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Scalar Relaxation of the Second Kind. A Potential Source of Information on the Dynamics of Molecular Movements. 3. A 13C Nuclear Spin Relaxation Study of CBrX3 (X = Cl, CH3, Br) Molecules Dominika Kubica,† Artur Wodyński,‡ Anna Kraska-Dziadecka,† and Adam Gryff-Keller†,* †

Faculty of Chemistry, Warsaw University of Technology, Noakowskiego 3, 00-664 Warszawa, Poland Faculty of Chemistry, University of Warsaw, Pasteura 1, 02-093 Warszawa, Poland

‡

ABSTRACT: Continuing studies based on measurements of the nuclear spin relaxation rates running via the SC2 mechanism (scalar relaxation of the second kind), we present in this work the results obtained for three bromo compounds: CBrCl3, (CH3)3CBr, and CBr4. A careful separation of saturation-recovery curves, measured for signals of 13C nuclei at 7.05 and 11.7 T on two components, has provided the longitudinal SC2 relaxation rates of carbon signals in 79Br and 81Br containing isotopomers of the investigated compounds. These data have enabled experimental determination of spin−spin coupling constants and relaxation rates of quadrupole bromine nuclei, both types of parameters being hardly accessible by direct measurements. Investigation of the relaxation behavior of these molecules, being of similar size and shape, has provided quite different practical and interpretational problems which are likely to be encountered in relaxation studies of many other carbon−bromine systems. In order to evaluate the quality of the obtained experimental results, advanced theoretical calculations of the indirect 1J(13C,79Br) coupling constants, magnetic shielding of carbon nuclei, and quadrupole coupling constants of bromines in the investigated compounds have been performed and compared with the experimental values. Relatively small divergences between experiment and theory have been found. The contributions of the relativistic effects to the values of the discussed parameters have been tentatively estimated.

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INTRODUCTION The data on nuclear spin relaxation rates carry a wealth of information on molecular movements and molecular parameters, which frequently cannot be easily obtained by other methods.1 In this paper we continue the exploration of possibilities, limitations and methodological problems of the studies based on measurements of the rates of the nuclear spin relaxation involving the mechanism named the scalar relaxation of the second kind (SC2),2−6 described first by Abragam.7 The SC2 relaxationbased methods have been exploited many times and provided many valuable results.2−5,8−13 Independent of the unquestionable successes of this method in the past, however, a careful reexamination of the methodology used seems to be desirable, taking into account substantial changes in the NMR instrumentation in the last decades. Such a discussion seems to be presently desired, also because the available state-of-the-art theoretical methods already enable calculating the values of the molecular parameters difficult to determine experimentally.6,14,15 Some divergences between theoretical and experimental values are being encountered, but actually, it is not always clear which of those values are more accurate. This paper reports the results of a relaxation study on three bromo compounds, CBrCl3, (CH3)3CBr, and CBr4, possessing molecules of the similar size and shape. For these objects the longitudinal relaxation of carbon signals has been investigated at 4.7, 7.05, and 11.7 T magnetic fields. The analysis of the obtained relaxation results allowed the values of the indirect 1J(13C,79Br) coupling constants and transverse relaxation rates of bromine nuclei to be determined. Monitoring and interpretation of the © 2014 American Chemical Society

relaxation behavior have provided quite different practical problems for each of these apparently similar objects. We have shown how these difficulties, which may be encountered also in relaxation studies of other carbon−bromine systems, can be overcome. The determined T2(Br) relaxation times have been used to estimate the reorientation rates of the investigated molecules in the solution. The quadrupole coupling constants of bromine nuclei, necessary for interpretation of these relaxation data, have been calculated theoretically. The experimentally obtained carbon magnetic shielding constants and 1J(13C,79Br) coupling constants have been compared with the values of these parameters calculated with the aid of the DFT methods. In spite of some divergences between experimental and theoretical values, an impact of the relativistic effects on the discussed parameters has been evidenced.

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EXPERIMENTAL SECTION

All materials and solvents used were commercial products. NMR measurements were performed for the solutions prepared from the equimolar mixtures of CBrCl3, (CH3)3CBr, and CBr4, which were diluted with CDCl3 to obtain finally the ca. 1 mol/dm3 concentration of each solute. A 0.6 mL portion of each solution was placed in 5 mm o.d. NMR tube, which was thoroughly degassed and sealed. Auxiliary measurements were performed for Received: January 30, 2014 Revised: March 27, 2014 Published: March 28, 2014 2995

dx.doi.org/10.1021/jp501064c | J. Phys. Chem. A 2014, 118, 2995−3003

The Journal of Physical Chemistry A

Article

1 mol/dm3 (monocomponent) solutions of CBrCl3 and CBr4 in DMSO-d6. These solutions were deoxygenized and saturated with argon. Then, their 0.5 mL portions were transferred under argon atmosphere into special 5 mm NMR tubes which could be closed with the airtight cups. The proton decoupled 13C NMR spectra were recorded using VNMRS, Bruker Avance II and Gemini 2000 NMR spectrometers working at 11.7, 7.05, and 4.7 T magnetic fields, respectively. The 13C NMR chemical shifts were determined using the solvent signal, δ(CDCl3) = 77.16 ppm, as an internal reference. The measurement temperature (25 °C) was monitored by the same ethylene glycol (DMSO-d6) sample at all three spectrometers. The deuterium signal of the solvent was used as a field/frequency lock. The extensive zero-filling of the FID signals and line broadening (lb =10) were applied prior to the FT operation. Shielding parameters for carbon nuclei in the investigated compounds, needed in this analysis (see eq 3), were calculated using the relationship: σ(13C) = σTMS − δ(13C), where σTMS = 186.6 ppm stands for the isotropic magnetic shielding constant of TMS carbons. The shielding of bromine nuclei: 2066, 1076, and 943 ppm for (CH3)3CBr, CBrCl3, and CBr4, respectively, were taken from the previous paper of this series.6 The longitudinal 13C relaxation times were measured using the saturation−recovery method16 at three magnetic fields. Each saturation−recovery experiment was composed of spectra recorded for at least 11 evolution delays distributed in such a way that adequate description of the recovery curve was ensured. The longest evolution delays (150−200 s) were more than three times longer than the maximum relaxation time measured. Only in the case of C81Br4 isotopomer was this condition not fulfilled. The inversion−recovery method16 was occasionally exploited to measure short relaxation times (see text). In these cases the relaxation and maximum evolution delays were chosen to be of the order of four relaxation times measured. The nuclear Overhauser enhancement factors for (CH3)3CBr carbons were measured by the standard gated decoupling method17 using 400 s period for magnetization equilibration. Each relaxation measurement was repeated at least 3 times for a slightly modified set of parameters defining the experiment. The standard spectrometer software was used to measure signal intensities. The set of recovery curves (each represented by 11−13 points) were analyzed by a nonlinear least-squares (l-s) procedure. Two relaxation parameters and, occasionally, a population parameter, common for the whole set, and pairs of linear parameters specific for each curve, were fitted simultaneously.18 The l-s method was also used to solve the overdetermined systems of equations connecting 13C relaxation data with 1J(13C,79Br), T2(79Br), and Roth (see eqs 1, 3, 5−7). The numbers in parentheses that follow the parameter values represent the error estimates and concern the last (or two last) decimal places of the value throughout the paper. The nonrelativistic quantum chemical calculations, which included geometry optimizations and calculations of NMR parameters and EFG tensors were performed with the aid of Gaussian03 program19 using DFT method with B3LYP20 hybrid functional and the standard 6-311++G(2d,p) basis set. The solvent effects were estimated on the basis of PCM21 solution model. The relativistic calculations of the indirect spin−spin coupling constants, which took into account both scalar and spin−orbit coupling terms, were performed using twocomponent ZORA Hamiltonian available in ADF program package.22−24 In these calculations the nonrelativistic molecular

geometries, B3LYP functional and jcpl (TZ2P basis set with additional tight functions) were used. Calculations of electric field gradient were also performed with so-ZORA Hamiltonian (with ZORA-4 approximation25), B3LYP functional and QZ4P basis set. The solvent effects were estimated on the basis of COSMO26 solution model. The theoretical calculations of magnetic shielding were performed for isolated molecules by the four-component Dirac−Coulomb Hamiltonian with the aid of DIRAC 12.7 computational package.27 These computations were performed with B3LYP functional, the quadrupole ζ type dyall.v4z24 basis set for bromine and aug-cc-pVTZ28−30 basis for other atoms. Methodological Remarks. SC2 in C−Br Systems. In the case of AX spin system (with fast-relaxing X possessing spin IX) the contribution of SC2 mechanism to the overall relaxation of nucleus A is given by the formulas:1,7,8 R1,SC2(A) = 1/T1,SC2(A) = (2/3)IX(IX + 1)(2πJAX )2 T2(X) /[1 + ΔωAX 2T2(X)2 ]

(1)

R 2,SC2(A) = 1/T2,SC2(A) = (1/3)IX(IX + 1)(2πJAX )2 {T1(X) + T2(X)/[1 + ΔωAX 2T2(X)2 ]}

(2)

where T1(X) and T2(X) are longitudinal and transverse relaxation times of nucleus X, respectively. ΔωAX is the difference of Larmor frequencies of coupled nuclei: ΔωAX = [γA(1 − σA ) − γX(1 − σX)]B0

(3)

The fast relaxation of nucleus X is most frequently dominated by the quadrupole mechanism. For systems involving chlorine, bromine or iodine nucleus, the complete erasure of the multiplet structure of the signal of the neighboring nucleus A is observed as a rule. Also the meaningful contribution of SC2 mechanism to the transverse relaxation for halogen-bonded nucleus A is common. Equations 1 and 2, which involve the Δω parameter representing the difference of Larmor frequencies, show that magnetic field induction affects the SC2 relaxation rates. However, for most of the heteronuclear pairs at B0 = 7.05 T or higher, the difference of Larmor frequencies of the coupled nuclei is large (107 − 1010 rad/s). Since this parameter occurs in eq 1 in the denominator, the longitudinal SC2 relaxation is in most cases inefficient.1,7,8 The bromine-carbon nuclear pair is a unique case, in which the magnetogyric ratios involved are close to each other, especially for 13C−79Br isotopes. As a result, the longitudinal relaxation of the bromine-bonded carbons usually involves a remarkable contribution of SC2 mechanism, and the signals of such carbons relax significantly faster than the signals of other quaternary carbons in the same molecule. Moreover, the signals of carbons bonded to 79Br and 81Br isotopes frequently relax at very different rates.1−13 This phenomenon can be conveniently monitored through the longitudinal relaxation experiments. The task is not so straightforward, however, as the signals of both isotopomers overlap each other almost exactly, which is obviously very important from the experimental standpoint. All the investigations in this work concern the longitudinal relaxation of 13C nuclei. Thus, from hereon the subscripts “1” and arguments “(13C)” at relaxation parameters are omitted in order to simplify the notation. 2996



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To prevent such accidents, we have found it useful to mix the list of the chosen recovery delays at random and to perform several possibly independent measurements. After appropriate intensity normalization, these independent sets of recovery data have been analyzed simultaneously,18 treating R79 and R81 as global parameters, and a and b for particular measurements as local parameters. As a result the number of degrees of freedom in the total analysis is reduced. An additional bonus of such a procedure is that it averages automatically the results of all particular measurements and allows avoiding the problem of weighting factors which tend to introduce some arbitrariness into the final result. Finally, it is to be kept in mind that measurements of long relaxation times can be additionally affected by the diffusive exchange of molecules between the active measurement volume and outside it. It is very difficult to avoid such effects and even more difficult to be sure that they are avoided. As a consequence, relaxation times on the order of 100 s or longer can be suspected to be of a low precision. The numerical analysis of the signal recovery curves provides for a given bromine-bonded carbon two relaxation parameters, R79 and R81, which include relaxation contributions to the observed relaxation rate also from the mechanisms other than SC2:

The most popular methods of investigating the longitudinal relaxation rates are classical inversion−recovery and saturation− recovery experiments.16 Both of them can also be used for investigating carbon−bromine systems, but it is to be stressed that in the case of long relaxation times, which are frequently encountered in the SC2 relaxation studies, the latter seems to be a method of choice rather than the former. If the longitudinal relaxation of a given carbon signal is affected by the relaxation of bromine atom via SC2 mechanism, the recovery curve can be described by the formula:8 Int(t ) = a[p79 exp(−R 79t ) + p81 exp(−R 81t )] + b

(4)

where Int(t) is an intensity of the carbon signal, R79 and R81 are relaxation rates (reciprocals of the appropriate relaxation times) of the signals of isotopomers containing 79Br and 81Br, a and b are adjustable parameters, and p79 and p81 (p79 + p81 = 1) are population parameters. It is to be kept in mind that the population parameters are equal to natural abundances of bromine isotopes, i.e., 0.506 and 0.494, respectively, only when at the beginning of the magnetization−recovery period of the monitored signal there is no differential saturation of the partial signals and when the recovery process is not affected by NOE.13 Taking into account that the relaxation times of the signal of 81Brbonded carbons are usually long (frequently several tens of seconds), the inversion−recovery experiment becomes very time-consuming. The saturation−recovery experiment is much more economical and the problem of the differential initial saturation of the isotopomer signals vanishes by definition. The problem caused by NOE remains, however. This complication will be addressed below, on the occasion of commenting results for (CH3)3CBr. Of course, the population parameter can essentially be retrieved from the numerical analysis of the intensity data as the additional adjustable parameter. Such a procedure is, however, error-prone due to the mathematical properties of the function to be analyzed (eq 4), and should better be avoided. Our practice has shown that the results of least-squares (l-s) analysis with fitting the population parameter can be dependent even on tiny data distortions arising when the spectra are affected by some noise. In the case of a large difference between relaxation times in isotopomers containing 79Br and 81Br nuclei an experimental procedure involving separate determination of the ‘short’ relaxation time in the inversion−recovery experiment and of the “long” relaxation time in the saturation−recovery experiment may be proposed. Namely, in the iterative numerical analysis of the intensity data of that composite experiment, one of the relaxation times would be fitted and the other would be kept constant. In the analysis of the inversion−recovery data, the population parameter has to be calculated at each iteration step from the current values of the fitted T1’s and the delay between the observing pulse and the inverting pulse in two subsequent scans. Our practice shows that this somewhat sophisticated procedure works, but its utility is rather limited. Moreover, similar positive effects can be probably gained by an appropriate choice of the evolution delays and enlargement of their number in the standard saturation−recovery procedure. Shortening of the measurement time is advantageous also because measurements of the long relaxation times are susceptible to various spectrometer instabilities in time. Such accidental instability can introduce a serious error to the results of numerical separation of the recovery curves into components because of numerical correlations between the fitted parameters.

R 79 = R 79,SC2 + R 79,DD + R oth

(5a)

R 81 = R 81,SC2 + R 81,DD + R oth

(5b)

The contributions of the bromine−carbon dipolar mechanism to the overall relaxation are frequently very small and can be neglected. The parameter Roth includes contributions from other relaxation mechanisms and also the effects of diffusion mentioned before. The number of unknown parameters is further reduced owing to the well-known relationships concerning parameters31 related to bromine isotopes: n

J (81Br, X)/nJ (79Br, X) = γ(81Br)/γ(79Br) = 1.078

(6)

T2(79Br)/T2(81Br) = [Q (81Br)/Q (79Br)]2 = 0.695

(7)

where Q(X) denotes the appropriate nuclear quadrupole moments. Thus, two relaxation parameters determined in a single relaxation experiment depend on three unknown parameters: JAX, T2(X) and Roth(A). The troublesome parameter Roth has to be determined independently. Of course, various possibilities are available, but their applicability depends mainly on the case in hand. Performing measurements at several magnetic fields is usually a way out. If one may assume that Roth is magnetic field independent, as it is in the case of our CBrX3 molecules, results from two magnetic fields are sufficient to solve the problem. A more difficult case is encountered when the chemical shift anisotropy dependent relaxation mechanism contributes to Roth.

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RESULTS CBrCl3. The results of analyses of saturation−recovery curves recorded for bromotrichloromethane molecule are given in Table 1. Unfortunately, the spectra recorded at 4.7 T were relatively noisy and it was because the performance and stability of the spectrometer used was lower than that of the other two spectrometers. The only reason for including these results into discussion was that 4.7 T was the lowest magnetic field accessible to us. Such a field was supposed to make the contribution of SC2

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Table 1. Longitudinal Relaxation Times T79 = 1/R79 and T81 = 1/R81 Determined from the Analysis of Magnetization Recovery Curves of 13C NMR Signals of 79Br- and 81BrContaining Isotopomers of CBrCl3 no.

B0 [T]

solvent

T79 [s]

T81[s]

1 2a 3 4 5a 6

4.7 4.7 7.05 11.7 11.7 11.7

CDCl3 CDCl3 CDCl3 CDCl3 CDCl3 DMSO-d6

1.81(22) 1.86(9) 1.82(12) 1.84(12) 1.98(3) 8.3(3)

51(12) 57(6) 59.2(4.6) 89(8) 88(6) 60(6)

a

A combined inversion−recovery and saturation−recovery experiment (see text).

mechanism to the observed relaxation rate the highest, which could help at least a qualitative interpretation of the results. Taking advantage of the large difference between the 13C relaxation times in C79BrCl3 and C81BrCl3, we have attempted to use the other experimental procedure mentioned above, composed of the inversion−recovery and saturation−recovery experiments. The “short” and “long” relaxation times obtained that way are given in rows 2 and 5 of Table 1. Positive outcomes are limited to some reduction of the error estimates and confirmation of the remaining results. Inspection of the data in Table 1 for CDCl3 solution shows that the 13C relaxation rate of the CBrCl3 signal for 79Br isotopomer is magnetic field independent in the 4.7−11.7 T range. It implies that in this case Δω(13C,79Br)2T2(79Br)2 ≪ 1. On the other hand (keeping in mind a limited reliability of the value of R81 measured at 4.7 T), it seems that for the other isotopomer the measured relaxation rate is field dependent. It also seems that the signal recovery for 81Br isotopomer is affected by other mechanisms and/or diffusion effects. The l-s analysis of the overestimated system of five equations based on the data of Table 1 and eqs 1−5 yielded the following values of unknowns: 1 13 79 J( C, Br) = 164 ± 11 Hz, T2(79Br) = 2.0(±0.3) × 10−7 s and Toth = 119 s. The values of the appropriate parameters concerning the 81Br isotope can be calculated using eqs 6 and 7. In the above analysis the Toth most likely plays a role of a parameter compensating for all inconsistencies of the data from Table 1, and so should not be interpreted. The l-s solution reproduces well all the data of Table 1, with the exception of T81 measured at 4.7 T, which, actually, has been omitted in the analysis (Figure 1). Additionally, we have performed the saturation−recovery measurements at 25 °C for CBrCl3 in DMSO, which is a more viscous solvent. The l-s analysis of the recovery curves has yielded the relaxation parameters given in row 6 of Table 1. It is apparent that the determined relaxation times T79 are longer and T81 are shorter in these conditions. These changes of the observed relaxation rates have been caused by the slower reorientation of the solute molecules, resulting in the faster quadrupole relaxation of bromine nuclei. Furthermore, one may hope that diffusive effects are in this case reduced as compared to CDCl3 solution. Assuming Roth = 0, one obtains for CBrCl3 in DMSO the upper limit of 1J(13C,79Br) = 178 Hz and T2(79Br) = 3.8 × 10−8 s. The shorter relaxation time of bromine in DMSO solution follows the expectation formulated above. Also the agreement between the estimates of spin−spin coupling constants obtained for both solutions is striking. (CH3)3CBr. Carbon-13 relaxation of tert-butyl bromide was investigated in the condition of proton decoupling. Performing

Figure 1. Reproduction of the experimental 13C NMR relaxation data for CBrCl3 (Table 1) by eq 1 assuming the values of 1J(13C,79Br) = 164 Hz, T2(79Br) = 2.0 × 10−7 s, and Toth = 119 s, obtained in the course of least-squares analysis.

the measurements in such a manner ensures a simple monoexponential recovery of the carbon signals of particular isotopomers. In the case of methyl groups the impact of the bromine-isotope-dependent relaxation mechanisms on the carbon signals is negligible, the relaxation rates in both isotopomers are identical and the recovery of the common methyl group signal is monoexponential. On the other hand, the signal of quaternary carbons recovers biexponentially. Moreover, decoupling introduces NOE enhancements, which for these carbons are different for isotopomers containing 79Br and 81Br: η79 = ηmax RDD/R 79

(8a)

η81 = ηmax RDD/R 81

(8b)

where from hereon RDD denotes the contribution of dipolar relaxation coming from the interaction of the carbon with all protons in the molecule. NOE had to be taken into account during numerical analysis of recovery curves by proper modification of the population parameter. This problem was first approached by Hayashi et al.13 who proposed measuring the value of NOE enhancement factor for the common signal and introducing it as additional data to the numerical analysis of the inversion recovery curves. However, in order to simplify the equations the authors assumed in their description the abundances of Br isotopes to be equal. Moreover, which is even more important, they also assumed that Int(0) = −Int(∞), where Int(0) and Int(∞) were the initial and equilibrium intensities of the signal under decoupling conditions, respectively. The Int(∞) parameter was taken directly from the experiment and was not adjusted during l-s analysis. Fulfillment of the experimental conditions ensuring validity of this procedure is rather difficult, especially when relaxation of the signal of interest is slow. This uncomfortable limitation can actually be overcome. Indeed, it can be shown that in the saturationrecovery experiment the population parameter from the eq 4 can 2998



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Table 2. Relaxation Times of the 13C Signal of Quaternary Carbons of 79Br- and 81Br-Containing Isotopomers of tert-Butyl Bromide and Their Interpretationa parameter T79 T81 T79 T81 T79 T81 T79 T81 parameter 1

13

79

J( C, Br) T2(79Br) τce Toth

B0 [T]

Unit

2.35 2.35 4.7 4.7 7.05 7.05 11.7 11.7 unit

s s s s s s s s

Hz μs ps s

(1)

(2)

(3)

lit. 8.6(7)b 61.5(2.0)b

(1)c 30.6(1.0) 1.3(2) 085e 64(4)

10.9(2.3) 61(12) 18.5(2.3) 66(13) 25.4(4.3) 109(57) (2)c

12.1(9) 67.6(4.1) 18.3(5) 65.5(1.9) 21.2(1.1) 74.7(3.9) (2)d

31.2(1.3) 1.1(2) 1.0e 62(3)

33.2(1.5) 0.67e 1.6 63

10.6(9) 59.2(2.6) 17.3(5) 60.6(1.3) 20.0(1.3) 69.5(2.9) (3)c 30.6(1.3) 1.0(3) 1.1e 66(4)

lit. 66b 0.27b 4.1 0

a

The relaxation times have been obtained by l-s analysis of the magnetization recovery curves assuming: (1) unrestricted adjustment of the population parameter, (2) the internal rotation of methyl groups being slow with respect to molecular reorientation in the solution, and (3) the internal rotation being 10 times faster than molecular reorientation. bReference 13. cFrom the least-squares analysis of the relaxation data fitting 1 13 79 J( C, Br), T2(79Br), and Toth. dT2(79Br) values were calculated from τc = 1.6 ps estimated on the basis of the measured T1,DD = 10.5 s for 13C of the methyl groups whereas 1J(13C,79Br), and Toth were obtained by l-s analysis. eThe relationship between τc and T2(79Br) was established adopting the calculated value of CQ(79Br) = 480 MHz.

be connected with the measurable NOE factor, η, and relaxation rates of particular isotopomers by the relationship: p79 = 0.506(ηR /R 79 + 1)/(η + 1)

rate of internal rotation of the methyl groups in (CH3)3CBr. Some literature data show that at room temperature the rate of internal rotation of these groups is slower than or comparable to the overall reorientation rate of the molecule.32,33 Thus, such a movement should influence the relaxation of the quaternary carbon only marginally. If this is the case, one can estimate RDD for the quaternary carbon using the relaxation data for the methyl group signal and molecular geometry. This contribution would amount to 0.0053 s−1, in a noticeable agreement with the value resulting from NOE. It can also be shown that the assumption about the internal methyl group rotation to be 10 times faster than the overall reorientation enhances the RDD contribution approximately two times only. The results of analyses of saturation−recovery curves recorded for the tert-butyl bromide molecule are given in Table 2. Obviously, some reservations concerning the spectra recorded at 4.7 T formulated in the preceding paragraph also apply in this case. On the other hand we could include into the table the relaxation data concerning 2.35 T magnetic field, which have been interpolated from the data reported by Hayashi et al.13 Examination of the data in Table 2 shows that the 13C relaxation rates of the signals of both 79Br and 81Br isotopomers are almost independent of the above methods of separation of two exponents composing the experimental recovery curves. Such a result confirms the reliability of the retrieved values of the relaxation parameters. Only the method in which the population parameter has been fitted simultaneously with R79 and R81 has yielded a poor result of R81 at 11.7 T, as judged from the error estimation. Contrary to the CBrCl3 in the discussed case, the R79 parameter is definitely magnetic field dependent while R81 is not. It implies that the relaxation of the signal of 81Br isotopomer involves a substantial contribution from mechanisms other than SC2 and probably includes the effect of diffusion. Thus, the field dependence of R79 carries information about the parameters governing SC2 relaxation, whereas the values of R81 determine, first of all, the Roth parameter. The l-s analysis of the data from Table 2 has yielded the values of 1J(13C,79Br) and T2(79Br) parameters collected in Table 2 and reproduced the experimental data satisfactorily (Figure 2). The values of the appropriate

(9)

where mean relaxation rate, R, and experimental NOE enhancement factor, η, are defined as follows: 1/R = 0.506/R 79 + 0.494/R 81

(10)

η = [Int(∞) − Int(0)]/Int(0) = ηmax RDD/R

(11)

Int(0) is the equilibrium intensity of the signal without decoupling. Thus, in the course of iterative l-s fitting of eq 4 to the set of the recovery data and the experimental value of η the values of four unknown parameters: a, b, R79, and R81 have to be found. Using the standard procedure we have measured the value of NOE enhancement factor for the quaternary carbon signal of (CH3)3CBr to be ca. 0.54 at 11.7 T. The least-squares analysis of the intensity data originating from this field yielded R79 and R81 values (Table 2), which allowed us to calculate the contribution of the dipolar mechanism to the overall relaxation (eq 11) to be: RDD = 0.0057 s−1. In the investigated case the NOE factor was magnetic field dependent (due to SC2 relaxation), whereas the RDD was not (due to extreme narrowing). Consequently, the value of RDD established at 11.7 T could be used in the numerical analysis of the saturation-recovery curves measured at two remaining magnetic fields. Thus, the population parameter was iteratively adjusted using the relationship: p79 = 0.506R(ηmax RDD + R 79)/[R 79(ηmax RDD + R )] (12)

Keeping in mind that in the case of slow relaxation the NOE enhancement factor measurements can be influenced by diffusive effects and tend to be not very precise, an independent procedure of establishing the value of the RDD parameter has been additionally applied. This alternative approach was based on the easily measurable contribution of the dipolar mechanism to the relaxation of the methyl group carbons. To use this information, it was necessary to adopt an assumption about the 2999



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was elegantly shown11,34 that, assuming equal populations of bromine isotopes, the multiexponential curve describing results of fully relaxed inversion−recovery experiment can be, after some coordinate transformation, expressed as a sum of two timedependent exponents. Using similar argumentation, the recovery curve for the saturation-recovery experiment can be expressed (without any additional assumptions) in the form similar to eq 4:

parameters concerning 81Br isotope can be calculated using eqs 6 and 7.

Int(t ) = a[p79 exp(−R 79t ) + p81 exp(−R 81t )]n + b

(13)

where n denotes the number of bromine atoms bonded to the involved carbon. At the same time, the parameter describing contributions of “other” relaxation mechanisms, incorporated into R79 and R81 parameters (cf. eq 5), has to be replaced in the above equation by Roth/n. The description formulated above, however, involves an implicit assumption of the lack of correlation between temporal behavior of spins of different bromine atoms bound to the same carbon, which is a serious problem actually.35−37 This assumption seems to be reasonable (vibrations, intermolecular collisions, extremely rapid bromine relaxation), but in fact is purely intuitive. A single example of the absence of crosscorrelation in a very different system36 cannot be treated as a proof of general unimportance of that phenomenon. A more rigorous description of the system involving more than one bromine atom would unavoidably introduce additional parameters to be adjusted during the relaxation data analysis.35−37 Assuming tentatively validity of eq 5, we have decomposed the saturation-recovery curves obtained from the measurements for CBr4 performed at 7.05 and 11.7 T magnetic fields. At B0 = 4.7 T, we have been able to determine only the relaxation time for the faster component by using the inversion−recovery method described in the paragraph concerning CBrCl3. The obtained results are given in Table 3. It is evident that the situation is

Figure 2. Reproduction of the experimental 13C NMR relaxation data for quaternary carbon of (CH3)3CBr (Table 2) by eq 1 assuming the values of 1J(13C,79Br) = 31.2 Hz, T2(79Br) = 1.1 × 10−6 s, and Toth = 62 s, obtained in the course of least-squares analysis.

The values of 1J(13C,79Br) and T2(79Br) obtained in this work are remarkably different than those reported by Hayashi et al.13 (see the last column of Table 2). There are several reasons of this disagreement. The measurements in ref 13 were done in one magnetic field and during the interpretation of their relaxation data the authors had to assume that Roth originated exclusively from 13C−1H and 13C−Br dipolar relaxation mechanisms. Such an assumption would have substantially changed our results as well. Moreover, during the calculation of Δω parameter (eq 3) the authors neglected the bromine shielding constants, which was a common practice in the past, and it is not clear which values of magnetogyric ratios for bromine they adopted. These factors may have affected their results.6 On the other hand, it is noticeable that the value of the composite parameter J2T2 calculated from the data of ref 13 (0.0012 s−1) remains in perfect agreement with that obtained in this work (0.0011 s−1). CBr4. Somewhat more complex is the case of the relaxation of carbon bonded to several bromine atoms. As before, the carbon signal of each isotopomer is characterized by its own relaxation rate, but the number of isotopomers is higher. The observed recovery curve is thus a sum of several exponential curves. In the case of carbon tetrabromide there are five isotopomers, C79Br4‑n81Brn (n = 0−4), in the molar ratio: 0.066:0.257:0.378:0.247:0.061, resulting from the natural abundances of bromine isotopes. Nevertheless, the recovery function is still dependent on three relaxation parameters only, namely, those describing the impact of a single 79Br nucleus, the impact of 81Br nucleus, and the contribution of other mechanisms to the observed carbon relaxation rate. It is to be stressed that two first parameters include both SC2 and dipolar (13C−79Br or 13C− 81Br) contributions to the relaxation rate. It

Table 3. Longitudinal Relaxation Times T79 = 1/R79 and T81 = 1/R81 Describing the Impact of a Single Bromine Nucleus on the Observed 13C Relaxation Rates, Determined from the l-s Analysis of Magnetization Recovery Curves of 13C NMR Signals of CBr4 in CDCl3 and DMSO-d6 Solutions no.

B0 [T]

solvent

T79 [s]

T81 [s]

1 2 3 4 5

4.7 7.05 11.7 11.7 11.7

CDCl3 CDCl3 CDCl3 DMSO-d6 DMSO-d6b

2.94(7)a 3.04(10) 2.97(3)a 18.9(1.2) 11.8(6)

− 113(34) 402(170) 58(11) 91(22)

a

A combined inversion−recovery and saturation−recovery experiment (see text). bMeasurement temperature 45 °C.

similar to that for CBrCl3, i.e. the short relaxation time is magnetic field independent contrary to the long relaxation time. However, the latter is only roughly estimated, especially at B0 = 11.7 T. Even taking into account the fact that the relaxation of the “laziest” isotopomer is four times faster than R81, it is still on the border of measurement possibility. It is not a surprise that such data have not allowed all three parameters to be determined simultaneously. Instead, we have assumed that the contribution of Roth to the overall relaxation is negligible and estimated the values of two remaining parameters (Table 4). Under the same assumption we have interpreted the relaxation data acquired for DMSO solution of CBr4 at two temperatures, obtaining very similar values of 1J(13C,79Br) (Table 4). The relaxation data measured for DMSO solution seem to be more reliable than 3000



Article

Table 5. Quadrupole Coupling Constants for 79Br Nuclei and Reorientation Rates of the Investigated Molecules in CDCl3 Solution at 25 °C Derived from SC2 Relaxation Data

Table 4. Parameters Concerning the CBr4 Molecule in CDCl3 or DMSO-d6 Solution Derived from the Relaxation Data of Table 3 dataa

1

J(13C,79Br) [Hz]

2 1+2+3 4 5

149.6 148.7(5.0) 151.1 147.4

T2(79Br) [μs]

τcb [ps]

0.15 0.16(1) 0.024 0.040

4.0 3.7 25 15

compound (CH3)3CBr CBrCl3 CBr4

512.0

c

650.4d

CQ(79Br) calcda [MHz]

τc [ps]

480 653 653

0.95, 1.6b 2.5 3.7

Values calculated by DFT B3LYP 6-311++G(2d,p) PCM(CHCl3) method using Gaussian 03 program. bThe value estimated on the basis of dipolar relaxation rate of methyl group carbons assuming slow internal rotation of these groups. cReference 38. dReference 39.

The values of 1J(13C,79Br) and T2(79Br) were calculated using the relaxation data contained in the specified below rows of Table 3. bThe reorientational correlation times were calculated using the quadrupole coupling constant CQ(79Br) = 653 MHz (see next chapter).

When molecular reorientation is described within the rotational diffusion model1,40 a single diffusion parameter (or correlation time) suffices to describe reorientation rate of the spherical top. Strictly speaking, such a model concerns the CBr4 molecule only. Two remaining molecules are symmetrical tops whose reorientations should be described by two diffusion coefficients, D⊥ and D||, only the former being connected with the relaxation rate of the appropriate bromine nucleus. However, taking into account very similar shapes of all three molecules, one may expect that D⊥ and D|| coefficients are not very different. The values of correlation times calculated by eq 14 are given in Table 5. The correlation time determined for tert-butyl bromide agrees reasonably well with the estimations based on the earlier NMR32 and Raman33 measurements (τc = 1.7−2.0 ps for neat liquid). Furthermore, the ordering of the reorientation rates within the series of the investigated molecules follows the prediction based on their moments of inertia. Evaluation and comparison of the computational methods is out of the scope of this paper. The theoretical calculations performed in this research have been used to evaluate the quality of the obtained experimental results only. Nevertheless, it seems worthwhile to discuss an impact of the relativistic effects on the NMR parameters in the investigated compounds. It has been proven elsewhere that such effects change remarkably the magnetic shielding of bromine nuclei6 and also the shielding of nuclei of bromine-bonded atoms.4,41−44 Indeed, comparison of the data for σ(13C) in Table 6 shows clearly that nonrelativistic calculation yields the results completely divergent from the experiment. The results of four-component relativistic calculation of σ(13C) values are much better. It is to be stressed that it also concerns the particularly challenging case of CBr4, for which an approximate relativistic calculation (one-electron SO correction using SOS-DFPT/PW91/BII)41 yielded σ(13C) = 157.9 ppm, which is rather a poor result. Second, we can compare the experimental 1J(13C,79Br) values with the ones calculated theoretically using two-component ZORA Hamiltonian. From previous researches it is known that this method should reproduce spin−spin coupling constants very well and usage of a four-component Dirac Hamiltonian is not mandatory. The experimental and theoretical data in Table 6 show a moderate agreement only and, at the same time, the nonrelativistic theoretical values do not seem to differ very much from the relativistic ones. However, it has been shown elsewhere45 that this fact does not mean unimportance of the relativistic effects. Rather, the scalar and spin−orbit relativistic terms, although sizable, are of opposite signs and compensate each other. This also holds for the investigated molecules, which has been confirmed by additional calculations performed at scalar ZORA level. It is still to be taken into account that the accuracy of

those obtained for CDCl3, since the more viscous solvent has caused the reorientation of CBr4 molecule to slow down and increased the quadrupole relaxation rate of its bromine nuclei. Indeed, the determined T2(79Br) for DMSO solution are significantly shorter than those for CDCl3 solution (Table 4). As a result, the measured R79 are smaller and R81 are larger (better measurable) for DMSO solution. The values of 1J(13C,79Br) obtained for both solutions are very similar, which validates the assumption about the negligible Roth. Moreover, they are in perfect agreement with those obtained earlier in ref 11 in different measurement conditions.

■

DISCUSSION The transverse relaxation rates of bromine nuclei determined in this work carry information about the reorientation rates of the investigated molecules in the solution. There is no doubt that in the case of bromine nuclei the quadrupole relaxation mechanism is dominant. For very fast movements (extreme narrowing) the transverse quadrupole relaxation time is connected to reorientational correlation time, τc, by the formula:1 1/T2(X) = 0.3π 2[(2IX + 3)/(IX 2(2IX − 1))]CQ 79

1.16 0.24 0.16

CQ(79Br) expt [MHz]

a

a

(X)2 (1 + χX 2 /3)τc

T2(79Br) [μs]

(14)

81

were X = Br or Br. Thus, in order to derive the reorientational correlation time, τc, the parameter describing molecular dynamics, the values of the appropriate quadrupole coupling constant, CQ(X), and asymmetry parameter, χX, are necessary. The values of CQ(79Br) can be calculated as the products of the effective quadrupole moment of 79Br nucleus and the electric field gradient (EFG) at this nucleus in a given molecule. Bailey and Gonzales15 have shown for a large set of bromo compounds that a cautiously selected nonrelativistic calculational method reproduces excellently the experimental CQ(79Br) values for gaseous state. The EFGs for the investigated compounds in CHCl3 solution have been calculated using DFT/B3LYP/6-311++G(2d,p)/PCM(CHCl3) method. The quadrupole coupling constants calculated for 79Br in the investigated compounds remain in a reasonable agreement with the experiment (Table 5). Some divergences may originate from the fact that the reported experimental values concern solid phase, whereas the theoretical ones have been calculated for CHCl3 solution. As it concerns the asymmetry parameters, let us note that in all three CBrX3 molecules bromine atoms lie on the molecular 3-fold symmetry axes. Such local symmetry results in the axial symmetry of the EFG tensors at bromine nuclei (χBr = 0), and causes that the nuclear spin relaxation of bromine is not affected by molecular rotation about the C−Br vector involved. 3001



Article

Table 6. Comparison of the Experimental and the Calculated Values of Magnetic Shielding Constants of the Central Carbons, σ(13C), Spin−Spin Coupling Constants, 1J(13C,79Br) and EFG Tensors at Bromine Nuclei Represented by Appropriate Components along the C−Br Directions, for the Investigated Compoundsa σ(13C) [ppm] b

compound

expt

(CH3)3CBr CBrCl3 CBr4

122.9 118.8 214.0

1

J(13C,79Br) [Hz]

c

B

B-PCM

4C

expt

97.2 30.5 21.8

92.5 30.6 22.1

104.8 97.1 216.7

32(1) 170(11) 150(5)

EFGzz [a.u.]

B-PCM

2C-COSMO

B-PCM

2C-COSMO

−50 −224 −229

−51.2 −212.9 −197.5

−6.770 −9.217 −9.213

−6.887 −9.630 −9.556

a

B-DFT with B3LYP functional and 6-311++G(2d,p) basis set, isolated species; B-PCM, method B for chloroform solution, using polarizable continuum model; 4C, 4-component Dirac−Coulomb Hamiltonian with B3LYP functional, dyall.v4z basis set for bromine and aug-cc-pVTZ basis set for other atoms, isolated species; 2C-COSMO, two-component SO-ZORA with B3LYP functional and jcpl basis set (TZ2P basis set with additional tight functions), for chloroform solution using a conductor-like screening model. bCalculated assuming σ(13C) = 186.6 − δ(13C). c Absolute values.

the experimental 1J(13C,79Br) values might be insufficient for making reliable comparisons. The experimental errors in Table 6 are the random statistical errors only. On the other hand, the agreement between the experimental CQ(79Br) for two of the investigated compounds and the values calculated by the nonrelativistic method does not seem to be fortuitous. The same result was obtained by Bailey and Gonzales15 for a large set of molecules. Moreover, the eigenvalues of EFG tensors obtained by relativistic and nonrelativistic calculations differ by ca. 4% only (Table 6). A rationalization of this fact does not seem to be straightforward.

experimental data on this parameter. It will be a goal of our nearfuture activity.

■

AUTHOR INFORMATION

Corresponding Author

*(A.G.-K.) E-mail: agryff@ch.pw.edu.pl. Telephone: +48 22 234 51 03. 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.

CONCLUSIONS The longitudinal relaxation of the 13C nuclei of the central carbons of three CBrX3 (X = Cl, CH3, Br) compounds investigated in this work is dominated by the scalar relaxation type 2 mechanism. Numerical analyses of the recovery curves after signal saturation have provided two relaxation parameters describing the longitudinal relaxation rates of 13C nuclei due to spin−spin interactions with two bromine isotopes and very fast relaxation of their nuclei. Of course, in the case of tert-butyl bromide, nuclear Overhauser effect had to be taken into account, whereas in the case of carbon tetrabromide the multiexponential recovery of the signal had to be considered during these analyses. A careful interpretation of these relaxation data and their dependence on the magnetic field have provided the values of the carbon-bromine spin−spin coupling constants involved and the transverse relaxation times of the bromine nuclei (Table 5). The latter parameters have been used to calculate the reorientation rates of the investigated molecules in the solution (Table 5). The necessary quadrupole coupling constants for bromine nuclei have been calculated theoretically. It has been found that ordering of the reorientation rates within the series of the investigated molecules follows the ordering of their moments of inertia. Considering an expected impact of the relativistic effects on the NMR parameters in the investigated series, it has been found that the experimental σ(13C) data definitely disagree with the results of nonrelativistic DFT calculations, whereas fourcomponent relativistic calculations have yielded much better results (Table 6). On the other hand, the experimental 1 13 79 J( C, Br) values have shown only a moderate agreement with the calculated ones. Probably it is connected with the fact that the contributions of the scalar and spin−orbit relativistic terms to the value of this parameter are of opposite signs and compensate each other.45 It seems, however, that a wider discussion of the effectiveness of theoretical calculations of 1 13 79 J( C, Br) values demands a broader set of the accurate

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REFERENCES

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