How to Tickle Spins with a Fourier Transform NMR Spectrometer

It is now half a century ago that Wes Anderson and Ray Freeman, then both at Varian Associates in Palo Alto, California, published two seminal papers about theoretical and experimental aspects of double resonance in high-resolution nuclear magnetic resonance (NMR). This was before the introduction of Fourier transform (FT) NMR, when the signal was measured by continuously sweeping the carrier and receiver frequencies (or the main magnetic field) in the manner of “slow passage”. In double resonance experiments, a second radio-frequency (rf) field is applied to characterize the energy levels of the spin systems. If the amplitude gB2/(2p) =w2/(2p) of the second rf field is equal or greater than the spin coupling constant J, a collapse of the multiplets onto the underlying chemical shifts can be observed, a phenomenon that has come to be known as decoupling. If the rf amplitude gB2/(2p) =w2/(2p) is smaller than the spin coupling constant J but larger than the line width, all transitions that have an energy level in common with the irradiated transition split into doublets. If the irradiation coincides exactly with the frequency of a single transition, and if all couplings are smaller than the chemical shift differences, all connected transitions split into symmetrical doublets with equal intensities, with a splitting equal to the amplitude of the second rf field. This phenomenon has later come to be known as “spin tickling”. 4] The increased complexity, which appears to achieve the opposite effect as decoupling, offers a means to study connectivities of transitions in a spin system. A tickling spectrum helps to discriminate between two possible types of connected transitions. If the energy levels that are not shared differ by Dm = 2 as shown in Figure 1, that is, if they span a double-quantum (DQ) transition, the connected transitions are called progressive (see the red transitions in the energy-level diagram of Figure 1 a). On the other hand, if these energy levels differ by Dm = 0, they span a zero-quantum (ZQ) transition and are called regressive (blue transitions in Figure 1 a). In the latter case, the splitting induced by tickling is usually well resolved, but in the case of progressively connected transitions, the lines are broader and the splitting is often hard to resolve, so that the two types can easily be distinguished. Freeman and Anderson explained the splitting of the lines by a partial admixture of allowed single-quantum (SQ) transitions and forbidden (ZQ or DQ) transitions. Both become equally allowed at exact resonance. The broadening in the progressive case can be attributed to the fact that DQ transitions are more sensitive than ZQ transitions to the inhomogeneity of the static magnetic field B0. To illustrate this effect, a J-coupled two-spin system IS was investigated by numerical simulations using the SIMPSON program. The chemical shifts were set at WI/(2p) = 5 Hz and WS/(2p) = 405 Hz, respectively, with a scalar coupling J = 10 Hz. The lowest-frequency line of the I spin doublet at In the long bygone days of continuous-wave nuclear magnetic resonance (NMR) spectroscopy, a selected transition within a multiplet of a high-resolution spectrum could be irradiated by a highly selective continuous-wave (CW) radio-frequency (rf) field with a very weak amplitude w2/(2p) J. This causes splittings of connected transitions, allowing one to map the connectivities of all transitions within the energy-level diagram of the spin system. Such “tickling” experiments stimulated the invention of two-dimensional spectroscopy, but seem to have been forgotten for nearly 50 years. We show that tickling can readily be achieved in homonuclear systems with Fourier transform spectrometers by applying short pulses in the intervals between the sampling points. Extensions to heteronuclear systems are even more straightforward since they can be carried out using very weak CW rf fields.


Introduction
It is now half a century ago that Wes Anderson and Ray Freeman, then both at Varian Associates in Palo Alto, California, published two seminal papers about theoretical and experimental aspects of double resonance in high-resolution nuclear magnetic resonance (NMR). [1] This was before the introduction of Fourier transform (FT) NMR, [2] when the signal was measured by continuously sweeping the carrier and receiver frequencies (or the main magnetic field) in the manner of "slow passage". In double resonance 1 experiments, a second radio-frequency (rf) field is applied to characterize the energy levels of the spin systems. If the amplitude gB 2 /(2p) = w 2 /(2p) of the second rf field 2 is equal or greater than the spin coupling constant J, a collapse of the multiplets onto the underlying chemical shifts can be observed, a phenomenon that has come to be known as decoupling. [1a] If the rf amplitude gB 2 /(2p) = w 2 /(2p) is smaller than the spin coupling constant J but larger than the line width, [3] all transitions that have an energy level in common with the irradiated transition split into doublets. If the irradiation coincides exactly with the frequency of a single transition, and if all couplings are smaller than the chemical shift differences, all connected transitions split into symmetrical doublets with equal intensities, with a splitting equal to the amplitude of the second rf field. This phenomenon has later come to be known as "spin tickling". [1b, 4] The increased complexity, which appears to achieve the opposite effect as decoupling, offers a means to study connectivities of transitions in a spin system.
A tickling spectrum helps to discriminate between two possible types of connected transitions. [5] If the energy levels that are not shared differ by Dm = AE 2 as shown in Figure 1, that is, if they span a double-quantum (DQ) transition, the connected transitions are called progressive (see the red transitions in the energy-level diagram of Figure 1 a). On the other hand, if these energy levels differ by Dm = 0, they span a zero-quantum (ZQ) transition and are called regressive (blue transitions in Figure 1 a). In the latter case, the splitting induced by tickling is usually well resolved, but in the case of progressively connected transitions, the lines are broader and the splitting is often hard to resolve, so that the two types can easily be distinguished. Freeman and Anderson explained the splitting of the lines by a partial admixture of allowed single-quantum (SQ) transitions and forbidden (ZQ or DQ) transitions. Both become equally allowed at exact resonance. [6] The broadening in the progressive case can be attributed to the fact that DQ transitions are more sensitive than ZQ transitions to the inhomogeneity of the static magnetic field B 0 .
To illustrate this effect, a J-coupled two-spin system IS was investigated by numerical simulations using the SIMPSON program. [7] The chemical shifts were set at W I /(2p) = 5 Hz and W S /(2p) = 405 Hz, respectively, with a scalar coupling J = 10 Hz. The lowest-frequency line of the I spin doublet at In the long bygone days of continuous-wave nuclear magnetic resonance (NMR) spectroscopy, a selected transition within a multiplet of a high-resolution spectrum could be irradiated by a highly selective continuous-wave (CW) radio-frequency (rf) field with a very weak amplitude w 2 /(2p) J. This causes splittings of connected transitions, allowing one to map the connectivities of all transitions within the energy-level diagram of the spin system. Such "tickling" experiments stimulated the in-vention of two-dimensional spectroscopy, but seem to have been forgotten for nearly 50 years. We show that tickling can readily be achieved in homonuclear systems with Fourier transform spectrometers by applying short pulses in the intervals between the sampling points. Extensions to heteronuclear systems are even more straightforward since they can be carried out using very weak CW rf fields.
[a] T. F. Segawa, Dr. D. Carnevale W I /(2p)ÀJ/2 = 0 Hz was irradiated with an average rf field strength hw 2 i/(2p) = 1 Hz. To simulate the inhomogeneity of the B 0 field, both rf carrier and receiver frequencies were shifted in 21 steps of 0.1 Hz from À1 to + 1 Hz around the resonance condition. Relaxation was not taken into account. Figure 2 shows the spectral lines of the off-resonance spin S. A few selected spectra with offsets of À1.0, À0.5, 0.0, 0.5, and 1.0 Hz are shown in Figures 2 a-e. The intensities are always symmetrical with respect to the chemical shift of spin S. The 1 Hz splittings in the on-resonance spectrum in Figure 1 c are equal to the amplitude of the tickling field. The sum of all 21 spectra is plotted in Figure 2 f. This nicely reproduces the experimental pattern of the broad progressively connected peaks and narrow regressively connected peaks. The differential broadening originates from the distinct slopes of the individual peaks across Figures 2 a-e. Spin tickling is a convenient method to assign spectral transitions to the energy levels, and thus to reconstruct the entire energy-level diagram. The approach could be applied to other forms of spectroscopy, not just to magnetic resonance. A similar splitting of lines in vibrational spectroscopy was explained by Fermi in 1931 [8] and is known today as Fermi resonance in infrared and Raman spectroscopy. [6a,b] In modern NMR, continuous-wave (CW) experiments have been almost fully replaced by pulsed experiments. To study connectivities, two-dimensional correlation spectroscopy (COSY) [9] has become popular. If the second pulse has a small flip angle (e.g. in COSY-45) one can readily determine the relative signs of J couplings from inspection of the multiplet structures. [9,10] The combination of FT NMR and tickling was discussed (in German) by F. Günther in 1971 3 , [11] but this was largely overlooked. In his calculations, Günther showed that tickling not only broadens the inhomogeneous line widths in a manner that depends on the connectivity, but the natural homogeneous line widths are also broadened in a differential manner. Such relaxation effects are reminiscent of the recently discovered long-lived coherences. [12] Wokaun and Ernst described a two-dimensional (2D) experiment where tickling is applied to multiple-quantum coherences in the evolution interval t 1 while the signal is observed in Fourier mode in the detection interval t 2 . [13] Figure 1. a) Normal proton spectrum at 500 MHz of the AX-system 2,3-dibromothiophene with the corresponding energy-level diagram, where the transitions are numbered from 1 to 4. The red transitions 1 and 4 are connected progressively, the blue transitions 1 and 3 regressively. b) Tickling spectrum obtained by irradiating resonance 1 with an amplitude of hw 2 i/(2p) = 1 Hz (see text). Peak 2 is not affected, since this transition is parallel to the irradiated one and is not connected. Lines 3 and 4 of the coupling partner are both split by tickling. The regressively connected resonance 3 splits into two narrow lines, while the progressively connected resonance 4 splits into a broadened doublet. c) Similar experiment with the same rf amplitude hw 2 i/(2p) = 1 Hz as in (b), but with the carrier frequency shifted to the middle of the left-hand doublet, half-way between transitions 1 and 2. This brings about incipient decoupling, leading to two additional lines coinciding with the chemical shifts. However, the rf amplitude is too weak to achieve complete decoupling. As usual, all spectra were processed by Fourier transform. No line-broadening was used. (Bruker Avance I with Topspin 2.1).

Results and Discussion
Today, tickling does not play a significant role in NMR spectroscopy, but there appear to be no obstacles to its renaissance. Recently, we developed a method dubbed window-acquired spin tailoring experiment (WASTE) [14] that allows one to eliminate the effects of a manifold of homonuclear scalar couplings by alternating the sampling of the free induction decay with brief rf pulses. This is closely related to the time-shared method suggested by Jesson et al., who also mention spin tickling. [15] If rf pulses and signal acquisition alternate, the effect of the second rf field can be described by an average Hamiltonian. [16] By using a similar time-shared sequence, but with a much weaker rf field, tickling experiments can be readily carried out. Thus, double resonance tickling and decoupling experiments can both be transferred from CW to FT NMR. [3,17] For the sake of illustration, consider the pair of weakly coupled protons in 2,3-dibromothiophene dissolved in [D 6 ]-dimethyl sulfoxide (DMSO). This AX-system has a homonuclear scalar coupling of J AX = 5.8 Hz and a chemical shift difference (W A ÀW X )/(2p) = 305 Hz at 500 MHz (see Figure 1 a). Figure 1 b shows the tickling spectrum obtained when an rf field with an average amplitude of hw 2 i/(2p) = 1 Hz is set on the left-most resonance 1, as evidenced by the interference pattern. The neighboring line 2 remains unperturbed. This confirms that these two transitions do not share any common energy level, as expected for parallel transitions. The left-hand peak 3 of the other doublet splits into two narrow lines, while the righthand peak 4 is broadened. The transition 3 is regressively connected, while 4 is progressively connected to transition 1. All of this is consistent with the energy-level diagram in Figure 1. If we irradiate half way between transitions 1 and 2, as we do for decoupling, but with the same very weak rf amplitude, all four transitions remain visible and no tickling effects are observed. Two new lines appear at the chemical shifts of the two nuclei, indicating the onset of decoupling, but the rf amplitude is too weak to obtain only two singlets, as shown in Figure 1 c.
In a three-spin system, an unambiguous assignment of transitions to the energy-level diagram by simple inspection of the multiplets is no longer possible. We considered 2,3-dibromopropionic acid, the same molecule as Freeman and Anderson [1b] chose to tickle. This molecule gave rise to a strongly coupled ABC-system in the 60 MHz spectrometer used for their original paper, but now appears as an ABX-system at 500 MHz. Figure 3 a shows a normal one-dimensional (1D) spectrum. For simplicity, we kept the same numbering of the spectral lines. In a three-spin system, the energy diagram can be represented by a cube where the 12 transitions correspond to the 12 edges, with four parallel transitions for each spin, which correspond to the doublet-of-doublets of each spin. Figure 3 b shows a tickling spectrum obtained by irradiating the left-most resonance C4 with an average amplitude of hw 2 i/(2p) = 1.5 Hz, showing the usual beat pattern. It can be seen that none of the three other C transitions are affected, since they are all parallel to C4. There are two pairs of connected transitions. A1 and B1 are both split into broad doublets since they are progressively connected, while A2 and one of the overlapping lines B2 or B3 are split into narrow doublets because of their regressive connectivities. The remaining lines A3, A4, B2 or B3, and B4 are not affected since they are not connected to C4. A single experiment is not sufficient to determine all connectivities. For a full analysis, we refer to the original work of Freeman and Anderson. [1b] In analogy to our homonuclear decoupling experiments, [14] the tickling field was generated by a sequence of n short pulses, each with a typical duration of t p = 1 ms, inserted in the n dwell times. If the latter have a duration of Dt = 50 ms as required for a spectral width of 20 kHz, this corresponds to a duty cycle of 2 % (see Figure 4). The peak rf amplitude of each tickling pulse was w 2 /(2p) = 50 Hz, resulting in an average amplitude of hw 2 i/(2p) = 1 Hz that satisfies the tickling condition hw 2 i/(2p) J. The A1 line is hardly visible due to broadening, B1 is also broadened, while A2 is split, and one of the overlapping lines B2 or B3 is also split. Lines A3, A4, B4, and one of the overlapping lines B2 or B3 are unaffected and must therefore correspond to transitions that are not connected to C4.
Since homonuclear tickling and decoupling can be achieved with the same pulse sequence, there is no need to introduce yet another acronym, and we prefer to stick to WASTE [14] and replace the original "T" for tailoring by "T" for tickling. We are not aware of any experimental demonstration of homonuclear spin tickling since the introduction of Fourier spectroscopy.
In heteronuclear spin systems, tickling can be readily carried out on Fourier transform spectrometers, since one can apply a weak CW field to one nucleus while observing the other in Fourier mode. Thus, one can observe a splitting of 13 C lines in a phosphine when some connected 31 P transitions are irradiated. [18] The heteronuclear version of tickling is actually quite easy to implement, since it is sufficient to apply a weak CW field to one of the transitions of the heteronucleus during acquisition (but not before the rf pulse to avoid inducing nuclear Overhauser effects). For illustration, we tickled the heteronuclear 1 H-13 C two-spin system in 13 C-enriched formate in D 2 O. Heteronuclear decoupling with a weak CW irradiation with w 2 / (2p) = 1 Hz was applied with the rf carrier set on the left line of the proton doublet while the 13 C signal was acquired.  Figure 5 b shows a 13 C{ 1 H}-tickling spectrum. Both carbon lines are split by 1 Hz. Note the differential broadening, with narrow lines to the left and broad lines to the right. Figure 5 c and d show similar experiments with stronger tickling fields with w 2 /(2p) = 5 and 33 Hz. In such heteronuclear systems, higher rf amplitudes can be employed than in homonuclear proton systems, since the tickling condition w 2 /(2p) J = 195 Hz is less restrictive. Nevertheless, the differential line broadening is most visible when the tickling fields are weak, which facilitates the identification of different types of connectivities, as can be seen in Figure 5 b. The variation of the tickling-field amplitude in Figure 5 b-d demonstrates that tickling can also be used to calibrate weak rf fields.

Conclusions
Can Fourier tickling be useful in modern NMR? Since it requires only one-dimensional experiments, tickling is a quick approach to characterize the progressive or regressive connectivity of particular resonances, probably faster than selective or soft COSY. [19] Another application might be the elimination of degeneracies in NMR spectra. In 1976, Courtieu et al. separated singlet and triplet lines in an AX 2 -system by combining tickling with a partially oriented medium. [20] Besides, we like dusting off the spider webs that have covered spin tickling over the last half century to expose this brilliant idea to the bright light of Fourier NMR. . Pulse scheme for window-acquired spin tickling (or tailoring) experiments (WASTE). The tall rectangle represents the initial 908 pulse, whereas the small rectangle represents a tickling pulse of duration t p . The black dot represents the acquisition of a data point, obtained in reality by averaging over the time interval during which the receiver is activated. The free induction signal is built up as usual by acquiring n data points through an nfold repetition of the loop, where each unit has the length of the dwell time Dt. One can use the same carrier frequencies for both the initial 908 pulse and the tickling pulse pulses. Alternatively, one can shift the effective carrier frequency of the tickling pulses with respect to the carrier of the initial 908 pulse by a time-proportional phase shift of f = nDf. [14] Figure 5. a) Normal 13 C spectrum at 11.7 T (500 MHz for protons) of 13 C-enriched formate showing the 13 C doublet J CH = 195 Hz. b) Tickling spectrum obtained with CW irradiation on the left proton peak with w 2 /(2p) = 1 Hz during 13 C signal acquisition. Both 13 C peaks are split by 1 Hz; the right signal being clearly broadened. c) and d) Same experiments as in (b) but with tickling amplitudes of w 2 /(2p) = 5 and 33 Hz.