Correlation spectroscopy
المؤلف:
Peter Atkins، Julio de Paula
المصدر:
ATKINS PHYSICAL CHEMISTRY
الجزء والصفحة:
ص544-547
2025-12-14
65
Correlation spectroscopy
Much modern NMR work makes use of techniques such as correlation spectroscopy (COSY) in which a clever choice of pulses and Fourier transformation techniques makes it possible to determine all spin–spin couplings in a molecule. The basic COSY experiment uses the simplest of all two-dimensional pulse sequences, consisting of two consecutive 90° pulses (Fig. 15.46). To see how we can obtain a two-dimensional spectrum from a COSY experiment, we consider a trivial but illustrative example: the spectrum of a compound containing one proton, such as trichloromethane (chloroform, CHCl3). Figure 15.47 shows the effect of the pulse sequence on the magnetization of the sample, which is aligned initially along the z-axis with a magnitude M0. A 90° pulse applied in the x-direction (in the stationary frame) tilts the magnetization vector toward the y-axis. Then, during the evolution period, the magnetization vector rotates in the xy-plane with a frequency ν. At a time t1 the vector will have swept through an angle 2πνt1 and the magnitude of the magnetization will have decayed by spin–spin relaxation to M = M0e−t1/T2. By trigonometry, the magnitudes of the components of the magnetization vector are:
Mx=Msin 2πνt1 My=Mcos 2πνt1 Mz=0
Application of the second 90° pulse parallel to the x-axis tilts the magnetization again and the resulting vector has components with magnitudes (once again, in the stationary frame)
Mx=Msin 2πνt1 My=0 Mz=Mcos 2πνt1
The FID is detected over a period t2 and Fourier transformation yields a signal over a frequency range ν2 with a peak at ν, the resonance frequency of the proton. The signal intensity is related to Mx, the magnitude of the magnetization that is rotating around the xy-plane at the time of application of the detection pulse, so it follows that the signal strength varies sinusoidally with the duration of the evolution period. That is, if we were to acquire a series of spectra at different evolution times t1, then we would obtain data as shown in Fig. 15.48a.
Fig. 15.46 The pulse sequence used in correlation spectroscopy (COSY). The preparation period is much longer than either T1 or T2, so the spins have time to relax before the next cycle of pulses begins. Acquisitions of free-induction decays are taken during t2 for a set of different evolution times t1. Fourier transformation on both variables t1 and t2 results in a two-dimensional spectrum, such as that shown in Fig 15.52.
Fig. 15.47 (a) The effect of the pulse sequence shown in Fig. 15.46 on the magnetization M0 of a sample of a compound with only one proton. (b) A 90° pulse applied in the x-direction tilts the magnetization vector toward the y-axis. (c) After a time t1 has elapsed, the vector will have swept through an angle 2πνt1 and the magnitude of the magnetization will have decayed to M. The magnitudes of the components of M are Mx = Msin 2πνt1, My =Mcos 2πνt1, and Mz = 0. (d) Application of the second 90° pulse parallel to the x-axis tilts the magnetization again and the resulting vector has components with magnitude Mx=Msin 2πνt1, My=0, and Mz=Mcos 2πνt1. The FID is detected at this stage of the experiment.
Fig. 15.48 (a) Spectra acquired for different evolution times t1 between two 90° pulses. (b) A plot of the maximum intensity of each absorption line against t1. Fourier transformation of this plot leads to a spectrum centred at ν, the resonance frequency of the protons in the sample.
Fig. 15.49 (a) The two-dimensional NMR spectrum of the sample discussed in Figs. 15.47 and 15.48. See the text for an explanation of how the spectrum is obtained from a series of Fourier transformations of the data. (b) The contour plot of the spectrum in (a).
A plot of the maximum intensity of each absorption band in Fig. 15.48a against t1 has the form shown in Fig. 15.48b. The plot resembles an FID curve with the oscillating component having a frequency ν, so Fourier transformation yields a signal over a frequency range ν1 with a peak at ν. If we continue the process by first plotting signal intensity against t1 for several frequencies along the ν2 axis and then carrying out Fourier transformations, we generate a family of curves that can be pooled together into a three-dimensional plot of I(ν1,ν2), the signal intensity as a function of the frequencies ν1 and ν2 (Fig. 15.49a). This plot is referred to as a two-dimensional NMR spectrum because Fourier transformations were performed in two variables. The most common representation of the data is as a contour plot, such as the one shown in Fig. 15.49b. The experiment described above is not necessary for as simple a system as chloroform because the information contained in the two-dimensional spectrum could have been obtained much more quickly through the conventional, one-dimensional approach. However, when the one-dimensional spectrum is complex, the COSY experiment shows which spins are related by spin–spin coupling. To justify this statement, we now examine a spin-coupled AX system.
From our discussion so far, we know that the key to the COSY technique is the effect of the second 90° pulse. In this more complex example we consider its role for the four energy levels of an AX system (as shown in Fig. 15.12). At thermal equilibrium, the population of the αAαX level is the greatest, and that of the βAβX level is the least; the other two levels have the same energy and an intermediate population. After the first 90° pulse, the spins are no longer at thermal equilibrium. If a second 90° pulse is applied at a time t1 that is short compared to the spin–lattice relaxation time T1, the extra input of energy causes further changes in the populations of the four states. The changes in populations of the four states of the AX system will depend on how far the individual magnetizations have precessed during the evolution period. It is difficult to visualize these changes because the A spins are affecting the X spins and vice-versa. For simplicity, we imagine that the second pulse induces X and A transitions sequentially. Depending on the evolution time t1, the 90° pulse may leave the population differences across each of the two X transitions unchanged, inverted, or somewhere in between. Consider the extreme case in which one population difference is inverted and the other unchanged (Fig. 15.50). Excitation of the A transitions will now gener ate an FID in which one of the two A transitions has increased in intensity (because the population difference is now greater), and the other has decreased (because the population difference is now smaller). The overall effect is that precession of the X spins during the evolution period determines the amplitudes of the signals from the A spins obtained during the detection period. As the evolution time t1 is increased, the intensities of the signals from A spins oscillate with frequencies determined by the frequencies of the two X transitions. Of course, it is just as easy to turn our scenario around and to conclude that the intensities of signals from X spins oscillate with frequencies determined by the frequencies of the A transitions. This transfer of information between spins is at the heart of two-dimensional NMR spectroscopy: it leads to the correlation between different signals in a spectrum. In this case, information transfer tells us that there is spin–spin coupling between A and X. So, just as before, if we conduct a series of experiments in which t1 is incremented, Fourier transformation of the FIDs on t2 yields a set of spectra I(t1,F2) in which the signal amplitudes oscillate as a function of t1. A second Fourier transformation, now ont1, converts these oscillations into a two-dimensional spectrum I(F1,F2). The signals are spread out in F1 according to their precession frequencies during the detection period. Thus, if we apply the COSY pulse sequence (Fig. 15.46) to the AX spin system, the result is a two-dimensional spectrum that contains four groups of signals in F1 and F2 centred on the two chemical shifts (Fig. 15.51). Each group consists of a block of four signals separated by J. The diagonal peaks are signals centred on (δA,δA) and (δX,δX) and lie along the diagonal F1 = F2. That is, the spectrum along the diagonal is equivalent to the one-dimensional spectrum obtained with the conventional NMR technique (Fig. 15.13). The cross-peaks (or off-diagonal peaks) are signals centred on (δA,δX) and (δX,δA) and owe their existence to the coupling between A and X. Although information from two-dimensional NMR spectroscopy is trivial in an AX system, it can be of enormous help in the interpretation of more complex spectra, leading to a map of the couplings between spins and to the determination of the bonding network in complex molecules. Indeed, the spectrum of a synthetic or biological polymer that would be impossible to interpret in one-dimensional NMR but can often be interpreted reasonably rapidly by two-dimensional NMR. Below we illustrate the procedure by assigning the resonances in the COSY spectrum of an amino acid.
Fig. 15.50 An example of the change in the population of energy levels of an AX spin system that results from the second 90° pulse of a COSY experiment. Each square represents the same large number of spins. In this example, we imagine that the pulse affects the X spins first, and then the A spins. Excitation of the X spins inverts the populations of the βAβX and βAαX levels and does not affect the populations of the αAαXandαAβXlevels. As a result, excitation of the A spins by the pulse generates an FID in which one of the two A transitions has increased in intensity and the other has decreased. That is, magnetization has been transferred from the X spins to the A spins. Similar schemes can be written to show that magnetization can be transferred from the A spins to the X spins.
Fig. 15.51 A representation of the two dimensional NMR spectrum obtained by application of the COSY pulse sequence to an AX spin system.
Fig. 15.52 Proton COSY spectrum of isoleucine. (Adapted from K.E. van Holde, W.C. Johnson, and P.S. Ho, Principles of physical biochemistry, p. 508, Prentice Hall, Upper Saddle River (1998).)
Our simplified description of the COSY experiment does not reveal some import ant details. For example, the second 90° pulse actually mixes the spin state transitions caused by the first 90° pulse (hence the term ‘mixing period’). Each of the four transitions (two for A and two for X) generated by the first pulse can be converted into any of the other three, or into formally forbidden multiple quantum transitions, which have |∆m|>1. The latter transitions cannot generate any signal in the receiver coil of the spectrometer, but their existence can be demonstrated by applying a third pulse to mix them back into the four observable single quantum transitions. Many modern NMR experiments exploit multiple quantum transitions to filter out unwanted signals and to simplify spectra for interpretation.
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