Quantitative analysis of zero-field splitting parameter distributions in Gd(iii) complexes

Abstract

The magnetic properties of paramagnetic species with spin S > 1/2 are parameterized by the familiar g tensor as well as "zero-field splitting" (ZFS) terms that break the degeneracy between spin states even in the absence of a magnetic field. In this work, we determine the mean values and distributions of the ZFS parameters D and E for six Gd(iii) complexes (S = 7/2) and critically discuss the accuracy of such determination. EPR spectra of the Gd(iii) complexes were recorded in glassy frozen solutions at 10 K or below at Q-band (∼34 GHz), W-band (∼94 GHz) and G-band (240 GHz) frequencies, and simulated with two widely used models for the form of the distributions of the ZFS parameters D and E. We find that the form of the distribution of the ZFS parameter D is bimodal, consisting roughly of two Gaussians centered at D and -D with unequal amplitudes. The extracted values of D (σD) for the six complexes are, in MHz: Gd-NO3Pic, 485 ± 20 (155 ± 37); Gd-DOTA/Gd-maleimide-DOTA, -714 ± 43 (328 ± 99); iodo-(Gd-PyMTA)/MOMethynyl-(Gd-PyMTA), 1213 ± 60 (418 ± 141); Gd-TAHA, 1361 ± 69 (457 ± 178); iodo-Gd-PCTA-[12], 1861 ± 135 (467 ± 292); and Gd-PyDTTA, 1830 ± 105 (390 ± 242). The sign of D was adjusted based on the Gaussian component with larger amplitude. We relate the extracted P(D) distributions to the structure of the individual Gd(iii) complexes by fitting them to a model that superposes the contribution to the D tensor from each coordinating atom of the ligand. Using this model, we predict D, σD, and E values for several additional Gd(iii) complexes that were not measured in this work. The results of this paper may be useful as benchmarks for the verification of quantum chemical calculations of ZFS parameters, and point the way to designing Gd(iii) complexes for particular applications and estimating their magnetic properties a priori.

Figure 1

Graphical representation of the models used in this work for the distributions of the ZFS parameters D and E (or E/D). (a) Model 1 assumes that P(D) and P(E) are described by two uncorrelated Gaussian distributions. (b) Reshuffling of the indices to correct for the inconsistencies of Model 1 with the conventional definitions of the D and E parameters results in a bimodal Gaussian distribution. (c) Model 2 assumes P(D) is a bimodal Gaussian distribution, where the positive (D > 0) and negative (D < 0) contributions have equal amplitude and width. (d) Model 3 adds an asymmetry parameter (denoted P(+D)/P(−D)) to Model 2, which allows the relative amplitudes of the positive and negative contributions to the P(D) distribution to vary. (e) For Models 2 and 3, P(E/D) follows a polynomial distribution given by P(E/D) ∝ (E/D) − 2 * (E/D)².

Figure 2

Structural formulae and naming of the Gd(III) complexes 1 - 7 which were studied in this work. Please note that in the case of Gd-TAHA (5) and Gd-PyDTTA (7) no crystal structures are available, and the dotted lines only indicate possible ligand atom-Gd(III) ion interaction.

Figure 3

Evolution of allowed EPR transitions as a function of field/frequency and temperature for an unimodal P(D) distribution with 〈D〉 = 1200 MHz, σ_D = 400 MHz, and P(E/D) as given in Equation 5. a) Q band and 10 K, b) W band and 10 K, c) G band and 5K.

Figure 4

Distribution of ZFS parameters for Model 1 as defined in Equation 5 (black) and after rearranging of the indexes (X, Y, Z) of the computed D_X, D_Y and D_Z values (light blue) for the Gd(III) complexes Gd-NO₃Pic (1) and Gd-PyDTTA (7). Gaussian distributions are overlaid over the rearranged P(D) distributions (red dashed lines). Distributions are scaled so that the area under the curves integrates to 1. (a, d) P(D) distributions, (b, e) P(E) distributions, and (c, f) P(E/D) distributions. The green line shows P(E/D) defined in Equation 8 [23], used in the simulations with Models 2 and 3 in this manuscript.

Figure 5

EPR spectra (black lines) and corresponding fits (light blue lines) obtained using Model 1 and the ZFS parameters given in Table 2 for the complexes Gd-NO₃Pic (1) and Gd-PyDTTA (7). Q band spectra at 10 K, W band spectra at 10 K, and G band spectra at approximately 5 K.

Figure 6

Contours of constant RMSD as a function of D and σ_D parameter values using Model 2 for the complexes Gd-NO₃Pic (1) and Gd-PyDTTA (7) in Q band and 10 K, W band and 10 K, and G band and 5 K. Simulated spectra were normalized to the experimental data using only the outer shoulders of the spectra. The asterisk denotes the set of parameter values available in the library of simulated spectra which has the minimum RMSD value for each measurement frequency. Each contour line represents a doubling of this minimum RMSD value.

Figure 7

Simulations using the best-fit ZFS parameters for Model 2, with and without the region of the central transition included in the RMSD error map analyses, for the complexes Gd-NO₃Pic (1) and Gd-PyDTTA (7).

Figure 8

Contours of constant RMSD as a function of P(+D)/P(−D) and σ_D parameter values using Model 3 and the complexes Gd-NO₃Pic (1) and Gd-PyDTTA (7) in G band and 5 K. The mean values of the ZFS parameter D were set to D = 500 MHz and D = 1800 MHz, respectively, corresponding to the closet D value available in the library of simulations to the D value as determined by Model 2 for these complexes (Table 2). The asterisk denotes the position of minimum RMSD.

Figure 9

Measured EPR spectra in Q band, W band, and G band for the Gd(III) complexes Gd-NO₃Pic (1), Gd-DOTA (2) (G-band spectra)/Gd-maleimide-DOTA (3) (Q-/W-band spectra), and iodo-(Gd-PyMTA) (4a) (G-band spectra)/MOMethynyl-(Gd-PyMTA) (4b) (Q-/W-band spectra). Overlaid are simulations with Model 3 using the best-fit ZFS parameters presented in Table 2. The faded regions indicate the portion of the spectra about the central transition which was excluded from the RMSD error map calculations.

Figure 10

Measured EPR spectra in Q band, W band, and G band for the Gd(III) complexes Gd-TAHA (5), iodo-(Gd-PCTA-[12]) (6), and Gd-PyDTTA (7). Overlaid are simulations with Model 3 using the best-fit ZFS parameters presented in Table 2. The faded regions indicate the portion of the spectra about the central transition which was excluded from the RMSD error map calculations.

Figure 11

Comparison of the extracted values for the mean (〈D〉) and width (σ_D) of the ZFS parameter D for the three models and each of the tested Gd(III) complexes. Structural formulae and naming for the Gd(III) complexes 1 - 7 are given in Figure 2. Model 1 was fit by visual inspection, and therefore error bars on the ZFS parameters D and σ_D were not computed. For Models 2 and 3, mean values and error bars for D and σ_D were computed by combining results from RMSD error maps which compare a library of simulated spectra to the data at the three measurement frequencies. Models 2 and 3 were fit with the region about the central transition excluded from analysis, and also with the full EPR spectra included in the analysis.

Figure 12

Comparison of distributions of anisotropy Δ (a, c, e, g) and axiality ξ (b, d, f, h) between fits to experimental data by Model 1 (blue) and Model 3 (green), as well as the prediction by superposition Model B with an isotropic standard deviation of atom positions σ_xyz = 0.05 Å (red). The orange curves are predictions by superposition Model B with an isotropic standard deviation of atom positions σ_xyz = 0.10 Å. (a, b) Gd-NO₃Pic (1). (c, d) Gd-DOTA (2). The grey curves are predictions by superposition Model B based on the crystal structure of the Ce(III)-DOTA. (e, f) Gd-PyMTA (4). (g, h) iodo-(Gd-PCTA-[12]) (6). The prediction for iodo-(Gd-PCTA[12]) is based on a crystal structure of the Ho(III) complex with a ligand that formally derives from PCTA-[12] by substitution of the carboxylate for phosphonate groups.

Figure 13

Structural formulae and naming of the Gd(III) complexes 8-14 considered for ZFS parameter value prediction with the superposition Model B.