NMR Spectroscopy: Principles
and Applications
Nagarajan Murali
2D NMR – Homonuclear 2D
Lecture 6
2D-NMR
Two dimensional NMR is a novel and non-trivial
extension of 1D NMR spectroscopy. In the
simplest form of understanding the (a) 1D
spectrum is a plot of intensity vs frequency,
whereas the (b) 2D NMR spectrum is a plot of
intensity vs two independent frequency axes.
NMR Signal in Two-Time Periods
We know that NMR signals are detected as a
function of time and 2D NMR thus implies we
have NMR signal as a function of two
independent time periods. Any 2D NMR
scheme can be represented in general as
below.
NMR Signal in Two-Time Periods
In the preparation period equilibrium
magnetization is built-up and transformed into
coherences that evolve during the evolution (t1)
period. The evolution period is incremented
systematically in successive experiments. During
the mixing period a coherence /magnetization
transfer is effected which then get detected
during the detection period.
NMR Signal in Two-Time Periods
The systematic incrementation of the t1 interval
and direct detection of NMR signal during t2
gives two dimensional time domain data.
1
1
( spctral width)1
1
t2 N *
( spctral width)2
NMR Signal in Two-Time Periods
The time domain signal S(t1,t2) up on two
dimensional Fourier transform yields two
dimensional spectrum S(1, 2).
Correlated Spectroscopy (COSY)
Let us now focus on the very basic 2D NMR experiment called COSY.
We analyzed a special case in the 1D section as a selective
correlation experiment. Here, we analyze the 2D version that
gives correlation between all J coupled protons. The pulse
sequence is give an as follows:
The experiment is repeated N – times with systematic
incrementation of the t1 time and the data collected is subjected
to double FT to yield 2D spectrum.
Correlated Spectroscopy (COSY)
A typical COSY spectrum will
have two types of peaks –
diagonal peaks that correlate
the shifts of the same spins in
both 1 and 2 dimensions,
and cross peaks that
correlate spin A with spin B
by the process of coherence
transfer by the second pulse
in the sequence.
Hamiltonian
Two Spins I=1/2 and J Coupling
Let us again use the two spin Hamiltonian in which each spin with
spin I=1/2 and J Coupling between them.
H 01I1z 02 I 2 z J12 I1z I 2 z in Hz for the case 01 02 J12
H 1 I1z 2 I 2 z 2J12 I1z I 2 z in rotating frame and rad s -1
In subsequent discussions we will drop the hat from the
operators and represent them as normal face italic
character for convenience.
COSY Experiment
Let us consider two protons coupled to each other and we apply
the COSY pulse sequence that has two non selective 90o x-
pulses. Let us just follow spin 1 and by induction we can write
for spin 2.
Ix
tI 2J t I I
I1z
2 I1 y 1 1
1z I
1 y cos(1t1 ) I1x sin( 1t1 )
12 1 1z
2z
cos(1t1 ) cos(J12t1 ) I1 y cos(1t1 ) sin(J12t1 )2 I1x I 2 z
sin( 1t1 ) cos(J12t1 ) I1x sin( 1t1 ) sin(J12t1 )2 I1 y I 2 z
Ix
2
cos(1t1 ) cos(J12t1 ) I1z cos(1t1 ) sin(J12t1 )2 I1x I 2 y
sin( 1t1 ) cos(J12t1 ) I1x sin( 1t1 ) sin(J12t1 )2 I1z I 2 y Detectable terms
COSY Experiment
Focusing just on the detectable terms, we have the signal at the
start of t2 as
sin(1t1 ) cos(J12t1 ) I1x sin(1t1 ) sin(J12t1 )2 I1z I 2 y
The I1x term will give a doublet at frequency 1 in the 2
dimension and is modulated in t1 by sin(1 t1) giving rise to the
diagonal peak.
The term 2I1zI2y will give an anti-phase doublet at frequency 2 in
the 2 dimension and is modulated in t1 by sin(1 t1) giving rise
to the cross peak.
In the 1 dimension the doublet structure of the diagonal peak is
in-phase (cosine function), whereas that of the cross peak is
anti-phase (sine function).
COSY Experiment
Focusing just on the detectable terms, we have the signal at the
start of t2 as
sin(1t1 ) cos(J12t1 ) I1x sin(1t1 ) sin(J12t1 )2 I1z I 2 y
2
1
1
1 2
2
COSY Experiment
Let us analyze the cross peak multiplet structure.
sin(1t1 ) cos(J12t1 ) I1x sin(1t1 ) sin(J12t1 )2 I1z I 2 y
1
sin(1t1 ) sin(J12t1 ) (cos(1 J12 )t1 cos(1 J12 )t1 )
2
2 I1z I 2 y cross peak structure
cos(1 J12 )t1
cos(1 J12 )t1
COSY Experiment
Let us analyze the diagonal peak multiplet structure.
sin(1t1 ) cos(J12t1 ) I1x sin(1t1 ) sin(J12t1 )2 I1z I 2 y
1
sin(1t1 ) cos(J12t1 ) (sin(1 J12 )t1 sin(1 J12 )t1 )
2
diagonal peak
I1x structure
sin( 1 J12 )t1
sin( 1 J12 )t1
COSY Experiment
The cross peaks and diagonal peaks will have different phases as
they are 900 out of phase in both 1 and 2 dimension.
diagonal peak
structure
2 I1z I 2 y
I1x
sin( I J12 )t1 cos( I J12 )t1
sin( I J12 )t1 cos( I J12 )t1
COSY Experiment
The diagonal peak will be dispersive mode in both dimension and
the cross peak will be in absorptive mode .
2D absorption mode lineshape
2D dispersion mode lineshape
COSY Experiment
The anti-phase structure of the cross peaks is a problem because,
if the line width exceeds the J coupling then the anti-phase
doublet overlap and the peak vanishes. Whereas, the diagonal
peaks have in-phase multiplets and add in signal strength in
overlap situations exacerbating the problem. (a) Simulation
shows the effect of overlap and in (b) a real situation with
added noise is shown. The line width is about 1/5 of J value on
the left most spectrum.
In (a) the smallest coupling
constant (Jmax/64) is still
visible, but in (b) due to noise
even (Jmax/32) is barly visible.
Thus the ability to see a cross
peak for small coupling
depends on linewidth and
noise.
COSY Experiment
The COSY spectrum below illustrate the usefulness of the
experiment in identifying the spin system in a molecule.
Azo-sugar
COSY Experiment
The COSY spectrum below illustrate the usefulness of the
experiment in identifying the spin system in a molecule.
Carbopeptoid
2D FT
We have seen in detail 1D FT in Lecture 3. We
summarize the result here as
S 0 A( ) iD( )
FT
S 0 exp(it ) exp( Rt )
Where A() is the absorption mode lineshape
and D() is the dispersion mode line shape.
Furthermore, we can also do cosine and sine
FT. cos in FT
S 0 A( )
S 0 cos(t ) exp( Rt )
sin FT
S 0 cos(t ) exp( Rt ) S 0 D( )
cos FT
S 0 sin( t ) exp( Rt ) S 0 D( )
sin FT
S 0 sin( t ) exp( Rt ) S 0 A( )
NMR Signal in Two-Time Periods
(a) Time domain signal (cosine modulated in t1
and t2). (b) cos-FT with respect to t2 and (c)
cos-FT with respect to t1.
2D FT – cosine Modulation
A typical 2D time domain signal will usually be
S t1 , t 2 S 0 cos At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
The 2D time domain signal is cosine modulated with
respect to t1. In t2 the signal is complex due to
quadrature detection.
We will now have to do two FTs with the 2D time
domain signal, one with respect to t2 and another
with respect to t1.
2D FT – cosine Modulation
FT with respect to t2 yields,
FT t 2 s (t , ) S cos t exp( R At ) A ( ) iD( )
S t1 , t 2 1 2 0 A1 1 2 B B
Then 2D we do a cosine FT with respect to t1.
cos FT t
2 s ( , ) S A ( ) A ( ) iD ( )
S t1 , 2 1 2 0 1 A 2 B 2 B
cos FT t
2 S A ( ) A ( ) iA ( ) D ( )
S t1 , 2 0 1 A 2 B 1 A 2 B
where A1(A) is the absorption mode lineshape along
1 axis. The real part of the above expression gives
a 2D spectrum with absorption lineshape in both
frequency axes.
2D FT
The 2D spectrum of the expression below is a
double absorption lineshape 2D peak.
ReS 1 , 2 S0 A1 ( A ) A2 ( B )
2D FT – sine Modulation
Sometime, we can also have a sine modulated t1 signal as
S t1 , t 2 S 0 sin At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
The FT along t2 gives then,
FT t 2 s (t , ) S sin t exp( R At ) A ( ) iD( )
S t1 , t 2 1 2 0 A1 1 2 B B
Then we do a sine FT with respect to t1.
sin FT t
2 s ( , ) S A ( ) A ( ) iD ( )
S t1 , 2 1 2 0 1 A 2 B 2 B
sin FT t
2 S A ( ) A ( ) iA ( ) D ( )
S t1 , 2 0 1 A 2 B 1 A 2 B
where A1(A) is the absorption mode lineshape along 1 axis. The real
part of the above expression gives, as before, a 2D spectrum with
absorption lineshape in both frequency axes.
2D FT – cosine/sine Modulation
The disadvantage of having just a cosine or sine
modulation in t1 is that there is no frequency
sign discrimination. We can see this from the
properties of FT. cos(t ) cos(t )
complex FT sin( t ) sin( t )
S 0 A( ) iD( )
S 0 exp( it ) exp( Rt )
cos in FT
S 0 A( )
S 0 cos(t ) exp( Rt )
sin FT
S 0 cos(t ) exp( Rt ) S 0 D( )
cos FT
S 0 sin( t ) exp( Rt ) S 0 D( )
sin FT
S 0 sin( t ) exp( Rt ) S 0 A( )
2D FT – cosine + sine Modulation
We can, however, generate 2D signals that
is both sine and cosine modulated in t1,
S s t1 , t 2 S 0 sin At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S c t1 , t 2 S 0 cos At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
It is useful to collect both signals, usually
in two experiments, so that frequency
sign discrimination can be achieved in
the t1 time domain.
2D FT – cosine + sine Modulation
A complex modulation in t1 can be generated
as P-type signal and N-type signal
S P (t1 , t 2 ) S c t1 , t 2 iS s t1 , t 2
S P t1 , t 2 S 0 cos( At1 ) i sin At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S 0 exp i At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S N (t1 , t 2 ) S c t1 , t 2 iS s t1 , t 2
S N t1 , t 2 S 0 cos( At1 ) i sin At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S 0 exp i At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
The resulting spectrum from these complex
signals will be frequency discriminated.
2D FT P-type Modulation
2D FT of the complex signal is written as,
S P (t1 , 2 ) S 0 exp i At1 exp( R At1 )[ A2 ( B ) iD2 ( B )]
A complex FT along t1 yields a spectrum
S P (1 , 2 ) [ A1 ( A ) iD1 ( A )][ A2 ( B ) iD2 ( B )]
[ A1 ( A ) A2 ( B ) D1 ( A ) D2 ( B )] i[ A1 ( A ) D2 ( B ) D1 ( A ) A2 ( B )]
Real Part Imaginary Part
A plot of either the real part or the imaginary
part will yield a phase twisted lineshape.
2D FT P-type Modulation
A typical phase twisted lineshape is
shown below.
Such a lineshape is undesirable in high
resolution work.
2D FT N-type Modulation
2D FT of the complex signal is written as,
S N (t1 , 2 ) S 0 exp i At1 exp( R At1 )[ A2 ( B ) iD2 ( B )]
A complex FT along t1 yields a spectrum
S N (1 , 2 ) [ A1 ( A ) iD1 ( A )][ A2 ( B ) iD2 ( B )]
[ A1 ( A ) A2 ( B ) D1 ( A ) D2 ( B )] i[ A1 ( A ) D2 ( B ) D1 ( A ) A2 ( B )]
Real Part Imaginary Part
A plot of either the real part or the imaginary
part will yield a phase twisted lineshape.
2D Hyper Complex Data
We can use the cosine and sine modulated data to form a hyper complex 2D data
that will yield a pure absorption spectrum data. Start with cosine modulated
signal and do a FT along t2
S c (t1 , 2 ) S 0 cos At1 exp( R At1 ) A2 ( B ) iD( B )
We then take the real part of the signal
S c, R (t1 , 2 ) S 0 cos At1 exp( R At1 ) A2 ( B )
We do the same process for the sine modulated signal.
S s, R (t1 , 2 ) S 0 sin At1 exp( R At1 ) A2 ( B )
Now we form a new complex signal from these two signals,
S (t1 , 2 ) S c, R (t1 , 2 ) iS s, R (t1 , 2 )
S 0 [cos At1 i sin At1 ] exp( R At1 ) A2 ( B )
S 0 exp( i At1 ) exp( R At1 ) A2 ( B )
2D Hyper Complex Data
The usual complex FT along t1will then yield the desired
spectrum.
S (t1 , 2 ) S 0 exp( i At1 ) exp( R At1 ) A2 ( B )
S (1 , 2 ) [ A1 ( A ) iD1 ( A )] A2 ( B )
S (1 , 2 ) A1 ( A ) A2 ( B ) iD1 ( A ) A2 ( B )
We then take the real part of the 2D FT to get double
absorption lineshape with frequency discrimination.
This method of data collection and Fourier transform is
known as States-Haberkorn-Ruben method or simply
States method.
Time Proportional Phase
Incrementation (TPPI)
We need complex data along t1 to discriminate positive frequency
from negative frequency. But by some means if we can set all
frequencies to be positive then we don not need complex signal.
One way to achieve this is to set the reference frequency at the end
of the spectrum and leave the negative frequency region empty.
But this wastes data space. In TPPI, we leave the reference
frequency at the center but move the offsets to look like all the
frequencies are positive.
Time Proportional Phase
Incrementation (TPPI)
Let us start with the cosine and sine modulated signals:
S c t1 , t 2 S 0 cos At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S s t1 , t 2 S 0 sin At1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
We know sine and cosine functions differ only in phase, a shift of
one quarter period or a phase of /2 radians converts one to the
other.
cos(t ) cos t cos sin t sin
sin t for
2
Time Proportional Phase
Incrementation (TPPI)
We can then simply add a phase to the cosine modulated function
that is defined as
add t1
S , t1 , t 2 S 0 cos At1 add t1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
S 0 cos[ A add ]t1 exp( R At1 ) exp(i B t 2 ) exp( R B t 2 )
If we set the phase =(2fmax)*t1 then we can shift all the
frequencies will be shifted to the right.
States –TPPI Method
Both States and TPPI
methods have advantages
and thus combining them
in a 2D experiments is
ideal. A COSY experiment
with States-TPPI method
can be represented as
below.
1
'1
(2 * spctral width)1
1
t2 N *
( spctral width)2
COSY Experiment
2D-FT of FID from such an experiment yields a spectrum with the diagonal
peak in dispersive mode in both dimension and the cross peak in
absorptive mode.
2D absorption mode lineshape
2D dispersion mode lineshape
Double Quantum Filtered COSY (DQFC)
The disadvantages of COSY can be alleviated by
double quantum filtered COSY experiment in which
the diagonal multiplets are also anti-phase like the
cross peaks and in same phase. This leads to better
resolution near diagonal and cleaner looking
spectrum. The pulse sequence is shown below.
DQFC
Up till the second pulse the sequence resembles
COSY, but between the second and third 90o pulse
double quantum (DQ) coherences are retained
which then converted in to observable single
quantum coherences by the third 90o pulse.
DQ - coherences
DQFC
If we follow the evolution of spin 1, from the analysis
of COSY we have right after the second 90o pulse
cos(1t1 ) cos(J12t1 ) I1z cos(1t1 ) sin(J12t1 )2 I1x I 2 y
sin(1t1 ) cos(J12t1 ) I1x sin(1t1 ) sin(J12t1 )2 I1z I 2 y
Only the term highlighted is retained and is sum of
double quantum and zero quantum term.
DQFC
After the second 90o pulse we retaiin only the DQC.
cos(1t1 ) sin(J12t1 )2 I1x I 2 y
DQ ZQ
1
cos(1t1 ) sin(J12t1 ) (2 I1x I 2 y 2 I1 y I 2 x ) (2 I1x I 2 y 2 I1 y I 2 x
2
Ix Third 90o pulse
2
cos(1t1 ) sin(J12t1 )
1
(2 I1x I 2 z 2 I1z I 2 x )
2
DQFC
The first term yields the diagonal peaks and the
second term yields the cross peaks.
cos(1t1 ) sin(J12t1 ) (2 I1x I 2 z 2 I1z I 2 x )
1
2
Also, both the diagonal and cross peaks are anti-
phase multiplets in both 1 and 2 dimensions and
have same phase characteristics in both
dimensions.
COSY vs DQFC
A comparison of properly phased (a) COSY and (b)
DQFC illustrates the advantage of DQFC. There is a
loss of signal by a factor of 2 in DQFC compared to
COSY.
COSY vs DQFC
A region of DQFC of andrographolide illustrates the
advantage of DQFC. Cross peaks close to diagonals
can be seen better.
Total Correlation Spectroscopy (TOCSY)
COSY and DQFC connect, via cross peaks, spins that
have coupling between them which can either be
short range or long range. In a spin network let’s
say spin A is coupled spin B but not to spin C and
spin B is coupled spin C (i.e. JAC=0, JAB 0, JBC 0 ).
Then in COSY or DQFC there will be a cross peak
between spin A and spin B, spin B and spin C, but
no cross peak between spin A and spin C. In a
TOCSY experiment spin A to spin C cross peak will
also appear and identify spins A, B, and C as a
unique group of coupled spins.
Total Correlation Spectroscopy (TOCSY)
COSY TCOSY
Total Correlation Spectroscopy (TOCSY)
A typical TOCSY pulse sequence is given below.
The key part of the experiment is isotropic mixing
caused by the spin locking field in the mixing
period. Isotropic mixing converts I1z, I1x, and I1y to
I2z, I2x, and I2y.
Total Correlation Spectroscopy (TOCSY)
Let us say we retain I1z at point A in the pulse
sequence. The evolution of I1z during isotropic
mixing yields at point B.
I1z 1 cos(2J12 mix )I1z 1 cos(2J12 mix )I 2 z
isotropic mixing 1 1
2 2
1
sin( 2J12 mix ) (2 I1 y I 2 x 2 I1x I 2 y )
2
Total Correlation Spectroscopy (TOCSY)
If we just focus on the Iz terms, the last 90o pulse would
rotate Iz to Iy which will produce in phase doublet in 2.
The Iz term arose from the Iy in t1 period that had
cosine modulations with respect to J coupling which
will is also be an in-phase doublet.
1
1 cos(2J12 mix )I1z 1 1 cos(2J12 mix )I 2 z
2 2
Ix
2
1
1 cos(2J12 mix )I1 y 1 1 cos(2J12 mix )I 2 y
2 2
Analytical solutions are not feasible for more than two
spin system and the qualitative analysis does not show
how a cross peak appear when the coupling is absent.
Total Correlation Spectroscopy (TOCSY)
In a extended coupled network of spins I1z at point A
in the pulse sequence transferred as to of I2z which
then get transferred as I3z and so on during
isotropic mixing. The transfer extends to more
remote spins as the mixing time is increased.
Total Correlation Spectroscopy (TOCSY)
TOCSY of the azo-sugar. On the right the COSY is
shown.
