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 Ilz cos( @2t1) cos( t1) polarization 2l1yl2x cos(Q21t1) sin(TJt) l1/I2 double/zero quantum coherence +liy sin(@2 t1) cos(t1) single quantum coherence 211212x sin(Q2jt1 sin(tu) I2 anti-phase single quantum coherent Phase cycling can be set up to select only the dQC part at this time, which is only present in the 211yl2x term(leaving the cos(221ty sin(wty part away for the moment 2I1x=2h2(1-1)2(2+12)=h2(12+I112-l112-ll2) DOC ZOCZOC DOC Only the DQC part survives(50% loss! )and yields(after convertion back to the Cartesian basis) vh2(1+12-li12)=2{(l1x+ily)(2x+i2)-(l1x-l1y)(I2x-l2y)}=h2(2l1xl2y+2I1yl2) However, this magnetization is not observable, only after another 90 pulse 90°y h2(2I1ly-2I1l2) 12(2 11zI2y +2 IvI2z) Since we still have the cos(22) sin(Tty) modulation from the t, time evolution, our complete signal at the beginning of t2 is /22 11z l2y cos(Q21t1)sin (TJt1)+12 2 lly I2z cos( 2]t1) sin(TJt1)
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 ¾¾® - I1z cos(W1 t1 ) cos(pJt1 ) I1 polarization + 2I1yI2x cos(W1 t1 ) sin(pJt1 ) I1 / I2 double/zero quantum coherence + I1y sin(W1 t1 ) cos(pJt1 ) I1 in-phase single quantum coherence + 2I1zI2x sin(W1 t1 ) sin(pJt1 ) 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(W1 t1 ) sin(pJt1 ) 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°y 1 /2 (-2 I1x I2y - 2 I1y I2x) ¾¾® 1 /2 (2 I1z I2y + 2 I1y I2z) Since we still have the cos(W1 t1 ) sin(pJt1 ) modulation from the t1 time evolution, our complete signal at the beginning of t2 is 1 /2 2 I1z I2y cos(W1 t1 ) sin(pJt1 ) + 1 /2 2 I1y I2z cos(W1 t1 ) sin(pJt1 )
After 2D FT, this translates into two signals both are antiphase signals at Q21 in FI (with identical absorptive/dispersive phase)and both are y antiphase signals (i.e, identical phase) in F2, the first one at Q22(cross-peak) and the second one at Q2(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: 211yI2z->211yl2x). Therefore, singulet signals-eg, 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 tI increment, i. e, after digitization To cope with the
45 After 2D FT, this translates into two signals: - both are antiphase signals at W1 in F1 (with identical absorptive/dispersive phase) and - both are y antiphase signals (i.e., identical phase) in F2, the first one at W2 (cross-peak) and the second one at W1 (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 ¾® 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
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 xg1o⊥lLL Time development of in-phase(cos Tty and antiphase(cos Tty) terms, with $2/=50 HE, J=2 H, 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 / 2j will clearly favor the in-phase over the antiphase signal in terms of S/N. This difference in sensitivity is further increased
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 pJt1) and antiphase (cos pJt1) terms, with W1 = 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
by fast T2(or T1) relaxation, leaving the antiphase signal not enough time to evolve into detectable magnetizatOn 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/: 10 True J value hz 20.0 6.0 2.0 0.7 -phase sp litter 20.0 6.0 1.8 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 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
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
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) l1xcos(πJ l1xcos(πJ2t)cos(πJt) +2 Ilyl3z cos(πJ2t)sin(πJt) 2llyl2z sin(TJ12t) 2Ilyl2z sin(TJ12t cos(πJ13t) sin(πJ12t)sin(πJ1t) The double antiphase term 41y2: 13: develops straightforward from the Ily factor in 211y12- according to the normal coupling evolution rules Ily--211x13sin(/13t) When we consider the time evolution of the single antiphase terms required for coherence transfer, such as III2- Sin(T//) cos(J13t) and 21/ 3- cos(TJ/2t)sin(/13t), we find that their trigonometric factors(the transfer amplidute )al ways assume the general form 2lnl2=Sin(J/n2)cos(πJ1cos(πJ4t)cos(πJ1st 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=72J, but at considerably shorter times, between ca. 16Jand /4(depending on the number of cosine factors elaxation
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 J13 I1x ¾¾® I1x cos(pJ12t) ¾¾® I1x cos(pJ12t) cos(pJ13t) + 2I1yI3z cos(pJ12t) sin(pJ13t) + 2I1yI2z sin(pJ12t) + 2I1yI2z sin(pJ12t) cos(pJ13t) - 4I1xI2z I3z sin(pJ12t) sin(pJ13t) The double antiphase term 4I1yI2z I3z develops straightforward from the I1y factor in 2I1yI2z , according to the normal coupling evolution rules I1y ¾® - 2I1xI3z sin(pJ13t). When we consider the time evolution of the single antiphase terms required for coherence transfer, such as 2I1yI2z sin(pJ12t) cos(pJ13t) and 2I1yI3z cos(pJ12t) sin(pJ13t) , we find that their trigonometric factors (the transfer amplidute) always assume the general form 2I1yI2z sin(pJ12t) cos(pJ13t) cos(pJ14t) cos(pJ15t) … 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)
1.0 sin(πJt) sin(πJt)cos(πJt) sin(πJt)cs2(πJt) 0.5 00 0. 0.020 0.040 0 0.100 -0.5 J,=12 Hz; T2=1s 1.0 However, in real spin systems the size of J varies considerably, for 'JHH from ca. 1 Hz up to ca 12 Hz(or even 16-18 Hz for J and trans in olefins ). The largest passive coupling determines when the transfer function becomes zero again for the first time(e.g,12J=35 ms for J=14 Hz), and the maximum of single antiphase coherence the occurs at or shortly before ca. /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
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 2 J and 3 Jtrans 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 1.0 sin(πJt)cos(πJ2t) ·-sn(πJt)cos(xJt)cos(πJ3t) 0.5 00 0. 0.020 0040…6060 0.080 0.100 -0.5 J, =4Hz. J=12 Hz. J=7 Hz:T=1 s 1.0 The same considerations as for the creation of 2I/ 12 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 ti 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 ontain data
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)
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 t It allows to correlate the chemical shifts of spins that are connected by a common coupling partner, as in the linear coupling network I1-12-I3, with the coupling constants J12 and J After the tI evolution period and the second 90 pulse we get(cf. COSY) I1z cos(Q2jt1) cos(TJ12tD) 211,2x cos(Q2jt1) sin(TJ12t1) Ily sin(Q21t1)cos(IJ12tD) +211zI2x sin(Q21t1)sin(TJi2t1) During the period A, chemical shift evolution is refocussed(180% pulse in the center!), but J12 coupling evolution continues I1zcos(Ω21t1)cos(πJt1) (no coupling evolution, Iz D) 211 2x cos(@ t1 sin(IJ12t1) (no coupling evolution, MQC!) ly sin(S2 t1) cos(IJ12t1) cos(TJ124) 2I1xI2zsin(2t1)cos(πJ1t1)sin(J12△) (evolution of antiphase) 211212x sin( Q2jt1) sin(TJ12t1) CoS(TJ124) L2y sin(21t1)sn(πJt)sin(πJ2A) (refocusing to in-phase)
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): ¾® - I1z cos(W1 t1 ) cos(pJ12t1 ) + 2I1yI2x cos(W1 t1 ) sin(pJ12t1 ) + I1y sin(W1 t1 ) cos(pJ12t1 ) + 2I1zI2x sin(W1 t1 ) sin(pJ12t1 ) During the period D, chemical shift evolution is refocussed (180° pulse in the center!), but J12 coupling evolution continues: J12 ¾® - I1z cos(W1 t1 ) cos(pJ12t1 ) (no coupling evolution, Iz !) + 2I1yI2x cos(W1 t1 ) sin(pJ12t1 ) (no coupling evolution, MQC!) + I1y sin(W1 t1 ) cos(pJ12t1 ) cos(pJ12D) - 2I1xI2z sin(W1 t1 ) cos(pJ12t1 ) sin(pJ12D) (evolution of antiphase) + 2I1zI2x sin(W1 t1 ) sin(pJ12t1 ) cos(pJ12D) + I2y sin(W1 t1 ) sin(pJ12t1 ) sin(pJ12D) (refocusing to in-phase)
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) I1z cos(Q2t1) cos(IJ12t1) (not affected by J23) 211y 2x cos(S21t1)sin(TJ12t1)Cos(TJ234) 211 2vI3z cos(@ t1) sin(TJ12t1) sin(IJ234) ly sin(S21t1)cos(IJ12t1)cos(IJ124) (not affected by J23) 211xI2z sin(@ t1)cos(J12t1) sin(TJ124) (not affected by J23) +2I1Zl2xSin(g1t1)sn(πJt1)cos(πJ2△)cos(πJ23△) +4112 3z sin(Q21t1)sin(T J12t1)cos(TJ124)sin(IJ234) 12y sin(g21t1)sn(rJt1)sin(πJ1△)c0s(πJ3△) 212x13z sin(@,) sin(J12t1) sin(TJ124) sin(TJ234) From the evolution of the second coupling, J23, we get a double antiphase term 41-2v32 (23 does not refocus the original 211zI2x antiphase of spin 2 relativ to spin /)) and another term 21213- 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 >-I1x cos(Q21t1) cos(TJ12tD) spin 1 in-phase ly2z cos(S2 t1)sin(TJ12t1)COs(TJ234) spin 1 antipha lyl2yI 3x cos(Q21t1)sin(TJ12t1) sin(T J234) ly sin(S2 t1) cos(TJ12t1)coS(TJ124) spin I in-phase 2l1Zl2xsin(g2t1)cos(πJ1t1)sin(J12△) spin 2 antipha
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 ¾® - I1z cos(W1 t1 ) cos(pJ12t1 ) (not affected by J23) - 2I1yI2x cos(W1 t1 ) sin(pJ12t1 ) cos(pJ23D) - 2I1yI2yI3z cos(W1 t1 ) sin(pJ12t1 ) sin(pJ23D) + I1y sin(W1 t1 ) cos(pJ12t1 ) cos(pJ12D) (not affected by J23) - 2I1xI2z sin(W1 t1 ) cos(pJ12t1 ) sin(pJ12D) (not affected by J23) + 2I1zI2x sin(W1 t1 ) sin(pJ12t1 ) cos(pJ12D) cos(pJ23D) + 4I1zI2yI3z sin(W1 t1 ) sin(pJ12t1 ) cos(pJ12D) sin(pJ23D) + I2y sin(W1 t1 ) sin(pJ12t1 ) sin(pJ12D) cos(pJ23D) - 2I2xI3z sin(W1 t1 ) sin(pJ12t1 ) sin(pJ12D) sin(pJ23D) 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 ¾® - I1x cos(W1 t1 ) cos(pJ12t1 ) spin 1 in-phase - 2I1yI2z cos(W1 t1 ) sin(pJ12t1 ) cos(pJ23D) spin 1 antiphase - 2I1yI2yI3x cos(W1 t1 ) sin(pJ12t1 ) sin(pJ23D) MQC + I1y sin(W1 t1 ) cos(pJ12t1 ) cos(pJ12D) spin 1 in-phase + 2I1zI2x sin(W1 t1 ) cos(pJ12t1 ) sin(pJ12D) spin 2 antiphase
