Rationalisation of anomalous pseudo-contact shifts and their solvent dependence in a series of C 3 -symmetric lanthanide complexes

Bleaney’s long-standing theory of magnetic anisotropy has been employed with some success for many decades to explain paramagnetic NMR pseudo-contact shifts, and has been the subject of many subsequent approximations. Here, we present a detailed experimental and theoretical investigation accounting for the anomalous solvent dependence of NMR shifts for a series of lanthanide(III) complexes, namely [LnL 1 ] (Ln = Eu, Tb, Dy, Ho, Er, Tm, and Yb; L 1 : 1,4,7-tris[(6-carboxypyridin-2-yl)methyl]-1,4,7-triazacyclononane), taking into account the effect of subtle ligand flexibility on the electronic structure. We show that the anisotropy of the room temperature magnetic susceptibility tensor, which in turn affects the sign and magnitude of the pseudo-contact chemical shift, is extremely sensitive to minimal structural changes in the first coordination sphere of L 1 . We show that DFT structural optimisations do not give accurate structural models, as assessed by the experimental chemical shifts, and thus we determine a magneto-structural correlation and employ this to evaluate the accurate solution structure for each [LnL 1 ]. This approach allows us to explain the counter-intuitive pseudo-contact shift behaviour, as well as a striking solvent dependence. These results have important consequences for the analysis and design of novel magnetic resonance shift and optical emission probes that are sensitive to the local solution environment.


INTRODUCTION
Complexes of lanthanide (Ln) ions are widely used in biochemical and medical applications of NMR spectroscopy including, for example, magnetic resonance imaging and structural and functional study of biological systems. [1][2][3][4][5][6] A cornerstone of this area has been the interpretation of chemical shift data via Bleaney's theory of magnetic anisotropy. 7,8 This theory states that, for remote nuclei -where the Fermi contact term δc is vanishingly small, as discussed by others 9the paramagnetic chemical shift is dominated by the pseudo-contact (dipolar) shift (δpc) and can be simply related to the crystal field (CF), the geometry, and a factor that relates to the identity of the specific Ln ion. For an axially symmetric complex, δpc is approximated by Equation 1. Here, θ and r are the polar coordinates of the NMR active nucleus with respect to the principal axes of the magnetic susceptibility tensor χ, is the second rank axial CF parameter of the Hamiltonian Equation 2 (where are the Steven's operator equivalents and 〈 ‖ ‖ 〉 are the operator equivalent factors) and CJ = gJ 2 J(J+1)(2J-1)(2J+3)〈 ‖ 2‖ 〉 is Bleaney's constant. Since CJ is a function of the total angular momentum J and the Landé factor gJ, its value depends only on the electronic configuration of the lanthanide ion.

3
(1) The crucial assumptions made by Bleaney were: (i) that the total CF splitting is << kT, and (ii) that J is a good quantum number. If these assumptions hold, only second order terms of temperature (T) are required to accurately describe the magnetic susceptibility. Furthermore, it is often assumed that the axial CF parameter and the geometric part do not vary across an isostructural series of complexes, in which case the relative order of for a given nucleus in an  7 While this simplistic description has been proven correct in many cases, 10,11 it has been found to be invalid in some recent works, 12,13,14 failing to reproduce even the trends in experimental shifts across isostructural series of Ln complexes. Discrepancies are often attributed to the many approximations given above, without specifying the main source. In certain cases, the "culprit" seems clear, as in the case reported by Piguet and co-workers where a sudden structural variation across the Ln series leads to abrupt change in the value of . 15 A relevant example, reported by some of us, concerns behaviour in the [LnL 1 ] family. 12  Based on CJ alone and assuming a constant value of , the expected order of the pseudo-contact shifts for the pyridyl protons should be Dy < Tb < Ho < Yb < Er < Tm, while experimentally, the order is found to be Tb < Ho < Er < Yb < Dy < Tm. In this series and two closely related isostructural series based on triazacyclononane, it was shown, with the aid of two/three nuclei plots devised by Reuben/Geraldes, that the pyridyl resonances located some 5.4 to 6.3 Å from the metal centre were not subject to any significant contact shift. 12 Here, we provide a detailed explanation of the origin of the peculiar paramagnetic NMR behaviour of [LnL 1 ], including the origin of a new and significant solvent dependence (D2O, MeOD and d6-DMSO). We demonstrate how the delicately balanced CF provided by the L 1 ligand renders the sense of magnetic anisotropy, i.e. easy axis ( ∥ ⏊ ) or easy plane ( ∥ ⏊ ), extremely responsive to seemingly trivial geometric changes in the first coordination sphere, ultimately controlling the sign and magnitude of the pseudo-contact paramagnetic NMR shift. This is not the first time that the tricapped trigonal prismatic geometry has been implicated in anomalous pseudo-contact shifts, [16][17][18] however, we rationalise the origins of such effects for the first time in terms of the underlying electronic structure of the lanthanide complexes.

RESULTS AND DISCUSSION
We focus on the NMR shifts of the three pyridyl 1 H nuclei (pyH3-5), which are quite distant from the lanthanide ion (Figure 1, left) and hence their paramagnetic shifts should be dominated by the pseudo-contact term; this is justified by Reuben/Geraldes plots in ref. 12 showing that the contact contribution is very small, and experimentally validated here (see below and Figure S7).
We first consider [DyL 1 ] which shows a striking departure from the simple assumptions given above, having (i) a pseudo-contact shift of pyH3-5 of the same sign (positive) as the late Ln   Figure S1). We note that this solvent dependence is not due to the change in diamagnetic shift, as these are negligible for the [YL 1 ] complex (Table S1) to which all our paramagnetic shift values are referenced.   20 with the SMD solvent model. 21 The electronic structure and resulting room temperature magnetic susceptibility tensor were then determined for this pseudo-solution structure using complete active space self-consistent field spin orbit (CASSCF-SO) calculations (see Supporting Information).
The theoretical were subsequently calculated using Equation 3,7,8 where ∥ is the anisotropy of the molar magnetic susceptibility in cm 3 mol -1 ( ∥ ⏊ ), NA is Avogadro's number, and r is the Ln···H distance in metres. The calculated with this method (M06/SMD,   Table S2) and importantly, does not change sign. Hence, it must be changes in the magnetic anisotropy term causing the large changes in the calculated . Therefore, the electronic structure of [DyL 1 ] must be very sensitive to the coordination geometry such that seemingly trivial structural variations, such as those described above, can induce large changes (including sign) in the magnetic anisotropy. This is not only important when assessing DFT methods for providing reliable structural models, but also reflects the intrinsic sensitivity of the electronic  In light of the similar effect for the O and Nax-atoms , we compared the relative DFT energies of the partially optimised structures for these distortions (Table S3, Figure S3). These data clearly show that movement of the O-atoms is much more facile than movement of the Nax-atoms within the relatively rigid 9-N3 ring, and that a variation of Δθ = ±2° (i.e. sufficient to change the sign of the magnetic anisotropy) for the O-donors is within kT at 298 K. Thus, we conclude that the structural distortion responsible of the variation of the susceptibility tensor, and hence is most likely to be associated with the movement of the axial O donor atoms.
Inspection of the partially optimised structures for different polar angles for the O-atoms shows that the main differences are in the rigid rotation of the pyridyl rings ( Figure S4): these can be parameterised by two torsion angles (labelled β) and (labelled α). The variation of α is three times larger than that of β ( Figure S5), suggesting that changes in θ can be adequately mapped through variation of α alone. Indeed, CASSCF-SO calculations of the magnetic susceptibility tensor as a function of α alone agree well with its dependence on θ ( Figure S6); hence we adopt α as the sole variable to study the effects of structural variations.
In order to understand the origin of the change in room temperature magnetic anisotropy under such a small structural change, we examined the electronic structure of the ground J = 15/2 multiplet as a function of α (Dy III has a 6 H15/2 ground state in the Russell-Saunders formalism).
The calculated electronic structure of the reference M06/SMD geometry (α = 40.4°, which gives a polar angle of θ = 49.6° for the O donors) gives two low-lying Kramer's doublets, very close in energy (ca. 11 cm -1 , Figure 3), which have characteristic g-tensors that are easy-axis ( || ) and easy plane ( || ) for the ground and first excited doublet, respectively ( Figure 3 and Table S4). . Kramers doublets between perfectly easy axis and isotropic will appear orange/yellow, and those between fully easy plane and isotropic will appear light blue/green. Upon decreasing α by only 1° from the reference geometry, hence increasing θ to a value closer to the magic angle, the two lowest doublets swap order, resulting in an easy-plane ground state and an easy-axis first excited state ( Figure 3 and Table S4). This change coincides with the change in sign of the calculated room temperature magnetic susceptibility anisotropy, although this necessarily results from contributions due to all Boltzmann-populated excited doublets at 298 K. Furthermore, decreasing α consolidates this trend with the two lowest Kramer's doublets progressively moving further apart and an increased easy-plane character of the ground doublet; the opposite trend is observed for increasing α from the reference geometry. Interestingly, the 13 optimised reference geometry is very close to the minimum overall CF splitting of the J = 15/2 multiplet (Figure 3), corresponding to a CF which does not favour any particular magnetic states and thus gives a near-isotropic magnetic susceptibility. In terms of the CF Hamiltonian Equation 2, we observe that only the second rank axial term changes sign as a function of α and that it has by far the largest variation of all the CF parameters ( Figure 4, Table S5; only , , , , , terms are allowed in C3 symmetry). Therefore, it is clear why an anomalous trend is observed for the pseudo-contact shifts of [LnL 1 ]: is very sensitive to very minor changes in geometry in this ligand system and cannot be assumed to be a constant.  Figures 5 and S7).
We now turn to the solvent dependence of the pseudo-contact shifts in [DyL 1 ]. Experimentally, we find that the measured for pyH3-5 become more positive, and have a larger spread, on moving from D2O to MeOD to d6-DMSO (Figures 1 and S1) Figure S8). Secondly, we have tested and observe that the structural part in Equation 3 varies very little across the D2O, MeOD and d6-DMSO optimised structures (≤ 3%, Table S7), or for variation in α (within a sensible range) for a given solvent (≤ 3%, Table S8). Hence, we hypothesise that the experimentally observed solvent dependence of is due to changes in the anisotropy of the magnetic susceptibility.
Under the assumption that for [DyL 1 ] the contact contribution is negligible and hence the paramagnetic shift is dominated by , 13 we can find the latter by plotting the experimental of protons as a function of the structural part of Equation 3: the slope then gives the magnetic susceptibility anisotropy ( ∥ ) for [DyL 1 ] in each solvent system ( Figure S7). Then, we can correlate these against the calculated angular dependence of ∥ to determine the structure   29 The luminescence spectra for this transition shows the 7 F1 splitting increasing as H2O < MeOH < DMSO ( Figure 6); circularly polarized luminescence does not increase the resolution of these spectra ( Figure S13).
Fitting the emission lines with a two component Gaussian model gives the expected 1:2 ratio (  Figure S7). 13 Variation of solvent for these complexes shows that of pyH3-5 has the same strong dependence as observed for [DyL 1 ] ( Figure S17) is approximately constant and it is the second rank axial CF parameter that can vary dramatically, including changing sign, upon minimal variation of the coordination geometry.
Thus, we conclude that cannot be considered a constant in this series of complexes. We have shown that significant variations in the NMR pseudo-contact shifts in different solvents are due to small structural variations, likely owing to solvent polarity and/or hydrogen bonding propensity, and have independently confirmed this with luminescence spectroscopy. These results have important consequences for the design of magnetic resonance shift agents and responsive optical probes. The ease of modulation of the size and sign of associated with this ligand type could be exploited in developing probes that respond to small physicochemical perturbations, e.g. from changes in the local environment, such as medium polarity.