Computational NMR spectroscopy of 205Tl

We have investigated the NMR chemical shift of 205Tl in several thallium compounds, ranging from small covalent Tl(I) and Tl(III) molecules to supramolecular complexes with large organic ligands and some thallium halides. NMR calculations were run at the ZORA relativistic level, with and without spin‐orbit coupling using few selected GGA and hybrid functionals, namely BP86, PBE, B3LYP, and PBE0. We also tested solvent effects both at the optimization level and at the NMR calculation step. At the ZORA‐SO‐PBE0 (COSMO) level of theory we find a very good performance of the computational protocol that allows to discard or retain possible structures/conformations based on the agreement between the calculated chemical shift and the experimental value.


| INTRODUCTION
Thallium has two NMR-active isotopes, 203 Tl and 205 Tl both with nuclear spin 1/2. 1 205 Tl is the preferred nuclide for NMR since it yields narrower resonance lines than 203 Tl and it is more abundant. 1,2 Thallium has a chemical shift range of more than 6000 ppm, from about +5000 ppm from the reference, Tl(I)NO 3(aq) at infinite dilution, to about À1000 ppm. A thorough investigation of the NMR reference of thallium and some issues related with this choice has been presented in Reference 3. It is, however, very sensitive to the chemical environment and it has been used as a probe also in biological studies. For example, 205 Tl NMR has been proposed as a probe to investigate the metal ion binding in transferrin 4 and ATPase 5 ; the complex of Tl(I) with the antibiotic valinomicyn was also investigated by thallium NMR 6 as well as the association with gramicidin. 7 These studies took advantage of the properties of thallium(I) which are quite similar to those of potassium, since the two ions have a similar ionic radius and charge, therefore the same charge density. This is also one of the reasons for the high toxicity of thallium which is exchanged with potassium in several metabolic pathways. 8 DFT calculations of NMR properties are now an established technique to complement experimental investigations aimed at structural determination of unknown organic natural substances, 9,10 organometallic systems containing heavy atoms, [11][12][13] as well as complex fluid matrices. 14 In the case of heavy atoms, relativistic effects need to be accounted for, since they strongly influence the chemical shift mainly through spin-orbit coupling. 12,[15][16][17][18][19][20] Several heavy nuclei have been the subject of extensive theoretical works, including the closely related 199 Hg [21][22][23] and 207 Pb [24][25][26] (the previous and successive element in the periodic table, with respect to thallium) as well as other heavy nuclei such as 125 Te, 27-29 119 Sn, 30,31 113 Cd, 32 129 Xe, 33-37 just to mention a few. In contrast, computational studies of thallium NMR are quite scarce. One of the most recent works is that one of Bashi and Rahnamaye Aliabad 38 who investigated Tl shielding properties of thallium halides TlX n (X = F, Cl, Br, and I) by the first principles calculation in solid state. The Perdew-Burke-Ernzerhof Generalized Gradient Approximation, Yukawa Screened-PBE0 hybrid functional, and modified Becke-Johnson (mBJ) functionals including relativistic effects and spin-orbit coupling were used. Few other papers dealing with computational Tl NMR can be found in the recent review by Krivdin. 39 There is, however, to the best of our knowledge, no systematic investigation of 205 Tl NMR by DFT methods. Therefore, in this work we present a relativistic DFT study of thallium chemical shifts in several covalent compounds, used as a calibration set, in some supramolecular complexes with relatively large organic ligands (up to more than 100 non-hydrogen atoms plus the metal ions) as well as some simple molecular models of solid thallium halides.
Several two-component relativistic DFT schemes have been proposed in the literature, and a thorough survey has been presented in a recent review paper by Rusakova. 18 Among these methodologies, however, the Zeroth-Order Regular Approximation (ZORA), 40,41 both at the scalar relativistic (ZSC) and spin-orbit relativistic (ZSO) levels 42 has gained much popularity and found widespread application also for the calculation of NMR properties. 43,44 Moreover, while indeed four-component relativistic methods have been found to improve the agreement with the experimental results, 45 the application to several large molecules as reported in this work is still computationally demanding. Therefore, the twocomponent ZORA approximation will be used in the present investigation to deal with relativistic effects.
F I G U R E 1 Structure formula of the compounds used for the calibration set.
LANL2DZpp and cc-pVDZpp basis sets with the corresponding relativistic pseudopotentials were taken from the Basis Set Exchange web site. 56,57 The PCM (Polarizable Continuum Model) 58 solvation was used in protocol G6 as implemented in Gaussian16. At the lowest level of geometry optimization, G3 (which also served as a starting point for the optimizations at higher levels of theory) we also checked the minimized structure of each calibration compound to be a true minimum by a frequency calculation. For all systems no imaginary frequencies were found. F I G U R E 2 Optimized geometries of the compounds used for the calibration set. Compound 19 is a Tl + ion surrounded by 60 water molecules. Graphical rendering with Molden. 68 The NMR isotropic shielding constant of Tl atoms, σ, was calculated using the energy-minimized geometries with the NMR module of the ADF2019 software. 11,[59][60][61][62] Both the ZORA Scalar level (ZSC) as well as the ZORA Spinorbit level (ZSO) were used with and without the inclusion of long range solvent effects through the COSMO model of solvation. 63 For the calculation of the NMR shielding constant the following functionals were used: BP86, 51,64 B3LYP, 50,52-54 PBE 65 and PBE0. 66,67 Basis set used for the NMR calculations are shown in Table 1, for thallium we always used the relatively large QZ4P all electron basis set. Although in this work we are interested only in the isotropic shielding constant, in Table S1 in Supporting Information we have listed also the three components, σ xx , σ yy , and σ zz , in the principal coordinate frame, of the shielding tensor obtained at the final level of theory (level xviii, see below). These data might be useful to the interested reader.
For each structure shown in Figure 1 and labeled from 1 to 19, For each computational protocol listed in Table 1 we have calculated the linear fit of the correlation between calculated shielding constants and the experimental chemical shifts the correlation coefficient, R 2 , and the Corrected Mean Absolute Error (CMAE), that is the average distance from the fitting line, defined by where, N is the total number of compounds in the calibration set, and σ i and δ i are the ith calculated shielding constant and experimental chemical shift, respectively.
The calculations for the larger supramolecular complexes shown in Figure 3 (structural formula of the ligands) and Figure 4 (3D rendering of the minimized thallium complexes' geometries) were run only at the level of theory xviii, which, as we will see in the Results Section, represents the best computational protocol for the prediction of the NMR shielding properties among the ones tested here.
Initial geometries were taken from the x-ray structure, where available, or built with Molden, 68 except for 19. Due to its relevance in thallium NMR as a reference compound, although it lacks a clear covalent geometry, Tl(I) + (aq) has been modeled by surrounding a Tl + ion with 60 water molecules with Packmol 69 to make a starting structure for the subsequent optimization.
For the supramolecular compounds 20-25, the chemical shifts, δ i , to be compared with the experimental value, has been calculated as where, σ i is the shielding constant of the thallium atom (or the For compounds 20-25, see Figure 3, the solvents used for the calculations, taken from the experimental conditions, are: Tl@20 + , there are no experimental data of 205 Tl chemical shift. 73 The authors measured 1 H spectra and from there some Tl coupling constants. We have also investigated the "through space" spin-spin coupling in such complex using relativistic DFT. 74 Here we use the selected computational protocol to predict the chemical shift using the x-ray structure in the gas phase and, for the sake of comparison, methanol.

| RESULTS AND DISCUSSION
We first discuss the results obtained from the calibration set (see For the various levels of theory employed and reported in Table 1 we have calculated the 205 Tl isotropic shielding constants, σ (for multinuclear compounds we calculated the average value), and we have correlated it with the corresponding experimental chemical shifts, δ. In Figure 5 and Table 2 we report the correlation coefficient of the linear fitting, R 2 , and the CMAE, see Equations (1) and (2). The graphs of such correlations can be found in Supporting Information, Figures S1-S20, while the experimental values and the results obtained at the level of theory xviii are in Table 3.
The analysis of both parameters highlights several important results.
First, not unexpectedly, the inclusion of spin-orbit coupling in the NMR calculation is mandatory. All calculations run at the ZSC level consistently show a poor correlation, with R 2 lower than 0.80 and a CMAE larger than 500 ppm. In contrast, all protocols including the SO coupling in the NMR calculation have R 2 higher than 0.90 and a CMAE lower than 400.
Second, optimization levels G1-G4 clearly produce geometries of lower qualities than the protocols G5 and G6, as indirectly judged from the correlation parameters of the calculated shielding constants at the same NMR levels.
Protocols G5 and G6 only differ by the inclusion of the solvent reaction field during the optimization. As can be seen the effect on  (3). For the level of theory xviii, the fitting parameters, also reported in Table 2 are b = 14,100.5 ppm, a = À1.1106.
It is interesting to analyze the performance also with respect to specific subsets of structurally similar compounds. For example, compounds 1-5 in Figure 1   Therefore, besides the quantitative estimation of the general performance of the computational protocol using the various statistical parameters discussed above, we can say that the predictive power when applied to a relatively small series of structurally related compounds, even differing by few ppm, is quite significant. This is a key feature if the protocol has to be used as an aid in structural elucidation.
We now turn our attention to some large supramolecular complexes of thallium(I) with organic ligands. They are shown in Figures 3 and 4 and the results of the calculations for these systems are reported in Table 4. As mentioned already, for these compounds we calculate the chemical shift using Equation (3). the di-nuclear complex is more shielded than the mono-nuclear Note: σ theo represent the theoretical prediction of the shielding constant based on the linear fit, that is σ theo À σ calc j j is the vertical distance of the point from the fitting line. The percent deviation is also referred to the theoretical shielding constant. complex by just 40 ppm, a value significantly smaller than the CMAE obtained from the calibration set. Nonetheless, the calculations correctly predict Tl 2 @22 2+ to be more shielded than Tl@22 + (by 38.2 ppm), in excellent agreement with the experiments.
Similarly, for ligand 23, the experimental data show a difference between the mono-and di-nuclear complex of 47 ppm, but in this case it is the mono-nuclear complex, Tl@23 + , which is more shielded than the di-nuclear Tl 2 @23 2+ . Again, the calculations correctly predict the sign of the relative shifts, and the resonance of Tl@23 + is calculated more shielded, by 22.3 ppm. It is therefore confirmed that the performance of the level of theory, when applied to similar structures is significantly higher than what can be deduced by the statistical parameters obtained for the full calibration set.
We believe that these small differences in chemical shift between the mono-and di-nuclear complexes are due to subtle dif- The last organometallic complex considered here is the dithallium cryptand Tl 2 @25 2+ . There is no experimental value, in the literature, of the 205 Tl chemical shift of this compound, though there is a clear indication from the 1 H NMR of a "through-space" spin-spin coupling with the protons of the aromatic ring. 73,74 Thus, for an estimation of the expected chemical shift we used the x-ray structure of the cryptand. 73 The predicted chemical shift (ca. 2200 ppm, see Table 3) is in a completely different region, compared to the set of supramolecular complexes of Tl(I) which all lie in the "shielded" range from 0 to À700 ppm. This can be explained considering the strong coordination of the thallium ions in the cryptand. For example, the Tl-Tl distance, in the x-ray structure, is 4.375 Å while the Tl-NR 3 and Tl-  F I G U R E 6 Correlation between experimental chemical shifts δ( 205 Tl) and calculated shielding constants σ at the level of theory xviii. Black circles refer to compounds 1-19 of Figure 1: red circles refer to the supramolecular complexes Tl@20 4À , Tl@21 + , Tl@22 + , Tl 2 @22 2+ , Tl@23 + , Tl 2 @23 2+ , Tl 2 @24 2+ ; magenta triangles refer to the thallium halides models 26-30. The black line is a linear fit of the data for compounds 1-19 only (calibration set). The empty markers represent results from additional structures/conformations discussed in the main text.
nitrogen donors and, as a consequence, the corresponding chemical shift falls in the typical range of covalent compounds of Tl(I).
Finally, we have analyzed some thallium halides for which experimental data have been reported. 75 The NMR chemical shift has been measured in the solid state and water solution, nonetheless we focus our attention here on the SSNMR data since these have a clearer geometry, while in water several species are in equilibrium depending on the halide concentration. 75  As we can see in Figure 6, the calculated shielding constants are well reproduced (the distance from the fitting line is comparable to all other compounds) except for Tl(III)Br 6 3À (empty triangular marker in Figure 6). The calculated chemical shift is off by about 1800 ppm, a very large discrepancy, significantly larger than the accuracy of the computational protocol as judged by the several statistical parameters discussed above. This point is calculated using the geometry experi-  Figure 6 and Table 5.
Two model TlI 4 À ions are considered here since there are two independent molecules in the cell. 79 The two calculated results are both shown in Figure 6 as empty markers, together with the average value.
The agreement is quite remarkable also for the ionic species whose NMR has been measured in solid state. In particular, the tri- After having investigated the three main sets of compounds (the calibration set, the supramolecular complexes and the thallium halides), it might be interesting to discuss more in detail the various relativistic contributions to the shielding constant. Within the ZORA approximation, the shielding constant at the spin-orbit level is given by the sum of three terms: the paramagnetic term, σ PARA , the diamagnetic term, σ DIA , and the spin-orbit term, σ SO . In contrast, at the ZORA Scalar level, only the paramagnetic and the diamagnetic terms contribute. It must be stressed, however, that the paramagnetic term is also affected by the inclusion of spin-orbit coupling in the Hamiltonian, therefore is it not exactly the same for a calculation run at the ZORA Scalar or ZORA spin-orbit level. However, the most important difference between the two levels (keeping all the rest the same, namely functional, basis set, integration accuracy, etc.) is the presence of an explicit spin-orbit contribution at the ZSO level.
In Figure 7 we show the paramagnetic and spin-orbit contributions at the level xviii (data can be found in Table S1  expected because both bromine and especially iodine are also heavy atoms for which relativistic effects play a prominent role. 17 Concerning the indirect effect of the spin-orbit coupling, that is the effect on the calculated NMR shielding constants due to the inclusion or neglect of the SO term in the Hamiltonian during the geometry optimization, it can be deduced by comparing the results at the levels of theory ii (no spin-orbit included in the geometry optimization) and iv (spin-orbit included in the geometry optimization) that this effect does not appear to be relevant here since the two sets of results are very similar concerning their statistical parameters (see Table 2).

| CONCLUSIONS
We propose a computational protocol based on relativistic DFT for the prediction of the 205 Tl NMR chemical shift. Some well-known issues have been confirmed by this investigation: on the one hand, the inclusion of spin-orbit coupling is mandatory for such heavy atoms; on the other hand, hybrid functionals generally performs much better than GGA functionals. Moreover, a significant improvement in the correlation between calculated and experimental data is obtained if solvent effects are included both in the optimization step as well as in the NMR calculation.
Besides these observations, we have also highlighted some relevant issues concerning 205 Tl NMR. First, the computational protocol labeled as xviii has a high predictive power. The CMAE obtained from the correlation of calculated shielding constants and experimental chemical shifts is below 200 ppm over a range of more than 6000 ppm. More important than that, however, is the fact considering sub-sets of chemically related compounds, the predictive power appears to be significantly higher allowing to reproduce the shielding/deshielding effect related to relatively small structural changes in the molecule such as the increasing bulkiness of alkyl substituents bonded to Tl, or the encapsulation of one or two Tl ions in a supramolecular complex. Therefore, the protocol can be used, for example, to discard some possible conformations as we have seen for compound Tl@21 + where the encapsulation of Tl by the four benzene rings not only has a higher energy but also a chemical shift very much in error compared with the experimental value. Similarly, the calculation using the ion Therefore, we believe that these results can provide the practicing chemist working with thallium compounds with an additional tool for structural elucidation based on the comparison of calculated and experimental chemical shifts.

ACKNOWLEDGMENTS
Calculations were run partly on the Linux cluster of the Computational Chemistry Community of the Department of Chemical Sciences of the University of Padova (C3P). CloudVeneto Consortium is acknowledged for the use of computing and storage facilities.

DATA AVAILABILITY STATEMENT
The data that supports the findings of this study are available in the supplementary material of this article.