1. Lecture 7 Two-dimensional NMR F2 F1 (  x,  X ) Diagonal (  A,  A ) (  A,  X ) Cross-peak (  X,  A )

2. Need mixing time to transfer magnetization to see cross peaks ! Interpretation of peaks in 2D spectrum

3. Experiment: Get a series of FIDs with incremental t 1 by a time . Thus, for an expt with n traces t 1 For the traces will be 0, , 2 , 3 , 4  ----- (n-1) , respectively. We will obtain a series of n 1D FID of S 1 (t 1, t 2 ). Fourier transform w.r.t. t 2 will get a series of n 1D spectra S 2 (t 1, 2 ). Further transform w.r.t. t 1 will get a 2D spectrum of S 3 ( 1, 2 ). Spectral width in the t 1 (F 1 ) dimension will be SW = 1/  General scheme: 1 H excitation To keep track of 1 H magnetization (Signal not recorded) Allows interaction to take place Signal contains info due to the previous three steps t 1 = 0 t 1 =  t 1 = 2  t 1 = 3  t 1 = 4  t 1 = 5  t 1 = n  F 2 =  1 F 2 =  2 F 2 =  1 F 2 =  3 F 2 =  4 F 2 =  N FT

4. Review on product operator formalism: 1. At thermal equilibrium: I = I z 2. Effect of a pulse (Rotation): exp(-i  I a )(old operator)exp(i  I a ) = cos  (Old operator) + sin  (new operator) 3. Evolution during t 1 : (free precession) (rotation w.r.t. Z-axis): = - I y for  1 t p = 90 o Product operator for two spins: Cannot be treated by vector model Two pure spin states: I 1x, I 1y, I 1z and I 2x, I 2y, I 2z I 1x and I 2x are two absorption mode signals and I 1y and I 2y are two dispersion mode signals. These are all observables (Classical vectors)

5. Coupled two spins: Each spin splits into two spins Anti-phase magnetization: 2I 1x I 2z, 2I 1y I 2z, 2I 1z I 2x, 2I 1z I 2y (Single quantum coherence) (Not observable) Double quantum coherence: 2I 1x I 2x, 2I 1x I 2y, 2I 1y I 2x, 2I 1y I 2y (Not observable) Zero quantum coherence: I 1z I 2z (Not directly observable) Including an unit vector, E, there are a total of 16 product operators in a weakly- coupled two-spin system. Understand the operation of these 16 operators is essential to understand multi-dimensional NMR expts.

6. Example 1: Free precession of spin I 1x of a coupled two-spin system: Hamiltonian: H free =  1 I 1z +  2 I 2z = cos  1 tI 1x + sin  1 tI 1y No effect Example 2: The evolution of 2I 1x I 2z under a 90 o pulse about the y-axis applied to both spins: Hamiltonian: H free =  1 I 1y +  1 I 2y

7. Evolution under coupling: Hamiltonian: H J = 2  J 12 I 1z I 2z Causes inter-conversion of in-phase and anti-phase magnetization according to the Diagram, i.e. in-phase  anti-phase and anti-phase  in-phase according to the rules: Must have only one component in the X-Y plane !!!

8. Coherence order: Only single quantum coherences are observables Single quantum coherences (p = ± 1): I x, I y, 2I 1z I 2y, I 1y I 2z, 2I 1x I 2z …. etc Zero quantum coherence: I z, I 1z2z Raising and lowering operators: I + = ½(I x + iI y ); I - = 1/2 (I x –i - I y ) Coherence order of I + is p = +1 and that of I - is p = -1  I x = ½(I + + I - ); I y = 1/2i (I + - I - ) are both mixed states contain order p = +1 and p = -1 For the operator: 2I 1x I 2x we have: 2I 1x I 2x = 2x ½(I 1+ + I 1- ) x ½(I 2+ + I 2- ) = ½(I 1+ I 2+ + I 1- I 2- ) + ½(I 1+ I 2- + I 1- I 2+ ) The double quantum part, ½(I 1+ I 2+ + I 1- I 2- ) can be rewritten as: Similar the zero quantum part can be rewritten as: ½(I 1+ I 2- + I 1- I 2+ ) = ½ (2I 1x I 2x – 2I 1y I 2y ) P = +2 P = -2P = 0 (Pure double quantum state) (Pure zero quantum state)

9. 2D-NOESY of two spins w/ no J-coupling: Consider two non-J-coupled spin system: 1.Before pulse:: I total = Let us focus on spin 1 first: 2. 90 o pulse (Rotation): 3. t 1 evolution: 4. Second 90 o pulse: 5. Mixing time: Only term with I z can transfer energy thru chemical exchange. Other terms will be ignored. This term is frequency labelled (Dep. on  1 and t 1 ). Assume a fraction of f is lost due to exchange. Then after mixing time (No relaxation): 6. Second 90 o pulse:

10. 7. Detection during t 2 : The y-magnetization = Let A 1 (2) = FT[cos  1 t 2 ] is the absorption signal at  1 in F 2 and A 2 (2) = FT[os  2 t 2 ] as the absorption mode signal at  2 in F 2. Thus, the y-magnetization becomes: Thus, FT w.r.t. t 2 give two peaks at  1 and  2 and the amplitudes of these two peaks are modulated by (1-f)cos  1 t 1 and fcos  1 t 1, respectively. FT w.r.t. t 1 gives: where A 1 1 = FT[cos  t] is the absorption mode signal at  1 in F 1.  Starting from spin 1 we observe two peaks at (F 1, F 2 ) = (  1,  1 ) and (F 1, F 2 ) = (  1,  2 )  Similarly, if we start at spin 2 we will get another two peaks at: (F 1, F 2 ) = (  2,  2 ) and (F 1, F 2 ) = (  2,  1 )  Thus, the final spectrum will contain four peaks at (F 1, F 2 ) = (  1,  1 ), (F 1, F 2 ) = (  1,  2 ), (F 1, F 2 ) = (  2,  1 ), and (F 1, F 2 ) = (  2,  2 )  The diagonal peaks will have intensity (1-f) and the off-diagonal peaks will have intensities f, where f is the fraction magnetization transferred, which is usually < 5%. (Diagonal)(Cross peak)

11. Experiment: Get a series of FIDs with incremental t 1 by a time . Thus, for an expt with n traces t 1 For the traces will be 0, , 2 , 3 , 4  ----- (n-1) , respectively. We will obtain a series of n 1D FID of S 1 (t 1, t 2 ). Fourier transform w.r.t. t 2 will get a series of n 1D spectra S 2 (t 1, 2 ). Further transform w.r.t. t 1 will get a 2D spectrum of S 3 ( 1, 2 ). Spectral width in the t 1 (F 1 ) dimension will be SW = 1/  General scheme: 1 H excitation To keep track of 1 H magnetization (Signal not recorded) Allows interaction to take place Signal contains info due to the previous three steps t 1 = 0 t 1 = , cos  1  t 1 = 2 , cos  1 2  t 1 = 3 , cos  1 3  t 1 = 4 , cos  1 4  t 1 = 5 , cos  1 5  t 1 = n , cos  1 n  F 2 =  1 F 2 =  2 F 2 =  1 F 2 =  3 F 2 =  4 F 2 =  N FT(t 1 ) FT(  ) FT cos  4 

12. 7.4. 2D experiments using coherence transfer through J-coupling 7.4.1. COSY: After 1 st 90 o pulse: t 1 evolution: J-coupling: Effect of the second pulse: (p=0, unobservable) (p=0 or ±2) (unobservable) (In-phase, dispersive) (Anti-phase) (Single quantum coh.)

13. The third term can be rewritten as: Thus, it gives rise to two dispersive peaks at  1 ±  J 12 in F 1 dimension Similar behavior will be observed in the F 2 dimension, Thus it give a double dispersive line shape as shown below. The 4th term can be rewritten as: Two absorption peaks of opposite signs (anti-phase) at  1 ±  J 12 in F 1 dimension will be observed.

14. Similar anti-phase behavior will be observed in F 2 dimension, thus multiplying F 1 and F 2 dimensions together we will observe the characteristic anti-phase square array.  Use double-quantum filtered COSY (DQF-COSY)

15. Double-quantum filtered-COSY (DQF-COSY): Using phase cycling to select only the double quantum term (2) can be converted to single quantum for observation. (Thus, double quantum-filtered) P = 2 P = -2P = 0 Rewrite the double quantum term as: The effect of the last 90 o pulse: Anti-phase absorption diagonal peak Anti-phase absorption cross peak

17. Heteronuclear correlation spectroscopy 1.Heteronuclear Multiple Quantum Correlation (HMQC): For spin 1, the chemical shift evolution is totally refocused at the beginning of detection. So we need to analyze only the 13 C part (spin 2) J-coupling 13 C evolution After 90 o 1 H pulse: At the end of  : - I 1y = = 2I 1x I 2z for  = 1/2J 12 After 2 nd 90 o pulse: The above term contains both zero and double quantum coherences. Multiple quantum coherence is not affected by J coupling. Thus, we need to consider only the chemical shift evolution of spin 2. J-coupling during 2nd  : J-coupling

18. Phase cycling: If the 1st 90 o pulse is applied alone –X axis the final term will also change sign. But those which are not bonded to 13 C will not be affected. Those do two expt with X- and –X-pulses alternating and subtract the signal will remove unwanted signal. 2. Heteronuclear Multiple-Bond Correlation (HMBC): In HMQC optimal  = 1/2J = 1/(2x140) = 3.6 ms. In order to detect long range coupling of smaller J one needs to use longer , say 30-60 ms (For detecting quaternary carbon which has no directly bonded proton).  Less sensitive due to relaxation. 3. Heteronuclear Single Quantum Correlation (HSQC)  Too complex to analyze in detail for every terms.  Need intelligent analysis, i.e. focusing only on terms that lead to observables.

20. W/ or w/o DCPL

21. 7.6 Lineshape and frequency discrimination: