44

4 DQF-COSY, Relayed-COSY, TOCSY © Gerd Gemmecker, 1999

Double-quantum filtered COSY

The phase problem of normal COSY can be circumvented by the DQF-COSY, using the MQC term

generated by the second 90° pulse:

90°y→ − I1z cos(Ω1t1) cos(πJt1) I1 polarization

+ 2I1yI2x cos(ΩΩ1t1) sin(ππJt1) I1 / I2

double/zero quantum coherence

+ I1y sin(Ω1t1) cos(πJt1) I1 in-phase

single quantum coherence

+ 2I1zI2x sin(Ω1t1) sin(πJt1) I2 anti-phase single

quantum coherence

Phase cycling can be set up to select only the DQC part at this time, which is only present in the

2I1yI2x term (leaving the cos(Ω1t1) sin(πJt1) part away for the moment):

2I1yI2x = 2 -i/2 (I1

+ - I1-) 1/2 (I2

+ + I2-) = -i/2 (I1

+ I2+ + I1

+ I2- - I1

- I2+ - I1

- I2-)

DQC ZQC ZQC DQC

Only the DQC part survives (50 % loss!) and yields (after convertion back to the Cartesian basis):

-i/2 (I1+ I2

+ - I1- I2

-) = -i/2 {(I1x + iI1y) (I2x + iI2y) - (I1x - iI1y) (I2x - iI2y)} = 1/2 (2 I1x I2y + 2 I1y I2x)

However, this magnetization is not observable, only after another 90° pulse:

90°y1/2 (-2 I1x I2y - 2 I1y I2x) → 1/2 (2 I1z I2y + 2 I1y I2z)

Since we still have the cos(Ω1t1) sin(πJt1) modulation from the t1 time evolution, our complete

signal at the beginning of t2 is 1/2 2 I1z I2y cos(ΩΩ1t1) sin(ππJt1) +

1/2 2 I1y I2z cos(ΩΩ1t1) sin(ππJt1) .

45

After 2D FT, this translates into two signals:

- both are antiphase signals at Ω1 in F1 (with identical absorptive/dispersive phase) and

- both are y antiphase signals (i.e., identical phase) in F2, the first one at Ω2 (cross-peak) and the

second one at Ω1 (diagonal peak).

Characteristics of the DQF-COSY experiment:

- the spectrum can be phase corrected to pure absorptive (although antiphase) cross- and diagonal

peaks in both dimensions

- both cross- and diagonal peaks are derived from a DQC term requiring the presence of scalar

coupling (since it can only be generated from an antiphase term with the help of another r.f.

pulse: 2I1yI2z → 2I1yI2x). Therefore, singulet signals – e.g., solvent signals like H2O! – should

be completely suppressed, even as diagonal signals.

Usually this suppression is not perfect (due to spectrometer instability, misset phases and pulse

lengths, too short a relaxation delay between scans etc.), and a noise ridge occurs at the frequency of

intense singulets. In addition, this solvent suppression occurs only with the phase cycling during the

acquisition of several scans with for the same t1 increment, i.e., after digitization! To cope with the

46

limited dynamic range of NMR ADCs, additional solvent suppression has to be performed before

digitization (i.e., presaturation).

If the DQ filtering is done with pulsed field gradients (PGFs) instead of phase cycling, then this suppresses

the solvent signals before hitting the digitizer. However, inserting PGFs into the DQF-COSY sequence causes

other problems (additional delays and r.f. pulses, phase distortions, non-absorptive lineshapes, additional

50 % reduction of S/N).

With the normal COSY sequence, they result in gigantic dispersive diagonal signals obscuring most

of the 2D spectrum.

Intensity of cross- and diagonal peaks

In the basic COSY experiment, diagonal peaks develop with the cosine of the scalar coupling, while

cross-peaks arise with the sine of the coupling. Theoretically, this does not make any difference (FT

of a sine wave is identical to that of a cosine function, except for the phase of the signal). While this

is normally true for the relatively high-frequency chemical shift modulations (up to several

1000 Hz), the modulations caused by scalar coupling are of rather low frequency (max. ca. 20 Hz for

JHH), with a period often significantly shorter than the total acquision time.

Time development of in-phase (cos πJt1) and antiphase (cos πJt1) terms, with Ω1 = 50 Hz, J = 2 Hz,

for T2 = 10 s (left) and T2 = 0.1 s (right).

While the total signal intensity accumulated over a complete (or even half) period is identical for

both in-phase and antiphase signals, an acquisition time much shorter than 1/2J will clearly favor the

in-phase over the antiphase signal in terms of S/N. This difference in sensitivity is further increased

47

by fast T2 (or T1) relaxation, leaving the antiphase signal not enough time to evolve into detectable

magnetization.

This phenomenon can also be explained in the frequency dimension: short acquisition times or fast

relaxation leads to broad lines, which results in mutual partial cancelation of the multiplet lines in

the case of an antiphase signal.

The simulation (next page) shows the dublet appearances for different ratios between coupling

constant J and linewidth (LW). The linewidths were set constant to 2 Hz (at half-height), so that the

different intensities of the dublet signal are only due to different J values.

Obviously, the apparent splitting in the spectrum can differ from the real coupling constant, if the

two dublet lines are not baseline separated: for in-phase dublets, the apparent splitting becomes

smaller, for antiphase dublets it is large than the true J value.

Ratio J/L: 10 3 1 1/3

True J value [Hz} 20.0 6.0 2.0 0.7

In-phase splitting 20.0 6.0 1.8 n/a

Antiphase splitting 20.0 6.0 2.2 1.3

In the basic COSY experiment the diagonal signals are in-phase and the cross-peaks antiphase, so

that signals with small J couplings and broad lines (due to short AQ or fast relaxation) will show

huge diagonal signals, but only very small or vanishing cross-peaks.

In the DQF-COSY, both types of signals stem from antiphase terms, so that both the cross- and

diagonal peak intensity depends on the size of the coupling constants.

48

49

Spins with more than one J coupling

For spins with several coupling partners, all couplings evolve simultaneously, but can be treated

sequentially with product operators (just as J coupling and chemical shift evolution).

J12 J13I1x → I1x cos(πJ12t) → I1x cos(πJ12t) cos(πJ13t)

+ 2I1yI3z cos(πJ12t) sin(πJ13t)

+ 2I1yI2z sin(πJ12t) + 2I1yI2z sin(πJ12t)

cos(πJ13t)

- 4I1xI2z I3z

sin(πJ12t) sin(πJ13t)

The double antiphase term 4I1yI2z I3z develops straightforward from the I1y factor in 2I1yI2z ,

according to the normal coupling evolution rules I1y → − 2I1xI3z sin(πJ13t).

When we consider the time evolution of the single antiphase terms required for coherence transfer,

such as 2I1yI2z sin(πJ12t) cos(πJ13t) and 2I1yI3z cos(πJ12t) sin(πJ13t) , we find that their

trigonometric factors (the transfer amplidute) always assume the general form

2I1yI2z sin(πJ12t) cos(πJ13t) cos(πJ14t) cos(πJ15t) …

with J12 being called the active coupling (that is actually responsible for the cross-peak) and all other

the passive couplings.

When all J couplings are of the same size, then the maximum of these functions is not at t = 1/2J , but

at considerably shorter times, between ca. 1/6J and 1/4J (depending on the number of cosine factors

and relaxation).

50

However, in real spin systems the size of J varies considerably, for 2, 3JHH from ca. 1 Hz up to ca.

12 Hz (or even 16-18 Hz for 2J and 3Jtrans in olefins). The largest passive coupling determines when

the transfer function becomes zero again for the first time (e.g., 1/2J = 35 ms for J = 14 Hz), and the

maximum of single antiphase coherence the occurs at or shortly before ca. 1/4J for this coupling

constant. With only one passive coupling constant and a very small active coupling, one could wait

till after the first zero passing to get more intensity. However, with a large number of passive

couplings of unknown size (as in most realistic cases), the only predictable maximum will occur at

20-30 Hz for most spin systems.

51

The same considerations as for the creation of 2I1yI2z terms out of in-phase magnetization apply to

the refocussing of these antiphase terms back to detectable in-phase coherence. In COSY

experiments, the single antiphase terms are generated during the t1 time and (after coherence

transfer) refocus to in-phase during the acquisition time t2 . Since both are direct and indirect

evolution times which are not set to a single value, but cover a whole range from t=0 up to the

chosen maximum values, the functions shown in the above diagrams will be sampled over this whole

range and always contain data points with good signal intensity (as well as some with zero intensity).

52

Relayed-COSY

The considerations about transfer functions become more important in experiments with fixed delay,

e.g., for coupling evolution. The simplest homonuclear experiment here is the Relayed-COSY, with

the following pulse sequence:

It allows to correlate the chemical shifts of spins that are connected by a common coupling partner,

as in the linear coupling network I1 — I2 — I3 , with the coupling constants J12 and J23 .

After the t1 evolution period and the second 90° pulse we get (cf. COSY):

→ − I1z cos(Ω1t1) cos(πJ12t1)

+ 2I1yI2x cos(Ω1t1) sin(πJ12t1)

+ I1y sin(Ω1t1) cos(πJ12t1)

+ 2I1zI2x sin(Ω1t1) sin(πJ12t1)

During the period ∆, chemical shift evolution is refocussed (180° pulse in the center!), but J12

coupling evolution continues:

J12→ − I1z cos(Ω1t1) cos(πJ12t1) (no coupling evolution, Iz !)

+ 2I1yI2x cos(Ω1t1) sin(πJ12t1) (no coupling evolution, MQC!)

+ I1y sin(Ω1t1) cos(πJ12t1) cos(πJ12∆)

− 2I1xI2z sin(Ω1t1) cos(πJ12t1) sin(πJ12∆) (evolution of antiphase)

+ 2I1zI2x sin(Ω1t1) sin(πJ12t1) cos(πJ12∆)

+ I2y sin(Ω1t1) sin(πJ12t1) sin(πJ12∆) (refocusing to in-phase)

53

The last two terms, however, are spin 2 coherence, and spin 2 has two couplings, J12 (which we have

just considered) and J23 , the effect of which we have to calculate now. Just as with chemical shift

and coupling, which evolve simultaneously, but can be calculated sequentially, we can here calculate

the effects of J12 and J23 one after the other (the order doesn't matter).

J23→ − I1z cos(Ω1t1) cos(πJ12t1) (not affected by J23)

− 2I1yI2x cos(Ω1t1) sin(πJ12t1) cos(ππJ23∆∆)

− 2I1yI2yI3z cos(Ω1t1) sin(πJ12t1) sin(ππJ23∆∆)

+ I1y sin(Ω1t1) cos(πJ12t1) cos(πJ12∆) (not affected by J23)

− 2I1xI2z sin(Ω1t1) cos(πJ12t1) sin(πJ12∆) (not affected by J23)

+ 2I1zI2x sin(Ω1t1) sin(πJ12t1) cos(πJ12∆) cos(ππJ23∆∆)

+ 4I1zI2yI3z sin(Ω1t1) sin(πJ12t1) cos(πJ12∆) sin(ππJ23∆∆)

+ I2y sin(Ω1t1) sin(πJ12t1) sin(πJ12∆) cos(ππJ23∆∆)

− 2I2xI3z sin(Ω1t1) sin(πJ12t1) sin(πJ12∆) sin(ππJ23∆∆)

From the evolution of the second coupling, J23, we get a double antiphase term 4I1zI2yI3z (J23

does not refocus the original 2I1zI2x antiphase of spin 2 relativ to spin 1!) and another term 2I2xI3z ,

which is spin 2 antiphase coherence with respect to spin 3.

The third 90° pulse has to be performed with the same phase setting as the second (i.e., either both

from x or both from y)! After this 90° pulse, we get the folowing terms at the beginning of t2 :

90°y→ − I1x cos(Ω1t1) cos(πJ12t1) spin 1 in-phase

− 2I1yI2z cos(Ω1t1) sin(πJ12t1 ) cos(πJ23∆) spin 1 antiphase

− 2I1yI2yI3x cos(Ω1t1) sin(πJ12t1) sin(πJ23∆) MQC

+ I1y sin(Ω1t1) cos(πJ12t1) cos(πJ12∆) spin 1 in-phase

+ 2I1zI2x sin(Ω1t1) cos(πJ12t1) sin(πJ12∆) spin 2 antiphase

54

− 2I1xI2z sin(Ω1t1) sin(πJ12t1) cos(πJ12∆) cos(πJ23∆) spin 1 antiphase

+ 4I1xI2yI3x sin(Ω1t1) sin(πJ12t1) cos(πJ12∆) sin(πJ23∆) MQC

+ I2y sin(Ω1t1) sin(πJ12t1) sin(πJ12∆) cos(πJ23∆) spin 2 in-phase

++ 2I2zI3x sin(ΩΩ1t1) sin(ππJ12t1) sin(ππJ12∆∆) sin(ππJ23∆∆) spin 3 antiphase

From the observable terms during t2 now, we get three types of peaks, all labeled with Ω1 in F1:

- diagonal peaks at Ω1 in F2 (mixture of terms with different phases)

- COSY peaks at Ω2 in F2 (mixture of terms with different phases)

- Relayed peaks at ΩΩ3 in F2 (pure antiphase in both dimensions)

Relayed-COSY spectrum, only the Relayed peaks (in boxes) show pure (anti-) phase behaviour

55

For the interesting Relayed peak, the transfer amplitude part from the fixed delay ∆ is

sin(πJ12∆) sin(πJ23∆) , which would be at a maximum for ∆ = 1/2J (for J12 = J23).

However, if there are more couplings to the relais spin 2, e.g., in a spin system topology

then the transfer function, for going from 2I1zI2x at the beginning of ∆ to – 2I2xI3z at its end

(and 2I2zI3x after the final 90° pulse) would be

sin(πJ12∆) sin(πJ23∆) cos(πJ24∆)

and, because of the cosine factor, it would be zero at 1/(2J24 ). Therefore, the delay ∆ should be set to

no more than 20-30 ms to avoid losing some Relayed peaks due to large passive couplings.

The Relayed-COSY can be easily extended to a Double-Relayed-COSY experiment, just by adding

another delay ∆ and another 90° pulse, to perform transfers I1 → I2 → I3 → I4 :

However, like in the simple Relayed-COSY, the phases of most of the peaks cannot be corrected to

pure absorption, and the sensitivity decreases further, due to the inefficient transfers and the

increasing length of the pulse sequence. Today, the Relayed experiments have been widely replaced

by the TOCSY experiment.

56

TOCSY / HOHAHA

The TOCSY / HOHAHA experiment does also create a multi-transfer step 1H,1H correlation:

It starts like any other 2D experiment so far, with a 90° pulse creating transverse magnetization

(coherence) which then evolves during an incremented t1 period to yield the indirect F1 dimension

after 2D FT. Between the two evolution times t1 and t2 for the two 1H dimensions, a mixing step has

to perform the coherence transfer. While this is done with simple 90° pulses in all COSY type

experiments, the TOCSY has a "black box" spinlock period instead.

What happens during this time cannot really be understood in terms of vector models or even the

product operators, because it relies on strong coupling. The term strong coupling applies to J

coupled systems where the coupling actually dominates the spectrum:

weak coupling strong coupling

J > ∆ Ω

all signals occur at their proper individual

chemical shift frequencies (dominating effect),

but are split into multiplets with equally

intense lines (small perturbation from J

coupling)

J coupling no more "minor disturbance", but

dominating: coherences of spins are

"coupled" together; instead of individual

resonance frequencies of individual spins,

combination lines occur which cannot be

assigned to just a single spin anymore

Under strong coupling conditions, chemical shift differences between different spins become

negligible, and in the energy level diagram for a two spin system the two states αβ and βα become

identical in energy. Instead of transitions of single spins, the coherences now involve transitions of

combinations of spins:

57

Under these conditions, a coherence / transition of one spin is actually in resonance with a coherence

of its coupling partner(s) (all with the same frequency / chemical shift), and will oscillate back and

forth between all coupled spins, like two (or more) coupled resonant oscillators (e.g., pendulums).

For a two-spin system, the evolution of strong coupling can be described as follows:

strong JI1x → 0.5 I1x {1 + cos(2πJ12τ)}

+ 0.5 I2x {1 – cos(2πJ12τ)}+ (I1yI2z – I1zI2y) sin(2πJ12τ)

So during the TOCSY spinlock, in-phase coherence of a spin is transferred directly into in-phase

coherence of its coupling partner, and back, in an oscillatory way. The frequency of this oscillation

is directly proportional to the coupling constant between the two spins, and complete transfer occurs

(for the first time) at t = 1/2J (see following diagramm).

As shown, another oscillatory component consists of "zero-quantum coherence" with respect to the

spinlock axis, of the form (I1yI2z – I1zI2y) , i.e., dispersive antiphase coherence of spins 1 and 2.

This term is the reason for the (usually rather small) dispersive contributions to the mainly in-phase

absorptive TOCSY cross-peaks.

58

-1,00

-0,80

-0,60

-0,40

-0,20

0,00

0,20

0,40

0,60

0,80

1,00

0 20 40 60 80 100 120 140 160 180 200t [ms]

Inte

nsi

ty

I1xI2x

I1yI2z – I1zI2y

Locking spins with a B1 field

The spinlock needed for converting a weakly coupled into a strongly coupled spin system consists

essentially of a continuing r.f. irradiation, e.g., in the x direction. Its field strength has to be much

stronger than the z component corresponding to the chemical shifts of the spins. These chemical

shifts usually cover a range of a few kHz (= precession frequency of the spins relativ to the

transmitter / receiver reference frequency). If the r.f. field strength is higher, then B1 will dominate

and the spins will start to precess about the B1 (i.e., x ) axis instead of the z axis: the magnetization

components aligned along the x axis (i.e., Ix) are frozen / spinlocked there, and no more chemical

shift evolution occurs in the xy plane. With all chemical shifts reduced to zero, their differences also

vanish, and the strong coupling condition J >> ∆ Ω is fulfilled.

In praxi, the high power transmitters (ca. 50 W for 1H, ca. 200 W for heteronuclei) can usually

generate field strengths of ca. 20-40 kHz, but only for a few hundred microseconds. Then the

spectrometer usually turns itself of automatically (thankfully), to prevent damage to the amplifiers or

the transmitting coils in the probe. Instead of just turning the transmitter on (CW mode), TOCSY

spinlocks are therefore performed with composite pulse trains, consisting of a repetitive series of

pulses with defined pulse lengths and phases. These spinlock sequences allow to effectively spinlock

spins within a wide range of chemical shifts, with reasonable transmitter powers of a few kHz. Some

often used sequences are, e.g., MLEV-17, WALTZ16, DIPSI-2.

59

There are two different classes of spinlocks: isotropic and anisotropic. Isotropic spinlocks (like

WALTZ or MLEV16) allow transfer of all magnetization components:

I1x →→ I2x / I1y →→ I2y / I1z →→ I2z

At the end of the t1 period in the TOCSY experiment, however, there are absorptive and dispersive

magnetization components (as a result of chemical shift and J coupling evolution, cf. COSY). All

these contribute to the TOCSY crosspeaks in the case of an isotropic spinlock, creating large

dispersive contributions.

Therefore, nowadays usually only anisotropic spinlock sequences are employed for TOCSY

experiments (MLEV-17, DIPSI), which can transfer only one transverse component (e.g., Ix ) and the

z component:

I1x →→ I2x / I1y ////→→ I2y / I1z →→ I2z

This anisotropic TOCSY version was initially named the homonuclear HARTMANN-HAHN experiment,

HOHAHA (today the terms TOCSY and HOHAHA are mostly used as synonyms). It leads to almost

absorptive cross-peaks (at least in the 2D plots, as long as one does not look at rows or columns, or a

1D TOCSY spectrum).

For really pure absorptive phases, a z-filtered TOCSY has to be performed:

Here, all magnetization components except for Iz are destroyed during the ∆ delays. This can be

achieved in different ways:

- by waiting. For larger molecules with T1 >> T2 , the z components will relax only slowly (with

T1), while all transverse components will decay fast (with T2).

60

- since the difference between T1 and T2 is usually not large enough for good suppression, the

effect can be enhanced by greatly increasing the B0 field inhomogeneities for a short time during

∆. This is done either with homospoil pulses (i.e., DC pulses on the z shim coils) or – with much

better performance – with pulsed field gradients (PFGs) from specifically designed z gradient

coils directly in the NMR probe.

- in additon, one can also vary the ∆ delays and then add scans acquired with different delay

lengths. This does not affect the z components, but all transverse magnetization terms will evolve

chemical shifts during ∆. With many different ∆ settings, their positions will be always at

different positions somewhere in the xy plane and cancel after coaddition. This requires,

however, a large enough variation of ∆ (at least over 10-20 ms), so that even the slowly rotating

zero-quantum coherences can go around at least once (they evolve only with the difference of the

chemical shifts of the two coupling protons). Also the coaddition of as many different ∆ settings

as possible (at least 6-8) is needed for good cancelation, thus increasing the minimum

experiment time considerably, since the ∆ variation has to be done on top of the phase cycling.

The additional 90° pulses at the end of t1 and beginning of t2 are needed to convert transverse

components into Iz and then back to detectable magnetization again:

90° SL 90°

→ → I1x cos Ω1t1 → I1z cos Ω1t1 → I2z cos Ω1t1 → I2x cos Ω1t1

Since all other magnetization components containing any transverse terms are quickly dephased

during the ∆ periods, the resulting spectrum shows pure in-phase absorptive lineshapes, for both the

cross-peaks and diagonal peaks.

Τ1ρ Relaxation

During the spinlock period, relaxation occurs according to a different mechanism, with a time

constant T1ρ which is neither T1 nor T2 , but somewhere in between. This means that in cases with

T2 >> T1 (slow tumbling limit) T1 ρ is longer than T2 (which is active, e.g., during the Relayed-

COSY mixing sequence). Due to this and the in-phase nature of its cross-peaks, TOCSY can still be

used for molecules up to ca. 10-20 kDa.