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to streaks extending up and down, and to the left and right from each diagonal peak or crosspeak. We will see that the DQF-COSY experiment avoids this problem since the phase of the diagonal is the same as that of the crosspeaks.

The magnitude mode lineshape is just:

Far from the peak maximum (v2 >> 1) we have Imagn ~ 1/v, just like the dispersive lineshape.

9.5.3 COSY-35: Simplifying Crosspeak Fine Structure

What about the COSY-35 experiment (Fig. 9.34)? We can now show with product operators why it simplifies the crosspeak fine structure. Consider again the AMX spin system of a peptide residue in D2O: ND-CHa-CHpHp'-Y. For the crosspeaks shown in Figure 9.33 (left) let's focus on the lower one: F1 = Hp/F2 = Ha (Hp — Ha coherence transfer). We start the t1 period with -Ip and write down the terms that result from J coupling, keeping in mind that there are two J couplings affecting Hp: Jap and Jpp/:

-IP — -IP cc' +2IpI^ sc' +2Ipif'cs' +4IpIazIp'ss' (Jevolution) A B CD

where c and s are cosine and sine of n Jap t1 and c' and s' are cosine and sine of nJpp' t1. As before, we advance the phase by 90°, multiply by 2Iz and change the cosine term to sine for each coupling that undergoes evolution from in-phase to antiphase. Only terms B and D above represent Hp coherence that is antiphase with respect to Ha, so we can ignore the A and C terms because they cannot give us coherence transfer from Hp to Ha. Now consider the effect on terms B and D of chemical-shift evolution of Hp coherence, which multiplies the starting terms by cos(^p t1) and then adds new terms with the phase of Hp coherence advanced by 90° and multiplying by sin(^pt1):

-t1 — 2IpIas c' c'' +4IpIaIp's s' c'' +2IpIas c's'' -4IpI^Ip's s's'' (shift evolution)

where s'' or c'' refer to sin(^p t1) or cos(^p t1), respectively. Terms E and G above come from the singly antiphase term B and terms F and H above come from the double antiphase term D. The final pulse is on xX, so the E and H terms (Hp coherence on X) cannot give us coherence transfer. We only need to consider the F and G terms:

4IpI^If s s' c'' +2IpI£s c's'' -©pulse on X — FG

Now we do the coherence transfer with our final 90° pulse on X, but we allow the pulse to be any rotation angle © (90° for COSY, 35° for COSY-35). This would generate two terms for each operator, for a total of 12 terms (!), but we need to worry only about those terms that represent coherence transfer from Hp to Ha. For this to happen, we need to have

If move from y' to the z axis and Ia move from z to the X-y' plane (antiphase to antiphase INEPT transfer). The "©" pulse on X converts to cos © - sin ©] and to [Iy cos © + Iz sin ©]. Just like with evolution, the cosine term goes with the unchanged operator and the sine term goes with the © = 90° result. Each term in the product is affected by the pulse, but we need only consider the results that have the Ia operator in the x'-y' plane and all others on z. Any terms with more than one operator in the x'-y' plane represent unobservable ZQC/DQC. Term G gives the standard INEPT coherence transfer result:

2IfyIaz 2(Ifcos © + If sin ©Xl^cos © - I^sin ©) = —2I^lfsin2© + 3 other terms

Only the sin2 © term represents coherence transfer; the three others can be ignored. The F term can give coherence transfer to Ha also, as long as Ia is the only operator in the x'-y' plane:

4IfI^If '©x 4(If sin ©)(—I^sin ©)(If cos ©) + 8 other terms = —4I^If If'sin2© cos ©

None of the other eight terms represents Hp ^ Ha coherence transfer. We also ignore the s s'c'' and s c's'' multipliers since they carry the F1 chemical shift and /-coupling information—we are only interested in the F2 slice. What do these two terms look like in F2? Remember that the Ha resonance (I^) is a doublet of doublets

(Fig. 9.40 A). 2I^lf is antiphase with respect to Hp only, so we get the pattern 1, -1,1, -1

(if Jap < Jap) (Fig. 9.40 B). 4I^lf Ip is antiphase with respect to both Hp and Hp, so we get an antiphase doublet on the left side (antiphase with respect to Jap) and another antiphase doublet on the right side that is opposite in phase (antiphase with respect to <): the pattern is 1, -1, -1,1 (Fig. 9.40 C). Now if we consider that the first pattern (2I^lf) is multiplied by sin2 © (0.329 for © = 35°) and the second pattern (4I^IpIf) is multiplied by sin2 © cos © (0.269 for © = 35°), we see that when we add them together there are two kinds of lines (Fig. 9.40 D): the first and second where the two add together (0.329 + 0.269 = 0.598) and the third and fourth where they are subtracted (0.329 - 0.269 = 0.060): an absolute value of sin2 © + sin2 © cos © for lines 1 and 2 and sin2 © - sin2© cos© for lines 3 and 4. If we look at the intensity ratio between these two types of lines we get:

Ratio = (sin2© - sin2© cos ©)/(sin2© + sin2© cos ©) = (1 - cos ©)/(1 + cos ©)

For © = 90°, this ratio is 1 and we see all four lines equally in each row (16 peaks in all in the crosspeak, Fig. 9.33, lower left). For © = 35°, the ratio is 0.1 (1 to 10): lines 3 and 4 are only 10% of the intensity of lines 1 and 2 (eight intense peaks in all in the crosspeak, Fig. 9.33 lower right). Some people use a 45° pulse ("COSY-45"), for a ratio of 0.17 (1 to 5.8). As we make © smaller, we pay a price in overall intensity since both types of line are multiplied by sin2 ©; although the ratio gets better as the overall sensitivity goes down. This analysis illustrates the power of product operators as well as the need to look ahead and anticipate which terms will be important to avoid an explosion of complexity.

9.6 2D TOCSY (TOTAL CORRELATION SPECTROSCOPY)

In Chapter 8 we saw how the TOCSY spin lock, a continuous string of medium-power pulses with carefully designed widths and phases, can transfer coherence in multiple jumps along a chain of /-coupled protons: a "spin system". In the selective 1D TOCSY experiment, the DPFGSE (a combination of shaped pulses and gradients) is used to selectively excite one resonance in the spectrum with the equivalent of a 90° pulse, and this coherence is then transferred to other protons in the spin system with the TOCSY mixing sequence (pulsed spin lock). To make this sequence (Fig. 8.42) into a 2D TOCSY (Fig. 9.41), we simply replace the selective 90° pulse (the DPFGSE) with a non-selective 90° pulse and insert a ti delay (evolution period) between this preparation pulse and the mixing sequence (Fig. 9.42). We already know how the 90° - t1 sequence produces four terms:

If we apply the TOCSY spin lock at this point on the xX axis, we will destroy the first and fourth terms (B1 field inhomogeneity) and "lock" the second and third terms. Because the TOCSY mixing sequence transfers coherence from in-phase to in-phase, only the third h)7,,H ' H" A,H' H^^t

Figure 9.41

term, Iaxsc7, leads to a crosspeak at F1 = Qa/F2 =

IXsc7 - TOCSY spin lock ^ lbs c7

We also saw that the s c7 encoding in t1 (sin(^a t1) cos(nJt1)) represents an in-phase doublet at the Ha chemical shift in F1. So the crosspeak is in-phase in both dimensions.

We saw in Chapter 8 that a continuous-wave spin lock is not effective for TOCSY transfer, giving efficient transfer only when the Hartmann-Hahn match is satisfied: Qa = ±^b. This corresponds to the diagonal and the "antidiagonal"—a line extending from the lower right corner of the 2D matrix to the upper left corner. In fact, TOCSY transfer crosspeaks do appear as artifacts in 2D ROESY spectra along the antidiagonal. To get efficient TOCSY transfer we use a specific sequence of medium-power pulses such as MLEV-17 or DIPSI-2. The holy grail of TOCSY mixing sequences is the "ideal isotropic mixing" sequence that completely eliminates the chemical shifts and leaves only the /-coupling interactions, just as if the Bo field were reduced to zero. In this ideal case if we start with Ha magnetization on the X axis, we get conversion to Hb magnetization on the X axis as follows:

IX ^ 0.5IX(1 + cos) + 0.5IX(1 - cos) + 0.5(2Iilb - 2l5IZ) sin where cos = cos(2n / rm), sin = sin(2n / rm), and rm is the mixing time. The derivation of this result will be shown in Chapter 10. The first two terms represent the transfer of in-phase coherence from Ha to Hb: after a time rm = 1/(4/) the cosine term equals zero and we have 0.5 Iax and 0.5 I£, i.e., 50% conversion. But we also have a combination of antiphase Ha coherence and antiphase Hb coherence (sin = 1)! This term decreases again and becomes zero when rm = 1/(2/). At this time we have complete conversion of IX to I£ since the cosine term equals -1. We saw this oscillation in Figure 8.43 but the antiphase terms were ignored at that time. These terms lead to distortion of peak shape unless the

sine term is zero (rm = 0, 1/(27), 1/J, etc.). This distortion is shown in Figure 9.43 (left) for a simulation of the Ha — Hb spin system with Jab = 7.5 Hz. You might think that the antiphase terms would disappear since they are perpendicular to the spin-lock axis and B1 field inhomogeneity should cause them to "fan out" over time. But these terms are actually immune to pulses on the X axis:

2iy ib — 21b ia — 90° ^ 2ia(—ib)—2ij(—iy)=2iy ij—2iy ia 2iy iz — 2iy iz —180° ^ 2(—iy)(—iz) — 2(—iy)(—1£) = 2iy iz — 2iy iz

M any tricks h ave been applied (trim pulses, z filters, gr adients, etc.) to remove these antiphase terms, leaving only the pure phase i\ and iJ terms (Fig. 9.43, right). For example, a "z filter" is a 90°y — A — 90° sequence that puts the desired magnetization on the z axis and then allows a bit of evolution to occur for the undesired terms:

—90—y ^ 0.5i^(1 + cos) + 0.5ib(1 — cos) + 0.5(—2iyiJ + 2iyiax) sin

The last term is now ZQC on the / axis, which undergoes chemical-shift evolution during the delay A at a rate of — Qb. The second 90° pulse puts the desired terms back on X and returns the undesired term to antiphase SQC on /. If we repeat the acquisition with different values of the delay A the antiphase terms will tend to cancel out due to different amounts of evolution during A. A variable delay for evolution of undesired ZQC terms can also be used in NOE mixing to remove ZQ artifacts. The strategy of "storing" desired terms on the z axis while taking care of other components of magnetization is also a common strategy we will encounter again later. Still, distortion of peak shape is commonly encountered in both 1D and 2D TOCSY spectra. In 2D TOCSY this appears as negative intensity in the center of a crosspeak or negative "ditches" on the sides of a crosspeak.

The MLEV-17 mixing sequence falls far short of achieving the "holy grail" of isotropic mixing, and the more complex DIPSI-2 sequence (Chapter 8) is superior in its tolerance of large resonance offsets (vo — vr) while avoiding high power and sample heating. But most people still use MLEV-17 out of blind tradition. For large biological molecules, there is a "clean" or "relaxation compensated" version of the DIPSI sequence (DIPSI-2rc) in which there are short delays separating all of the pulses. The magnetization is on the z axis for the delays, and this allows an NOE to develop that is opposite in sign to the ROE that develops in the spin lock along with TOCSY transfer. This positive NOE (for large molecules) cancels the negative ROE and leaves pure coherence transfer (TOCSY) mixing.

We saw a 2D TOCSY spectrum in Figure 9.22 and compared it to a COSY spectrum: in the TOCSY spectrum, we have more peaks because starting from any proton in the spin system we can see correlations to all other members of the spin system, not just to the protons connected by a single J coupling. We saw the same thing in a real example by comparing the COSY spectrum of 3-heptanone (Fig. 9.23) with the TOCSY spectrum of the same sample (Fig. 9.24).

### 9.6.1 Examples of 2D TOCSY

The 600 MHz 2D TOCSY spectrum of cholesterol (Chapter 8, Fig. 8.35) is shown in Figure 9.44. Note that all peaks (diagonal and crosspeaks) are positive (black) and in-phase.

The t\ noise streaks can be seen extending downward from the two singlet methyl peaks on the diagonal (H-18 and H-19) and to a lesser extent from the methyl doublet peaks between them. An F2 slice at the H6 resonance (5.35 ppm) on the diagonal shows efficient TOCSY transfer to H7ax and H7eq and then on to H8, and weak transfer to H4ax and H4eq (long-range coupling) and on to H3. Another F2 slice at the H3 resonance (3.5 ppm) on the diagonal shows transfer to H4ax, H4eq and then on weakly to H6 and H7eq, as well as strong crosspeaks to H2ax, H2eq, H1ax, and H1eq. These F2 slices are almost identical to the two separate selective 1D TOCSY experiments shown in Figure 8.45. The advantage of 2D TOCSY is that we only have to do one experiment to get all possible TOCSY correlations, and we do not rely on selecting resolved peaks in the 1D spectrum.

Figure 9.45 shows the amide NH region of the TOCSY spectrum of the glycopeptide Tyr-Thr-Gly-Phe-Leu-Ser(O-Lactose) in 90% H2O/10% D2O. In F2, we see the amide proton (Hn) resonances in the range of 8-8.5 ppm, and in F1 we see the entire region from the water resonance upfield. We can see the entire spin system of each amino acid residue stretching upwards from its HN chemical-shift position on the horizontal (F2) ppm scale. Coherence was transferred to the HN proton by TOCSY mixing from the Ha proton, the Hp protons, the HY protons, and so on, and each of these was labeled during t1 with its proton chemical shift. Just by looking at the patterns of chemical shifts in each vertical line, we can identify the amino acid or at least narrow it down to a group of amino acids. The farthest left spin system (HN = 8.42 ppm) is a threonine (side chain CHOH-CHy3), since the Ha and Hp shifts are close together and the y-methyl is far upfield (~1.0 ppm). The next residue (Hn = 8.33 ppm) has only one crosspeak, so it must be a glycine (no side chain) with only the Ha peak. Note that these two HN protons exchange more rapidly with water, leading to an exchange crosspeak at the H2O resonance (F1 = 4.6 ppm). Moving to the right, the next system (HN = 8.25 ppm) is a leucine (side chain CHp2-CHY(CH53)2). The HY signal is overlapped with the Hp, and we see two different 5-methyl crosspeaks due to the chiral environment. Just upfield of this system is a serine (side chain CHp2-OH), with both Hp and Hp close to Ha in chemical shift. So far every one of these patterns of crosspeaks is unique, leading to the identification of a single amino acid among the 20 naturally occurring possibilities. The farthest upfield HN (8.01 ppm) is a classic AMX ("three-spin") system: Ha plus two Hp signals in the 2.5-3.5 ppm region. There are many possible amino acids that give this pattern (side chain CHpHp-Y): all of the aromatic amino acids (Phe, His, Trp, Tyr) plus Cys, Asp, and Asn. In this case, however, there are only two residues left: Phe and Tyr. Because the tyrosine is at the unprotected N terminus, there is no amide HN. The H3N+ protons at the amino terminus are exchanging so rapidly with water that they are never observed in NMR. That leaves phenylalanine (side chain CHp2-C6H5) for the most upfield Hn. Note that TOCSY mixing does not penetrate the aromatic ring because there is no J coupling between the Hp protons and the aromatic ring protons: these are two separate spin systems.

### 9.7 DATA SAMPLING IN T1 AND THE 2D SPECTRAL WINDOW

The hardware and data-processing details of 1D NMR data were discussed in Chapter 3: data sampling in the ADC, quadrature detection, the spectral window, weighting (window) functions, and phase correction. We will have to revisit each of these topics in the second (t1, F1) dimension and some of them will take on added significance.

The evolution delay (t1) is usually started at zero for the first FID and increased by the same amount, the t1 increment At1, for each successive FID. We are trying to describe the evolution of the nucleus A magnetization as it precesses during t1, so the same digital sampling limitations (Nyquist theorem) apply as they do in direct (analog-to-digital converter) sampling of the t2 FID. The rule is that we need to have a minimum of two samples per cycle to define a frequency: we can think of it as a sample in each crest and a sample in each trough of the wave. This fundamental limitation defines the maximum frequency that we can observe without aliasing. Before the advent of quadrature detection (real and imaginary FIDs), the audio frequency scale ran from zero on the right side edge to the maximum frequency (sw) on the left side edge of the spectral window. The spectral window is still defined by this maximum frequency, which is determined by the sampling rate. The sampling delay At1 is half of the time of one full cycle (1/sw) of the maximum frequency since we have to have two samples per cycle:

Ai1 = 1/(2 x swl) whereswl = the spectral width in F1 in Hz

In 2D NMR the spectral window is now a rectangle (Fig. 9.46), with horizontal width sw defined by the sampling rate in t2 (the dwell time of the ADC, At2) and the vertical "width" defined by the sampling rate in t1 (the t1 increment At1). Any F1 frequency greater than swl (above the upper edge or below the lower edge of the rectangle) will alias, folding back vertically into the rectangle.

Quadrature detection in t2 gives us two FIDs (real and imaginary) by sampling both the Mx component and the My component of the net magnetization as it precesses. This allows us to put zero audio frequency in the center of the spectral window and defines the left side

edge of the window as sw/2 and the right side edge as — sw/2. The maximum detectable frequency is now sw/2, but the width of the spectral window is still sw. Recall that there are two ways to sample the real (Mx component) and imaginary (My component) audio channels: the "Bruker" or alternating method (real - At2 - imag. - At2 - real-At2 - imag. - At2, etc.) or the "Varian" or simultaneous method (real & imag. - 2At2 - real & imag. -2At2, etc.). As long as we define the sampling delay in this way (acquisition time divided by the total number of samples) we can say in either case that the width of the spectral window is 1/(2 At2), based on the need for 2 samples per cycle. The radio frequency center of the spectral window in F2 is the reference frequency vr, which is subtracted out by analog mixing in the detector of the NMR receiver to give zero audio frequency at the center of the spectral window.

9.7.1 Phase-Sensitive 2D NMR: Quadrature Detection in F1

In phase-sensitive 2D NMR, the same kind of strategy is used. To create an imaginary data point in t1, the phase of the preparation pulse is advanced by 90o and the FID is recorded again. This means that the magnetization component of interest, the one that will be transferred in the mixing step, is evolving during t1 according to a sine function instead of a cosine function. For example, if only the yf component Iy can be transferred (e.g., in a 2D TOCSY with the spin lock on yf), we have:

"real" FID : Iz — 90° ^ —Iy ^ —Iycos(^at1) + Ixsin(^at1) ^ ^ —cos(^at1) x FIDb(t2) "imag." FID : Iz — 90° ^ Ix ^ Ixcos(^at1) + Iysin(^at1) ^ ^ sin(^at1) x FIDbfe)

Both FIDs are acquired with the same t1 value, and both are encoded with the same frequency Qa in t1, but they are 90° out of phase (cosine vs. sine modulation in t1), just as the real and imaginary channels of the receiver (Mx and My) are 90o out of phase. This gives us our quadrature detection in F1, allowing us to put zero F1 audio frequency in the center of the F1 spectral window.

There are two methods of encoding the phase information, just as there are for 1D spectra in t2—alternating and simultaneous—except that the alternating method is done a little

differently in the indirect dimension (Fig. 9.47). The "simultaneous" method is called States or States-Haberkorn after the inventor(s), and the "sequential" or "alternating" method is called TPPI, for time proportional phase incrementation. Instead of an ADC choosing between sampling of the real receiver channel or the imaginary receiver channel, the real FID is created with a preparation pulse with phase "x" and the imaginary FID is created with a preparation pulse with phase "y". There is a difference between TPPI and the alternating 1D method: the second pair of data points is recorded with opposite sign of the pulse phase. This means that the preparation pulse has phase 0o, 90°, 180°, 270° (x', y', — X, — y') for the first four t1 values, and then repeats this pattern. TPPI data is processed in the F1 dimension with another Fourier transform algorithm called the "real Fourier transform." The end result is the same, and peaks which alias ("fold") in F1 will alias (vertically) from the same side of the spectral window, just as they do in a 1D alternating ("Bruker") spectrum. The States method is not really "simultaneous" in t1, since t1 is not a direct or "real-time" variable. You simply repeat the acquisition with the preparation pulse on the y' axis using the same t1 value, and record this FID as an "imaginary" FID in t1. Then you increment t1 by twice the sampling delay (2At1, where At1 = 1/(2 x swl)) and repeat the process, first with an X pulse and then with a y' pulse. The data is processed with a standard complex Fourier transform, just like 1D simultaneous ("Varian") data, and peaks outside the spectral window in F1 will alias vertically from the opposite side of the 2D spectrum.

Although these methods are not spectrometer specific, Varian's software (VNMR) is biased toward States mode. The States method is implemented in VNMR by setting the parameter phase to an array of two integers: 1,2. This means that for each value of t1 you acquire two FIDs, one with the preparation pulse applied along the xX axis (phase = 1) and one with the pulse applied along the y' axis (phase = 2). This array is in addition to the t1 array (using d2 as the t1 delay), which has ni values starting with zero and incrementing by 1/swl (or 2 x At1) each time. This can be confusing, because the actual number of FIDs acquired will be twice the value of ni in States mode. If you want to acquire 512 FIDs, for example, you will set ni to 256 if phase is set to 1,2. TPPI mode is accomplished by setting phase to the single value of 3. Bruker uses the parameter MC2 to define the phase encoding method in F1: it can be set to States or TPPI. In either case, the number of FIDs acquired will be equal to td(Fi) so there is no confusion.

Bruker uses an odd parameter called nd0 (number of d-zeroes) to calculate the t1 increment At1. Technically, nd0 is 1 if t1 is a single delay and 2 if t1 is split into two delays of t1/2 each (usually by a 180° pulse in the middle of t1). But if we use TPPI mode these numbers are 2 and 4, respectively, so that the increment in d0 can be calculated as:

in0 = 1/(nd0 x swh(F1)), where swh(F1)is theF1 spectral width in Hz

Thus if there is in fact only one d0 in the pulse sequence (d0 = t1), we have an increment of 1/swh(F1) for States and 1/(2 x swh(F1)) for TPPI. If there are two d0's in the sequence (d0 = t1/2) the increment of d0 is cut in half. Setting nd0 wrong will completely mess up the experiment!

Because this "sampling" is just the incrementation of a delay and changing the phase of a pulse, we have complete control at the software level of how we want to sample the real and imaginary data in t1.

9.7.2 Weighting (Window) Functions and Zero-Filling in ti

Massaging the FID with multiplier functions and increasing the digital resolution with zero filling take on much more importance in 2D NMR because we typically sample the FID for a much shorter time (the acquisition time). In t2, acquisition times are usually cut from 1-2 s for 1D spectra to 100-400 ms for 2D spectra to limit total experiment time and file size (we are collecting hundreds of FIDs) and because resolution is not as important in a crosspeak "blob." In t1, we are even more parsimonious with acquisition time because each data point in the t1 FID represents a complete 1D acquisition, which may involve many scans (transients). Consider a 2D COSY with a 1H spectral width in F1 of 6 ppm on a 600 MHz spectrometer: to cover the spectral width of 3600 Hz we need a sampling delay At1 of 139 /s (1/(2 x swl)), and if we acquire 512 FIDs the final t1 value will be 512 x 139 /s = 71 ms. If we consider that T2 for a typical proton might be 0.5 s, we have only lost about 13% of our FID intensity by the time we stop collecting data (Fig. 9.48). If we do a Fourier transform of this FID, we will have two big problems. First, the digital resolution of our spectrum will be very low. With only 512 data points in the FID, we have 256 data points in our real spectrum and 256 data points in our imaginary spectrum. After phase correction, we discard the imaginary spectrum and keep the real (absorptive) spectrum, which now has only 256 data points to cover a range of 6 ppm (3600 Hz). That is one data point every 14.1 Hz or 0.023 ppm. The details in the spectrum will be lost since even a large J coupling is smaller than the distance between two data points. To solve this problem we simply extend

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