## P33 cc4 a4 b4ia4 b4i a b2 a b4

which is also a positive real number between 0 and 1, such that P(aa) + P(ftft) = 1. That is, it has to be in either the aa or the ftft state because the coefficients for the other states (aft and fta) are zero. If we had a enormous number N of spin pairs (like a mole) in the exact same spin state we could say that there are N-P(aa) spin pairs in the aa state and N-P(ftft) spin pairs in the ftft state, but we still could not be sure which state any individual spin pair is in.

Now we get to the interesting part that leads to our expanded definition of coherence. We would like to describe the degree of overlap of the aa and ftft spin states for a particular individual spin pair. Consider the product c*4c1 = (a4 - b4i)(a1 + b1i) = degree of mixing or overlap

Note that if c1 = 1 and c4 = 0 (pure aa state), this product is zero; and if c1 = 0 and c4 = 1 (pure ftft state), the product is also zero. So only if there is mixing of the two states for this spin pair will the product be nonzero. Consider the real and imaginary parts of this product, and let's call them x and y:

If we associate the real part with the x axis and the imaginary part with the y axis, we can think of the "degree of mixing" parameter (c*4ci) as a vector with the property of phase: It points in a particular direction in the x-y plane. In fact, this is not a stable ("stationary") energy state and it "precesses" in the x-y plane (the "complex plane") at a rate of vdq = vH + vC. If we average together the degree of mixing between the aa and ftft states for all of the identical spin pairs in the sample, we have coherence:

<c4ci> av = Coherence between the ftft and the aa states = DQC

This is analogous to adding up the individual magnetic dipole vectors of individual single spins to get the net magnetization; the part of this net magnetization that is in the x-y plane is what we called coherence, actually SQC. Just like with SQC, if the phase of the individual degree of mixing parameters (c4 c1) is random over the population of spin pairs, they will average to zero and there will be no coherence (no DQC). It is only when these phases are coherent or tend to have the same phase over the entire population of spin pairs that we have coherence (DQC in this case).

So, in general, in order to have coherence the individual spins or spin pairs have to have mixing or overlap or superposition of two energy states. Because two different energy states define a transition, we can say that coherence is associated with a particular NMR transition between energy states. Furthermore, in order to have coherence this mixing or overlap has to add together in a phase coherent manner over all of the spins or spin systems in the sample. The coherence is this sum (actually an average) of the degree of overlap of two energy states over the entire sample. As long as we associate the real and imaginary parts of the overlap product (c*cj) with the x and y axes, we can talk about the phase of this product and whether these phases are coherent in a population ("ensemble") of spin systems.

Just for completeness, let's define all of the coherences for a two-spin (1H-13C) system. If we consider the most general spin state for this system

We can describe the populations of the four stable energy states:

N(c*ci) = Population in aa state N(c*c3) = Population in fa state The coherences are as follows:

<c*ci> av = 1H SQC(13C = a) <c*ci> av = 13C SQC(1H = a)

N(c4c2) = Population in afi state N(C4C4) = Population in fifi state

<c*4€2> av = XH SQC(13C = fi) <c4c3> av = 13C SQC(1H = fi)

Note that each of these coherences corresponds to a transition between two energy levels (e.g., c3 c1 corresponds to the fiHaC to aHaC transition) in the two-spin energy diagram. The four sQCs correspond to the four peaks in the 13 C and 1H spectra (two doublets), and the MQCs are not observable. Later, we will see that all of these numbers (the four populations, the four single-quantum coherences, and the ZQC and DQC) can be fit into a 4 x 4 matrix that provides a succinct summary of everything we could ever want to know about the spin state of this ensemble of N spin pairs. This matrix is called the density matrix.

### 10.4.3 Raising and Lowering Operators

The "Cartesian" operators (Ix, Iy, Iz, Sx, Sy, Sz) are easy to use because they relate to the vector model and we can easily figure out what happens to them with a pulse of a particular phase or a delay. For example, Iy will become Iz under the influence of a 90° pulse on the X axis on the 1H channel, and Iy will become — Ix after chemical-shift evolution for t = n/(2 ^H). But when we talk about DQC and ZQC, the Cartesian operators become more cumbersome:

{ZQJy = 2IxSy — 2IySx —V' 2IXS y — 2IZ Sx = 0.5({DQ}y + {ZQ}y) — 2SXIZ

{ZQ}y = 2IxSy — 2Iy Sx T—Vy 2(Ixcos + Iysin)(Sy cos' — Sx sin')

Furthermore, the coherence order (sensitivity to twisting of a coherence by a gradient) is ambiguous, at least with respect to the sign

We can define another type of operator that describes the spin state without reference to the x and y (Cartesian) axes. These are the raising and lowering operators that refer to coherence in terms of the transitions between spin states. For example, I+ refers to the transition of spin I from the a to the f state, while the S spin does not change state. For the 1H, 13C spin system, this is a transition from the aHaC state to the iHaC state, or from the aHiC state to the fHfC state (Fig. 10.27). Likewise, the S— coherence refers to the transition from

PiPs s+ Figure 10.27

aHpC to aHaC or from pHpC to pHaC. These are the S (13C) single-quantum transitions in the energy diagram. The double-quantum transitions give rise to the coherences I+ S+ (aa to PP) and I-S- (PP to aa), and the zero-quantum transitions give rise to the coherences I-S+ (Ph aC to aH Pc) and I+ S- (aH Pc to Ph aC).

The effect of the spherical operators on individual spin states is actually opposite to this: for example \ft > ^ \a > . It is the magnetic quantum number that is "raised" by the operator: -1/2 (ft state) to +1/2 (a state). In this book, we will reverse the definition for convenience so that the operators make intuitive sense: I+ "raises" the spin state from a to ft.

Note that these are product operators just like 2IxSz; they are two single-spin operators multiplied together. We can also have antiphase SQC states such as I+ Sz , S Iz, and so on. The formal definition of and I- is

By adding and subtracting them, we get the definition of Ix and Iy in terms of the raising and lowering operators

These are extremely useful for the math but they do not give us any feel for their physical meaning. We are sacrificing the comfort level of visualizing operators in terms of the vector model in order to focus on the transitions that are associated with a coherence. The phase of pulses and careful tracking of the location of vectors in the x-y plane will be ignored completely, freeing us to focus on the overall processes of coherence transfer, evolution, and so on.

The most important thing about the raising and lowering (or "spherical") operators is the way they react to gradients, which is to say their coherence order. The coherence order is no longer ambiguous. For the heteronuclear system

I+ : p = 4; I- : p = -4; S+ : p = 1; S- : p = -1 I+ S+ : p = +5; I-S- : p = -5; I+ S- : p = +3; I-S+ : p = -3

Note that the coherence orders simply add in the product operators: i+s- , which represents heteronuclear ZQC, has a coherence order of 3 (+4 -1). The XH SQC corresponds to p = ±4 because the 1H nuclear magnet is four times as strong as the 13 C nuclear magnet (YH/YC = 4) and this makes its precession rate four times as sensitive to changes in the magnetic field (Avo = y ABo). As gradients are simply a position-dependent change in the magnetic field (Bg = Bo + Gz -z), this means that 1H coherence is twisted four times as much as 13 C coherence by the same strength and duration of gradient. For a homonuclear two-spin system (I = Ha, S = Hb), we use p = ± 1 for SQC for both of the protons because they are equally sensitive to gradients. Thus, the coherence order is context dependent and the 1H SQC coherence order reflects the relative magnetogyric ratio (yh/yx) in comparison to the other nucleus (X) in the spin system.

The effect of a 180° pulse on spherical operators is very simple: it reverses the coherence order of the affected spin. For example, a 1H 180° pulse converts I+ to I-, I- to I+, and I+S+ to I-S+ (S = 13C). This is easy to prove using the definitions

Note that the phase of the pulse does have an effect: It introduces a "phase factor" (in this case 1 or -1) in front of the operator, but it does not change the coherence order. Most of the time, we will be ignoring these phase factors.

The effect of delays is even simpler: the coherence order does not change during a delay. For example, I- remains I- after a delay t:

I- = Ix - iIy — [Ixcos + Iysin] - i[Iycos - Ixsin]

= (Ix - iIy)cos + i(Ix - iIy)sin = (Ix - iIy)e^HT = I-e^HT

where sin = sin(^HT) and cos = cos(^Ht) and e10 = sin © + i cos ©. The important thing is that we get back I- multiplied by a phase factor. Chemical-shift evolution in the x-y plane has been reduced to the rotation of a unit vector (e1^) in the complex (real, imaginary) plane.

We can also have /-coupling evolution from I+ to I+Sz or from I-Sz to I-, but the coherence order (1 and -1, respectively) does not change. Because ZQC and DQC do not undergo /-coupling evolution, I+S+ will stay as I+S+ and I+S- will stay as I+S- during a delay (times a phase factor for DQ or ZQ chemical-shift evolution) and the coherence order (5 and 3, respectively) will not change.

This allows us to diagram the coherence order pathway of an NMR experiment in a very simple way. For example, in an INEPT experiment with intermediate DQC we have

We can add gradients to this pulse sequence for selection of the coherence pathway outlined above by applying a gradient of relative strength 5 during the I+ ^ I+S z delay (p = 4) and another gradient of strength -4 during the I+S+ period (p = 5). The amount of coherence

"twist" (position-dependent phase shift) produced by the two gradients is 20 (5-4) and -20 (-4-5) for an overall twist of zero for the desired coherence pathway. Other pathways will lead to nonzero twist and will be scrambled during the acquisition of the FID, so they will not give any signal in the probe coil.

The coherence order p can be understood precisely if we consider the effect of a gradient on one of the spherical operators. A gradient is just like chemical-shift evolution except that the amount of evolution depends on the position in the tube (z coordinate) rather than the chemical shift.

I- = Ix - iIy G-4-T) [Ixcos + Iysin] - i[Iycos - Ixsin]

= (Ix - iIy)cos + i(Ix - i!y)sin = (Ix - iIy)eiYzGzT = I-eiYzGzT

where sin = sin(y z Gz t), cos = (y z Gz t), and e1© = sin© + i cos©. The gradient preserves the coherence order but it is now multiplied by a phase factor that depends on the position within the NMR tube. A similar analysis shows that I+ is twisted in the opposite direction

where e-1© = sin© - i cos©. Notice that the exponent is now negative, so that coherence is twisted in the opposite direction as a function of position (z). If we associate the imaginary term with the y' axis, we see that the helix is reversed because the imaginary term is reversed in sign. In general, if we focus on the exponent © in e1© as the "twist" induced by the gradient, we have

"twist" = © =- pYozGzt where yo is the smallest y in the system analyzed. For example, in the heteronuclear (13C, XH) system I- has p =-4 and we get © = 4yc z Gz t = yH z Gz t. The phase factor is el©=e'YzGzT, exactly as derived above. For an operator product like i+s- , each operator undergoes twisting by the gradient and gains its own phase factor

I+S- G-ir) I+[e-iYHzGzT]S-[e7YCzGzT] = I+s-e-i(YH-Yc)zGzT

The resulting "twist" is © = —(yh - YC) z Gz t = -(4 - 1) yc z Gz t =-3 Yo z Gz t. Because we defined the twist as © = —p Yo z Gz t, it is clear that p = 3 for the coherence i+s- . Because exponents add when we multiply the terms, the coherence orders of each term in the product simply add together. The conclusion is simple and we never have to look into the detailed math again: the sensitivity to gradient twisting is precisely defined, right down to the direction of twisting, by the coherence order p. The phase factors accumulate as we apply more and more gradients, and the "twists" add up according to the sum of p Gz (assuming that all gradients have the same duration t). We can only observe coherence at the end if the twist is zero: SpGi = 0.

One more thing we can do with spherical operators: We can easily derive the expressions given in Chapter 8 for pure ZQC and DQC. Start with the spherical product I+S+ and express it in terms of the "Cartesian" (Ix, Iy, etc.) operators

I+S+ = (Ix + iIy)(Sx + iSy) = IxSx - Iy Sy + ilySx + iIxSy

= [1(2IxSx - 2IySy)] + i[1(2IySx + 2IXSy)] = {DQ}x + i{DQ}y

The final step uses the analogy with I+ = Ix + iIy to define pure {DQ}x and {DQ}y. From this we have: {DQ}X = 1/2(2IXSX - 2IySy) and {DQ}y = l/2(2IySx + 2IXSy). Starting with i+s- ', we get

I+S- = (Ix + iIy )(Sx - iSy) = IxSx + Iy Sy + iIy Sx - iIxSy

= [1(2IxSx + 2IySy)] - i[1(2IxSy - 2IySx)] = {ZQ}x - i{ZQ}y by analogy to S- = Sx - i Sy .Equating the real and imaginary parts, this gives us: {ZQ}X = 1/2(2IxSx + 2Iy Sy) and {ZQ}y = 1/2(2IxSy - 2Iy Sx). If you are not convinced yet, you can try to show that {ZQ}x turns into {ZQ}y after evolution for a time t, for which (^H -^c)t = n/2 and that {DQ}x turns into {DQ}y after evolution for a time t, for which (^H + ^c)t = n/2. The skeptic will be rewarded after a few happy hours and many pages of paper.

10.5 DOUBLE-QUANTUM FILTERED COSY (DQF-COSY) 10.5.1 The Double-Quantum Filter

We saw in Chapter 9 that the homonuclear "front end" 90x —i1—90x gives us four terms for the Ha-Hb system:

One problem with the simple COSY sequence is that the crosspeak term (—2IbI^) is on the i axis whereas the diagonal term (IX) is on the x! axis of the rotating frame. This means that there is no way to phase the crosspeaks to absorption in the F2 dimension without having the diagonal peaks in dispersion mode. The same is true in the F1 dimension, as the crosspeak has cosine modulation in t1 and the diagonal peak has sine modulation (Chapter 9, Section 9.5.2). These strong dispersive signals on the diagonal extend out much farther than absorptive peaks and give rise to long streaks stretching out in both F1 and F2, interfering with the observation of crosspeaks.

In the DQF-COSY experiment, a third 90° pulse is added immediately after the 90°-t1-90° COSY sequence (Fig. 10.28). Instead of transferring the antiphase Ha magnetization directly into antiphase Hb magnetization with the second pulse, it is first converted into DQC (H®, Hg ^ Hf, Hf) as an intermediate state in coherence transfer. This is "filtered" by destroying all other coherences (ZQC, I2, SQC) with a phase cycle, and then the DQC is immediately converted into antiphase Hb magnetization by the third pulse and the FID is recorded. The filter assures that only magnetization that passes briefly through the double-quantum state between the second and third pulse can be observed in the FID. The "filtration" is accomplished by varying the phase of the third pulse by 90° on each successive transient (e.g., x', y', —x', —y', etc.) and varying the receiver phase (i.e., Figure 10.28

the reference axis) to select only magnetization that is converted from DQC to SQC by the final pulse.

The DQ filter destroys all terms from the homonuclear front end except for the one with both operators Ia and Ib in the x-y plane: the second term — 2I£Ib c s'. This term will become the diagonal and the crosspeak in the DQF-COSY spectrum. To understand its fate we will express it in terms of pure ZQC and DQC

Now we apply the final 90o pulse, but in four consecutive scans we cycle the phase of the pulse as follows: x, y, —x, —y. The receiver phase is cycled in the reverse sense: x, —y, —x, y. This is the essence of the DQ filter: We will see that only the DQ part of the — 2I^Ib term will survive this phase cycle. The other three terms from the homonuclear front end, as well as the ZQC portion of the third term, will be canceled in the phase cycle. Let's write out the result of the phase cycle for the DQ term first

first scan: 900 ^ 2IXIa + 2I£Ib (crosspeak and diagonal peak) second scan: 90y ^ — 2IyIb — 2tfj(diagonal peak and crosspeak) third scan: 90° ^ — 2IbIZ — 2IXIb (crosspeak and diagonal peak)

fourth scan: 90°y ^ 2iyIb + 2I1?IZ (diagonal peak and crosspeak)

Note that both the diagonal peak and the crosspeak are antiphase and that they have the same phase (x' or y'), unlike in the simple COSY where the diagonal peak is antiphase on y' and the crosspeak is in-phase on x'. Also, note the effect of the phase cycle on the phase of the crosspeak: x', —y ', —x', y ' (examining the left term, right term, left term, and right term above). If we cycle the receiver phase so that the reference axis is x', then —y', —x', and finally y ' for the four scans above, the crosspeak terms will all add together in the receiver. Now focus on the diagonal peaks: their phase is also x', —y', —x' and y' for the four scans of the phase cycle. So with the receiver phase set to x', —y', —x', and y' these terms will also add together in the receiver.

Another way to view the receiver phase is as a shifting of the reference axes. With receiver phase set to x' we do not alter anything in the result, but with the receiver phase (reference axis) set to y' we read an Iy operator as Ix (retarding its phase by 90o). All the other operators are retarded in phase accordingly: — Ix becomes Iy, — Iy becomes —Ix, and Ix becomes — Iy. The effect of all four receiver phase settings are summarized below:

 Receiver phase ix iy -ix -iy / . x: ix iy -ix -iy y : -iy ix iy -ix -x : -ix -iy ix iy -y : iy -ix -iy ix

For example, a final result of Ix (first column) will be received as Iy if the receiver phase is set to —y' (fourth row). Now we can "convert" our results for the final 90° pulse, taking into account the effect of the receiver phase cycle x', —y', —X, y':

90 90 9090!

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