NMR Spectroscopy:
A Quantum Phenomena
Pascale Legault
Département de Biochimie
Université de Montréal
Outline
1)
Energy Diagrams and Vector Diagrams
2) Simple 1D Spectra
3) Beyond Simple 1D Spectra
4) The Spin Echo
5) Selective Population Transfer
6) The INEPT Experiment
1
1)
Energy Diagrams
and Vector Diagrams
Energy Diagram and NOE

Importance of NOE (nuclear Overhauser Effect):
- Through-space transfer of magnetization
in NOESY-type experiments
- Provide distance constraints for structure
calculations

Like T1 and T2, NOE is a relaxation phenomenon:
Key: A fluctuating interaction is capable of causing
transition, just like the RF field. Example of
interactions: chemical shift anisotropy, dipoledipole, etc. NOEs result from dipole-dipole
interactions.
2
Energy Diagrams
EXTREMELY useful for understanding energy transfer in certain
experiments (e.g.: NOESY, SPT, INEPT, HSQC)

At equilibrium for single spins: In the presence of Bo the spin states α
and β have energies ≠ 0 and related frequencies:
E = - γIzBo = -mћγB o = mћωo

The population of spins in each state is given by the Boltzmann
equation. At T ≈ 273 K:
N m/N ≈ [1+(mћγBo /kBT)]/(2I+1)
where N m is the number of nuclei in the mth state (e.g. α)
N is the total number of spins, kB is the Boltzmann
constant


For 1H at 500 MHz, 273 K:
N α/N ≈ 1/2+2.2*10-5; N β/N ≈ 1/2-2.2*10-5
There is a small excess in the α state ...
Energy Diagram at Equilibrium

In detail:

More simply:
3
Steady-State NOE
Lets take an AX 1 H spin system
with A and X close in space, but not
connected through bonds (J = 0).
Lets saturate X with an RF field to
equilibrate the population.

At equilibrium:
( AX spin system)


After saturation:
Steady-State NOE

Now, allow relaxation
W0 , W1, W2 are transition
probabilities or rate
constants
W0 and W2 are 2-spin
transitions ALLOWED for
relaxation
Lets say W0 is most efficient
(large molecules)


Now, “A“ transition decreases by Δ:
4
NOE and Molecular Size

For macromolecules W0 >> W2
This leads to negative NOE in steady-state NOE and
positive crosspeaks in 2D NOESY spectra

For small molecules W2 > W0
This leads to positive NOE in steady-state NOE and
negative crosspeaks in 2D NOESY spectra

When W2 ≈ W0 small or no NOE observed.
NOE and Molecular Size
ssNOE: main use today:
13C detection with 1 H BB
decoupling
5
Vector Diagrams


They are EXTREMELY useful, but it is important to know that they
have certain limitations i.e. difficult to explain NOE, 2nd order
spectra, population transfer, zero or multiple quantum coherence,
etc.
For ease of representation, usually in the rotating frame (x', y', and
z) instead of the laboratory frame (x, y, and z) . Very important to
know what is the frequency (ν) of the rotating frame.
Vector Diagrams:
Effect of a Pulse on the Longitudinal Magnetization (Mz)
At equilibrium (longitudinal magnetization Mz):
♦Bulk magnetization along z caused by B0
♦Excess population in the α state
After 90˚, 270˚ pulses:
♦B1 field brings Mz to the transverse x'-y'
plane
After 180˚ pulses:
♦B1 field inverts Mz
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Vector Diagrams:
Effect of 90˚ and 270˚ Pulses on Mz
Energy Diagrams:
Effect of 90˚ and 270˚ Pulses on Mz
The populations of the two states are now equal:
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Vector Diagrams:
Effect of a 180˚ Pulse on Mz
Energy Diagrams:
Effect of a 180˚ Pulses on Mz
The populations of the two states are now
inverted:
8
Vector Diagrams:
Effect of a Pulse on the Transverse Magnetization (Mx', My')
The transverse magnetization (Mx', My') is not at
equilibrium :
♦ Bulk magnetization in the x'-y' plane
♦ Equal populations in the α and β states
Vector Diagrams: Effect of 90˚ and 180˚ Pulses on
Transverse Magnetization (My' only)
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Vector Diagrams: Effect of 90˚ and 180˚ Pulses on
Transverse Magnetization (Mx' only)
2) Simple 1D Spectra
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Vector Diagrams:
Transverse Magnetization: Where Does it Come From ?
Lets consider a simple 1H 1D experiment
Vector Diagrams:
Simple 1H 1D Experiment
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Vector Diagrams: Effect of 90˚x Pulses on Transverse
Magnetization with Mx' and My'
Vector Diagrams: Effect of 180˚ Pulses on Transverse
Magnetization with Mx' and My'
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3) Beyond Simple 1D Spectra
Beyond Simple 1D Spectra
♦ Simple 1D spectra are not always sufficient for assigning
spectra and determining structure even for small organic
compounds. The main problems are:
1) Resonance assignment
2) Low S/N in insensitive nuclei with low natural abundance
(e.g. 13C and 15N)
3) No correlation information
Example: Neuraminic acid derivative 1
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Beyond Simple 1D Spectra
♦ We would also like to use the following information:
1) 13C-1H correlations
2) The number of protons attach to one carbon
3) 1H-1H correlations (through-bond and through-space)
4) 13C-13C correlations
etc.
♦ Solution:
Complex pulse sequences, which use multiple
pulses, delays and decoupling schemes to transfer
magnetization
Various pulses: hard pulses: 90˚x, 90˚y, 180˚x, 180˚y, etc.
selective pulses: 90˚x, 90˚y, 180˚x, 180˚y, etc.
pulse field gradients
Various delays: fixed or variable delays
Decoupling:
for selective or broadband decoupling
Magnetization Transfer
♦ Via J-coupling (Through-bond)
♦ Via NOE (Through-space)
♦ Via chemical exchange (dynamics)
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Analyzing the Effect of Complex Pulse Sequences
Various Tools to Represent Magnetization Transfer
♦ Block Equations
√
♦ Energy Diagrams
√
♦ Vector Diagrams
√
♦ Density Matrix
♦ Product Operator
√
4) The Spin Echo
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The Spin-Echo
♦ Spins echoes are widely used as
part of larger pulse sequences to
refocus the effects of:
1) unwanted chemical shift
precession
2) magnet inhomogeneity
3) heteronuclear J coupling
♦ The spin-echo does not refocus
homonuclear J coupling.
♦ The spin-echo pulse sequence
can be used to measure the
relaxation parameter T2; it does
not refocus the effect of T2
relaxation.
The Spin-Echo in Vector Diagram:
the Non-Coupled Single Spin Case
♦ Example: 1 H in CHCl3 (not 13C-labeled) with νH = νrf + 100 Hz
1H: 90˚x- τα - 180˚y - τα (echo)
♦ Detected Signal after FT:
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The Spin-Echo in Vector Diagram:
the Non-Coupled Single Spin Case
♦ Example: 1 H in CHCl3 (not 13C-labeled) with νH = νrf + 100 Hz
1H: 90˚x- τα - 180˚x - τα (echo)
♦ Detected Signal after FT:
Note that the intensity is plotted relatively to the positive signal on
the previous page. In practice, this signal would be drawn as a
positive signal by adjusting the zero order phase correction by 180˚.
The Spin-Echo in Vector Diagram:
the Non-Coupled Single Spin Case
♦ Conclusions:
1) Chemical shift evolution (precession) is refocused by the
spin-echo
2) Similarly the spin-echo refocuses magnet inhomogeneity
(ΔBo ):
• The magnetic field Bo is not perfectly
homogeneous throughout the volume of the sample,
therefore not all nuclei experience the same
magnetic field.
• The small differences in magnetic field (ΔB o )
across the sample volume causes nuclei that are
chemically equivalent to precess at different rate.
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The Spin-Echo in Vector Diagram:
Simple Case of Heteronuclear Coupling
♦ Example:
a two-spin AX system with A = 1H and X = 13C
in CHCl3 (13C-labeled) with νrf = νH . 1JAX = 209 Hz
1H: 90˚x- τα - 180˚x - τα (echo)- Acquisition time
♦ Detected Signal after FT:
More on AX Spin Systems by Energy Diagrams
♦ Example:
a two-spin AX system with A = 1H and X = 13C
CHCl3 (carbon is 13C-labeled) with νrf = νH . 1 J AX = 209 Hz
♦ Essentially equal population differences for the α and β
Population
diferences:
αα
βα
αα
αβ
to αβ
to ββ
to βα
to ββ
transition: (N
transition: (N
transition: (N
transition: (N
13C
transitions
+ ΔH + ΔC) - (N + ΔH) = ΔC
+ ΔC) - (N ) = ΔC
+ ΔH + ΔC) - (N + ΔC) = ΔH
+ ΔH) - (N ) = ΔH
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The Spin-Echo in Energy Diagram:
Simple Case of Heteronuclear Coupling
♦ Two different Larmor frequencies as a result of C-H coupling
ν (13CHαCl3) = νc - 1/2*JCH
ν (13CHβCl3) = νc + 1/2*JCH
with JCH = 209 Hz and δ = 77.7 ppm (center of the doublet)
77.7
δ (ppm)
♦ In the first delay τ of the spin-echo experiment, a phase angle Θ is
created between these two vectors
Θ = 2πJCH*τ
Examples: If τ = 0 than Θ = 0, if τ = 1/(4J) than Θ = π/2 = 90˚, etc.
The Spin-Echo: Simple Case of Heteronuclear Coupling
♦ Conclusions:
Heteronuclear coupling is refocused by the spin-echo
(180˚x -> with inversion of magnetization)
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5) Selective Population Transfer
Sensitivity Problem in NMR
♦ Sensitivity problem in NMR:
ε = electromagnetic induction force in detection coil
ε ∝ Nγ3h 2Bo 2 I( I+1)/(3kBT)
Small S/N in spectra of insensitive nuclei with low natural abundance (e.g.
13C, 15N) is a main problem in NMR spectroscopy of organic molecules.
Example:
[ ε ( 13C)/ ε (1H)] = (1.1% * 1) / (100% * 43 ) = 1/5818
One would need to record ~33 million (58182 ) more scans
in a 1D 13C spectrum to get equal S/N than in a 1D 1H
spectrum!
♦ Solutions to this problem are:
1) Get more sample
2) Isotope labeling (may be expensive and not practical)
3) Record spectrum at higher field (Bo)
4) Record spectrum at lower temperature (not significant)
5) Special NMR experiments
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Selective Population Transfer (SPI Experiment)
♦ Advantage of SPI: Very useful to explain the principle
of Population Transfer that
provides a means to "recover" one
of the γ factor.
♦ Disadvantage of SPI: Not very practical because
selective pulses are used.
Selective Population Transfer (SPI Experiment)
♦ Lets consider the two-spin AX system (13CHCl3)
with A=1 H = sensitive nuclei
and X=13C = insensitive nuclei
A) At equilibrium:
N4 = N
N3 = N + ΔC
N2 = N + ΔH
N1 = N + ΔC + ΔH
N2 - N4 ≈ N1 - N3 = ΔH
N3 - N4 ≈ N1 - N2 = ΔC
ΔH = 4 * ΔC
For 13C spectrum:
X1 transition: N3 – N4 = ΔC
X2 transition: N1 – N2 = ΔC
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Selective Population Transfer (SPI Experiment)
B) After a selective 180˚ pulse exciting the A2 transition:
The populations of N1 and N3
are inverted:
N4 = N
N3 = N + ΔC + ΔH
N2 = N + ΔH
N1 = N + ΔC
X1 transition: N3 – N4 = ΔC + ΔH = 5ΔC
X2 transition: N1 – N2 = ΔC - ΔH = -3ΔC
Selective Population Transfer (SPI Experiment)
C) After a selective 180˚ pulse exciting the A1 transition:
The populations of N2 and N4
are inverted:
N4 = N + ΔH
N3 = N + ΔC
N2 = N
N1 = N + ΔC + ΔH
X1 transition: N3 – N4 = ΔC - ΔH = -3ΔC
X2 transition: N1 – N2 = ΔC + ΔH = 5ΔC
After selective inversion of the A1 or A2 transition, the signal
amplification factors for the spectra of X are given by:
1 + γA / γX and 1 - γA / γX
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6) The INEPT Experiment
The INEPT Experiment
♦ INEPT: Insensitive Nuclei Enhanced by Polarization Transfer
Polarization transfer achieved using non-selective pulses
♦ A) Pulse sequence in the 1H and 13C channels
(Note: without carbon pulses, this is a spin-echo experiment on 1H!)
23
The INEPT Experiment
♦ B) Vector diagrams showing the 1 H magnetization vectors (CHCl3 )
The INEPT Experiment
♦ B) Vector diagrams showing the 1 H magnetization vectors (CHCl3 )
a: MHCα and MHCβ are of approximately equal populations
b: ν (13CαHCl3) = νH – JCH/2
and ν (13CβHCl3) = νH + JCH/2
c- d: until then just like beginning of a spin-echo experiment on 1H
e: Effect of 13C 180˚:
- phase of 180˚ doesn’t matter (x or y), MC from z to –z
- inverts population between N1 and N2 and between N3 and N4
- MHCα becomes MHCβ and MHCβ becomes MHCα
f: JCH continue to evolve instead of being refocused during the next
τ delay
g: 1 H 90˚ pulse rotates MHCα to +z and MHCβ to –z
24
The INEPT Experiment
♦ Limitation of vector diagrams:
We can’t pursue our analysis at this poing (g) without
trying to understand what happens in terms of the
energy diagram . . .
The INEPT Experiment: Vector and Energy Diagrams
♦ C) Energy diagrams showing the population transfer (CHCl3)
g: 1 H 90˚ pulse rotates MHCα to +z and MHCβ to –z
The populations of N2 and N4 are inverted:
N4 = N + ΔH
N3 = N + ΔC
N2 = N
N1 = N + ΔC + ΔH
X1 transition: N3 – N4 = ΔC - ΔH = -3ΔC
X2 transition: N1 – N2 = ΔC + ΔH = 5ΔC
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The INEPT Experiment: Vector and Energy Diagrams
♦ D) Vector diagrams showing the 13C magnetization vectors
g’: Note that MCHα is in its original position, but that MCHβ is inverted
h: The 90˚x pulse on 13C create transverse magnetization components
which are observable
The INEPT Experiment: Vector and Energy Diagrams
♦ E) The natural I spin magnetization in the INEPT experiment:
In many applications, the contribution from the natural
is unwanted. There are multiple ways to remove it:
1) Presaturate
13C
13C
magnetization (ΔC)
at the start of the pulse sequence
2) Apply a 90˚ 13C pulse followed by a gradient pulse at the start of the
pulse sequence
In cases 1) and 2) the populations at point a are:
N4 = N + ΔC/2;
N3 = N + ΔC/2
N2 = N + ΔC/2 + ΔH;
N1 = N + ΔC/2 + ΔH
The populations at point g are (N2 and N4 inverted):
N4 = N + ΔC/2 + ΔH;
N3 = N + ΔC/2
N2 = N + ΔC/2;
N1 = N + ΔC/2 + ΔH
X1 transition: N3 – N4 = -ΔH = -4ΔC
X2 transition: N1 – N2 = ΔH = 4ΔC
3) By phase cycling
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The INEPT Experiment: Phase Cycling
The INEPT Experiment: Vector and Energy Diagrams
♦ F) Signal enhancement in the INEPT spectra
Nucleus
Maximum NOE
Polarization Transfer
31P
2.24
2.47
13C
2.99
3.98 (~4)
15N
3.94
9.87 (~10)
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Exercises
(due in a week from now)
1)
Steady State NOE and energy diagrams. For small molecules W2 is the
most efficient relaxation mechanism. Use an energy diagram to explain
relaxation in a small molecule after excitation of spin A of an AX system,
where A and X are close in space but not J coupled. Also draw the
expected 1D spectra before and after saturation of the A resonance.
2)
Vector diagrams and spin echo. Using vector diagrams, show that the spin
echo sequence 90x-τ-180x- τ refocuses Bo field inhomogeneity. To
represent inhomogeneity, use three vectors to represent 3 spins (one that
rotates with the rotating frame, one a little faster, and one a little
slower than the rotating frame). Label you Cartesian axes, indicate the
direction of rotation of the spins, and show all the steps explicitly.
1)
Use vector and energy diagrams
to show the fate of magnetization
in the modified INEPT sequence:
(label your axes and show each step)
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