large errors are observed only for 33 principal components from sites featuring Cd–I bonds. For systems featuring Cd–I or Cd–Se bonds, the SO contribution to the principal components of the cadmium magnetic shielding tensors is in the range of 501 to 1094 ppm (Figure 5). The large SO contribution to the shielding of these sites likely arise from heavyatom-to-heavy atom (HAHA) effects.128 For all cadmium sites that are bonded only to light atoms, the average SO contribution to the principal components of the cadmium magnetic shielding tensors is between 468 and 593 ppm; this narrow distribution of SO contributions to the shielding, which is observed when the Cd–ligand bond does not contain a second heavy atom, is likely the reason why other reports have suggested that the inclusion of SO effects is not necessary for the calculation of cadmium magnetic shielding tensors (in the current work, comparisons with DFH calculations suggest that SO effects are necessary in all cases, vide infra). 3.4. Performance of hybrid functionals combined with spin-orbit coupling. GGA functionals are typically employed for relativistic calculations of magnetic shielding tensors because of the efficient scaling of these methods in the SCF cycle and calculation of magnetic shielding tensors, and because the accuracy of such calculations is often sufficient for many applications. The introduction of an admixture of HFX, through use of hybrid functionals, in NR calculations can substantially increase agreement with experiment for light nuclides, although with the tradeoff of increased computational cost.85-86,122-123,126-127,129-130 For NR calculations of



cadmium magnetic shielding tensors, hybrid functionals perform more similarly to CCSD(T) calculations than do pure DFT methods.63 The use of hybrid functionals in DFT/ZORA calculations of magnetic shielding and spin-spin coupling tensors has also been explored.4344,85,131

At the ZORA SO level, calculations of lead magnetic shielding tensors using hybrid

functionals outperform those obtained using GGA functionals;50-51 similar, although smaller, effects have been observed for tin60 and tellurium.61 Here, we explore the effects of including an admixture of HFX on the calculation of cadmium magnetic shielding tensors using the hybrid DFT functionals PBE0 (25% HFX), B3LYP (20% HFX), and BH&HLYP (50% HFX). The calculation of cadmium magnetic shielding tensors using the PBE0 and BH&HLYP functionals, combined with SO coupling, results in a strong correlation with experiment for all cadmium sites (Figure 4, Table 2). HF/SO calculations also bring the problematic tensor components into agreement with experiment, although with larger rms and rmsʹ errors than are obtained at the PBE0 and BH&HLYP levels. The largest discrepancies in the B3LYP calculations result from the prediction 33 principal components for sites featuring Cd–I bonds. Calculation using the BH&HLYP functional predicts that the shielding of the reference system is higher by 159 ppm than predicted using the BLYP functional (Table 2). The likely cause for this observation is that use of hybrid functionals increases the HOMO-LUMO energy gap, thereby decreasing the role of occupied-unoccupied interactions on the calculated magnetic shielding tensor. The differences in cadmium magnetic shielding between the BH&HLYP and BLYP functionals is also observed for the Cd(CH3)2 molecule, where the isotropic shielding using the BH&HLYP functional is 202 ppm higher than calculated at the BLYP level; the higher value is in closer agreement with results obtained at the DHF level (Figure 3).63 (N.B.: For Cd(CH3)2, HF SO calculations result in the closest agreement with DHF calculations).


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BH&HLYP calculations also result in slopes of the correlation line closer to unity (BH&HLYP: A = -1.06 ± 0.02; BLYP: A = -1.09 ± 0.03). This fact is reflected in the BH&HLYP rms and rmsʹ errors of 48 ppm and 41 ppm, respectively. Using the BLYP functional, these errors are 87 ppm and 76 ppm, respectively. BH&HLYP calculations predict an increase between 68 and 257 ppm for all principal components (relative to BLYP), except for the 33 components of sites bound to iodine, which are decreased between 178 and 216 ppm. This has the effect of reducing the spans of the shielding tensors, bringing the tensors for CdI2(C5H4NCOOPrn)2 and CdI2(C5H4NCOOMe)2·MeOH into better agreement with experiment than is attainable through any other protocol examined in this work (Table 4). For these two cadmium sites, the differences between the BH&HLYP and BLYP functionals are also evident when examining the orientations of the cadmium magnetic shielding tensors within the molecular frame (Figure 6). In the CdI2(C5H4NCOOPrn)2 model, the cadmium site resides at the center of a distorted octahedron, with vertices consisting of two oxygen, two nitrogen, and two iodine atoms. At the BLYP SO level, 11 and 22 are oriented such that they both fall nearly along the two Cd–O bonding axes (within 8 – 12°), and 33 is located approximately perpendicular to the I–Cd–I plane. At the BH&HLYP SO level, 11 is approximately in the I–Cd–I plane, 22 is in the O–Cd–O plane, and 33 is located approximately in N–Cd–N plane. In contrast, calculations at the BLYP/SO and BH&HLYP/SO levels for cadmium sites bonded to light atoms (e.g. Cd(acetate)2·2H2O) result only in minimal differences in the orientations of the magnetic shielding tensors. 3.5. Theoretical Investigation of Cd(I) Sites. Cadmium tetrachloroaluminate, Cd2(AlCl4)2, was the first reported compound containing cadmium in the +1 oxidation state. This material features a Cd–Cd bond of 2.576 Å, as measured by single-crystal XRD66 and 15 ACS Paragon Plus Environment


substantiated through Raman spectroscopy.132 To the authors’ knowledge, 111/113Cd SNMR spectroscopy has not been used to study the Cd22+ ion in this or any other material. The effects of SO coupling on the cadmium magnetic shielding tensor are substantially smaller (314 – 319 ppm) for the σ11 and σ22 components oriented perpendicular to the Cd–Cd bonding axis (Table 5, Figure 5); in contrast, the SO contribution to σ33, which is oriented along the bonding axis, is 518 ppm, a value that is consistent with the majority of systems studied herein. As a consequence, the span of the magnetic shielding tensor is predicted to be ca. 200 ppm larger when SO effects are included in the calculation. The predicted span of this site is larger than the quantities predicted for the Cd(II) species, with the exception of CdCl2·18-crown6 (which features a linear CdCl2 structural unit). For the cluster model of Cd2(AlCl4)2, the HOMO level originates largely from the linear combination of Cd 5s and 4p states, and can be viewed as a bonding orbital between adjacent cadmium atoms. The LUMO level originates largely from Cd 5s states, and can be viewed as an antibonding orbital. In contrast, the HOMO and LUMO levels for cluster models of systems featuring cadmium in the +2 oxidation state largely describe Cd–ligand and/or ligand–ligand interactions. Thus, the anomalous results for the SO contribution to the shielding of the Cd(I) species likely correspond to the description of occupied-unoccupied interactions between Cd 5s states in the expression for the magnetic shielding. Use of the hybrid functional BH&HLYP results in higher shielding for each of the principal components, and further increases the span of the shielding tensor ( = 871 ppm at the BLYP SO level versus  = 931 at the BH&HLYP SO level). As with the other cadmium species, the higher shielding likely results from the increased HOMO-LUMO gap when the BH&HLYP functional is employed in the calculation of magnetic shielding.


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Conclusions This work has assessed the roles of various relativistic and electronic-state approximations for the calculation of cadmium magnetic shielding tensors in crystalline solids. In particular, we have compared periodic and cluster models of the crystal structures, assessed the role of the ZORA Hamiltonian at the SC and SO levels, and compared the performance of several density functionals for calculating cadmium magnetic shielding tensors. Cadmium magnetic shielding tensors were computed at the ZORA SC level using the GIPAW approach (including two pseudopotential approximations) and an AE cluster-based approach. The AE and NCPP calculations typically result in reasonable agreement with experiment for cadmium sites that are not bound to heavy atoms; in contrast, calculations employing USPPs result in poor agreement with experiment in all cases, and fail to reproduce trends relating shielding and the type of atom involved in Cd–ligand bonds. The results of these calculations can be improved upon by employing relativistic effects in combination with certain hybrid DFT functionals. Cadmium magnetic shielding tensors are influenced strongly by the effects of SO coupling. When cadmium sites are bound only to light atoms, the average SO contribution to the shielding tensor is ca. 540 ppm, with similar contributions to each of the principal components. The SO contribution to an individual principal component can be as high as 1069 ppm when the cadmium atoms is bound to another heavy atom. The inclusion of SO coupling is critical for evaluating the magnetic shielding tensors of cadmium sites featuring bonds with heavier atoms (e.g., Cd–I, Cd–Se, Cd–Cd).



Hybrid functionals, combined with relativistic treatment at the SO level, result in increased agreement with experiment, relative to GGA functionals and HF theory. The best agreement with experiment was found using the BH&HLYP functional, which resulted in an rmsʹ error of only 41 ppm for all cadmium sites. The increased accuracy of calculations employing this functional is most apparent for the two cadmium sites bound to iodine, where the rms errors in the principal components are only between 18% and 27% of that predicted using the BLYP functional. We have also undertaken a theoretical investigation of cadmium magnetic shielding for a material featuring cadmium in the +1 oxidation state. The anomalous results indicate that the SO contribution to the shielding of this site is much smaller (particularly for 11 and 22) than observed for the thirty cadmium sites in the +2 oxidation state, probably because of the role played by Cd 5s atomic states in both the HOMO and LUMO levels of this system. Although this work has focused exclusively on calculations of the magnetic shielding tensors of cadmium atoms, these insights may prove useful for future calculations of the magnetic shielding tensors of other heavy atoms such as platinum, which have proven difficult to calculate.131133 Reliable predictions of the magnetic shielding tensors of heavy elements may have an impact on the emerging field of NMR crystallography, which combines experimental SSNMR spectroscopy and quantum mechanical calculations to refine crystal structures determined by single-crystal XRD, to assist in the solution of crystal structures in tandem with powder XRD, or to aid first-principles crystal structure predictions. In particular, 111/113Cd SSNMR can be used to aid in structural predictions of diverse materials such as metal-organic frameworks or nanoparticles. To assist in such applications, high-throughput screening of structures via rapid calculations of cadmium magnetic shielding tensors is a prerequisite. The


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results of this work demonstrate that calculations employing the ZORA SC Hamiltonian are suitable (i.e., the calculations are inexpensive and provide reasonable agreement with experiment for particular cases) for such applications when the cadmium site is bound only to light atoms.

Supporting Information Summary of all calculated magnetic shielding tensors. Correlation plots for all model chemistries. Demonstration of basis set convergence. Atomic coordinates for all cluster models in the form of .xyz files.

Acknowledgements We thank Genentech and the Natural Sciences and Engineering Research Council of Canada (NSERC) for funding this research (NSERC in the form of a Discovery Grant). R.W.S. is grateful for a 50th Anniversary Golden Jubilee Chair from the University of Windsor. This work was made possible, in part, by the facilities of the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca). We thank Dr. Stanislav Veinberg and Dr. Fahri Alkan for helpful discussions.

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122. Hartman, J. D.; Monaco, S.; Schatschneider, B.; Beran, G. J. O., Fragment-based 13C nuclear magnetic resonance chemical shift predictions in molecular crystals: an alternative to planewave methods. J. Chem. Phys. 2015, 143, 102809. 123. Hartman, J. D.; Kudla, R. A.; Day, G. M.; Mueller, L. J.; Beran, G. J. O., Benchmark fragment-based 1H, 13C, 15N and 17O chemical shift predictions in molecular crystals. Phys. Chem. Chem. Phys. 2016, 18, 21686-21709. 124. Beran, G. J. O., Modeling polymorphic molecular crystals with electronic structure theory. Chem. Rev. 2016. 125. Hartman, J. D.; Balaji, A.; Beran, G. J. O., Improved electrostatic embedding for fragment-based chemical shift calculations in molecular crystals. J. Chem. Theory Comput. 2017, 13, 6043-6051. 126. Holmes, S. T.; Iuliucci, R. J.; Mueller, K. T.; Dybowski, C., Density functional investigation of intermolecular effects on 13C NMR chemical-shielding tensors modeled with molecular clusters. J. Chem. Phys. 2014, 141, 164121. 127. Holmes, S. T.; Iuliucci, R. J.; Mueller, K. T.; Dybowski, C., Critical analysis of cluster models and exchange-correlation functionals for calculating magnetic shielding in molecular solids. J. Chem. Theory Comput. 2015, 11, 5229-5241. 128. Lantto, P.; Romero, R. H.; Gómez, S. S.; Aucar, G. A.; Vaara, J., Relativistic heavy-atom effects on heavy-atom nuclear shieldings. J. Chem. Phys. 2006, 125, 184113. 129. Holmes, S. T.; Bai, S.; Iuliucci, R. J.; Mueller, K. T.; Dybowski, C., Calculations of solid-state 43Ca NMR parameters: a comparison of periodic and cluster approaches and an evaluation of DFT functionals. J. Comput. Chem. 2017, 38, 949-956. 130. Lu, M.; Sarkar, S.; Wang, M.; Kraus, J.; Fritz, M.; Quinn, C. M.; Bai, S.; Holmes, S. T.; Dybowski, C.; Yap, G. P. A.; Struppe, J.; Sergeyev, I. V.; Maas, W.; Gronenborn, A. M.; Polenova, T., 19F magic angle spinning NMR spectroscopy and density functional theory calculations of fluorosubstituted tryptophans: integrating experiment and theory for accurate determination of chemical shift tensors. J. Phys. Chem. B 2018, 122, 6148-6155. 131. Lucier, B. E. G.; Johnston, K. E.; Xu, W.; Hanson, J. C.; Senanayake, S. D.; Yao, S.; Bourassa, M. W.; Srebro, M.; Autschbach, J.; Schurko, R. W., Unravelling the structure of Magnus’ pink salt. J. Am. Chem. Soc. 2014, 136, 1333-1351. 132. Corbett, J. D., The cadmium(I) Ion Cd22+. Raman spectrum and relationship to Hg22+. Inorg. Chem. 1962, 1, 700-703. 133. Lucier, B. E. G.; Reidel, A. R.; Schurko, R. W., Multinuclear solid-state NMR of square-planar platinum complexes — cisplatin and related systems. Can. J. Chem. 2011, 89, 919-937.


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Table 1. Crystallographic information and experimental cadmium chemical shift tensor parameters.a,b Compound

CdF2 Cd(ClO4)2·6H2O K2[Cd(CN)4] CdCl2 Cd(OH)2 CdSO4·H2O 3CdSO4·8H2O (Site 1) 3CdSO4·8H2O (Site 2) Cd(NH4)2(SO4)2·6H2O Cd(NO3)2·4H2O Cd(thiourea)2(NO3)2 Cd(2-aminomethylpyridine)2(NO3)2 Cd(formate)2 Cd(formate)2·2H2O (Site 1) Cd(formate)2·2H2O (Site 1) Cd(acetate)2·2H2O Cd(maleate)·2H2O Cd(diethylphosphate)2 Cd(ethylxanthate)2 Cd(n-butylxanthate)2 Cd(glycinate)2·H2O Cd(alaninate)2·3H2O Cd(histidinate)2 Cd(acetate)2(imidazole)2 CdCl2·18-crown-6 Cd[N(Pri2PSe)2]2 CdSe2N2C40H54 (Site 1) CdSe2N2C40H54 (Site 2) CdI2(C5H4NCOOMe)2·MeOH CdI2(C5H4NCOOPrn)2

Crystallographic Information Space Cd Group Coord.

Ref.

22

33

(ppm)

(ppm)

(ppm)

(ppm)

(ppm)

-223 0 569 173 96 -69 -77 -85 70 -156 308 224 -42 17 30 -70 4 -111 386 430 200 266 354 101 385 663 570 547 175

-223 0 569 173 96 -128 -93 -96 14 -174 -340 -128 -120 -19 -56 -119 -47 -217 262 334 -116 -56 88 -40 -700 540 93 100 122

-223 0 569 211 150 -53 -48 -61 61 -103 98 218 -54 15 20 -52 10 -112 410 440 113 177 286 103 33 629 472 460 196

186

47

167

102 121 120 103 119 104 105 105

Pccn Pccn P21/n

F8 O6 C4 Cl6 O6 O6 O6 O6 O6 O8 O4S2 N2O4 O7 O6 O6 O5 O7 O6 S4 S4 N2O4 N2O4 N4O2 N2O4 O6Cl2 Se4 N2Se2 N2Se2 N2O2I2

106 107 118 100 108 109 109 110 111 92 112 113 25 114 115 94 116 117 99 99 96

-223 0 569 288 257 39 26 -3 98 22 327 558 0 46 85 34 73 -7 583 556 256 321 416 249 414 683 752 733 291

P21/c

N2O2I2

96

268

Fm3m P3m1 Fd3m R3m P3m1 P21/c C2/c C2/c P21/a Fdd2 P21/n P21/c C2/c P21/c P21/c P212121 R3 P1 Pa P21/c I2/a P3121 P43212 P21/c R3 P1

SSNMR Data iso

11






Ref.

0 0 0 115 161 167 119 93 84 196 667 686 120 65 141 153 120 210 321 222 372 377 328 289 1114 143 659 633 169

-1.00 -1.00 -0.29 -0.73 -0.76 0.33 -0.82 0.94 0.03 0.30 0.11 0.22 -0.36 -0.15 0.01 -0.23 -0.14 0.70 0.71 0.62 -0.02 0.95 0.72 0.45 0.41 -0.37

88 93 97 88 93 89 90 90 91 90 101 100 38 91 91 91 91 92 93 93 25 25 38 94 95 98 99 99 96

221

0.26

96


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a

The principal components of the chemical shift tensors are ranked using the frequency-ordered convention such that 11 ≥ 22 ≥ 33. The isotropic chemical shifts, spans, and skews are defined by iso = ⅓(11 + 22 + 33),  = 11 - 33, and  = 3(22 - iso)/, respectively. b 3CdSO ·8H O, Cd(formate) ·2H O, and CdSe N C H each have two crystallographically 4 2 2 2 2 2 40 54 and magnetically distinct cadmium sites.

Table 2. Statistical data for the relationships between calculated principal components of cadmium magnetic shielding tensors and experimental principal components of cadmium chemical shift tensors.a,b Method

B (ppm)

PBE PW91 BLYP RPBE

4486 ± 8 4485 ± 8 4490 ± 8 4506 ± 7

PBE PW91 BLYP RPBE

4114 ± 11 4110 ± 11 4106 ± 11 4137 ± 10

PBE PW91 BP86 BLYP RPBE HF

3676 ± 10 3671 ± 10 3663 ± 10 3670 ± 10 3715 ± 10 3980 ± 9

PBE PW91 BP86 BLYP RPBE PBE0 B3LYP BH&HLYP HF

4202 ± 8 4200 ± 8 4193 ± 8 4209 ± 10 4242 ± 7 4295 ± 6 4276 ± 7 4368 ± 5 4496 ± 6

R2

A

Class A (GIPAW, USPP) -0.40 ± 0.03 0.74 -0.40 ± 0.03 0.74 -0.38 ± 0.03 0.72 -0.40 ± 0.02 0.76 Class B (GIPAW, NCPP) -1.05 ± 0.03 0.91 -1.06 ± 0.03 0.91 -1.07 ± 0.03 0.91 -1.05 ± 0.03 0.91 Class C (AE, ZORA SC) -1.17 ± 0.03 0.94 -1.18 ± 0.03 0.94 -1.18 ± 0.03 0.94 -1.17 ± 0.03 0.94 -1.17 ± 0.03 0.94 -1.08 ± 0.03 0.94 Class D (AE, ZORA SO) -1.12 ± 0.03 0.96 -1.12 ± 0.03 0.95 -1.12 ± 0.03 0.95 -1.09 ± 0.03 0.93 -1.12 ± 0.02 0.96 -1.11 ± 0.02 0.97 -1.10 ± 0.02 0.96 -1.06 ± 0.02 0.98 -1.03 ± 0.02 0.97 28 ACS Paragon Plus Environment

rms (ppm)

rms' (ppm)

194 195 199 211

161 149 168 169

90 91 95 89

84 84 86 84

99 100 100 99 99 76

72 71 71 71 71 66

74 78 79 87 73 60 66 48 52

57 61 63 76 56 44 54 41 49

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a

Data used in the statistical analysis presented here are found in Tables S1 – S5. The parameters A and B are obtained by fitting the correlation between calculated principal components of cadmium magnetic shielding tensors (m,ii) and experimental cadmium chemical shift tensors (m,ii) using linear regression, σ𝑚,𝑖𝑖 = 𝐴δ𝑚,𝑖𝑖 +𝐵. The parameter R2 is the coefficient b

of determination. Errors are defined by 𝑟𝑚𝑠 = 1

∑ 𝑀 ― 2 𝑚,𝑖𝑖

(

𝐵 ― 𝜎𝑚,𝑖𝑖 |𝐴|

1

∑

𝑀 ― 1 𝑚,𝑖𝑖

(𝐵 ― 𝜎𝑚,𝑖𝑖 ― 𝛿𝑚,𝑖𝑖)2 and 𝑟𝑚𝑠′ =

2

)

― 𝛿𝑚,𝑖𝑖 , where M is the total number of cadmium magnetic shielding

principal components.

Table 3. Comparison of calculated isotropic magnetic shieldings and spans for isolated cadmium-containing molecules.a,b Molecule

Class A Class B Class C GIPAW, USPP GIPAW, NCPP AE, ZORA SC iso  iso  iso  (ppm) (ppm) (ppm) (ppm) (ppm) (ppm) Cd(CH3)2 4798 359 2936 3088 3034 2685 Cd(NH2)2 4674 563 3483 2102 3333 2117 Cd(OH)2 4643 656 3885 1645 3617 1822 CdF2 4658 604 4182 1276 3858 1537 a Shielding constants were calculated at the BLYP level. b See Table 1 for definitions of chemical shift tensor and magnetic shielding tensor parameters.



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Table 4. Experimental and calculated cadmium chemical shift tensors for sites featuring Cd–I or Cd–Se bonds.a,b 11 (ppm)

22 (ppm)

33 (ppm)

iso (ppm)

 (ppm)



rms (ppm)

CdI2(C5H4NCOOPrn)2 Exp. 268 186 47 167 221 0.26 BLYP/SC 606 529 175 437 431 0.64 275 BLYP/SO 238 197 -380 18 618 0.87 292 BH&HLYP/SO 317 219 -6 177 323 0.39 80 CdI2(C5H4NCOOMe)2·MeOH Exp. 291 175 122 196 169 -0.37 BLYP/SC 592 555 135 427 457 0.84 280 BLYP/SO 243 129 -272 33 515 0.56 231 BH&HLYP/SO 334 159 65 186 269 -0.30 42 i Cd[N(Pr 2PSe)2]2 Exp. 683 663 540 629 143 0.72 BLYP/SC 805 790 780 792 25 -0.20 172 BLYP/SO 771 736 511 673 260 0.73 68 BH&HLYP/SO 716 702 569 662 147 0.81 34 CdSe2N2C40H54 (Site 1) Exp. 752 570 93 472 659 0.45 BLYP/SC 803 761 87 550 716 0.88 114 BLYP/SO 711 589 125 475 586 0.58 32 BH&HLYP/SO 770 625 120 505 650 0.55 37 CdSe2N2C40H54 (Site 2) Exp. 733 547 100 460 633 0.41 BLYP/SC 772 728 57 519 715 0.88 110 BLYP/SO 673 536 90 433 583 0.53 36 BH&HLYP/SO 756 584 118 486 638 0.46 27 a See Table 1 for definitions of chemical shift tensor and magnetic shielding tensor parameters.


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b

Calculated chemical shift tensor parameters are reported as the difference between the shielding of the reference system (B) and the calculated magnetic shielding tensor parameters. Error is defined by 𝑟𝑚𝑠 =

1

∑

3 𝑚,𝑖𝑖

(𝐵 ― 𝜎𝑚,𝑖𝑖 ― 𝛿𝑚,𝑖𝑖)2.

Table 5. Calculated cadmium chemical shift tensors for Cd2(AlCl4)2.a,b 11 22 33 iso   (ppm) (ppm) (ppm) (ppm) (ppm) BLYP/SC 296 282 -376 67 672 0.96 BLYP/SO 516 507 -355 223 871 0.98 BH&HLYP/SO 569 560 -362 256 931 0.98 a See Table 1 for definitions of chemical shift tensor and magnetic shielding tensor parameters. b Calculated chemical shift tensor parameters are reported as the difference between the shielding of the reference system (B) and the calculated magnetic shielding tensor parameters.



ToC Graphic

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Figure 1. Examples of clusters used in the calculations of cadmium magnetic shielding tensors. (a) CdF2; (b) Cd(NO3)2·4H2O; (c) Cd(glycinate)2·H2O; (d) CdCl2·18-crown-6. Colors: purple (Cd), pink (H), black (C), red (O), blue (N), dark green (F), light green (Cl).



Figure 2. Correlations between calculated principal components of cadmium magnetic shielding tensors and experimental cadmium chemical shift tensors, as determined with different computational protocols. Computed shielding constants were obtained using (a, class A) the GIPAW method with ultrasoft pseudopotentials generated on the fly, (b, class B) the GIPAW method with norm-conserving pseudopotentials generated on the fly, and (c, class C) the all-electron cluster-based approach. Calculations were performed using the BLYP functional and incorporated relativistic effects at the ZORA/SC level.


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Figure 3. Calculated cadmium isotropic shielding constants for the isolated Cd(CH3)2 molecule (red markers). Errors in calculated shielding relative to the Dirac-Hartree-Fock level (blue bars). The shielding constants plotted on this graph are as follows: sDFH = 4093 ppm; sHF/SO = 3892 ppm; sBH&HLYP/SO = 3702 ppm; sB3LYP/SO = 3570 ppm; sBLYP/SO = 3500 ppm; sHF/SC = 3329 ppm; sBLYP/SC = 3034 ppm.



Figure 4. Correlations between calculated principal components of cadmium magnetic shielding tensors and experimental cadmium chemical shift tensors, as calculated at the (a, class C) BLYP ZORA/SC level, the (b, class D) BLYP ZORA/SO level, the (c, class D) PBE0 ZORA/SO level, the (d, class D) B3LYP ZORA/SO level, the (e, class D) BH&HLYP ZORA SO level, and the (f, class D) HF ZORA/SO level. These calculations were performed using cluster-based models of the solids.


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Figure 5. (a) Correlations between cadmium magnetic shielding principal components computed using the ZORA SC and SO Hamiltonians. (b) SO contribution (sZORA/SO – sZORA/SC) to the principal components of the cadmium magnetic shielding tensors for (1) Cd[N(Pri2PSe)2]2, (2 - 3) the two cadmium sites in CdSe2N2C40H54, (4) CdI2(C5H4NCOOMe)2·MeOH, (5) CdI2(C5H4NCOOPrn)2, and (6) Cd2(AlCl4)2.



Figure 6. Orientations of the principal components of the cadmium magnetic shielding tensor (a) in CdI2(C5H4NCOOPrn)2, and (b) in Cd(acetate)2·2H2O, as calculated at the BH&HLYP/SO (yellow vectors) and BLYP/SO (green vectors) levels. Some atoms have been removed for clarity. Colors: white (C), gray (H), blue (N), red (O), purple (I).


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ZORA Study of Cadmium Magnetic Shielding Tensors: Analysis

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