Two-Dimensional NMR Spectroscopy

Recall that the general time dependent NMR experiment can be broken up into three events,
preparation, evolution, and detection. In the two-dimensional experiment a series of spectra are
collected where the time of the evolution period is varied.
General Two-Dimensional NMR Pulse Sequence
Relaxation
Delay
Preparation

Indirect time dimension
Direct time dimension
t1
t2
Evolution
Detection
The resulting data is composed of a series of FID each differing by a phase, which is a function
of the evolution period. In other words there are two inherent time dependencies that of the FID
itself, and that of the phase factor. Thus each FID represents a time point in the indirect
dimension. The combined set of FID can be thought of being a two-dimensional FID which is
time dependent along both dimensions.
A Two-dimensional Spectrum is collected as a series of FID at different
evolution times, these FID differ by a time-dependent phase factor

The two-dimensionally time dependent function can be Fourier-transformed in each dimension,
giving a two-dimensional frequency plot representing the two-dimensional spectrum. These can
be represented as a stack plot of the individual 1-D spectra, or as a contour plot.
A Stack Plot of a Two-Dimensional NMR spectrum.
The Contour Plot Corresponding to the Stack Plot Above.
 A two-dimensional spectrum can be performed where the nucleus is the same or different in both
dimensions. The former is referred to as a homonuclear correlation spectrum whilst the latter is
referred to as a heteronuclear correlation experiment.

In any two-dimensional spectrum correlations between two nuclei are made based on some type
of interaction, depending on the experiment, where this interaction transfers coherence from on
spin onto another.

During the evolution period the magnetization can be attributed to one nucleus and evolves
according to its chemical shift at couplings, it is then transferred to another nucleus through the
coherence transfer mechanism and resonates according to the chemical shift and couplings of
that other nucleus. This magnetization will give rise to a signal at the frequency of the first
nucleus in the indirect dimension and that the second nucleus in the direct dimension, and is
referred to as a cross-peak.

When no transfer occurs the magnetization will precess according the chemical shift and
coupling of the same nucleus in both dimensions and thus will occur at the same frequency in
both the direct and indirect dimension. A signal of this type is referred to as an auto-correlation
peak, or a diagonal peak.
The Two-Dimensional COSY spectrum of an AX spin system
The COSY spectrum of an Azo-sugar.
O
R
N3
HO
OH
Correlation Spectroscopy (COSY)
 The COSY experiment is a 2D NMR experiment where the correlation is based on J coupling
interactions. Thus the cross peak intensity is proportional to the strength of the coupling
interaction.
 This technique unambiguously determines which nuclei are coupled to each other, something
that can not be determined from by 1D experiments with absolute certainty. It thus allows the
experimenter distinguish between groups of coupled spins. Thus sub-spectra indiidul
components of a multi-component solution can be assigned.
 Another experiment related to COSY is Total Correlation Spectroscopy, TOCSY, which is able
to base correlation on J couplings over long range. (several bonds)
The General COSY Pulse Sequence
Relaxation
90x
90x
Evolution

Acquire
t1
t2
 The system start with the equilibrium density matrix, which is allowed to evolve under the
influence of chemical shift and coupling terms for the evolution period t1. The evolution period
ends with a final 900 pulse, where the coherence transfer takes place, which will be explained
further. Finally the density matrix evolves for the period t2 during which signal is collected.
 The time dependence in the indirect domain is based on chemical shift and coupling evolution,
and the degree of coherence transfer is determined by the efficiency of the evolution under the J
coupling term. As a result the intensity of the cross peaks will be determined by the coupling
strength and evolution time, and consequently often an additional fixed evolution delay is
inserted after the first pulse to enhance the cross peak intensity from weakly coupled nuclei.
COSY Pulse Sequence Optimized for Long Range Couplings
Relaxation

90x
90x
J
Evolution
Acquire
t1
t2
Spin Dynamics for a Two-spin system during the COSY Experiment.

Starting with the equilibrium density matrix of a two-spin system, reflecting the z-magnetization of
the two nuclei as follows:
Iz  E  E  Iz

After the initial 900x excitation pulse the density matrix becomes:
 Iy E  EIy

Allowing the system to evolve under the chemical shift of spin 1 we get:


 I y cos1t1  I x sin 1t1  E  E  I y

Now letting the system evolve under the chemical shift of the second nucleus:



 I y cos1t1  I x sin 1t1  E  E  I y cos2t1  I x sin 2t1

which can be simplified as:
 I y  E cos1t1  I x  E sin 1t1  E  I y cos2t1  E  I x sin 2t1

These are all single quantum coherences that will evolve between detectable and undetectable terms
under the influence of the J coupling term.

At this stage, in the interest of simplicity, the terms involving spin 2 will be dropped since the
treatment is identical. Thus the density matrix under consideration is:
 I y  E cos1t1  I x  E sin 1t1

Now applying scalar coupling evolution the density matrix becomes:




 I y  E cosJt1  2I x  I z sin Jt1 cos1t1  I x  E cosJt1  2I y  I z sin Jt sin 1t1
which simplifies to:
 I y  E cosJt1 cos1t1  2I x  I z sin Jt1 cos1t1  I x  E cosJt1 sin 1t1  2I y  I z sin Jt sin 1t

The amplitude factors can be repressed as periodic functions of sums and differences of the angular
frequencies as:
1
 I y  Ecos1  J t1  cos1  J t1   I x  I z sin 1  J t1  sin 1  J t1 
2
1
 I x  Esin 1  J t1  sin 1  J t1   I y  I z cos1  J t1  cos1  J t1 
2

Factoring out 2, thus working in frequency units we get:


1
J
J 
J
J 




 I y  Ecos 2 1  t1  cos 2 1  t1   I x  I z sin 2 1  t1  sin 2 1  t1 
2
2
2 
2
2 








1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I y  I z cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 







The above expression says that the observable term,
Iy E
has amplitude factor
J
J
J
J




cos 2 1   + cos 2 1   and I x  E has factor sin 2 1   + sin 2 1   . The first
2
2
2
2




term, if detected about the y-axis would give a signal composed of two in-phase doublets, split by J,
centered at +1 and -1. The second term if detected about the x-axis would give a signal composed
of an anti-phase doublet centered about +1 and -1, split by J.
Splitting patterns in the frequency domain due to amplitude factors in the indirect
time dimension
cos(Jcos(J


sin(Jsin(J


cos(Jcos(J


sin(Jsin(J



Now lets apply the final 900x pulse:


1
J
J 
J
J 




 I z  Ecos 2 1  t1  cos 2 1  t1   I x  I y sin 2 1  t1  sin 2 1  t1 
2
2
2 
2
2 








1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I z  I y cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 






i)
ii)
iii)
iv)
The first term is not observable and does not evolve into an observable terms and thus will be
ignored.
The second terms is a double quantum coherence and does not evolve into an observable
term.
The third term is observable.
The last term is not observable but evolves into an observable term under the scalar coupling
term.
In other formulations of the COSY experiment additional pulses can be employed to make the other
terms observable. This is done in the double quantum filtered COSY where the fourth term is
converted to an observable term. This you will be doing in your assignment.

At this stage the first and second terms are dropped and the calculation continues with just the third
and fourth terms, which are single quantum coherences, as:


1
J
J 
J
J 




 I x  Esin 2 1  t1  sin 2 1  t1   I z  I y cos 2 1  t1  cos 2 1  t1 
2
2
2 
2
2 







Notice that the second term will evolve into a term observable on spin 2, with frequency 2, but has a
phase factor with a time dependence of 1, thus it will give rise to a cross peak. The first term is
observable on spin 1 and has time of 1 in its phase factor thus it will give rise to an autocorrelation
peak on the main diagonal.

Lets consider evolution on spin 1:


1
I x  E cos 21t2  I y  E sin 21t2
2
sin 2 1  J2 t1  sin 2 1  J2 t1




 

J
J 


I z  I y cos 2 1  t1  cos 2 1  t1 
2
2 




Continuing with evolution on spin 2:


1
I x  E cos 21t2  I y  E sin 21t2
2

sin 2 1  J2 t1  sin 2 1  J2 t1






 

J
J 


 I z  I y cos 22t2  I z  I x sin 22t2 cos 2 1  t1  cos 2 1  t1 
2
2 



Now considering scalar evolution:


1  I x  E cos Jt2  2I y  I z sin Jt2 cos 21t2  
J
J 


 
 sin 2 1  t1  sin 2 1  t1 
2  I y  E cos Jt2  2I x  I z sin Jt2 sin 21t2  
2
2 






 I z  I y cos Jt2  0.5I x  E sin Jt2 cos 22t2  
J
J 



 cos 2 1  t1  cos 2 1  t1 
2
2 


 I z  I x cos Jt2  0.5I y  E sin Jt2 sin 22t2  


which simplifies to:
0.5I x  E cos 21t2 cos Jt2  I y  I z cos 21t2 sin Jt2  


 sin 2 1 

 0.5I y  E sin 21t2 cos Jt2  I x  I z sin 21t2 sin Jt2  
I z  I y cos 22t2 cos Jt2  0.5I x  E cos 22t2 sin Jt2  


 cos 2 1 

 I z  I x sin 22t2 cos Jt2  0.5I y  E sin 22t2 sin Jt  

J
J 

t1  sin 2 1  t1 
2
2 

J
J 

t1  cos 2 1  t1 
2
2 

In terms of periodic functions of sums and differences of the angular velocities this becomes:


J
J  


0.25I x  E cos 2 1  t2  cos 2 1  t2  
2
2  






J
J 


 0.5I y  I z sin 2 1  t2  sin 2 1  t2  
2
2   



J
J 




 sin 2 1  2 t1  sin 2 1  2 t1 



 
 0.25I  E sin 2   J t  sin 2   J t   
y
1
2
1
2



2
2  






J
J  



 0.5I x  I z cos 2 1  2 t2  cos 2 1  2 t2  



 




J
J  


0.5I z  I y cos 2  2  t2  cos 2  2  t2  
2
2  






J
J  


 0.25I x  E sin 2  2  t2  sin 2  2  t2  
2
2   



J
J 




 cos 2 1  2 t1  cos 2 1  2 t1 



 
 0.5I  I sin 2   J t  sin 2   J t   
z
x
2
2
2
2



2
2  






J
J  



 0.25I y  E cos 2  2  2 t2  cos 2  2  2 t2  



 



The first term is an auto-correlation peak, present on the main diagonal, and can be simplified to,
considering only the detectable terms:

J
J  
J
J 




0.25I x  Esin 2 1  t1  sin 2 1  t1  cos 2 1  t2  cos 2 1  t2 
2
2  
2
2 






J
J  
J
J 




 0.25I y  Esin 2 1  t1  sin 2 1  t1  sin 2 1  t2  sin 2 1  t2 
2
2  
2
2 





Assuming simultaneous detection in the direct frequency domain, t2, with receiver phase x. The
signal can be represented as:


J
J


cos 2 1  2 t2  i sin 2 1  2 t2 

J
J 







0.25sin 2 1  t1  sin 2 1  t1  

2
2
J
J 



 



 cos 2 1  t2  i sin 2 1  t2 
2
2 



which is:


J
J   
J 
J  




0.25sin 2 1  t1  sin 2 1  t1  exp  i 2 1  t2   exp  i 2 1  t2 
2
2   
2 
2  






Applying a Fourier transform in just the direct domain there is an in phase doublet centered at 1,
with a time dependence in the indirect dimension of sin 2 1  0.5J t1  sin 2 1  0.5J t1, which
with a non-phase sensitive Fourier transformation gives an anti-phase doublet of in-phase doublets.
Auto-correlation signal on the main diagonal from the basic COSY pulse sequence.

sin(Jt1sin(Jt1
f1



f2

The second term is an cross-peak, present off the main diagonal, and can be simplified to,
considering only the detectable terms:

J
J  
J
J 




 0.25I x  Ecos 2 1  t1  cos 2 1  t1  sin 2  2  t2  sin 2  2  t2 
2
2  
2
2 






J
J  
J
J 




 0.25I y  Ecos 2 1  t1  cos 2 1  t1  cos 2  2  t2  cos 2  2  t2 
2
2  
2
2 





Assuming simultaneous detection, with receiver phase x, in the direct frequency domain, t2. The
signal can be represented as:

J
J  


 sin 2  2  t2  i cos 2  2  t2  
2
2  



J
J   


 0.25cos 2 1  t1  cos 2 1  t1  

2
2   


J
J 



 sin 2  2  t2  i cos 2  2  t2 
 
2
2   



which can be shown to be:


J
J   
J 
J  




 0.25i cos 2 1  t1  cos 2 1  t1 exp  i 2  2  t2   exp  i 2  2  t2 
2
2   
2 
2  






Applying a Fourier transform in just the direct domain there is an anti-phase doublet centered at 2,
with a time dependence in the indirect dimension of,  0.25cos 2 1  0.5J t1  cos 2 1  0.5J t1
which with a non-phase sensitive Fourier transformation gives an anti-phase doublet of anti-phase
doublets. (This signal is 900 out of phase with respect to the diagonal signal since it is pre-multiplied
by i.)
Cross-peak signal off the main diagonal from the basic COSY pulse sequence.

cos(Jt1cos(Jt1
f1



f2

Notice that the auto-correlation signal is purely sine modulated in the indirect domain and the crosspeak signal is purely cosine modulated. One way to remove the signals corresponding to the
negative frequencies is to find a modification of the pulse phases to give complementary data, being
cosine modulated in t1 for the auto-correlation peaks and sine modulated for the cross-peak, since
these. Thus there would be one set of data that is sine modulated which is added to another set that
is cosine modulated for both types of peaks giving signal just at the positive frequency, just like
Quadrature detection in the 1D case.

Changing the pulse phase of the first pulse from x to –y in the basic COSY pulse sequence this can
be achieved, as follows:
Relaxation
90x
90-y
Evolution


Acquire
t1
t2
The signal arises from the terms:
I x  E cos 21t1 cos J1t1  I z  I y cos 21t1 sin J1t1
which occur just after the second pulse, which evolve to giver the observable terms, namely the first
term gives the diagonal peak and the second term gives the cross-peak. Evolution in the second time
domain is neglected. The above expression can be simplified to give:
I x  Ecos 2 1  0.5J t1  cos 2 1  0.5J t1   I z  I y sin 2 1  0.5J t1  sin 2 1  0.5J t1 

The first term gives an in-phase pattern of in-phase doublets in the indirect domain and an in-phase
doublet in the direct domain and is on the main diagonal. The second term give an in-phase doublet
of anti-phase doublets in the indirect domain and an anti-phase doublet in the direct domain, and is
off-the main diagonal. The signal from the cross-peak is 900 out of phase with respect to the diagonal
peak.
Auto-correlation and cross-peak signal from the modified COSY sequence.

cos(Jt1cos(Jt1
f1

sin(Jt1sin(Jt1



f2
f1



f2

Sensitivity to the sign of the signal is achieved by adding the data from the basic COSY and
modified COSY experiments using an appropriate phase cycle.

In the processes of achieving sign sensitivity phase information is lost in the indirect domain.
Sine modulated data is being added to Cosine modulated data without correcting the phase,
which would require two separate data sets to be store in memory.

The resulting peak shape upon has both absorptive and dispersive character, in the indirect
dimension. This line shape is known as the phase twist line shape.
The phase twist line shape

Instead of phasing the data at all, which is still possible in the direct dimension, the magnitude of
each data point is calculated and plotted.

This is known as the magnitude spectrum, which is characterized by long tails in both dimensions,
and is thus not amenable to high-resolution studies. Often apodization with a sine bell is used to
remove these tails at the expense of sensitivity.

This type of COSY experiment is referred to as the magnitude COSY. It was most commonly used
in the past due to requiring much less computer memory. It was the only way to obtain fast sign
sensitive data, since the phase cycle involved is minimal.

Today since computer memory is no real issue and upon the development of pulsed field gradient,
removing the necessity of complex phase cycles, the phase sensitive experiment has become most
common.
The phase twist line-shape, compared to the corresponding magnitude spectrum
and that apodized by a sine bell function.

The data from the COSY and Modified COSY pulse sequence can be separately stored in
memory, in what is know as a hyper-complex data set, which is a complex data set in both
dimensions. This data can be Fourier-transformed in both dimensions and corrected in phase
correction in both dimensions, allowing for pure absorption or pure dispersion line shapes line
shapes in both dimension. This approach to phase sensitivity is know as the States-Haberkorn
method.

The resulting phase sensitive COSY spectrum will have cross-peaks the aforementioned antiphase pattern and diagonal peaks that are in-phase. Since this a phase sensitive experiment the
diagonal peaks are seen to be 90o out of phase with respect to the cross-peaks. Either can be
phase to pure absorption.

It is not possible to phase both cross-peaks and diagonal peaks in pure absorption
simultaneously.
Cross peak structure in a phase sensitive COSY experiment with the cross peaks
phase in pure absorption mode.
Diagonal peak structure in a phase sensitive COSY experiment when the cross
peaks phase in pure absorption mode.

One way to obtain the same phase for both cross-peaks and diagonal peaks is to ensure that both
signal arise from the same term from the indirect dimension.

One way to do this is to select the double quantum coherence after the second pulse and converting it
to terms that evolve into detectable signal during the second evolution period. This is known as the
Double-Quantum-Filtered COSY, which has an additional conversion pulse the end.

J
J 


I x  I y sin 2  1  t1  sin 2  1  t1 
2
2 



The Double Quantum Filtered COSY Pulse Sequence.

2
 
2 2 




t1
1
receiver
2
t2
3 4 5
The Phase Cycle for the DQF-COSY Pulse Sequence
Scan



reciever
1
x
x
x
x
2
x
x
y
-y
3
x
x
-x
-x
4
x
x
-y
y