A procedure for total lineshape analysis of DNMR spectra of energetically strongly asymmetrical not-coupled two-site systems. 19F DNMR spectra of 2-chloro-6-fluoroisopropylbenzene

JOURNAL

OF MAGNETIC

RESONANCE

64,375-383

(1985)

A Procedurefor Total LineshapeAnalysis of DNMR Spectra of Energetically Strongly Asymmetrical Not-Coupled Two-Site Systems. 19FDNMR Spectra of 2-Chloro+fluoroisopropylbenzene R. LAATIRAINEN Department

of Chemistry,

University

of Kuopio.

POB

6, 70211

Kuopio

21, Finland

Received December 7,1984; revised April 9, 1985 Properties and analyxabilities of the thermodynamic parameters of exchange-broadened NMR spectra of energetically strongly asymmetrical two-site systems are discussed and the program DNMRS is described. The optimization of the thermodynamic and spectral parameters is based on the iterative fitting of a set of simultaneously given digitixed spectra measured at up to 20 temperatures or in different magnetic field strengths. Because the information about the entropies and the temperature dependence of the chemical shifts lies in the temperature dependence of some spectral properties, especially if the signal of the energetically less favored site is not observed, the procedure makes the analyzability of the parameters possible in a most efficient way. o 1985Academic PW, hc. INTRODUCTION

The most common DNMR spectroscopic problem is analysis of not-coupled Ati B exchange systems. Several approximate methods (1, 2) and many computer programs based on the TLS (Total Lineshape) method have already been described (2-7).’ The methods have been discussed in a few reviews (2, 5-7). However, a procedure and its advantages, in which the spectra from various temperatures and/or a few magnetic field strengths are simultaneously fitted during one single run, has not been described. The aim of this study was to develop such a program and to apply it for an examination of the analyzabilities of the spectral and thermodynamic parameters in the case of energetic nondegenerary of the two sites. Very reasonable estimates of the absolute rate theory activation free energy AC* at the coalescence temperature and the free energy difference AC between the sites at temperatures below the coalescence temperature (if the signal of the less populated site is detected) are easily obtained by the traditional methods (1, 2). The temperature dependencies of the free energies arise from molecular entropy differences and temperature-dependent solvent-solute interactions (8) and may significantly increase the information available from the DNMR analysis. For example, different degrees of degeneracy (n) of the sites result in the energy term RT ln(ni/nj), which can be combined to the entropy component of the free energy. The separation of the enthalpy and entropy components is known to be very difficult and the approximate methods have acquired a reputation for giving erroneous results (2). The entropy-type com’ See also the references cited in Ref. (2-6). 375

QO22-2364185 $3.00 CopyrigJtt Q 1985 by Academic F’ms. Inc. All rights of reprhv2tion in any form reSmed.

376

R. LAATIKAINEN

ponents arising from association reactions can be separated by studying the thermodynamics at various concentrations. When the other site signal is too weak or broad (for example, in cases like XC1 = X+ + Cl- (9), where the other site is broadened by quadrupolar interactions) to be observed, the analysis based on the traditional TLS approach is inconvenient. EXPERIMENTAL

2-Chloro-64uoroisopropylbenzene was prepared from the corresponding benzoic acid via a previously described method (20). The slow and badly balanced esterification was performed by the toluene-ethanol azeotrope method. In spite of the calculated amount of hydrogen in the hydrogenation of the ethylene stage the reaction gave a mixture of dechlorinated by-products. A part of the dechlorinated products was removed by azeotropic distillation with methanol. The NMR sample was prepared from 50 mg of material containing about 30% of the impurities by dissolving it in 1 ml of CD2Clz/isopentane (1: 1) with 1% of fluorobenzene as the internal i9F standard. The analysis of the coupled spectrum is reported elsewhere (I I). A typical VAX 1 l/780 cpu time of DNMRS for a set of 10 spectra of about 60 spectral points each is about 3 s for one iteration cycle, when all the thermodynamic and spectral parameters are adjusted. The convergence is usually found within 20-50 iterative cycles. RESULTS AND DISCUSSION

The Program DNMRS

The lineshape of not-coupled A = B DNMR

spectra can be given by

Z(u) = F(T, AH*, AS*, AH, AS, W(T), AI+‘(T), Tf).

111

The Gutowsky-Holm equation (I) was used for the function F, which defines the amplitude I(v) vs the frequency Y. AH* and AH are the temperature (T)-independent components of the free energies (enthalpies), AS* and AS reflect the corresponding temperature dependencies (entropies). W(T) and A W( T) are the temperature-dependent average chemical shift and the shift difference of the two sites. Tf is the effective T2 time. In the traditional TLS procedure each experimental digitized spectrum is independently fitted and the thermodynamic parameters are obtained from the Arrhenius or Eyring plot (I, 2). An advantage of DNMRS over the other programs is that the thermodynamic parameters are obtained without plotting vs temperature. The procedure offers a simple way for examination and accounting for correlations between the thermodynamic and spectral parameters. The program was realized by using a Newton-Raphson-type algorithm. The principal difficulty in accounting for the temperature dependence of the spectral parameters is that their behaviors are unknown and in spectra there is very little information about them, and that which is present is correlated with other information, around the coalescence temperature. In DNMRS the spectral parameters can be described by the polynomials AR’(T) = Y, + Y,AT+

Y3AT2 + Y,(lOO/T) + Y5(10000/T2),

PI

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with AT = (TO - T)/lOO. A similar polynomial (with X,-X,) can be used for W(T). Either the polynomial coefficients and temperature TOcan be directly optimized, or each W(T) is independently refined and constrained to follow a smooth least-squares fit. For example, it can be specified that an rrms (residual root mean square) of 0.1 Hz of W(T) from a smooth least-squares shape is as serious as a 1% rrms of the observed and computed Z(V)S.The chemical shifts outside the exchange-broadening range can be included in the fit. If the chemical shifts are not strongly temperature dependent, the constraining procedure leads to a substantial improvement in the confidence limits of the thermodynamic parameters, in comparison with the procedure in which each W(T) is independently optimized. In the case of the strong temperature dependence, the former procedure leads easily to a poor convergence and fit. The TT times of the sites A and B can be related with the reference T? through the linewidths (A) in two alternative ways: AA = Arer + dAA AA

= Ku&f

131 [4]

and the parameters d& , dAn, Z&, , and K,n can be optimized. The observed and calculated lineshapes can be related by Z(U),~ = AZ( ZJ)~~+ B + C(Y - v,,),

151

where A, B, and C are optimizable least-squares coefficients. Properties of Energetically Asymmetrical

Spectra: An Example

Some characteristics of the problem can be drawn by inspecting the analysis of the spectra shown in Fig. 1. The following equilibrium is assumed (11) and AG of about 5 kJ/mol is obtained from the spectra below 180 K.

If each clearly exchange broadened spectrum is independently fitted assuming AS of zero, the plots shown in Figs. 2-3 are obtained. A W(T) was extrapolated from temperatures 160 to 180 K. The plots strongly suggest that only AH of 5.1 kJ/mol gives a smooth and a continuous behavior of AG* and W(T)‘s: the full available information about the thermodynamics is obtained only if the shapes of the plots are taken into account. The detailed results are given in the caption to Fig. 2. The Analyzabilities

of the Thermodynamic

and Spectral Parameters

For this purpose a synthetic set of spectra was created with AH* = 40.0 kJ/mol, ASS = -20.0 J/K mol, AH = 5.0 kJ/mol, AS = 0.0 J/K mol, and 250 and -10 for Y, and Y4 in Eq. [3]. The spectra were simulated at 190,210,220,230,240,250,260,

R. LAATIKAINEN

378

240 K

)(

305K

c__

10 x

--/,

180 K

A

_...... -

I\

-.----Jd

FIG. 1. The proton-decoupled 19Fspectra of 2-chloro-6-fluoroisopropylbenzene 1) at 84.670 MHz (by Bruker WH-90, The University of Manitoba, Canada).

-

in CD,Cl,/isopentane

( 1:

AG'. kJ/mot AH 5.2 5.1 5.0

42s1 /

/ 41

i :,,[,,,, 180

200

220

240

260

280

T. K FIG. 2. AG* of the spectra in Fig. 1, when each temperature-broadened spectrum is independently fitted. The I(V) rrms values of the three values of AH are practically equal and the fitting AG* for AH of 5.1 kJ/ mol by the program ACTPAR (7) gave AH* = 38.43(80) kJ/mol and AS* = - 13.18(336) J/K mol. In the full analysis with DNMRS, AS was taken as zero, AIF was given directly at two temperatures and Y, and Y, were optimized. The analysis gave AH = 38.07(26) kJ/mol, AS = -14.06(112) J/K mol, AH = 5.135(10) kJ/mol, I(V) rrms = 1.52%, and the W(T) rrms := 0.19 Hz. A four-parameter fit (Xi, X2. X,, X,) was applied for the W( 7) function, the W( 7) values constrained as in Table 1, footnote d, and the baselines optimized. According to Table 1, the real standard deviations of the parameters may be two to three times higher than given above.

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AH

5.2 5.1 5.0

“1 160

200

220

240

260

260

T. K FIG. 3. W(T) vs T when each spectrum of 2-chloro-6~fluoroisopropybenzene is fitted independently. Insert: the behavior of the 19Fchemical shiR of 24luoroisopropylbenzene in the same sample shows that the behavior of the 19Fchemical shift is typical of the system.

270, 280, 290, and 310 K. The digitized spectra of about 50 points (automatically chosen around the spectral maxima) of a 200-280 K range were used in the analysis. The rest of the spectrum was given as signal positions. The signal of the less populated site was given at the two lowest temperatures. A set of 20 spectra was simulated as in two magnetic field strengths, the other being five times higher than the lower one. It is well known (7, 13, 14) that the accuracies of the entropies are poor. Statistically this follows from the strong internal correlation between AH and AS components and appears as large correlation coefficients and large confidence limits of the parameters. For example, if A HS of the model case is changed by 0.1 kJ/mol, the rrms value between the observed and calculated spectra is as large as 0.90% and would be easily observable in presence of random noise of 1%. When W( T)‘s at each temperature are independently adjusted, the I(Y) rrms is reduced to 0.55%, due to the correlation between AGS and W( T)‘s. If W( T)‘s are fitted by a five-term polynomial, the W( 7’) rrms from the smooth fit is 0.09 Hz and discontinuity similar to Fig. 3 is observed. When both W( T)‘s and AS* are optimized keeping A Hf as 40.1, the rrms values are reduced to 0.05% and 0.01 Hz, proving the very strong correlation between AH* and AS*. In the same way it can be shown that a wrong value of AH cannot be compensated by AH’ or AS* alone. However, the ratio of the AH* and ASS is rather sensitive to the value of AH. The same is seen from Fig. 1 and means that an accurate value of AS* cannot be obtained without a good estimate of AH. In principle the analyzability of the W(T) and A W( T) formula is a most serious problem. The maximum exchange broadening is proportional to PAW(T), where p is the molar fraction of the less populated site (12), p and A W( T) being fully correlated at that temperature. This and the Gutowsky-Holm equation suggest a strong correlation

380

R.

LAATIKAINEN

between the A W( 2”) coefficients and the thermodynamic parameters. Test calculations showed that the coefficients become analyzable only if the W(T) fit is considered or some direct information from A W( T) is given (for example, by signal positions below the exchange broadening temperature range). If A W( T) is approximately known, the value of AG* can be derived from the coalescence temperature and AG from maximum broadening (12). AS* and AS must be derived from the temperature dependence of the exchange broadening and the line positions. Because, in principle, W(T) and A W(T) may obey any mathematical function, AS* and AS are not analyzable without some constraints. Experiments in which AS* was set +5.0 J/mol K and AHS, AH, Y, , Y, , Y3, Y, , and Y, were optimized simultanously constraining W(T), did not lead to a substantial improvement in the fit, in comparison with the case in which only the thermodynamic parameters were optimized. Although the convergence was rather difficult to obtain and there may be false minima, the experiments suggest that if A W( T) and W(T) behave in some simple ways, the correlations between the entropies and the models are not fatal. If the chemical shifts can be decided to be temperature independent or linearly dependent on T or l/T, the most optimistic confidence limits of the present example can be applied to the results, Usually Tt is taken as equal for both sites and estimated from the linewidth of an internal standard. This approach may lead to systematical errors in the thermodynamics. For example, if the linewidth estimates of the model case were changed from 0.5 to 0.6 Hz, the optimized AHS was changed by +0.49 kJ/mol and AS* by -2.0 J/ K mol. The problem is partly removed if the spectra of the least broadening are neglected. The problem is less serious for the two-magnetic-field data. The analyzabilities of the thermodynamic parameters were examined in presence of random noise of l%, 0.1 K, and 0.1 Hz in the lineshapes, the temperatures, and the linewidth estimates. The changes were introduced into 15 sets of spectra, the spectra analyzed, and the deviations from the correct values averaged. These “observed’ standard deviations are given in Tables 1 and 2. The calculated standard deviations are too small. There are some reasons for this. First, the equations solved are strongly nonlinear and, thus, the first-order confidence limits are poor (1.5). Second, the calculated standard deviations are more reasonable if the uncertainties in the linewidth estimates can be removed. If ASt and A W( T)'s are not optimized, the calculated and “observed” standard values are equal. More reliable limits are obtained by using the method used for “observed” values, a ready procedure being included in the program. If both the AG components are optimized, the calculated standard deviations are far too optimistic and the convergence is slow, for the reasons discussed above. In practice, only one AG component can be derived from one field experiment. If Z& (Ku = K&B) is totally unknown, only one AGS component can be estimated. A most interesting question was whether the use of the constraining procedure for W( T)‘s really improves the result. The experiments show that both the calculated and “observed” standard deviations are reduced by up to 50%. The confidence limits of the A W( T) coefficients appear to be large, but including them (and &) in the fitting does not seriously increase the standard deviations of the thermodynamic parameters; in fact, the “observed” standard deviations are slightly improved. It can be shown, by removing the uncertainties in A and T, that the opti-

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TABLE 1 Some Calculations to Probe the Analyzability of the Thermodynamic Parameters and the Realiability of the Standard Deviations in the Presence of Random Noise’ for the Case of One Field Strength rrms values

Calculatedstandarddeviation@ (“observed” value)* 10’ Optimized setsof parametersc

w-9e weight

AH’, AS*, AH, AH*, AS’, AH, AW(T) AH*, AS’, AH, AW(T) AH? AH’, AH*,

AS’, AH, AW(T) AS*, AH, AW(T) AS’, AH, AW(T)

AH*,

AS’, AH, AW(T),

0’ 0’ 0’

K,,

n The standarddeviations of (0.5 Hz), and temperatures,

Y,

Y,

8000 (9100) 19 (32)

Y5

-

-

3200(3600)

320 (360)

4(6)

-

-

-

AH*

AS*

AH

20(47) 40(65) 14(49)

80 (190) 184 (280) 54 (200)

3 (2) l(2)

45 (160) 71 (130) 50(130)

10/2Gd 10/2”d 10/2c++

320 (I LOO) 18 (37)

130 (470) 3 (8)

13(40) -

I I (40) 17(31) 12(33)

10/2”d

330(1800)

130(430)

13 (43)

39 (73)

I%, 0.1 Hz,

and 0.1

K were induced

in the lineshape,

Mo(300) estimate

I(V)

W(T)

1 (2) 0.99

0.07 0.19

0.98 0.98

0.08

I (1) 0.99

0.06

1 (I)

0.99 0.99

0.04 0.06

1 (2)

0.99

0.04

l(1)

of the reference

linewidth

respectively.

’ The normalized (correspnnding to 1% noise) standard deviations given by the program. Tbe number in parentheses givesan “observed” number ascalculatedby the procedure describedin the text. ’ Each W(T) wasindependently refined. d The weight wassetequal to 10 for chemical shiftsoutsidethe exchange-broadeningtemperature rangeand 2 for inside the range. The value 10 meansthat a 0.1 Hz W( 7) rnns (including the direct W(T) and the AW( T) information) is as seriousas I(v) rrms of 1%.

mization of the A W( T) coefficients offers a route to release “strain” between the observed and calculated spectra of less broadening, otherwise released by maintaining AS* and AH*. The origin of the large standard deviations is the correlation between TABLE 2 Some Calculations to Probe the Analyzability of the Thermodynamic Parameters and the Realiability of the Standard Deviations in the Presence of Random Noise” for the Case of Two Magnetic Fields (for Example, 60 and 300 MHz) Calculated standard deviatic& Optimized set of pxameters” AH’, AS’, AWT) AH’, as’, AWT) AH’, AS’, AWT),

W(T) weight

Y,

(“observed” value)* I@

mm values

Y4

Y,

AH*

AS*

AH

AS

I(4

W(T)

AH, OC

24cnl(5700)

I100 (2800)

120 (334)

18 (33)

71 (130)

I(5)

-

0.98

0.70

OC

2600 (13oM))

1200 (5900)

I30 (660)

19(56)

77 (220)

14(36)

52(M)

1.03

1.23

OC

2900 (12000)

l3cm (5200)

150 (580)

22(51)

90(200)

I7 (35)

62(130)

0.97

I.13

10/5/2”*

160(1400)

7 I (770)

8(80)

9 (20)

33 (76)

I(2)

-

1.00

0.10

10/S/2””

420 (2200)

I80 (WOO)

20 (120)

9 (34)

35 (130)

6(24)

24(100)

1.00

0.17

10/S/2’.+

390 (1500)

170 (710)

I8 (82)

I I (19)

43 (78)

6 (19)

22 (74)

0.99

0.11

AH, AS, AH, AS, &

AH*, ASS’,AH, AWT) AH*, AS*, AH, AS, AW-) AH’, ast, AH, AS, AW(T), &

n~b.rSee footnotes (I, b, and c in Table I. ‘The weight was set equal to IO (or 5 for the higher field) for chemical shifts outside the exchangbbroadeniag temperature range and 5 (or 2) Inside the range. The smaller weights for the hiir field s$ectra were used because the variation of the chemical shifis may be partly due to inaccuracies in tbe temperatures and, thus, may be field dependent. See also footnote d, in Table I.

382

R. LAATIKAINEN

Y, and Y,. The large numbers mean, that a reasonable description of the correct A W( 7’) is obtained with many polynomials. When the signal of the less populated site is unobservable, only one AI+‘(T) coefficient can be optimized. The last point to be discussed is the analyzability of both the AG components by using spectra from two magnetic field strengths. In the case described in Table 2 the standard deviation of AS is 1 J/K mol. It also appeared that the convergence was slow, in comparison to the other experiments. In practice it may be better to estimate the value of AS on the basis of iterations with AS = 0, +2, +4, etc. An experiment was carried out to examine this: the model spectra were analyzed keeping AS = 5.0 J/K mol and optimizing AH*, AS*, AH, Y, , and Y4. As the result of 15 such trials the average I(V) rrms increased from about 0.99 to 1.03%, the W(T) rrms from about 0.10 to 0.40 Hz, and the weighted total rrms from about 1.00 to 1.07% (in the I(Y) units). The differences may seem insignificant, but one must remember that the rrms is an average value of all the spectral points, including many points describing the wings of the signals and the baseline, and there may exist rather observable deviations in some spectra or parts of spectra. Anyway, if W(T) and A W(T) can be assumed to be temperature independent, the analysis of AS is possible well within R*ln 2. When AS was set 5.0 J/K mol, the rest of the parameters were optimized, and the optimization was then continued including also AS, the program was able to find the original solution within 50 iteration cycles. This shows that the procedure works also in this very complicated situation and that AS is not fully correlated with the other parameters. The baseline, the spectral phasing, and the lineshape of the spectrometer were assumed to be correct in the present experiments, but they may contribute the quality of the statistics in real cases. If the coefficients B and C in Eq. [5] are optimized, the standard deviations of the thermodynamic parameters are not changed substantially, but the convergence is slow. CONCLUSION

The present procedure worked well in all the conditions tried. The greatest advantages are that all the experimental data can be properly accounted for in the estimation of the thermodynamics and that consequences of different approximations are easily tested. The present procedure is convenient in cases where the other signal is not observed. When the spectra are measured at one magnetic field strength, AH*, AS*, and AH can be rather reliably estimated if I+‘( 7’) and A W( T) are not very strongly temperature dependent or their behavior can be assumed to obey a known type of function. By measuring the spectra in several magnetic field strengths, the accuracy of the results can be substantially improved and the value of AS becomes analyzable in optimal cases. ACKNOWLEDGMENTS I thank Professor Ted Schaefer for his comments and the Academy of Finland and the Natural Sciences and Engineering Research Concil of Canada (Grant A 1296) for financial support.

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REFERENCES

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