## Unv

acquiring the FID immediately and Iz is not observable, so this term is not important. The second term represents a mixture of ZQC and DQC, which is also not observable. We will use this term in the double-quantum filtered (DQF) COSY experiment as an intermediate state in coherence (INEPT) transfer, but in the simple COSY we can ignore it. The third term represents in-phase Ha magnetization that is labeled with the chemical shift of Ha (s = sin(^atO) and with the J coupling (c' = cos(nJtO). This will be our diagonal peak in the 2D spectrum: since it is Ha coherence that will be observed in the FID, it will give an in-phase doublet at chemical-shift position in F2. Since it is labeled with the Ha chemical shift in t1, the second Fourier transform will place it at chemical-shift position in F1. This is the Ha peak on the diagonal. The fourth term is the important one: it represents coherence transfer (INEPT transfer) from Ha to Hb. It will be observed in the FID as Hb antiphase coherence, leading to an antiphase peak at the position in F2. But this peak is modulated in F1 according to the chemical shift of Ha (s = sin (^a Jt1)), so it carries along the information about the spin it came from. The second Fourier transform places the peak at the chemical shift in F1, so it is the crosspeak at F1 = F2 = Now we see in detail how 2D NMR works!

9.5.2 Untangling the /-Coupling Patterns in Fi

Understanding how the chemical shift and /-coupling modulation in t1 works (s s' for the crosspeak and s c' for the diagonal peak) takes a bit of mathematical manipulation. The sin(^at1) sin(n/t1) term (s s') can be written as a sum rather than a product using the trigonometric identity cos(a+ft) = cosa cosft—sina sinft:

cos((^a + n/)t1) = cos(^at1)cos(n/t1) - sin(^at1)sin(n/t1) cos((^a — n/)t1) = cos(^a t1)cos(—n/t 1) — sin(^a t1)sin(-n/t1) = cos(^a t1)cos(n/t1) + sin(^a t1)sin(n/t1)

Subtracting the second equation from the first, cos((^a + n/)t1) — cos((^a — n/)t1) = —2sin(^a t1)sin(n/t1)

ss' = sin(^at1)sin(n/t1) = 0.5 [—cos((^a + n/)t1) + cos((^a — n/)t1)]

Thus Fourier transformation in F1 will yield two peaks: a positive peak at F1 = — n/ and a negative peak at F1 = + n/. This is an antiphase doublet in F1 centered at frequency and separated by 2n/ rad or / Hz. So we have a crosspeak that is an antiphase doublet in F2 (—21^ia observed in the FID) and an antiphase doublet in F1 (sin(^ai1)sin(n/i1)), with both doublets showing a separation of /ab. This is the crosspeak fine structure shown in Figure 9.29.

For the diagonal peak F1 fine structure, we have the t1 modulation sc' = sin(^at1) cos(n/i1). We start with the trigonometric identity: sin(a + ft) = sin a cos ft + cos a sin ft, applied to the sum and difference frequencies:

sin((^a + n/)t1) = sin(^a t1)cos(n/t1) + cos(^a t1)sin(n/t1) sin((^a — n/)t1) = sin(^a t1)cos(—n/t1) + cos(^a t1)sin(—n/t1) = sin(^a t1)cos(n/t1) — cos(^a t1)sin(n/t1)

Since we are looking for sc;, we add the two equations together to get:

sin((^a + nJ )t1) + sin((^a — nJ )t1) = 2sin(^a ii)cos(nJii)

sc' = sin(^at1)cos(nJt1) = 0.5 [sin((fta + nJ)t1) + sin((^a — nJ)t1)]

Fourier transformation in F1 will yield two peaks, both of them positive: one at F1 = Qa — nJ and one at F1 = Qa + nJ. This is an in-phase doublet in F1 centered at frequency Qa and separated by 2nJ rad or J Hz. So we have a 2D peak that is an in-phase doublet at frequency Qa in F2 (+I£ observed in the FID) and an in-phase doublet at frequency Qa in F1 (s C = sin(^at1) cos(nJt1)), with both doublets showing a separation of Jab. The diagonal peak has a fine structure of four peaks in a square pattern, all with the same phase. How does its phase compare to the crosspeak? It is 90o out of phase with the crosspeak in F2 (IX vs. — 2iyI£) and it is 90o out of phase with the crosspeak in F1 (0.5 [—cos((^a + nJ)t1) + cos((^a — nJ)t1)] for the diagonal peak vs. 0.5 [sin((^a + nJ)t1) + sin((^a — nJ)t1)] for the crosspeak. Note that a sine function in time is always 90o out of phase with a cosine function. This is a significant conclusion because it means we cannot phase correct the whole 2D spectrum: either we have absorptive crosspeaks and dispersive diagonal peaks or vice-versa. The dispersive lineshape does not go to zero quickly as we move away from the center of the resonance (Fig. 9.39) as the absorptive shape does, so this will lead to large streaks extending above and below and to the left and right of the diagonal peaks.

An exponentially decaying FID gives a Lorentzian lineshape upon Fourier transformation. The general form of the absorptive Lorentzian line is Iabs = 1/(1 + v2), whereas the dispersive line has the form Idisp = v/(1 + v2), where I is the intensity at each point in the spectrum. Far from the peak maximum (v2 >> 1), we have Iabs ~ 1/v2 and Idisp~ 1/v. This is the reason that the dispersive lineshape extends much further from the peak maximum.

The COSY phase differences can be eliminated by presenting the data in "magnitude mode," as we did for the HETCOR spectrum. But we lose useful information such as the distinction between active and passive couplings, and more importantly the magnitude mode peaks are broader and extend farther outward at their bases (Fig. 9.39), again leading

 Peak shapes Imaginary Real (absorptive) Imaginary \ (dispersive) 'V
0 0