Direct magnetic-field dependence of NMR chemical shift

Nuclear shielding and chemical shift are considered independent of the magnetic-field strength. Ramsey proposed on theoretical grounds in 1970 that this may not be valid for heavy nuclei. Here we present experimental evidence for the direct field dependence of shielding, using Co shielding in Co(acac)3 [tris(acetylacetonate)cobalt(III)] as an example. We carry out NMR experiments in four field strengths for this low-spin diamagnetic Co(III) complex, which features a very large and negative nuclear shielding constant of the central Co nucleus. This is due to a magnetically accessible, low-energy eg ’ t2g orbital excitation of the d system. The experiments result in temperature-dependent magnetic-field dependence of 5.7 to 5.2 ppb T 2 of the Co shielding constant, arising from the direct modification of the electron cloud of the complex by the field. First-principles multiconfigurational non-linear response theory calculations verify the sign and order of magnitude of the experimental results.


Introduction
Nuclear magnetic resonance (NMR) parameters, 1 nuclear shielding, its associated chemical shift, spin-spin coupling and quadrupole coupling, are normally considered independent of the applied magnetic field. Due to the increasing field strengths used in contemporary NMR instrumentation, currently reaching 1.2 GHz proton resonance frequency, the validity of this approximation should be investigated. In 1970, Ramsey 2 published a theoretical study predicting a quadratic, O(B 0 2 ), leading-order dependence on the field strength, B 0 . Ramsey concluded that ''in some molecules, especially where the magnetic shielding is abnormally large, it is possible that the field dependence of the shielding could be large enough to be observable''. Inspired by this prediction, Bendall and Doddrell 3 carried out 59 Co NMR experiments of diamagnetic cobalt compounds. The authors claimed a field dependence of 59 Co shielding constant in Co( 15 NH 3 )Cl 3 and Co(acac) 3 , with values À70 (30) and À80(50) ppb T À2 , respectively. Such Co(III) complexes feature a broad chemical shift range and show a sensitivity of the NMR signal on temperature, pressure and solvent, 4 which can be ascribed to the existence of low-lying, magnetically accessible d-d excitations 5 at the metal centre and, correspondingly, a large tendency of the electron cloud to accommodate to changing interactions with surroundings.
About 25 years later, Manninen and Vaara 6 presented a theory and computed from first principles the field-dependence of 59 Co shielding in the same compounds, with results that are by an order of magnitude smaller than the experimentally derived values in ref. 3. This effect can be termed direct field dependence, arising from the modification of the molecular electron cloud by the magnetic field. In contrast, the indirect effect results from a slight influence on the orientational distribution of molecules in liquid samples due to the interaction of magnetic field with the anisotropy of the molecular susceptibility, causing incomplete rotational averaging of NMR parameters. 7 Further computational work on the direct field dependence of nuclear shielding has been reported at both semi-empirical 8,9 and first-principles 10-12 levels. The field dependence of spinspin coupling has been elaborated in ref. 13 and 14. So far, the only both experimentally observed and theoretically confirmed case of direct field dependence of an NMR parameter is that of 131 Xe quadrupole coupling in atomic xenon. [15][16][17][18] The 131 Xe NMR spectrum consists of a 3 : 4 : 3 triplet when the nuclear electric quadrupole moment interacts with a non-vanishing electric field gradient (EFG) at the nuclear site. In this case, the non-zero EFG stems from the slight deformation of the atomic electron cloud induced by the magnetic field. 16 In the present work, we report experiments carried out at four magnetic fields, as well as improved first-principles nonlinear response theory computations on the direct field effect on the 59 Co shielding in Co(acac) 3 . We eliminate the difficulty of precise temperature control of the experiments in different NMR instruments by using the gaseous and solution-state 129 Xe NMR signal as a chemical shift and temperature reference, respectively. 19 Our experiments demonstrate the existence of the direct field dependence of shielding and, hence, verify the prediction by Ramsey.

Results and discussion
Due to the requirement of time-reversal symmetry, terms in the molecular energy expression have to contain the nuclear spin I K and external magnetic field B 0 appearing to even total power. 2,6 The nuclear Zeeman term in the standard NMR spin Hamiltonian 1 is O(I 1 K B 1 0 ) (total power 2) and the next term in increasing powers of the magnetic field is O(I 1 K B 0 3 ) (total power 4). Consequently, to the leading order, the B 0 -dependent NMR shielding tensor component s K,ab for nucleus K can be expressed as 2,6 where s (0) K is the standard field-independent shielding and the 4-index tensor t parameterises the leading-order field dependence. In a liquid or gas-phase sample, isotropic molecular tumbling renders observable the average 8 which is a combination of the tensor components of t expressed in a molecule-fixed Cartesian frame. A sample consisting of saturated solution of tris(acetylacetonate)cobalt(III) [Co(acac) 3 ] in CDCl 3 and natural xenon gas of about 2 atm was prepared into a 5 mm (o.d.) NMR tube. Resonance frequency of natural abundance gaseous 129 Xe (2 atm in a 5 mm tube) was used as an external chemical shift reference for both 59 Co and 129 Xe resonance frequencies in the solution. Experiments were carried out at four magnetic fields, 18.8, 14.1, 9.4 and 7.0 T, and five temperatures at each field strength, on Bruker Avance III instruments. Following Ramsey,59 Co resonance frequency can be presented in the form where g Co is the gyromagnetic ratio of 59 Co, s s Co is the fieldindependent part of the isotropic shielding constant in solution and t Co follows from eqn (2).
There is a strong temperature dependence of 59 Co chemical shift in Co(acac) 3 , e.g., reported in ref. 20 at +2.79 ppm K À1 in CHCl 3 solution. Therefore, one must guarantee that the 59 Co frequencies in different magnetic-field instruments are obtained at the same temperature. To this end, we used the 129 Xe chemical shift in solution, d s Xe , as an internal temperature reference (see ESI †), and ensured that d s Co at each magnetic field corresponds to the same d s Xe value. This procedure is based on the extreme sensitivity of d s Xe upon temperature 19 and is done to gain a precise value for t Co . The resonance frequencies of xenon in gas (g) and in solution (s) can be approximately written as Consequently, the 129 Xe chemical shift in solution, referenced to 129 Xe gas, is Eqn (4) and (5) neglect any field dependence of shielding in atomic 129 Xe, which has been shown by correlated ab initio response theory calculations to be of the order of 10 À6 ppb T À2 , i.e., hardly observable. 11 The minute field dependence can be understood to be due to the large difference between the occupied and vacant energy levels of the noble gas atom, a situation vastly different from that of Co(acac) 3 with low-energy magnetically accessible excitations. The 129 Xe gas frequency has a negligible temperature dependence compared to that of both 59 Co and 129 Xe in solution. [20][21][22][23] The 59 Co resonance frequency relative to the chosen shift reference, xenon gas, is leading to the chemical shift with the field-dependence coefficient appearing in the second term. The function in eqn (7) was least-squares fitted to the experimental data at different magnetic fields using two adjustable parameters: c ¼ 1 À s s Co À At this point we may assess the significance of the temperature dependence of s s Xe , which we neglected in the above analysis. As this effect does not depend on the external magnetic field, it could be absorbed in the field-independent fitting parameter c in eqn (7). Using the ideal gas law to approximate the number density of a 2 atm Xe gas at room temperature and the temperature-dependent second virial coefficient, s Xe,1 , of 129 Xe shielding constant reported in ref. 22, one can arrive at a (linearized) contribution to the temperature dependence of c amounting to just À0.0019 ppm K À1 . This change would be inside the error margins of the present fit results for c and those of the previous experiment of ref. 20.
Another physical influence that should be considered is the correction due to the difference of the bulk magnetic susceptibilities of the chloroform solution, w s , and that of the external xenon gas reference, w r . At the limit of an infinitely long cylindrical sample oriented along B 0 , the associated chemical shift contribution (in SI units) to eqn (7) equals approximately Using the molar susceptibilities w m listed in ref. 24, À58.9 Â 10 À6 Â 4p and À45.5 Â 10 À6 Â 4p cm 3 mol À1 for chloroform and xenon, respectively, we obtain w s = À9.19 and w r = À0.02509 ppm, corresponding to d b = À3.07 ppm from eqn (8). This is an additional, magnetic field-independent contribution to the parameter c. We are not aware of data on the temperature dependence of the susceptibility of either chloroform or xenon, which hinders us from accurately evaluating the contribution of the d b term to the temperature dependence of c. Presumably the susceptibility of liquid CHCl 3 , which is bigger than that of gaseous Xe by three orders of magnitude, dominates also the temperature dependence of d b . Ref. 25 reported a close-to-linear temperature dependence of the relative mass susceptibility of water at temperatures in the liquid range as w(T)/w(20C) E 1 + 1.39 Â 10 À4 (T -20C). Using w m (H 2 O) = À12.96 Â 10 À6 Â 4p cm 3 mol À1 , 24 hence w(H 2 O) = À9.01 ppm, corresponds to 1 3 Presumably the same property for chloroform does not deviate very much from this result for liquid water. Therefore, also the bulk susceptibility effect falls well within the error margins obtained in the present fit of the c parameter. Fig. 1 shows the principal result of the paper: the fielddependence coefficient t resulting from the slope of 59 Co shift with respect to B 0 2 [ Fig. 1(a)]. The magnitude of the field Xe chemical shift of xenon in the solution, d s Xe , is used to approximately detect the temperature T. The given temperatures were determined from the equation given in ref. 19. In this case, the sample contained only D-chloroform and xenon gas. Therefore, the tabulated temperatures may not be correct. One should note that the knowledge of exact temperatures is not essential in this case, but that of the 129 Xe chemical shift is. b Parameters c and t Co appearing in eqn (7) are listed for a fit of a series of measurements at different magnetic fields. Each series is performed at a constant value of d s Xe . Fig. 1 (a) (7)] as functions of temperature. Colour coding is used to indicate corresponding data points in panel (a) and panels (b and c). The error margins resulting from the statistical uncertainty of the fit are in panel (b) smaller (circa AE0.2 ppm) than the used symbols and in (c) circa AE1 ppb T À2 .
dependence is around À5.5 AE 0. 9 ppb T À2 , which is significant in view of the NMR instruments with 1.2 GHz proton frequency (28.2 T field strength) in production use soon. This result deviates clearly from the experimental data reported by Bendall and Doddrell, 3 where a distinctly larger magnitude of the field dependence, at t = À80 AE 50 ppb T À2 , was found. The underlying reasons of the difference include the limitation of the 40-year-old experiment to the modest maximum field of 2.1 T in contrast to the 18.8 T field strength used in the present work. In a similar vein, the precise temperature control enabled by using 129 Xe signal in solution is instrumental to the present success. Despite the sizable statistical error margins, t from the present measurements shows evidence of temperature dependence: its magnitude decreases with temperature with a least-squares fitted slope of 2.29 Â 10 À2 ppb T À2 K À1 . Theoretical electronic-structure formulation of the fielddependence tensor t K was presented in the form of four distinct linear and nonlinear response functions 26 in ref. 6: where the last, fourth-order perturbation theory term 6 t para K;abgd ¼ was found in ref. 6 to overwhelmingly dominate for 59 Co nucleus (see ESI † for the other terms). Indeed, the other contributions remained three or more orders of magnitude smaller than t para . In eqn (10), hhA;B,C,Dii 0,0,0 is a cubic response function 26 involving h PSO K,a , the Cartesian a-component of the orbital hyperfine interaction and h OZ B 0 ;b etc. the corresponding components of the Zeeman interaction of the orbital motion of the electrons with the external field, 27 as static perturbations. The symbol P bgd denotes a sum over all six permutations of its three indices.
Computations of eqn (9) were carried out for Co(acac) 3 in 2004 using Hartree-Fock (HF) and density-functional theory (DFT) levels. 6 The result of HF calculation, t para Co B O(1) ppb T À2 agrees with the experiment carried out presently. In view of the importance of electron correlation effects on high-order valence-like response properties such as t a question remains whether this agreement is coincidental. Since then, cubic response functions at electron-correlated ab initio levels have become feasible for systems of the present size in the DALTON software. 28 We employ the multiconfigurational self-consistent field (MCSCF) level of theory 29 to address this question here, on models of Co(acac) 3 described in the ESI. † All the tensorial components of the four terms in eqn (9) were computed using HF and MCSCF response theory. In the latter, restricted active-space (RASSCF)-type wave functions 30 correlating 18, 32 or 38 electrons [denoted RASSCF (18/32/38 el.)] were used for a truncated molecular model. The ESI † is referred to for details of the used MCSCF wave functions (Table S7, ESI †) and basis sets. For calculating the magnetic properties of a lowgap Co(acac) 3 complex, multireference-type wave functions are preferred over single configuration-based methods such as DFT, second-order many-body perturbation theory, or coupledcluster theory at low excitation level. 31 Even the present truncated model of Co(acac) 3 is currently prohibitively large for beyond-singles and doubles (CCSD) coupled-cluster theorybased nonlinear response theory calculations. 32 Table 2 lists the computational results for the dominating t para contribution to the field-dependence coefficient, as well as the total t (the latter using three of the methods: HF, RASSCF (32 el.) and DFT/BHandHLYP 33,34 ). While the results depend on the method used, both HF and the different RASSCF wave functions lead to t in an order-of-magnitude agreement with the experimental datum, with the largest result, À1.4 ppb T À2 obtained at the HF level. Increasing the number of correlated electrons tends to lower the final result as evidenced by the 10% decrease between RASSCF (18 el.) and RASSCF (32 el.), and further to RASSCF (38 el.), whereas increasing the virtual orbital space, e.g., from the RASSCF (32 el.) to the RASSCF' (32 el.) wave function has the opposite effect. In contrast, the DFT result is at a much smaller value. The small difference between the t para and t total results reassures us of the earlier finding that the 4th-order term dominates in the field dependence in Co(acac) 3 . The small contributions of the other terms are listed in Table S9 of the ESI. † The present calculations of t at optimised, static molecular geometries lead to negative field-dependence coefficient of 59 Co nuclear shielding constant in agreement with the experimental findings, and at the ab initio level similar order of magnitude of the coefficient as in the experimental data. At the same time, the new calculations result also in smaller t para than what was found earlier 6 at the HF level. The well-known sensitivity of 59 Co chemical shift on temperature in this complex suggests that both thermal rovibrational and solvation effects, which are neglected in the present computations, may significantly influence t. To investigate this possibility, we repeated the HF calculations of t para in modified models obtained by varying the Co-O bond distance from the value 1.885 Å of the optimised truncated model (for reference, 1.880 Å was obtained for the optimised full model). The results ( Fig. 2 and Table S10 in the ESI †) indicate, expectedly, a strong dependence of t on the Co-O bond lengths of the coordination shell of the Co ion. Indeed, coincidence with the range of present experimental data for t is reached at the HF level by relatively modest modification of r(Co-O). In addition to the direct magnetic field effect, molecular properties may also depend upon the indirect magnetic-field effect. 7 It arises from partial alignment of molecules because of the interaction of the anisotropy of the molecular susceptibility with the magnetic field. In partially aligned molecules, the NMR properties become dependent on the orientational order parameter. 35 In the case of Co(acac) 3 , the 59 Co chemical shift, when excluding the direct field effect, can be presented in the form 36,37 Ds Co S k ðTÞ: Here d iso Co is the isotropic average of the standard, fieldindependent chemical shift, Ds Co is anisotropy (relative to the molecular symmetry axis) of the shielding tensor and S 8 (T) is the orientational order parameter of the symmetry axis. S 8 (T) can be shown to be 7 when higher than second-order terms in B 0 are omitted. Here, Dw is the anisotropy of the molecular susceptibility tensor, k is the Boltzmann constant and T temperature. Substitution of the right-hand side of eqn (12) for S 8 (T) to (11) results in Consequently, the indirect (anisotropic) effect, the latter term in eqn (13), on the 59 Co chemical shift is proportional to the square of the magnetic flux density similarly to the direct fielddependence effect. This means that the indirect contribution may confuse the determination of the direct contribution, except if it is negligible. We computed the anisotropies of the shielding and susceptibility tensors at the DFT/BHandHLYP level, on the DALTON program. The computations resulted in Ds Co E À619 ppm and Dw E À188 Â 10 À30 J T À2 . With these values, the indirect contribution to the 59 Co chemical shift is about three orders of magnitude smaller than the direct contribution within the present temperature and magnetic field ranges, and therefore is insignificant in the present analysis.

Conclusions
We have both experimentally and computationally verified the 50-year-old prediction of Ramsey of direct magnetic-field dependence of nuclear magnetic shielding and the associated NMR chemical shift. The present experiments on 59 Co NMR in Co(acac) 3 resulted in significant field-dependence coefficient by using a range of high-field instruments and both external and internal 129 Xe reference in solution-state experiments. The external xenon gas, with its well-known and modest temperature and pressure dependence of the NMR frequency acted as the shift reference for 59 Co. On the other hand, the xenon resonance in the liquid state is extremely sensitive to the experimental conditions, which was used to adjust the temperature to constant value in NMR instruments of different field strengths. This way, the effects due to the temperature dependence of the 59 Co signal could be effectively eliminated. Similarly, the bulk magnetic susceptibility and field-induced molecular orientation (the indirect field dependence) were shown to be insignificant in the present results. Ab initio nonlinear response theory computations confirmed the order of magnitude of the field dependence and indicated its strong dependence on the coordination distance of the Co(III) ion, as expected based on the well-known temperature dependence of the 59 Co chemical shift. The magnitude of the observed field dependence in Co(acac) 3 , circa À5.5 ppb T À2 , corresponds to an experimentally significant frequency shift in, e.g., the 28.2 T field of a 1.2 GHz 1 H frequency instrument, and the phenomenon is expected to gain importance in pursuit of even higher fields and resolution in contemporary NMR. The verification of the fundamental field dependence of shielding and chemical shift adds a new parameter to the panoply of NMR observables, for which one can envision applicability in future high-field experiments designed to specifically make use of the explicit field dependence. Questions for future work remain also in finding the specific molecular characteristics that predict a significant field dependence.

Conflicts of interest
There are no conflicts to declare.