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Application of power spectra to lineshape analysis in NMR spectroscopy
JOURNAL
OF MAGNETIC
RESONANCE
Application
ANDRZEJ Centre
sf
Molecular
40, 469-474 (1980)
of Power Spectra to Lineshape Analysis in NMR Spectroscopy EJCHART
AND KRZYSZTOF
and Macromolecular
Studies,
WR~BLEWSKI
Boczna
5, 90-362Rhdi,
Poland
Received July 9, 1979; revised February 29, 1980 The advantage of using power spectra instead of absorption spectra for lineshape analysis is discussed. It is demonstrated that values of fitted parameters and their accuracy are worse for absorption spectra exhibiting phase shift than for power spectra when the least-squares method is used. This is because the phase distortions which occur in absorption spectra are eliminated in power spectra. On tbe other hand, results obtained for absorption spectra without phase shift are comparable to those obtained for power spectra. The influence of noise on the results is also discussed.
Changes in signal shape in NMR spectroscopy caused by chemical exchange (I) or spin-spin coupling to a fast-relaxing nucleus (2) can be used in the quantitative description of these processes by applying lineshape analysis (3, 4). All possible experimental distortions of signals should be avoided or minimalized, so that signal broadening reflects only the process investigated, and lineshape analysis is efficient (I, 4). NMR spectra of liquids are commonly run in the absorption mode (5). This mode has many advantages; e.g., the proportionality of integration to the number of auclei and the fast decay of signals in the frequency domain (5, 6). On the other hand, there is a certain disadvantage in that a small amount of dispersion mode might possibly distort the signal; this is the so-called phase shift (5). In pulsed Fourier transform spectrometers, as a result of Fourier transformation of free-induction decay, one can simultaneously obta.in the absorption mode, II, and the dispersion mode, U. Nevertheless, r and 1, two components obtained in the experiment, are not pure absorption and dispersion modes, but for various reasons they exhibit the phase shift and are a linear combination of the u and u modes. The amplitudes of absorption and dispersion modes, corresponding to the frequency vi, are given by (5)
Z.Li= Yi Sill@ - Ei COS&
[II
where ri and li are the amplitudes of the real and imaginary parts of the spectrum after Fourier transformation. 0, is the phase shift, which is approximately a linear function of frequency. 0, = 00 f 01Vi; PI 469 W22-2364/80/120469-06$02.00/O Copyright 8 1980 by Academic Press. Inc. All rights of reproduction in any form reserved.
470
EJCHART
AND WR6BLEWSKI
0, and O1are constants which are usually adjusted manually by the operator, who visually checks the shape of the spectrum. This procedure is inaccurate because the evaluation by the operator can be erroneous. Moreover, relationship [2] is approximate, and the phase shift can show other frequency characteristics (5, 7). It is possible to avoid the problem of phase adjustment in calculating the power spectrum, which does not show a phase dependence and is defined by the following equation (5): Pi = uf + ZJ;= vf + I;.
131
IJse of the power spectra eliminates the phase distortions frequently present in absorption spectra. For this reason, power spectra can be more convenient than absorption spectra when lineshape analysis is applied to describe the results. It should be also pointed out that the shape of the power signal is Lorentzian, similar to the absorption signals (5). The best proof of the usefulness of power spectra for lineshape analysis is a comparison of the accuracy of parameter fits describing the lineshapes for absorption and power spectra, when the least-squares method is used. Such a comparison was carried out in the case of signals broadened by the exchange process and broadened by spin-spin coupling to a fast-relaxing nucleus in the presence of random disturbance. For this purpose typical electronic noise of the NMR spectrometer (1K points) was recorded and stored after elimination of the DC component. This noise is presented in Fig. 1. Then the reference spectra in absorption and dispersion modes (each up to 512 points) were calculated by use of the known set of parameters. These spectra could be distorted by the phase shift according to Eq. [2], when 0, and O1values were introduced. The amplitude of the highest peak in the simulated spectrum could be adjusted. Thus, after addition of noise to the simulated lineshape one could obtain the assumed signal-to-noise ratio, SIN, defined in our work in the commonly accepted way (5): SIN =
2.5 x signal height peak-to-peak noise ’
Absorption reference spectra with noise thus obtained were used in further calculations. At the same time these spectra, together with spectra in the dispersion
A
B
FIG. 1. Eiectronic noise obtained by the use of a Bruker HX-72 NMR spectrometer as a result of averaging eight scans. DC bias was removed from the noise. Peak-to-peak amplitude is equal to 160 units. Five hundred twelve data points added to the absorption mode (A) and to the dispersion mode (B).
POWER SPECTRA IN LINESHAPE
ANALYSIS
FIG. 2. Reference spectra for exchange between two nonequally populated sites calculated by Eq. [lS] taken from Ref. (I). The spectra are composed of 512 points and cover a range of 30 Hz. Other parameters used in calculations were as follows: vA = 10 Hz, vg = 20 Hz, k,, = 10 set-‘, T,, = T2a = 1 see, andp, = 0.45. Absorption and power spectra calculated without phase shift are presented in A and C, respectively, whereas those with a zero-order phase shift, 8, = Zo”, are presented in B and D. The amplitude of higher signal in spectrum A is equal to 1000 units, giving a signal-to-noise ratio, S/N = 16.
mode similarly obtained, were used to calculate power spectra according to [3]. Afterward the standard deviation, RMS, was calculated point by point for differences in shapes between a reference spectrum and a noiseless one, fitted by change of the parameter values describing the shape of the lines. Amplitudes in these spectra which were fitted to the power spectra and to the absorption spectra with phase shift were matched to obtain a minimum of RMS deviation prior to fitting other parameters. In the case of absorption spectra without phase shift the minimum was always obtained for the value of amplitude used in calculation of the reference spectrum. For the exchange-broadened signals, the reference spectra used in calculations with assumed signal-to-noise ratio, SIN = 16, are presented in Fig. 2. One pair of these spectra (Figs. 2B and D) was obtained after introducing a zero-order phase shift, Bi = 20”. The accuracy of fit was evaluated for two out of five parameters which characterize these lineshapes; namely, for rate constant, kAB, and population, pA. Figure 3 shows the values of RMS deviation as a function of these parameters. Two out of three curves correspond to absorption spectra: curve A to the spectrum without phase shift and curve AS to the spectrum with phase shift. The third curve, PS, corresponds to the power spectrum obtained from 20” phase-shifted absorption and dispersion modes. The curve corresponding to the power spectrum obtained from pure absorption and dispersion spectra was not plotted, because it is nearly identical to the PS curve. The differences between curves A and PS are insignificant. The accuracy of parameter fit obtained in
EJCHART
472
AND WR6BLEWSKI
RMS 140
120
100
60
60
60
40
40
20
20 5
7
9
11
13
kAE
0.40
0.42
0.44
A
0.46
0.46
PA
B
FIG. 3. Values of standard deviation, RMS, in arbitrary units calculated for differences between the reference (Fig. 2) and fitted spectra as a function of the rate constant, k,, (A), and population in site A, pA (B). Curves AS, A, and PS were obtained for the reference spectra shown in Figs. 2B, 2A, and ZD, respectively. Amplitudes of fitted spectra were matched to obtain the minimum of RMS value. Thus, values of 940 for 2B, 1000 for 2A, and 1004 for 2D were used.
lineshape analysis by the least-squares method can be evaluated from the slo of lines A and PS in Fig. 3, and is comparable. The character of the relation discussed changes dramatically in the case of the phase-shifted absorption spectrum. The appropriate AS curve in Fig. 3 exhibits not only significantly smaller slope in the residue of the minimum which is wider than those for curves A and PS, but also the position of the minimum is shifted in comparison with the parameter value used in calculation of the reference spectrum. In consequence, phase shift in absorption spectra causes the choice of unsuitable parameter values as well as diminution of the fit accuracy.
60
60
20
20 5
7
11
9 A
13
km
0.40
0.42
0.44
0.46
0.46 p*
0
Fro. 4. Values of RMS obtained in a way similar to that in Fig. 3 for the reference spectra exhibiting signal-to-noise ratio, SIN = 8. Amplitude values of fitted spectra were 470, 500, and 484 for AS, A, and PS curves, respectively.
POWER SPECTRA IN LINESHAPE
A
B
ANALYSIS
C
FIG. 5. Calculated lineshapes in the absorptron mode of a dipolar nucleus coupled to a fast relaxing nucleus with spin I = 1, using Eq. [3.97] from Ref. (2) under assumption of Jax = 7.5 Hz and Tzx = 0.32 sec. (A) T, = 0.025 set, range of 35 Hz, 351 points; (B) T, = 0.040 set, range of 50 Hz, 501 points; as well as (C) exchange-broadened spectrum above the coalescence point with the same parameters as those presented in Fig. 2A, with the exception of the rate constant, which in this case is k *B = 20 set-‘.
Curves similar to those discussed above, but obtained for the reference spectra exhibiting only one-half as large a signal-to-noise ratio, SIN = 8, are presented in Fig. 4. The interrelation between curves A, AS, and PS does not change. The curve obtained for the phase-distorted absorption spectrum shows a shifted and significantly wider minimum than the curve obtained for power spectra. Comparison of Figs. 3 and 4 leads to the conclusion that the minima are wider when the signal-to-noise ratio is lower. As could be expected, the lowering of signal to noise ratio reduces the accuracy of parameter fit. Similar results were obtained for the exchange-broadened spectrum above the coalescence point as well as for signals broadened by spin-spin coupling to a fast relaxing nucleus. In the latter case, similarly as in the case of exchange, signals before and after coalescence of the spin multiplet were calculated. All these hneshapes in the absorption mode without phase shift are presented in Fig. 5. One should note that before calculating the power spectrum, the baseline anomalies must be eliminated. Unfortunately, the automatic cubic method for cor-
FIG. 6. Noise of the power spectrum presented in Fig. 2D. This figure was obtained by subtracting the spectrum calculated for the best fit of parameters (including amplitude) from the spectrum shown in Fig. 2D. It should be stressed that this noise does not exhibit a DC component.
EJCHART AND WR6BLEWSKI
474
recting the baseline (8) cannot be used in the case of broadened signals because it can partly remove these signals. It seems that assuming a linear dependence of the baseline (Legrange’s two first polynomials in Ref. (8)) should sufIiciently compensate baseline anomalies without significant distortions of the broadened signals. On the other hand, one need not correct the baseline in power spectra, because the difference between the reference spectrum with noise and the noiseless spectrum, both calculated for the same parameter set, does not have a DC bias, although the noise exhibits variable amplitudes, as seen in Fig. 6. It can be concluded from the comparison of Fig. 6 and Fig. 26 (or 2D) that random turbance in the power spectrum depends on the signal amplitude. In contrast, one can see from Figs. 1 and 6, both being in the same vertical scale, that, on average, noise in the power spectrum is smaller than in the absorption spectrum. This can be also concluded from the RMS value in the minimum for curve PS lower than that for curve A in Figs. 3 and 4. Therefore, we can conclude that the use of power spectra for lineshape analysis, if possible, seems to be advantageous mainly because of the elimination of phase distortions which can induce an unexpected change of fitted parameters as well as lower their accuracy. REFERENCES 1. G. BINSCH, in “Topics in Stereochemistry” Wiley-Interscience, New York, 1968.
(N. L. Allinger and E. L. EIiel, Eds.), Voi. 3, p” 97,
LEHN AND J. P. KINTZINGER, in “Nitrogen NMR” (M. Witanowski and @. A. Webb, Eds.), p. 79, Plenum, London, 1973. Nuclear Magnetic Resonance” (L. M. Jackman and F. A. Cotton, G. BINSCH, in “Dynamic Eds.), Chap. 3, Academic Press, New York, 1975. D. S. STEPHENSON AND G. BINSCH, J. Magn. Reson. 32, 145 (1978). D. SHAW, “Fourier Transform NMR Spectroscopy,” Elsevier, Amsterdam, 1976. T. C. FARRAR AND E. D. BECKER, “Pulse and Fourier Transform NMR. Introduction to Theory and Methods,” Academic Press, New York, 1971. B. L. NEFF, J. L. ACKERMAN, AND J. S. WAUGH, J. Magn. Reson. 25, 335 (1977).
2. J. M. 3. 4. 5. 6.
7. 8. 6.
A. PEARSON,
J. Magn.
Reson.
27, 26.5 (1977).
OF MAGNETIC
RESONANCE
Application
ANDRZEJ Centre
sf
Molecular
40, 469-474 (1980)
of Power Spectra to Lineshape Analysis in NMR Spectroscopy EJCHART
AND KRZYSZTOF
and Macromolecular
Studies,
WR~BLEWSKI
Boczna
5, 90-362Rhdi,
Poland
Received July 9, 1979; revised February 29, 1980 The advantage of using power spectra instead of absorption spectra for lineshape analysis is discussed. It is demonstrated that values of fitted parameters and their accuracy are worse for absorption spectra exhibiting phase shift than for power spectra when the least-squares method is used. This is because the phase distortions which occur in absorption spectra are eliminated in power spectra. On tbe other hand, results obtained for absorption spectra without phase shift are comparable to those obtained for power spectra. The influence of noise on the results is also discussed.
Changes in signal shape in NMR spectroscopy caused by chemical exchange (I) or spin-spin coupling to a fast-relaxing nucleus (2) can be used in the quantitative description of these processes by applying lineshape analysis (3, 4). All possible experimental distortions of signals should be avoided or minimalized, so that signal broadening reflects only the process investigated, and lineshape analysis is efficient (I, 4). NMR spectra of liquids are commonly run in the absorption mode (5). This mode has many advantages; e.g., the proportionality of integration to the number of auclei and the fast decay of signals in the frequency domain (5, 6). On the other hand, there is a certain disadvantage in that a small amount of dispersion mode might possibly distort the signal; this is the so-called phase shift (5). In pulsed Fourier transform spectrometers, as a result of Fourier transformation of free-induction decay, one can simultaneously obta.in the absorption mode, II, and the dispersion mode, U. Nevertheless, r and 1, two components obtained in the experiment, are not pure absorption and dispersion modes, but for various reasons they exhibit the phase shift and are a linear combination of the u and u modes. The amplitudes of absorption and dispersion modes, corresponding to the frequency vi, are given by (5)
Z.Li= Yi Sill@ - Ei COS&
[II
where ri and li are the amplitudes of the real and imaginary parts of the spectrum after Fourier transformation. 0, is the phase shift, which is approximately a linear function of frequency. 0, = 00 f 01Vi; PI 469 W22-2364/80/120469-06$02.00/O Copyright 8 1980 by Academic Press. Inc. All rights of reproduction in any form reserved.
470
EJCHART
AND WR6BLEWSKI
0, and O1are constants which are usually adjusted manually by the operator, who visually checks the shape of the spectrum. This procedure is inaccurate because the evaluation by the operator can be erroneous. Moreover, relationship [2] is approximate, and the phase shift can show other frequency characteristics (5, 7). It is possible to avoid the problem of phase adjustment in calculating the power spectrum, which does not show a phase dependence and is defined by the following equation (5): Pi = uf + ZJ;= vf + I;.
131
IJse of the power spectra eliminates the phase distortions frequently present in absorption spectra. For this reason, power spectra can be more convenient than absorption spectra when lineshape analysis is applied to describe the results. It should be also pointed out that the shape of the power signal is Lorentzian, similar to the absorption signals (5). The best proof of the usefulness of power spectra for lineshape analysis is a comparison of the accuracy of parameter fits describing the lineshapes for absorption and power spectra, when the least-squares method is used. Such a comparison was carried out in the case of signals broadened by the exchange process and broadened by spin-spin coupling to a fast-relaxing nucleus in the presence of random disturbance. For this purpose typical electronic noise of the NMR spectrometer (1K points) was recorded and stored after elimination of the DC component. This noise is presented in Fig. 1. Then the reference spectra in absorption and dispersion modes (each up to 512 points) were calculated by use of the known set of parameters. These spectra could be distorted by the phase shift according to Eq. [2], when 0, and O1values were introduced. The amplitude of the highest peak in the simulated spectrum could be adjusted. Thus, after addition of noise to the simulated lineshape one could obtain the assumed signal-to-noise ratio, SIN, defined in our work in the commonly accepted way (5): SIN =
2.5 x signal height peak-to-peak noise ’
Absorption reference spectra with noise thus obtained were used in further calculations. At the same time these spectra, together with spectra in the dispersion
A
B
FIG. 1. Eiectronic noise obtained by the use of a Bruker HX-72 NMR spectrometer as a result of averaging eight scans. DC bias was removed from the noise. Peak-to-peak amplitude is equal to 160 units. Five hundred twelve data points added to the absorption mode (A) and to the dispersion mode (B).
POWER SPECTRA IN LINESHAPE
ANALYSIS
FIG. 2. Reference spectra for exchange between two nonequally populated sites calculated by Eq. [lS] taken from Ref. (I). The spectra are composed of 512 points and cover a range of 30 Hz. Other parameters used in calculations were as follows: vA = 10 Hz, vg = 20 Hz, k,, = 10 set-‘, T,, = T2a = 1 see, andp, = 0.45. Absorption and power spectra calculated without phase shift are presented in A and C, respectively, whereas those with a zero-order phase shift, 8, = Zo”, are presented in B and D. The amplitude of higher signal in spectrum A is equal to 1000 units, giving a signal-to-noise ratio, S/N = 16.
mode similarly obtained, were used to calculate power spectra according to [3]. Afterward the standard deviation, RMS, was calculated point by point for differences in shapes between a reference spectrum and a noiseless one, fitted by change of the parameter values describing the shape of the lines. Amplitudes in these spectra which were fitted to the power spectra and to the absorption spectra with phase shift were matched to obtain a minimum of RMS deviation prior to fitting other parameters. In the case of absorption spectra without phase shift the minimum was always obtained for the value of amplitude used in calculation of the reference spectrum. For the exchange-broadened signals, the reference spectra used in calculations with assumed signal-to-noise ratio, SIN = 16, are presented in Fig. 2. One pair of these spectra (Figs. 2B and D) was obtained after introducing a zero-order phase shift, Bi = 20”. The accuracy of fit was evaluated for two out of five parameters which characterize these lineshapes; namely, for rate constant, kAB, and population, pA. Figure 3 shows the values of RMS deviation as a function of these parameters. Two out of three curves correspond to absorption spectra: curve A to the spectrum without phase shift and curve AS to the spectrum with phase shift. The third curve, PS, corresponds to the power spectrum obtained from 20” phase-shifted absorption and dispersion modes. The curve corresponding to the power spectrum obtained from pure absorption and dispersion spectra was not plotted, because it is nearly identical to the PS curve. The differences between curves A and PS are insignificant. The accuracy of parameter fit obtained in
EJCHART
472
AND WR6BLEWSKI
RMS 140
120
100
60
60
60
40
40
20
20 5
7
9
11
13
kAE
0.40
0.42
0.44
A
0.46
0.46
PA
B
FIG. 3. Values of standard deviation, RMS, in arbitrary units calculated for differences between the reference (Fig. 2) and fitted spectra as a function of the rate constant, k,, (A), and population in site A, pA (B). Curves AS, A, and PS were obtained for the reference spectra shown in Figs. 2B, 2A, and ZD, respectively. Amplitudes of fitted spectra were matched to obtain the minimum of RMS value. Thus, values of 940 for 2B, 1000 for 2A, and 1004 for 2D were used.
lineshape analysis by the least-squares method can be evaluated from the slo of lines A and PS in Fig. 3, and is comparable. The character of the relation discussed changes dramatically in the case of the phase-shifted absorption spectrum. The appropriate AS curve in Fig. 3 exhibits not only significantly smaller slope in the residue of the minimum which is wider than those for curves A and PS, but also the position of the minimum is shifted in comparison with the parameter value used in calculation of the reference spectrum. In consequence, phase shift in absorption spectra causes the choice of unsuitable parameter values as well as diminution of the fit accuracy.
60
60
20
20 5
7
11
9 A
13
km
0.40
0.42
0.44
0.46
0.46 p*
0
Fro. 4. Values of RMS obtained in a way similar to that in Fig. 3 for the reference spectra exhibiting signal-to-noise ratio, SIN = 8. Amplitude values of fitted spectra were 470, 500, and 484 for AS, A, and PS curves, respectively.
POWER SPECTRA IN LINESHAPE
A
B
ANALYSIS
C
FIG. 5. Calculated lineshapes in the absorptron mode of a dipolar nucleus coupled to a fast relaxing nucleus with spin I = 1, using Eq. [3.97] from Ref. (2) under assumption of Jax = 7.5 Hz and Tzx = 0.32 sec. (A) T, = 0.025 set, range of 35 Hz, 351 points; (B) T, = 0.040 set, range of 50 Hz, 501 points; as well as (C) exchange-broadened spectrum above the coalescence point with the same parameters as those presented in Fig. 2A, with the exception of the rate constant, which in this case is k *B = 20 set-‘.
Curves similar to those discussed above, but obtained for the reference spectra exhibiting only one-half as large a signal-to-noise ratio, SIN = 8, are presented in Fig. 4. The interrelation between curves A, AS, and PS does not change. The curve obtained for the phase-distorted absorption spectrum shows a shifted and significantly wider minimum than the curve obtained for power spectra. Comparison of Figs. 3 and 4 leads to the conclusion that the minima are wider when the signal-to-noise ratio is lower. As could be expected, the lowering of signal to noise ratio reduces the accuracy of parameter fit. Similar results were obtained for the exchange-broadened spectrum above the coalescence point as well as for signals broadened by spin-spin coupling to a fast relaxing nucleus. In the latter case, similarly as in the case of exchange, signals before and after coalescence of the spin multiplet were calculated. All these hneshapes in the absorption mode without phase shift are presented in Fig. 5. One should note that before calculating the power spectrum, the baseline anomalies must be eliminated. Unfortunately, the automatic cubic method for cor-
FIG. 6. Noise of the power spectrum presented in Fig. 2D. This figure was obtained by subtracting the spectrum calculated for the best fit of parameters (including amplitude) from the spectrum shown in Fig. 2D. It should be stressed that this noise does not exhibit a DC component.
EJCHART AND WR6BLEWSKI
474
recting the baseline (8) cannot be used in the case of broadened signals because it can partly remove these signals. It seems that assuming a linear dependence of the baseline (Legrange’s two first polynomials in Ref. (8)) should sufIiciently compensate baseline anomalies without significant distortions of the broadened signals. On the other hand, one need not correct the baseline in power spectra, because the difference between the reference spectrum with noise and the noiseless spectrum, both calculated for the same parameter set, does not have a DC bias, although the noise exhibits variable amplitudes, as seen in Fig. 6. It can be concluded from the comparison of Fig. 6 and Fig. 26 (or 2D) that random turbance in the power spectrum depends on the signal amplitude. In contrast, one can see from Figs. 1 and 6, both being in the same vertical scale, that, on average, noise in the power spectrum is smaller than in the absorption spectrum. This can be also concluded from the RMS value in the minimum for curve PS lower than that for curve A in Figs. 3 and 4. Therefore, we can conclude that the use of power spectra for lineshape analysis, if possible, seems to be advantageous mainly because of the elimination of phase distortions which can induce an unexpected change of fitted parameters as well as lower their accuracy. REFERENCES 1. G. BINSCH, in “Topics in Stereochemistry” Wiley-Interscience, New York, 1968.
(N. L. Allinger and E. L. EIiel, Eds.), Voi. 3, p” 97,
LEHN AND J. P. KINTZINGER, in “Nitrogen NMR” (M. Witanowski and @. A. Webb, Eds.), p. 79, Plenum, London, 1973. Nuclear Magnetic Resonance” (L. M. Jackman and F. A. Cotton, G. BINSCH, in “Dynamic Eds.), Chap. 3, Academic Press, New York, 1975. D. S. STEPHENSON AND G. BINSCH, J. Magn. Reson. 32, 145 (1978). D. SHAW, “Fourier Transform NMR Spectroscopy,” Elsevier, Amsterdam, 1976. T. C. FARRAR AND E. D. BECKER, “Pulse and Fourier Transform NMR. Introduction to Theory and Methods,” Academic Press, New York, 1971. B. L. NEFF, J. L. ACKERMAN, AND J. S. WAUGH, J. Magn. Reson. 25, 335 (1977).
2. J. M. 3. 4. 5. 6.
7. 8. 6.
A. PEARSON,
J. Magn.
Reson.
27, 26.5 (1977).