Article pubs.acs.org/JPCA
Scalar Relaxation of the Second Kind A Potential Source of Information on the Dynamics of Molecular Movements. 2. Magnetic Dipole Moments and Magnetic Shielding of Bromine Nuclei Adam Gryff-Keller,*,† Sergey Molchanov,† and Artur Wodyński‡ †
Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
‡
S Supporting Information *
ABSTRACT: In this paper, we continue the exploration of possibilities, limitations, and methodological problems of the studies based on measurements of the nuclear spin relaxation rates running via the scalar relaxation of the second kind (SC2) mechanism. The attention has been focused on the 13C−79Br and 13C−81Br systems in organic bromo compounds, which are characterized by exceptionally small differences of Larmor frequencies, ΔωCBr, of the coupled nuclei. This unique property enables experimental observation of longitudinal SC2 relaxation of 13C nuclei, which makes investigation of the SC2 relaxation rates an attractive experimental method of determination of spin− spin coupling constants and relaxation rates of quadrupole bromine nuclei, both types of parameters being hardly accessible by direct measurements. A careful examination of the methodology used in SC2 relaxation studies of carbon−bromine systems reveals, however, some disturbing facts that could burden the results with systematic inaccuracies. Namely, the way of calculating the Larmor frequency differences between 13C and bromine isotopes, ΔωCBr, may cause some reservations. In this work, the values of 79Br and 81Br magnetogyric ratios have been rechecked using bromine NMR data for the KBr·Kryptofix 222 complex in acetonitrile solution and the results of the advanced calculations of the magnetic shielding of the bromine nucleus in the Br− anion. Moreover, it has been pointed out that in the case of 13C−79Br, the magnetic shielding of the bromine nucleus in the investigated molecule must not be neglected during the calculation of the ΔωCBr parameter. Some recommendations concerning the exploitation of available theoretical methods to calculate bromine shielding constants for bromo compounds have also been formulated, keeping in mind relativistic effects.
1. INTRODUCTION Investigation of the rates of the nuclear spin relaxation running via the mechanism dependent on the spin−spin coupling, known as the scalar relaxation of the second kind (SC2), is an attractive experimental method of determination of spin−spin coupling constants and relaxation rates of quadrupole nuclei, both types of parameters being hardly accessible by direct measurements.1−4 The scalar spin−spin coupling constant, which is the orientation-independent part of the tensor describing indirect (transmitted by electrons) interaction between a pair of nuclear spins, is a very important NMR parameter, interesting for spectroscopists, structural chemists, and theoreticians.5,6 Strong and well-recognized dependence of coupling constants on the number of bonds separating coupled nuclei in the molecule has been the basis for spectral assignments since the first applications of high-resolution NMR spectroscopy. Most of the popular contemporary multidimensional correlation techniques are actually developments of that method. The high sensitivity of these parameters to space relationships has been widely exploited in the studies of stereochemistry. At the same time, coupling constants carry important information on electronic structures of molecules. For theoreticians, calculations of sufficiently precise values of coupling constants are © 2013 American Chemical Society
still a challenging task, especially when a heavy nucleus participates in the coupling.6 As it concerns the other parameter mentioned before, measurable via SC2 relaxation data interpretation, it is to be noticed that the data on nuclear spin relaxation rates carry a wealth of information on molecular movements, the investigation of which by other methods is difficult.3 The nuclei of most magnetic isotopes possess spin higher than 1/2 and electric quadrupole moments. Such nuclei interact with the local electric field gradients generated by their electronic surroundings, which couples the nuclear spin orientation with the orientation of the molecule.1,3,7 This relaxation mechanism is most frequently very efficient for quadrupole nuclei of other than the first-row elements, which simplifies substantially interpretation of such relaxation data. On the other hand, rapid relaxation causes the NMR signals of these nuclei to be extremely broad, which most frequently prevents measurement of their NMR spectra and direct determination of appropriate chemical shifts, coupling constants, and relaxation parameters. It is to be added that interpretation of the relaxation data for Received: October 13, 2013 Revised: December 13, 2013 Published: December 13, 2013 128
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much broader 79Br NMR and 81Br NMR signals were determined by performing the least-squares line shape analysis of bromine NMR spectra, assuming the Lorentzian shape of these spectral lines. The nonrelativistic calculations, which included geometry optimization and NMR shielding calculation, were performed with the aid of the Gaussian03 program12 using the DFT method with the B3LYP13 or PBE1PBE hybrid functional (also known as PBE0)14 and the standard 6-311++G(2d,p) basis set. Additional nonrelativistic computations at the CCSD level were performed with the aug-cc-pVTZ basis set using the CFOUR program.15 The relativistic shielding calculations, which took into account both scalar and spin−orbit coupling terms, were performed by using the two-component ZORA Hamiltonian available in the ADF program package.16−18 In these calculations, the nonrelativistic molecular geometries, B3LYP or PBE0 functionals, and QZ4P basis set were used. The final results of theoretical calculations were achieved by the fourcomponent Dirac−Coulomb Hamiltonian with the aid of the DIRAC 12.7 computational package.19 These computations were performed again with B3LYP and PBE0 functionals, the quadrupole zeta-type Dyall.v4z basis set18 for bromine, and the aug-cc-pVTZ basis set20,21 for other atoms. The unrestricted kinetic balance (UKB) ansatz combined with London atomic orbitals (LAOs) in the simple magnetic balance (sMB)22 approach was used during the computations. The solvent effects were estimated on the basis of PCM23 and COSMO16 solution models in calculations performed by Gaussian and ADF programs, respectively.
quadrupole nuclei demands knowledge of the quadrupole coupling constants involved, which are products of the nuclear quadrupole moments and the electric field gradient tensors. Determination of the latter is another exploration field for quantum chemistry methods. The SC2 mechanism can appear for a magnetic nucleus A that is coupled to rapidly relaxing nucleus X. Most frequently, such a situation occurs when the nucleus X is of a higher spin and relaxes via the quadrupole mechanism. The rapid relaxation of X erases partially or completely the spin−spin splitting of the signal of A. At the same time, this relaxation causes random fluctuation of the energy of the spin−spin interaction in the AX spin system, which affects the behavior of nuclei coupled to the quadrupole nucleus. Under certain conditions, this phenomenon may be described as an additional relaxation path, named the SC2 mechanism. The contributions of this mechanism to the overall longitudinal and transverse relaxation of the nucleus A are given by well-known formulas1−4 R1,SC2(A) =
⎛2⎞ R 2(X) 2 ⎜ ⎟I (I + 1)(2πJ ) AX ⎝3⎠ X X [R 2(X)2 + ΔωAX 2] (1)
R 2,SC2(A) =
⎛1⎞ 2 ⎜ ⎟I (I + 1)(2πJ ) AX ⎝3⎠ X X ⎧ 1 ⎫ R 2(X) ⎨ ⎬ + [R 2(X)2 + ΔωAX 2] ⎭ ⎩ R1(X)
(2)
R1(X) and R2(X) are the rates of longitudinal and transverse relaxation of X, respectively, and ΔωAX is the difference of Larmor frequencies of the coupled nuclei. The above equations show why investigation of the SC2 relaxation rates can be an experimental method of determination of spin−spin coupling constants and relaxation rates of quadrupole nuclei. Continuing studies based on SC2 relaxation measurements4,8−11 in this paper, we have focused our attention on the 13C−79Br and 13C−81Br spin systems, occurring in organic bromo compounds. These systems, especially the first of them, are characterized by exceptionally small Larmor frequency differences, which is their well-known, unique property, essential for their relaxation behavior. This advantageous feature, however, causes the precise evaluation of these differences to become an important issue. This apparently trivial problem has appeared to be worth discussing, which we undertake below.
3. RESULTS AND DISCUSSION 3.1. Unique Carbon−Bromine Spin System. Exploitation of higher magnetic fields in NMR spectroscopy of liquid samples enhances the potential of this method in almost every case. On the other hand, in relaxation studies, the access to lower magnetic fields is still desirable. It is because the relaxation rates are Larmor-frequency-dependent, and this dependence is characteristic for various relaxation mechanisms.1,3 In the case of SC2 relaxation, eqs 1 and 2 involve the ΔωAX parameter representing the difference of Larmor frequencies of the coupled nuclei, showing that the magnetic field can affect the SC2 relaxation rate. For most of the heteronuclear pairs, however, at B0 = 7 T or higher, the difference of Larmor frequencies of the coupled nuclei is large (107−1010 rad/s). As a result, the longitudinal SC2 relaxation is, in most cases, inefficient, and the frequency-dependent part of the transverse SC2 relaxation vanishes.1−3 The bromine− carbon nuclear pair is a unique case because the magnetogyric ratios involved are close to each other, especially for the 79Br isotope. As a result, the SC2 relaxation mechanism operates for the bromine-bonded carbons, which usually relax significantly faster than other quaternary carbons in the same molecule. Moreover, carbons bonded to 79Br and 81Br isotopes relax at different rates. This phenomenon can be monitored by longitudinal relaxation measurements, which is very attractive from the experimental standpoint. Indeed, it was observed many times that for 13C−79Br and 13C−81Br nuclear pairs, the Δω-dependent terms in relaxation equations were relevant.8−11,24−28 Thus, in the experimental works dealing with 13 C magnetic relaxation of brominated carbons, it is important to calculate the exact value of the difference of the involved Larmor frequencies for each molecular object.
2. EXPERIMENTAL AND THEORETICAL METHODS Deuterated acetonitrile, Kryptofix 222 (K222), and analytically pure KBr were commercial products. The solutions of the KBr·K222 complex were prepared by adding the appropriate weighted amounts of K222 and KBr to 3 mL of CD3CN and stirring them at room temperature to obtain transparent liquid. Then, 0.6 mL portions of these solutions were transferred to 5 mm o.d. NMR tubes used subsequently in measurements. The 13C NMR, 79Br NMR, and 81Br NMR spectra were recorded using VNMRS NMR spectrometer working at B0 = 11.7 T. The deuterium signal of the solvent was used as a field/ frequency lock. The frequencies of the 13C NMR signal of the acetonitrile methyl group in 13C NMR spectra were measured directly using the spectrometer software. The precise positions of the 129
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3.2. Larmor Frequency Difference Parameter. The Larmor frequency difference parameter ΔωAX = ωA − ωX
potassium bromide in heavy water. The authors underlined that nuclear magnetic dipole moments calculated in their work had not been corrected for magnetic shielding of bromine nuclei by electrons. Apparently, the UPAC recommendations of 200933 concerning the method of chemical shift referencing of bromine NMR signals were also based on the results of ref 32. The extensive table of nuclear magnetic dipole moments published recently under the auspices of the International Atomic Energy Agency34 cites the same reference as the data source; however, the values themselves, μ79Br = 2.106400(4) and μ81Br = 2.270562(4), differ from the original ones. On the other hand, they agree (or almost agree) with the values given in some other tables and important publications.35−37 Probably the values of ref 32 had been corrected, although we have not been able to establish how and when it was done. On the other hand, the values of the nuclear magnetic dipole moment μ79Br from the range of 2.1055−2.1059, published in 1976 by Fuller,38 were corrected for diamagnetic shielding of the bromine nucleus using the earlier computational data concerning the Br7+ ion.39 Our computation for Br− done by state-of-the-art methods, however, has yielded a remarkably different value of the overall correction. In order to remove any ambiguity, we have decided to repeat the determination of magnetogyric ratios of the bromine isotopes, exploiting the relationship
(3)
can be calculated by using the formula connecting the Larmor frequency of a given nucleus with the induction of the spectrometer magnetic field B0. This formula, basic for NMR spectroscopy, in the case of isotropic liquids reads1,3 ωZ = 2πνZ = γZ(1 − σZ)B0
(4)
where σZ is the isotropic shielding constant involved. The magnetogyric ratio, γ, appearing in this relationship is linked directly to the nuclear magnetic dipole moment, μ, the parameter preferred in physics29,30
γ=
⎛ 2π ⎞⎛ μ ⎞ ⎜ ⎟⎜ ⎟μ ⎝ h ⎠⎝ I ⎠ N
(5)
The constants I, h, and μN denote nuclear spin, Planck’s constant (h = 6.62606957 × 10−34 J·s), and the nuclear magneton (μN = 5.05078317 × 10−27 J/T), respectively. It has to be stressed that nuclear magnetic moments and magnetogyric ratios are true nuclear parameters, independent of electronic surrounding of the nucleus or any intermolecular effects. For the 13C nucleus, the precise values of these parameters are known (γ13C = 6.72787928 × 107 rad/T, μ13C = 0.702369417).31 In the case of bromine isotopes, however, the situation is not so obvious. Different values of nuclear parameters for the 79Br nucleus (as well as for 81Br) can be encountered in the literature. At the same time, some of the papers dealing with nuclear spin relaxation do not specify which values of bromine nuclear parameters have been used. Moreover, shielding constants appearing in eq 4 used to be generally ignored when calculating the ΔωAX parameter. While such an approximation seems to be rational for most nuclear pairs, it can hardly be justified in the case of the 13C−79Br system. In this case, the use of imprecise values of nuclear parameters and omission of the bromine shielding constants could yield false values of the Δω parameter, as illustrated in Table 1. This is why we decided to devote some space here to a discussion of this problem. 3.3. Magnetogyric Ratios of 79Br and 81Br. It seems that the primary source of the older values of bromine parameters, μ79Br = 2.099047 and μ81Br = 2.262636, was the work published in 1972.32 They were based on the known value of μ2H and on the NMR frequencies of bromine isotopes and deuterium measured in an identical magnetic field for dilute solutions of
⎛ ν ⎞ (1 − σ 13C) γ13 γBr = ⎜ Br ⎟ ⎝ ν13C ⎠ (1 − σBr) C
where νBr and ν13C frequencies concern the same external magnetic field. Unfortunately, the elegant methodology proposed by Jackowski and Jaszuński,29,30 based on the above relationship, has to be compromised as the bromine NMR signals for gaseous-state samples are hardly measurable. The authors of ref 32 used the NMR data for KBr dissolved in water, extrapolated to the infinite dilution. The extrapolated position of the signal was, however, still affected by hydration, as has been evidenced by our theoretical calculations (Table 2). Table 2. Theoretically Calculated [B3LYP/6-311++G(2d,p) PCM] Medium Effects on Magnetic Shielding of the Bromine Nucleus of a Bromide Anion Br− species −
Br Br− Br− [Br−*H2O]a [Br−*4H2O]b
Table 1. Larmor Frequency Differences Δν79 = ν13C − ν79Br and Δν81 = ν13C − ν81Br at B0 = 11.7 T Calculated for Bare Nuclei and Selected Compounds, Using “Old”32 and “New”34 Values of Bromine Magnetogyric Ratiosa compound
σBr (ppm)
σC (ppm)
Δν79,old (kHz)
Δν79,new (kHz)
Δν81,new (kHz)
bare nuclei Br−, TMS CH3Br (CH3)3CBr CBrCl3 CBr4
0 3432 2778 2136 1198 1073
0 186 176 122 118 213
479 884 804 731 614 587
42 449 368 294 177 150
−9718 −9278 −9365 −9445 −9571 −9600
(6)
KBr in CH3CN
conditions
σ (ppm)
K+−Br− distance (Å)
σ (ppm)
isolated in CH3CN in H2O in H2O in H2O
3126.3 3126.6 3126.7 3075.5 3029.6
5.0 4.0 3.5 3.2 3.021c
3122.7 3099.6 3063.1 3019.9 2981.1
a
Optimized structure; one Br···H hydrogen bond at r = 2.426 Å. Each water molecule forms one hydrogen bond with a Br− anion. c Optimal distance. b
In order to overcome this difficulty, we have used the data measured for acetonitrile-d3 solution of KBr, the cation of which has been caged by a K222 molecule. The 79Br NMR, 81Br NMR, and 13C NMR (of the solvent) signals have been recorded for samples of different concentrations and at various temperatures. The results are given in Table 3. Examination of these data shows that conditions supporting dissociation (lower concentration and higher temperature)
The values of magnetogyric ratios: γ79Br,old = 6.702140 × 107, γ79Br,new = 6.725618 × 107, γ81Br,new = 7.249778 × 107 in rad·s−1·T−1.
a
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follows, μ79Br = 2.10642(5), γ79Br = 6.7257(2) × 107, μ81Br = 2.27058(5), and γ81Br = 7.2498(2) × 107. Their errors have been calculated assuming ±20 ppm uncertainty of the magnetic shielding constant estimated theoretically for the bromide anion in our sample 3. Thus, nuclear magnetic dipole moments determined in this work remain in a very good agreement with those recommended in ref 35. 3.4. Magnetic Shielding Calculations. In order to determine magnetogyric ratios for bromine isotopes with the aid of eq 6, the magnetic shielding constant of the bromine nucleus for the KBr·K222 complex in acetonitrile solution had to be evaluated. First, the shielding constant for the isolated bromide anion was calculated by the DFT method, neglecting relativistic effects, using four different functionals (B3LYP, PBE0, PBE, and BVP86) and the QZ4P basis set. These calculations yielded a σBr value of about 3126 ppm. Interestingly, this value was almost independent of the functional used (differences less than 1 ppm). Also, the CCSD method with the aug-cc-pVTZ basis set resulted in practically the same value (3127 ppm). Thus, one may suppose that the nonrelativistic DFT calculation should provide a proper simulation of the medium effects (see Table 2). On the other hand, it is believed that reliable calculation of magnetic shielding tensors for nuclei of the third and higher row elements of the periodic table requires including relativistic effects into the theoretical model.6,41−43 We have found out that, indeed, inclusion of the relativistic spin−orbit and scalar effects at the DFT zeroth-order regular approximation of theory (SO-ZORA) yielded a different value of σBr = 3309 ppm. A more rigorous DFT method based on the four-component Dirac−Coulomb Hamiltonian14,42,43 has still changed the theoretical σBr value to 3434 ppm. Apparently, the twocomponent description of relativistic effects in the case of magnetic shielding calculations of bromine nuclei is still insufficient. It is not a surprise as magnetic shielding calculations based on the two-component ZORA Hamiltonian are usually less accurate than those based on the fourcomponent Dirac−Coulomb Hamiltonian, which has been discussed extensively in the literature.17,42,43 We have found that also the values obtained in the above relativistic calculations are roughly functional-independent (again, differences less than 1 ppm for the four functionals used). This feature provides evidence in favor of reliability of our theoretical estimation of σBr for an isolated bromide anion, despite the known limitations of the DFT method in the case of halogen derivatives. To use the above results for establishing the nuclear magnetic moments of 79Br and 81Br, the influence of the
Table 3. NMR Frequencies for 79Br and 81Br Signals of the KBr·K222 Complex Dissolved in CD3CN and the Resonance Frequency of the 13C Methyl Group Signal of the Solvent Measured at the Same Magnetic Field B0 ≈ 11.7 Ta sample
temp. (°C)
ν(79Br) (Hz)
ν(81Br) (Hz)
ν(13C) (Hz)
1 1 1 2 2 2 3 3b 3 3
0 25 50 0 25 50 25 50 75 100
125265254.6 125264325.1 125263450.6 125265068.5 125264114.3 125263255.0 125263400.8 125262761.7 125262263.6 125261209.9
135027811.8 135026816.1 135025870.1 135027591.6 135026586.4 135025653.1 135025836.9 135025052.0 135024619.1 135023472.8
125708493.2 125708462.9 125708434.5 125708492.2 125708460.9 125708432.6 125708435.6 125708416.0 125708394.5 125708373.0
a
Samples: 1, 0.1 M KBr·K222 in CD3CN; 2, 0.02 M KBr·K222 in CD3CN; 3, 0.01 M KBr + 0.015 M K222 in CD3CN. bAfter removing this most diverging result, one obtains ⟨ν(81Br)/ν(79Br)⟩ = γ(81Br)/ γ(79Br) = 1.0779351(2).
result in shielding of bromine nuclei in Br− ions. This property agrees well with the calculated dependence of the shielding of the bromine nucleus in potassium bromide on the K−Br distance (Table 2). One can admit that the total experimentally observed range of the bromine signal positions for our samples was on the order of 30 ppm (Table 3), albeit very small with respect to the value of the shielding constant itself. Nevertheless, it seems that the most shielded bromine signal (observed for our sample 3 at 100 °C) is still affected by deshielding intermolecular interactions and that the position of the signal of the isolated bromide anion is probably shifted upfield by some 20 ppm. In accord with expectations, we have observed that the ratio of frequencies measured for two bromine isotopes, equal to their magnetic dipole moments, is sample- and temperature-independent. The determined mean value of this ratio, ν81Br/ν79Br = 1.0779351(2), remains in perfect agreement with the value 1.0779355(3) reported by Lutz.40 These facts confirm indirectly the high quality of our experimental results. The value of the magnetic shielding constant for the Br− anion in our solution, σBr = 3414 ppm, has been calculated by diminishing the theoretical value for the isolated Br− anion (see the next section) by 20 ppm. This tentative correction has been proposed on the basis of the data in Table 2. This value and the literature value of σ13C = 185.3 ppm for the methyl group carbon of acetonitrile31 have been used to calculate magnetic dipole moments and magnetogyric ratios for the nuclei of 79Br and 81Br isotopes using eq 6. The obtained values are as
Table 4. Comparison of the Magnetic Shielding Constants (ppm) of Bromine Nuclei Calculated for a Set of Simple Molecular Objects Using the Standard Nonrelativistic Methods and the Advanced Relativistic Methodsa species
σ(P)
σ(B)
σ(4CP)
σ(4CB)
Δσ(4CP − P)
Δσ(4CB − B)
Br− CH3Br (CH3)3CBr CBrCl3 CBr4 [BrO4]−
3126.5 2511.1 1917.7 973.8 855.2 322.4
3126.3 2415.7 1795.8 742.3 606.1 166.1
3432.2 2777.5 2135.6 1198.3 1073.1 409.6
3433.9 2718.2 2065.8 1076.4 942.9 331.5
305.7 266.4 217.9 224.5 217.8 87.2
307.6 302.5 270.0 334.1 336.8 165.4
a
P: DFT PBE1PBE (PBE0) 6-311++G(2d,p) as isolated species using the Gaussian 03 program;12 B: the same as P but using the B3LYP functional; 4CP: DFT four-component Dirac−Coulomb Hamiltonian computation with the PBE0 functional, Dyall.v4z basis set for bromine, and aug-cc-pVTZ basis set for other atoms calculated as isolated species using DIRAC12 program19; 4CB: the same as 4CP but using the B3LYP functional. 131
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intermolecular interactions experienced by the Br− anion in the investigated solution on σBr had to be estimated. We found out that both the nonrelativistic and relativistic results were only marginally solvent-dependent within the polarizable continuum (PCM)23 or COSMO16 solution models. The interaction with the counterion does decrease σBr, but this effect is expected to be small for dilute solutions, as shown by the shielding constant dependence on the anion−cation distance (Table 2). On the other hand, formation of hydrogen bonds by a bromide anion in H2O solutions seems to cause quite a pronounced deshielding effect. Equations 3 and 4 show that for evaluation of the ΔωAX parameter, the values of the magnetic shielding constants of bromine nuclei in the investigated bromo compounds are needed. For small molecules, it is still possible to perform σBr calculation by methods using a high level of theory. For larger molecules, however, explicit treatment of the relativistic effects in computation becomes more and more difficult. Unfortunately, in the case of σBr, the efficient SO-ZORA method does not reproduce the four-component results. Keeping in mind that for interpretation of SC2 relaxation data a rough estimation of the bromine shielding would be quite sufficient, in difficult cases, we suggest using the nonrelativistic methods and correcting the obtained values by an empirical correction. In order to estimate roughly such a correction, we have performed some test calculations and collected their results in Table 4. These data suggest that in the case of bromine bound to carbon, the value of σBr calculated by the DFT/PBE0/6-311+ +G(2d,p) method should be increased by some 234 ppm. In the case of DFT/B3LYP/6-311++G(2d,p) results, this relativistic correction should be a bit larger, about 311 ppm. It seems that this procedure should yield the σBr values, deviating by less than 100 ppm from those calculated by a fourcomponent relativistic method. For organic bromo compounds, such a deviation would cause an error of ΔωAX of ∼5% in the case of 79Br and negligible for 81Br, which can be tolerated in most cases. Also, the differences between the results calculated with the use of PBE0 and B3LYP functionals seem to be unimportant for interpretation of SC2 relaxation data. These differences can be an effect of a crude treatment of electron correlations in DFT calculations (see the Supporting Information). A wider discussion of this problem, however, is out of the scope of this research. Finally, one may suppose that using PCM23 or COSMO16 models in σBr calculations to include solvent effects can additionally improve the results.
eqs 3 and 4, one obtains the following relationships for carbon−bromine systems: Δω13C, 79Br = 22613.3(1 − 2975σ 13C + 2974σ 79Br)B0
(7)
Δω13C,81Br = −5218985(1 + 12.9σ 13C − 13.9σ 81Br)B0
(8)
The influence of the values of shielding constants on the ΔωAX parameter, especially in the case of the 13C−79Br system, is striking. It is well-illustrated in Table 1, which contains Larmor frequency differences between carbon-13 and two bromine isotopes, calculated for bare nuclei and for five arbitrarily selected bromo compounds.
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ASSOCIATED CONTENT
* Supporting Information S
Additional remarks on σBr calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
■ ■
AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was financially supported by the National Science Centre (Poland) within Grant No. 2466/B/H03/2011/40. The MPD/2010/4 project, realized within the MPD programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, is acknowledged for a fellowship to A.W.
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REFERENCES
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4. CONCLUSIONS Our study on an acetonitrile solution of the KBr·K222 complex has confirmed that the most precise values of nuclear magnetic moments for bromine isotopes are provided in the table by Stone.32 It has also been shown that approximate evaluation of σBr parameters is quite feasible, and actually easy, by using standard quantum chemical methods. On the other hand, a reliable calculation of the shielding constants for bromine nuclei demands application of the four-component Dirac−Coulomb approach42,43 to take properly into account the relativistic effects, which can be troublesome for larger molecules. We have proposed the method to overcome this difficulty. The importance of proper selection of γ values and taking into account bromine shielding constants during ΔωAX parameter calculation is evident. Indeed, by substituting the numerical values of gyromagnetic ratios (in rad·s−1·T−1) into 132
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(33) Harris, R. K.; Becker, E. D.; Sonia, M.; Cabral de Menezes, S. M.; Granger, P.; Hoffman, R. E.; Zilm, K. W. Further Conventions for NMR Shielding and Chemical Shifts (IUPAC Recommendations 2008). Pure Appl. Chem. 2008, 80, 59−84. (34) Stone, N. J. Table of Nuclear Magnetic Dipole and Electric Quadrupole Moments, IAEA Nuclear Data Section; Vienna International Centre: Vienna, Austria, 2011; see www-nds.iaea.org/nsdd/indcnds-0594.pdf. (35) Mason, J. Multinuclear NMR; Plenum Press: New York, 1987; p 623. (36) Dove, M. F. A.; Sanders, J. C. P.; Appelman, E. H. Comparative Multinuclear (35Cl, 79Br, 81Br, 127I and 17O) Magnetic Resonance Study of the Perhalate Anions XO4− (X = Cl, Br or I) in Acetonitrile Solution. Magn. Reson. Chem. 1995, 33, 44−58. (37) Chapman, R. P.; Widdifield, C. M.; Bryce, D. L. Solid-State NMR of Quadrupolar Halogen Nuclei. Prog. Nucl. Magn. Reson. Spectrosc. 2009, 55, 215−237. (38) Fuller, G. H. Nuclear Spins and Moments. J. Phys. Chem. Ref. Data 1976, 5, 835−1092. (39) Feiock, F. D.; Johnson, W. R. Relativistic Evaluation of Internal Diamagnetic Fields for Atoms and Ions. Phys. Rev. Lett. 1968, 21, 785− 786. (40) Lutz, O. The Hyperfine Structure Anomaly of 79Br and 81Br. Phys. Lett. A 1970, 31, 384−384. (41) Helgaker, T.; Jaszunski, M.; Ruud, K. Ab Initio Methods for the Calculation of NMR Shielding and Indirect Spin−Spin Coupling Constants. Chem. Rev. 1999, 99, 293−352. (42) Komorovsky, S.; Repisky, M.; Malkina, O. L.; Malkin, V. G.; Ondik, I. M.; Kaupp, M. A Fully Relativistic Method for Calculation of Nuclear Magnetic Shielding Tensors with a Restricted Magnetically Balanced Basis in the Framework of the Matrix Dirac−Kohn−Sham Equation. J. Chem. Phys. 2008, 128, 104101−104115. (43) Repisky, M.; Malkina, O. L.; Malkin, V. G. Fully Relativistic Calculations of NMR Shielding Tensors Using Restricted Magnetically Balanced Basis and Gauge Including Atomic Orbitals. J. Chem. Phys. 2010, 132, 154101−154109.
(12) Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Montgomery, J. A., Jr.; Vreven, T.; Kudin, K. N.; Burant, J. C.; et al. Gaussian 03; Gaussian, Inc.: Wallingford, CT, 2004. (13) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652. (14) Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0 Model. J. Chem. Phys. 1999, 110, 6158−6170. (15) CFOUR, Coupled-Cluster Techniques for Computational Chemistry, a Quantum-Chemical Program Package written by Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G.; with contributions from Auer, A. A.; Bartlett, R. J.; Benedikt, U.; Berger, C.; Bernholdt, D. E.; Bomble, Y. J. et al. http://www.cfour.de (2013). (16) Klamt, A.; Jonas, V. Treatment of the Outlying Charge in Continuum Solvation Models. J. Chem. Phys. 1996, 105, 9972−9981. (17) Autschbach, J. On the Accuracy of the Hyperfine Integrals in Relativistic NMR Computations Based on the Zeroth-Order Regular Approximation. Theor. Chem. Acc. 2004, 112, 52−57. (18) Dyall, K. G. Relativistic Quadruple-Zeta and Revised Triple-Zeta and Double-Zeta Basis Sets for the 4p, 5p, and 6p Elements. Theor. Chem. Acc. 2006, 115, 441−447. (19) DIRAC, a Relativistic ab initio Electronic Structure Program, Release DIRAC12 written by Jensen, H. J. Aa.; Bast, R.; Saue, T.; Visscher, L. with contributions from Bakken, V.; Dyall, K. G.; Dubillard, S.; Ekström, U.; Eliav, E.; Enevoldsen, T. et al. http://www. diracprogram.org (2012). (20) Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. I. The Atoms Boron Through Neon and Hydrogen. J. Chem. Phys. 1989, 90, 1007−1023. (21) Woon, D. E.; Dunning, T. H., Jr. Gaussian Basis Sets for Use in Correlated Molecular Calculations. III. The Atoms Aluminum Through Argon. J. Chem. Phys. 1993, 98, 1358−1371. (22) Olejniczak, M.; Bast, R.; Saue, T.; Pecul, M. A Simple Scheme for Magnetic Balance in Four-Component Relativistic Kohn−Sham Calculations of Nuclear Magnetic Resonance Shielding Constants in a Gaussian Basis. J. Chem. Phys. 2012, 136, 014108−014120. (23) Tomasi, J.; Mennucci, B.; Cammi, R. Quantum Mechanical Continuum Solvation Models. Chem. Rev. 2005, 105, 2999−3093. (24) Levy, G. C. Carbon-13 Spin-Lattice Relaxation: Carbon− Bromine Scalar and Dipole−Dipole Interactions. J. Chem. Soc., Chem. Commun. 1972, 6, 352−354. (25) Levy, G. C.; Cargioli, J. D.; Anet, F. A. L. Carbon-13 SpinLattice Relaxation in Benzene and Substituted Aromatic Compounds. J. Am. Chem. Soc. 1973, 95, 1527−1535. (26) Yamamoto, O.; Yanagisawa, M. Spin−Spin Coupling Constants Between Carbon-13 and Bromine in Bromomethanes. J. Chem. Phys. 1977, 67, 3803−3807. (27) Yanagisawa, M.; Yamamoto, O. Scalar Coupling and Spin− Rotation Interactions in the 13C Nuclear Magnetic Relaxation of Methyl-d3 Bromide. Bull. Chem. Soc., Jpn 1979, 52, 2147−2148. (28) Hayashi, S.; Hayamizu, K.; Yamamoto, O. Spin−Spin Coupling Constants Between Carbon-13 and Bromine and Their Correlation with the S Character of the Carbon Hybridization. J. Magn. Reson. 1980, 37, 17−29. (29) Jackowski, K.; Jaszuński, M. Nuclear Magnetic Moments from NMR Spectra Experimental Gas Phase Studies and Nuclear Shielding Calculations. Concepts Magn. Reson. 2007, 30A, 246−260. (30) Jaszuński, M.; Antusek, A.; Garbacz, P.; Jackowski, K.; Makulski, W.; Wilczek, M. The Determination of Accurate Nuclear Magnetic Dipole Moments and Direct Measurement of NMR Shielding Constants. Prog. Nucl. Magn. Reson. Spectrosc. 2012, 67, 49−63. (31) Jackowski, K.; Jaszuński, M.; Wilczek, M. Alternative Approach to the Standardization of NMR Spectra. Direct Measurement of Nuclear Magnetic Shielding in Molecules. J. Phys. Chem. A 2010, 114, 2471−2475. (32) Blaser, J.; Lutz, O.; Steinkilberg, W. Nuclear Magnetic Resonance Studies of 35Cl, 37Cl, 79Br, and 81Br in Aqueous Solution. Z. Naturforsch., A: Phys. Sci. 1972, A27, 72. 133
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Scalar Relaxation of the Second Kind A Potential Source of Information on the Dynamics of Molecular Movements. 2. Magnetic Dipole Moments and Magnetic Shielding of Bromine Nuclei Adam Gryff-Keller,*,† Sergey Molchanov,† and Artur Wodyński‡ †
Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warsaw, Poland Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warsaw, Poland
‡
S Supporting Information *
ABSTRACT: In this paper, we continue the exploration of possibilities, limitations, and methodological problems of the studies based on measurements of the nuclear spin relaxation rates running via the scalar relaxation of the second kind (SC2) mechanism. The attention has been focused on the 13C−79Br and 13C−81Br systems in organic bromo compounds, which are characterized by exceptionally small differences of Larmor frequencies, ΔωCBr, of the coupled nuclei. This unique property enables experimental observation of longitudinal SC2 relaxation of 13C nuclei, which makes investigation of the SC2 relaxation rates an attractive experimental method of determination of spin− spin coupling constants and relaxation rates of quadrupole bromine nuclei, both types of parameters being hardly accessible by direct measurements. A careful examination of the methodology used in SC2 relaxation studies of carbon−bromine systems reveals, however, some disturbing facts that could burden the results with systematic inaccuracies. Namely, the way of calculating the Larmor frequency differences between 13C and bromine isotopes, ΔωCBr, may cause some reservations. In this work, the values of 79Br and 81Br magnetogyric ratios have been rechecked using bromine NMR data for the KBr·Kryptofix 222 complex in acetonitrile solution and the results of the advanced calculations of the magnetic shielding of the bromine nucleus in the Br− anion. Moreover, it has been pointed out that in the case of 13C−79Br, the magnetic shielding of the bromine nucleus in the investigated molecule must not be neglected during the calculation of the ΔωCBr parameter. Some recommendations concerning the exploitation of available theoretical methods to calculate bromine shielding constants for bromo compounds have also been formulated, keeping in mind relativistic effects.
1. INTRODUCTION Investigation of the rates of the nuclear spin relaxation running via the mechanism dependent on the spin−spin coupling, known as the scalar relaxation of the second kind (SC2), is an attractive experimental method of determination of spin−spin coupling constants and relaxation rates of quadrupole nuclei, both types of parameters being hardly accessible by direct measurements.1−4 The scalar spin−spin coupling constant, which is the orientation-independent part of the tensor describing indirect (transmitted by electrons) interaction between a pair of nuclear spins, is a very important NMR parameter, interesting for spectroscopists, structural chemists, and theoreticians.5,6 Strong and well-recognized dependence of coupling constants on the number of bonds separating coupled nuclei in the molecule has been the basis for spectral assignments since the first applications of high-resolution NMR spectroscopy. Most of the popular contemporary multidimensional correlation techniques are actually developments of that method. The high sensitivity of these parameters to space relationships has been widely exploited in the studies of stereochemistry. At the same time, coupling constants carry important information on electronic structures of molecules. For theoreticians, calculations of sufficiently precise values of coupling constants are © 2013 American Chemical Society
still a challenging task, especially when a heavy nucleus participates in the coupling.6 As it concerns the other parameter mentioned before, measurable via SC2 relaxation data interpretation, it is to be noticed that the data on nuclear spin relaxation rates carry a wealth of information on molecular movements, the investigation of which by other methods is difficult.3 The nuclei of most magnetic isotopes possess spin higher than 1/2 and electric quadrupole moments. Such nuclei interact with the local electric field gradients generated by their electronic surroundings, which couples the nuclear spin orientation with the orientation of the molecule.1,3,7 This relaxation mechanism is most frequently very efficient for quadrupole nuclei of other than the first-row elements, which simplifies substantially interpretation of such relaxation data. On the other hand, rapid relaxation causes the NMR signals of these nuclei to be extremely broad, which most frequently prevents measurement of their NMR spectra and direct determination of appropriate chemical shifts, coupling constants, and relaxation parameters. It is to be added that interpretation of the relaxation data for Received: October 13, 2013 Revised: December 13, 2013 Published: December 13, 2013 128
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much broader 79Br NMR and 81Br NMR signals were determined by performing the least-squares line shape analysis of bromine NMR spectra, assuming the Lorentzian shape of these spectral lines. The nonrelativistic calculations, which included geometry optimization and NMR shielding calculation, were performed with the aid of the Gaussian03 program12 using the DFT method with the B3LYP13 or PBE1PBE hybrid functional (also known as PBE0)14 and the standard 6-311++G(2d,p) basis set. Additional nonrelativistic computations at the CCSD level were performed with the aug-cc-pVTZ basis set using the CFOUR program.15 The relativistic shielding calculations, which took into account both scalar and spin−orbit coupling terms, were performed by using the two-component ZORA Hamiltonian available in the ADF program package.16−18 In these calculations, the nonrelativistic molecular geometries, B3LYP or PBE0 functionals, and QZ4P basis set were used. The final results of theoretical calculations were achieved by the fourcomponent Dirac−Coulomb Hamiltonian with the aid of the DIRAC 12.7 computational package.19 These computations were performed again with B3LYP and PBE0 functionals, the quadrupole zeta-type Dyall.v4z basis set18 for bromine, and the aug-cc-pVTZ basis set20,21 for other atoms. The unrestricted kinetic balance (UKB) ansatz combined with London atomic orbitals (LAOs) in the simple magnetic balance (sMB)22 approach was used during the computations. The solvent effects were estimated on the basis of PCM23 and COSMO16 solution models in calculations performed by Gaussian and ADF programs, respectively.
quadrupole nuclei demands knowledge of the quadrupole coupling constants involved, which are products of the nuclear quadrupole moments and the electric field gradient tensors. Determination of the latter is another exploration field for quantum chemistry methods. The SC2 mechanism can appear for a magnetic nucleus A that is coupled to rapidly relaxing nucleus X. Most frequently, such a situation occurs when the nucleus X is of a higher spin and relaxes via the quadrupole mechanism. The rapid relaxation of X erases partially or completely the spin−spin splitting of the signal of A. At the same time, this relaxation causes random fluctuation of the energy of the spin−spin interaction in the AX spin system, which affects the behavior of nuclei coupled to the quadrupole nucleus. Under certain conditions, this phenomenon may be described as an additional relaxation path, named the SC2 mechanism. The contributions of this mechanism to the overall longitudinal and transverse relaxation of the nucleus A are given by well-known formulas1−4 R1,SC2(A) =
⎛2⎞ R 2(X) 2 ⎜ ⎟I (I + 1)(2πJ ) AX ⎝3⎠ X X [R 2(X)2 + ΔωAX 2] (1)
R 2,SC2(A) =
⎛1⎞ 2 ⎜ ⎟I (I + 1)(2πJ ) AX ⎝3⎠ X X ⎧ 1 ⎫ R 2(X) ⎨ ⎬ + [R 2(X)2 + ΔωAX 2] ⎭ ⎩ R1(X)
(2)
R1(X) and R2(X) are the rates of longitudinal and transverse relaxation of X, respectively, and ΔωAX is the difference of Larmor frequencies of the coupled nuclei. The above equations show why investigation of the SC2 relaxation rates can be an experimental method of determination of spin−spin coupling constants and relaxation rates of quadrupole nuclei. Continuing studies based on SC2 relaxation measurements4,8−11 in this paper, we have focused our attention on the 13C−79Br and 13C−81Br spin systems, occurring in organic bromo compounds. These systems, especially the first of them, are characterized by exceptionally small Larmor frequency differences, which is their well-known, unique property, essential for their relaxation behavior. This advantageous feature, however, causes the precise evaluation of these differences to become an important issue. This apparently trivial problem has appeared to be worth discussing, which we undertake below.
3. RESULTS AND DISCUSSION 3.1. Unique Carbon−Bromine Spin System. Exploitation of higher magnetic fields in NMR spectroscopy of liquid samples enhances the potential of this method in almost every case. On the other hand, in relaxation studies, the access to lower magnetic fields is still desirable. It is because the relaxation rates are Larmor-frequency-dependent, and this dependence is characteristic for various relaxation mechanisms.1,3 In the case of SC2 relaxation, eqs 1 and 2 involve the ΔωAX parameter representing the difference of Larmor frequencies of the coupled nuclei, showing that the magnetic field can affect the SC2 relaxation rate. For most of the heteronuclear pairs, however, at B0 = 7 T or higher, the difference of Larmor frequencies of the coupled nuclei is large (107−1010 rad/s). As a result, the longitudinal SC2 relaxation is, in most cases, inefficient, and the frequency-dependent part of the transverse SC2 relaxation vanishes.1−3 The bromine− carbon nuclear pair is a unique case because the magnetogyric ratios involved are close to each other, especially for the 79Br isotope. As a result, the SC2 relaxation mechanism operates for the bromine-bonded carbons, which usually relax significantly faster than other quaternary carbons in the same molecule. Moreover, carbons bonded to 79Br and 81Br isotopes relax at different rates. This phenomenon can be monitored by longitudinal relaxation measurements, which is very attractive from the experimental standpoint. Indeed, it was observed many times that for 13C−79Br and 13C−81Br nuclear pairs, the Δω-dependent terms in relaxation equations were relevant.8−11,24−28 Thus, in the experimental works dealing with 13 C magnetic relaxation of brominated carbons, it is important to calculate the exact value of the difference of the involved Larmor frequencies for each molecular object.
2. EXPERIMENTAL AND THEORETICAL METHODS Deuterated acetonitrile, Kryptofix 222 (K222), and analytically pure KBr were commercial products. The solutions of the KBr·K222 complex were prepared by adding the appropriate weighted amounts of K222 and KBr to 3 mL of CD3CN and stirring them at room temperature to obtain transparent liquid. Then, 0.6 mL portions of these solutions were transferred to 5 mm o.d. NMR tubes used subsequently in measurements. The 13C NMR, 79Br NMR, and 81Br NMR spectra were recorded using VNMRS NMR spectrometer working at B0 = 11.7 T. The deuterium signal of the solvent was used as a field/ frequency lock. The frequencies of the 13C NMR signal of the acetonitrile methyl group in 13C NMR spectra were measured directly using the spectrometer software. The precise positions of the 129
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3.2. Larmor Frequency Difference Parameter. The Larmor frequency difference parameter ΔωAX = ωA − ωX
potassium bromide in heavy water. The authors underlined that nuclear magnetic dipole moments calculated in their work had not been corrected for magnetic shielding of bromine nuclei by electrons. Apparently, the UPAC recommendations of 200933 concerning the method of chemical shift referencing of bromine NMR signals were also based on the results of ref 32. The extensive table of nuclear magnetic dipole moments published recently under the auspices of the International Atomic Energy Agency34 cites the same reference as the data source; however, the values themselves, μ79Br = 2.106400(4) and μ81Br = 2.270562(4), differ from the original ones. On the other hand, they agree (or almost agree) with the values given in some other tables and important publications.35−37 Probably the values of ref 32 had been corrected, although we have not been able to establish how and when it was done. On the other hand, the values of the nuclear magnetic dipole moment μ79Br from the range of 2.1055−2.1059, published in 1976 by Fuller,38 were corrected for diamagnetic shielding of the bromine nucleus using the earlier computational data concerning the Br7+ ion.39 Our computation for Br− done by state-of-the-art methods, however, has yielded a remarkably different value of the overall correction. In order to remove any ambiguity, we have decided to repeat the determination of magnetogyric ratios of the bromine isotopes, exploiting the relationship
(3)
can be calculated by using the formula connecting the Larmor frequency of a given nucleus with the induction of the spectrometer magnetic field B0. This formula, basic for NMR spectroscopy, in the case of isotropic liquids reads1,3 ωZ = 2πνZ = γZ(1 − σZ)B0
(4)
where σZ is the isotropic shielding constant involved. The magnetogyric ratio, γ, appearing in this relationship is linked directly to the nuclear magnetic dipole moment, μ, the parameter preferred in physics29,30
γ=
⎛ 2π ⎞⎛ μ ⎞ ⎜ ⎟⎜ ⎟μ ⎝ h ⎠⎝ I ⎠ N
(5)
The constants I, h, and μN denote nuclear spin, Planck’s constant (h = 6.62606957 × 10−34 J·s), and the nuclear magneton (μN = 5.05078317 × 10−27 J/T), respectively. It has to be stressed that nuclear magnetic moments and magnetogyric ratios are true nuclear parameters, independent of electronic surrounding of the nucleus or any intermolecular effects. For the 13C nucleus, the precise values of these parameters are known (γ13C = 6.72787928 × 107 rad/T, μ13C = 0.702369417).31 In the case of bromine isotopes, however, the situation is not so obvious. Different values of nuclear parameters for the 79Br nucleus (as well as for 81Br) can be encountered in the literature. At the same time, some of the papers dealing with nuclear spin relaxation do not specify which values of bromine nuclear parameters have been used. Moreover, shielding constants appearing in eq 4 used to be generally ignored when calculating the ΔωAX parameter. While such an approximation seems to be rational for most nuclear pairs, it can hardly be justified in the case of the 13C−79Br system. In this case, the use of imprecise values of nuclear parameters and omission of the bromine shielding constants could yield false values of the Δω parameter, as illustrated in Table 1. This is why we decided to devote some space here to a discussion of this problem. 3.3. Magnetogyric Ratios of 79Br and 81Br. It seems that the primary source of the older values of bromine parameters, μ79Br = 2.099047 and μ81Br = 2.262636, was the work published in 1972.32 They were based on the known value of μ2H and on the NMR frequencies of bromine isotopes and deuterium measured in an identical magnetic field for dilute solutions of
⎛ ν ⎞ (1 − σ 13C) γ13 γBr = ⎜ Br ⎟ ⎝ ν13C ⎠ (1 − σBr) C
where νBr and ν13C frequencies concern the same external magnetic field. Unfortunately, the elegant methodology proposed by Jackowski and Jaszuński,29,30 based on the above relationship, has to be compromised as the bromine NMR signals for gaseous-state samples are hardly measurable. The authors of ref 32 used the NMR data for KBr dissolved in water, extrapolated to the infinite dilution. The extrapolated position of the signal was, however, still affected by hydration, as has been evidenced by our theoretical calculations (Table 2). Table 2. Theoretically Calculated [B3LYP/6-311++G(2d,p) PCM] Medium Effects on Magnetic Shielding of the Bromine Nucleus of a Bromide Anion Br− species −
Br Br− Br− [Br−*H2O]a [Br−*4H2O]b
Table 1. Larmor Frequency Differences Δν79 = ν13C − ν79Br and Δν81 = ν13C − ν81Br at B0 = 11.7 T Calculated for Bare Nuclei and Selected Compounds, Using “Old”32 and “New”34 Values of Bromine Magnetogyric Ratiosa compound
σBr (ppm)
σC (ppm)
Δν79,old (kHz)
Δν79,new (kHz)
Δν81,new (kHz)
bare nuclei Br−, TMS CH3Br (CH3)3CBr CBrCl3 CBr4
0 3432 2778 2136 1198 1073
0 186 176 122 118 213
479 884 804 731 614 587
42 449 368 294 177 150
−9718 −9278 −9365 −9445 −9571 −9600
(6)
KBr in CH3CN
conditions
σ (ppm)
K+−Br− distance (Å)
σ (ppm)
isolated in CH3CN in H2O in H2O in H2O
3126.3 3126.6 3126.7 3075.5 3029.6
5.0 4.0 3.5 3.2 3.021c
3122.7 3099.6 3063.1 3019.9 2981.1
a
Optimized structure; one Br···H hydrogen bond at r = 2.426 Å. Each water molecule forms one hydrogen bond with a Br− anion. c Optimal distance. b
In order to overcome this difficulty, we have used the data measured for acetonitrile-d3 solution of KBr, the cation of which has been caged by a K222 molecule. The 79Br NMR, 81Br NMR, and 13C NMR (of the solvent) signals have been recorded for samples of different concentrations and at various temperatures. The results are given in Table 3. Examination of these data shows that conditions supporting dissociation (lower concentration and higher temperature)
The values of magnetogyric ratios: γ79Br,old = 6.702140 × 107, γ79Br,new = 6.725618 × 107, γ81Br,new = 7.249778 × 107 in rad·s−1·T−1.
a
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follows, μ79Br = 2.10642(5), γ79Br = 6.7257(2) × 107, μ81Br = 2.27058(5), and γ81Br = 7.2498(2) × 107. Their errors have been calculated assuming ±20 ppm uncertainty of the magnetic shielding constant estimated theoretically for the bromide anion in our sample 3. Thus, nuclear magnetic dipole moments determined in this work remain in a very good agreement with those recommended in ref 35. 3.4. Magnetic Shielding Calculations. In order to determine magnetogyric ratios for bromine isotopes with the aid of eq 6, the magnetic shielding constant of the bromine nucleus for the KBr·K222 complex in acetonitrile solution had to be evaluated. First, the shielding constant for the isolated bromide anion was calculated by the DFT method, neglecting relativistic effects, using four different functionals (B3LYP, PBE0, PBE, and BVP86) and the QZ4P basis set. These calculations yielded a σBr value of about 3126 ppm. Interestingly, this value was almost independent of the functional used (differences less than 1 ppm). Also, the CCSD method with the aug-cc-pVTZ basis set resulted in practically the same value (3127 ppm). Thus, one may suppose that the nonrelativistic DFT calculation should provide a proper simulation of the medium effects (see Table 2). On the other hand, it is believed that reliable calculation of magnetic shielding tensors for nuclei of the third and higher row elements of the periodic table requires including relativistic effects into the theoretical model.6,41−43 We have found out that, indeed, inclusion of the relativistic spin−orbit and scalar effects at the DFT zeroth-order regular approximation of theory (SO-ZORA) yielded a different value of σBr = 3309 ppm. A more rigorous DFT method based on the four-component Dirac−Coulomb Hamiltonian14,42,43 has still changed the theoretical σBr value to 3434 ppm. Apparently, the twocomponent description of relativistic effects in the case of magnetic shielding calculations of bromine nuclei is still insufficient. It is not a surprise as magnetic shielding calculations based on the two-component ZORA Hamiltonian are usually less accurate than those based on the fourcomponent Dirac−Coulomb Hamiltonian, which has been discussed extensively in the literature.17,42,43 We have found that also the values obtained in the above relativistic calculations are roughly functional-independent (again, differences less than 1 ppm for the four functionals used). This feature provides evidence in favor of reliability of our theoretical estimation of σBr for an isolated bromide anion, despite the known limitations of the DFT method in the case of halogen derivatives. To use the above results for establishing the nuclear magnetic moments of 79Br and 81Br, the influence of the
Table 3. NMR Frequencies for 79Br and 81Br Signals of the KBr·K222 Complex Dissolved in CD3CN and the Resonance Frequency of the 13C Methyl Group Signal of the Solvent Measured at the Same Magnetic Field B0 ≈ 11.7 Ta sample
temp. (°C)
ν(79Br) (Hz)
ν(81Br) (Hz)
ν(13C) (Hz)
1 1 1 2 2 2 3 3b 3 3
0 25 50 0 25 50 25 50 75 100
125265254.6 125264325.1 125263450.6 125265068.5 125264114.3 125263255.0 125263400.8 125262761.7 125262263.6 125261209.9
135027811.8 135026816.1 135025870.1 135027591.6 135026586.4 135025653.1 135025836.9 135025052.0 135024619.1 135023472.8
125708493.2 125708462.9 125708434.5 125708492.2 125708460.9 125708432.6 125708435.6 125708416.0 125708394.5 125708373.0
a
Samples: 1, 0.1 M KBr·K222 in CD3CN; 2, 0.02 M KBr·K222 in CD3CN; 3, 0.01 M KBr + 0.015 M K222 in CD3CN. bAfter removing this most diverging result, one obtains ⟨ν(81Br)/ν(79Br)⟩ = γ(81Br)/ γ(79Br) = 1.0779351(2).
result in shielding of bromine nuclei in Br− ions. This property agrees well with the calculated dependence of the shielding of the bromine nucleus in potassium bromide on the K−Br distance (Table 2). One can admit that the total experimentally observed range of the bromine signal positions for our samples was on the order of 30 ppm (Table 3), albeit very small with respect to the value of the shielding constant itself. Nevertheless, it seems that the most shielded bromine signal (observed for our sample 3 at 100 °C) is still affected by deshielding intermolecular interactions and that the position of the signal of the isolated bromide anion is probably shifted upfield by some 20 ppm. In accord with expectations, we have observed that the ratio of frequencies measured for two bromine isotopes, equal to their magnetic dipole moments, is sample- and temperature-independent. The determined mean value of this ratio, ν81Br/ν79Br = 1.0779351(2), remains in perfect agreement with the value 1.0779355(3) reported by Lutz.40 These facts confirm indirectly the high quality of our experimental results. The value of the magnetic shielding constant for the Br− anion in our solution, σBr = 3414 ppm, has been calculated by diminishing the theoretical value for the isolated Br− anion (see the next section) by 20 ppm. This tentative correction has been proposed on the basis of the data in Table 2. This value and the literature value of σ13C = 185.3 ppm for the methyl group carbon of acetonitrile31 have been used to calculate magnetic dipole moments and magnetogyric ratios for the nuclei of 79Br and 81Br isotopes using eq 6. The obtained values are as
Table 4. Comparison of the Magnetic Shielding Constants (ppm) of Bromine Nuclei Calculated for a Set of Simple Molecular Objects Using the Standard Nonrelativistic Methods and the Advanced Relativistic Methodsa species
σ(P)
σ(B)
σ(4CP)
σ(4CB)
Δσ(4CP − P)
Δσ(4CB − B)
Br− CH3Br (CH3)3CBr CBrCl3 CBr4 [BrO4]−
3126.5 2511.1 1917.7 973.8 855.2 322.4
3126.3 2415.7 1795.8 742.3 606.1 166.1
3432.2 2777.5 2135.6 1198.3 1073.1 409.6
3433.9 2718.2 2065.8 1076.4 942.9 331.5
305.7 266.4 217.9 224.5 217.8 87.2
307.6 302.5 270.0 334.1 336.8 165.4
a
P: DFT PBE1PBE (PBE0) 6-311++G(2d,p) as isolated species using the Gaussian 03 program;12 B: the same as P but using the B3LYP functional; 4CP: DFT four-component Dirac−Coulomb Hamiltonian computation with the PBE0 functional, Dyall.v4z basis set for bromine, and aug-cc-pVTZ basis set for other atoms calculated as isolated species using DIRAC12 program19; 4CB: the same as 4CP but using the B3LYP functional. 131
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intermolecular interactions experienced by the Br− anion in the investigated solution on σBr had to be estimated. We found out that both the nonrelativistic and relativistic results were only marginally solvent-dependent within the polarizable continuum (PCM)23 or COSMO16 solution models. The interaction with the counterion does decrease σBr, but this effect is expected to be small for dilute solutions, as shown by the shielding constant dependence on the anion−cation distance (Table 2). On the other hand, formation of hydrogen bonds by a bromide anion in H2O solutions seems to cause quite a pronounced deshielding effect. Equations 3 and 4 show that for evaluation of the ΔωAX parameter, the values of the magnetic shielding constants of bromine nuclei in the investigated bromo compounds are needed. For small molecules, it is still possible to perform σBr calculation by methods using a high level of theory. For larger molecules, however, explicit treatment of the relativistic effects in computation becomes more and more difficult. Unfortunately, in the case of σBr, the efficient SO-ZORA method does not reproduce the four-component results. Keeping in mind that for interpretation of SC2 relaxation data a rough estimation of the bromine shielding would be quite sufficient, in difficult cases, we suggest using the nonrelativistic methods and correcting the obtained values by an empirical correction. In order to estimate roughly such a correction, we have performed some test calculations and collected their results in Table 4. These data suggest that in the case of bromine bound to carbon, the value of σBr calculated by the DFT/PBE0/6-311+ +G(2d,p) method should be increased by some 234 ppm. In the case of DFT/B3LYP/6-311++G(2d,p) results, this relativistic correction should be a bit larger, about 311 ppm. It seems that this procedure should yield the σBr values, deviating by less than 100 ppm from those calculated by a fourcomponent relativistic method. For organic bromo compounds, such a deviation would cause an error of ΔωAX of ∼5% in the case of 79Br and negligible for 81Br, which can be tolerated in most cases. Also, the differences between the results calculated with the use of PBE0 and B3LYP functionals seem to be unimportant for interpretation of SC2 relaxation data. These differences can be an effect of a crude treatment of electron correlations in DFT calculations (see the Supporting Information). A wider discussion of this problem, however, is out of the scope of this research. Finally, one may suppose that using PCM23 or COSMO16 models in σBr calculations to include solvent effects can additionally improve the results.
eqs 3 and 4, one obtains the following relationships for carbon−bromine systems: Δω13C, 79Br = 22613.3(1 − 2975σ 13C + 2974σ 79Br)B0
(7)
Δω13C,81Br = −5218985(1 + 12.9σ 13C − 13.9σ 81Br)B0
(8)
The influence of the values of shielding constants on the ΔωAX parameter, especially in the case of the 13C−79Br system, is striking. It is well-illustrated in Table 1, which contains Larmor frequency differences between carbon-13 and two bromine isotopes, calculated for bare nuclei and for five arbitrarily selected bromo compounds.
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ASSOCIATED CONTENT
* Supporting Information S
Additional remarks on σBr calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
ACKNOWLEDGMENTS This work was financially supported by the National Science Centre (Poland) within Grant No. 2466/B/H03/2011/40. The MPD/2010/4 project, realized within the MPD programme of Foundation for Polish Science, cofinanced from European Union, Regional Development Fund, is acknowledged for a fellowship to A.W.
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REFERENCES
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