Jan 7 kBT and other basic concepts
Homework #1 out
Jan 9 Basic Quantum Review
Jan 14 Basic Quantum Review
Jan 16 Many electron atoms (Blake Farrow
lectures)
Homework #1 due
Jan 21 Molecular QM
homework #2 out
Jan 23 Molecular QM
Jan 28 Molecular orbitals
Homework #2 due
Jan 30 Group Theory
Homework #3 out
Feb 4 Group Theory; light & matter
2 Papers and roles assigned
Feb 6 light and matter: classical
Feb 11 light and matter: IR & Raman
Homework #3 due; midterm out
Feb 13 light & Matter: electronic
Feb 18 biomolec spect: labeling molecules
Midterm due; homework #4 out
Feb 20 biomolec spect: x-ray
Paper discussions#1/papers due
Feb 25: biomolec spect: x-ray
Feb 27 biomolec spect: microscopies
Homework #4 due paper discussions #2 due
Mar 4: NMR basic
Homework #5 out Blake Farrow lectures
Mar 6 2D NMR
Mar 11: biomolecular spectroscopy via super
resolution microscopy of cells Homework #5 due
Final, end of quarter, etc.
papers
Biophysical Magnetic Resonance
Theory and Applications
Major points to learn today
What is NMR? – basic physics, transition dipole
why are NMR signals weak?
Relaxation times (t1 and t2)
NMR spectroscopy of simple molecules (Pascal’s triangle, chemical shifts)
NMR- Why is it useful?
•
•
•
•
•
•
Unparalleled chemical specificity.
Quantitatively interpretable at molecular level.
Structures without crystals.
Diverse dynamic probes from 10-12 to 102 s.
Nondestructive.
Penetrates nearly all samples.
Spin and Intrinsic Magnetism
• Fundamental particles have intrinsic angular momentum
which is proportional to a magnetic moment.
• Spin angular momentum and the magnetic moment are
proportionally related by the gyromagnetic ratio, which can
be positive or negative.
μ   S.
μ = magnetic moment (North/South vector)
S = spin angular momentum, an intrinsic
quantity of the particle
γ = gyromagnetic ratio, also intrinsic quantity
associated with the particle
NMR Active Nuclei
• There are a number of biologically relevant nuclei with
spin
• The most sensitive is 1H
– Sensitivity is determined both by γ and natural abundance
There’s a lot of it, and it’s got a
big γ, meaning it makes a big μ.
Spin States are Quantized
• Spin is a quantum mechanical phenomenon, so it is
quantized.
• For a given spin quantum number S, the spin state mS can
only take values S, S-1, S-2, …, -S.
• The number of states available for a given spin is given by
2S+1.
• We generally only deal with spin ½ particles, since
other’s have quadrupole angular momenta, instead of
dipole (complicated and unimportant to you guys).
So, for a spin ½ particle, there are two spin states:
mS= +½, -½. Or  and .
Spin in a Magnetic Field
• At zero field, the spin states mS = S, S-1, …, -S all have the
same energy. (I.e., they are degenerate.)
• When spins experience an applied magnetic field, the spin
states separate.
N
 E  hv 
h B
2
For H1 nuclei at 1 Tesla, this
ΔE is about 10-7 eV. Very low
energy.
This means that at room temperature the nuclei tend to be in both  and 
states more or less equally. This makes the signal very weak.
Vector Interpretation
• This can also be visualized as spin “vectors” partially aligning
with the magnetic field.
• A spinning top might process around its axis of spinning.
• A bunch of these tops will process at different angles at a
given time (spinning out of phase), but will add up to one net
direction (M0)
Spin Precession
• Physically, this is the external field producing a “torque” on
nuclei (similarly to that of gravity on a top) that causes them to
precess around the field.
• The direction of spin follows the right hand rule, and the
frequency of spin is known as the Larmor frequency.
Larmor frequency

0
  B
0
NMR Spectroscopy
• Like any other spectroscopy, we need to detect the
absorption/emission of radiation to obtain a signal which only
happens when the emitted radiation matches the energy gap.
• Originally this was done by irradiating the sample with a fixed
radiowave frequency, then sweeping the applied field until
resonance was observed (continuous wave NMR).
• Nowadays a constant magnetic field is applied, and a spectrum of
radiowave frequencies are applied all at once.
NMR Signal
• NMR doesn’t directly look at longitudinal magnetization, rather it
measures magnetization perpendicular to the field.
• A radiofrequency pulse along the x-direction rotates the net
magnetization vector into the x-y plane.
B0
• As discussed previously, once off-resonance the net magnetization
precesses around the applied field at the Larmor frequency.
  B
Signal Detection
• Signal precession is detected by the receiver coil, which is
often the same coil used to pulse the sample.
• The oscillating magnetic field induces a current in the
receiver coil, which is recorded as a function of time and
is called the Free Induction Decay (FID).
Relaxation
• There are two main modes of relaxation in NMR, T1
and T2
• Magnetization re-establishes an equilibrium value
along the z-axis according to T1
• Free Induction Decay is called so because its
amplitude decreases with time according to T2
• For small molecules freely rotating in solution T1  T2
• For larger molecules or viscous solutions T1 >>> T2
T1
• T1 is called the longitudinal relaxation time, and is the
time required for the net magnetic vector to return to
the z-axis after being tipped into the x-y plane.
• This is what dictates how long one must wait between
scans, thus determining the practical limit of S/N
improvement by obtaining multiple FIDs.
T1 Mechanisms
• Another name for T1 is the spin-lattice relaxation time
• Intermolecular mechanisms
– Unpaired electrons
– Extramolecular nuclei
• Intramolecular mechanisms
–
–
–
–
–
Dipole-dipole
Chemical shift anisotropy
Spin rotation
Scalar coupling
Quadrupolar relaxation
• Solution state T1’s for 1H is usually sub second, for 13C on
the order of seconds, and for 15N on the order of a minute.
T2
• T2 is the transverse relaxation time, and is the time it
takes for the tipped net magnetization vector to
decohere.
• T2 is the amount of time during which there is
measurable coherence, so it determines the sampling
time, or acquisition time, of an experiment
T2 Mechanisms
• T2 differs from T2*, which is a combination of inherent T2
and magnetic field inhomogeneity
• Inherent T2 is caused by the loss of phase coherence that
results from each resonance possessing a slightly different
Larmor frequency.
• Considerations:
– Viscosity
– Spin exchange
• T2 can never be longer than T1, but if the molecular
motions are slow it can be much shorter
Signal to Noise
• The signal from a
thermally polarized NMR
experiment is weak.
• S/N can be improved by
taking multiple transients.
S/N proportional to (# transients)1/2
Extracting Meaning from NMR Signal
• The time dependent FID contains a wealth of
information about the chemical environment
experienced by the detected spins.
• However, it’s difficult to visually interpret.
• So, we transform the data into a frequency spectrum
and use tricks to help extract observable quantities like
chemical shift and spin coupling.
• These observables give us information about the
molecular structure and dynamics of a sample.
Fourier Transform
• The Fourier transform is used to change a time
dependent NMR signal to a frequency dependent
spectrum.
• Continuous Fourier transform

A ( ) 

A ( t )[cos( t )  i sin( t )] dt

• Discrete Fourier transform:
1 
cos(2  kn )
sin(2  kn ) 
N 1
x(n) 
A(n)
 iA ( n )



n0
N 
N
N

Discrete FT
• Sampling rate: distance
between time steps
• Nyquist frequency: highest
frequency that can be
defined by a particular
sampling rate
N 
1
2t
So, if a FID is sampled
every 0.01 sec, the
frequency range that can
be detected is -50 -> 50
Hz.
FID Manipulation Tricks
• Zero filling
– Adds zero points to the end of
the FID.
– Increases resolution without
adding extra acquisition time or
noise.
• Truncation and apodization
– Increases S/N by emphasizing
the initial points of the FID
– Truncation can create baseline
distortion due to abrupt change
in signal amplitude
– Apodization adds non-zero
points to more gradually
decrease the FID
Tricks Continued
• Exponential multiplication
– The FID is multiplied by an exponential of the form exp(At–Bt2)
– An increasing exponential will improve resolution
– A decreasing exponential will improve S/N
These cannot be achieved simultaneously!
– The most common use of this technique is when B=0 and A < 0.
(line broadening)
– If both A, B > 0 a Lorentz to Gauss transformation is achieved
Chemical Shift
• Caused by local magnetic field distortions from nearby electrons,
which change the Larmor frequency for nuclei in different
chemical environments.
• Both the Larmor frequency and the chemical shift are field
dependent, so the adjusted chemical frequencies are reported in
field-independent ppm units relative to some standard.
ppm =
(  peak   ref )
 spectrom eter
• ppm on a 700 MHz/16.4 T NMR would be 700 Hz, but on a
50 MHz/1.1 T NMR would be 50 Hz. So given the same sensing
mechanism, bigger field is way better.
Chemical Shift
• Nuclei are said to be “shielded” if surrounded by
a high density of electrons, and are progressively
“deshielded” as the electrons around the nucleus
decrease.
Deshielded
Shielded
Spin-spin coupling
• Also known as j-coupling.
• Measured in Hz.
• Indirect “communication” between neighboring spins via
surrounding electrons allows spins to “feel” the spin state
of their neighbor.
• For neighboring spin ½ nuclei, the rule is that there are n +
1 peak splittings for n neighboring nuclei, with amplitudes
given by the nth row of Pascal’s triangle.
Dipolar Coupling
• Direct coupling between spin “dipoles”.
• Rapid tumbling in solution averages the dipolar coupling to zero.
• In viscous solutions or with large molecules, partial alignment leads
to incomplete averaging of the dipolar coupling.
– Residual Dipolar coupling
D IS 
 I S
4 r
3
IS
1  3 co s 2  


• Useful in structural biology to determine long range distance
constraints and in protein dynamics for long scale motions
Decoupling
• Splitting increases the complexity of a spectrum
• Can negate its effects with decoupling
• Generally accomplished with selective irradiation of
splitting nucleus
– Constant irradiation induces rapid spin flips, which
averages the effect of the neighboring spin states
• A side consequence of 1H decoupling in 13C spectra is
signal enhancement by the Nuclear Overhauser Effect
Example Assignment
1D Experiments
• Simple one pulse experiments are sometimes
sufficient for determining small structures
• Current Varian software has standard parameters
built into the software that cover most general
experiments.
• Generally 1H spectra, sometimes in combination
with 13C spectra, are taken and peak integrations
along with chemical shifts and j-coupling
information are used to assign peaks.
Proton Chemical Shifts
downfield
TMS=0
•Electronegative groups are "deshielding" and tend to move
NMR signals from neighboring protons further "downfield"
•Protons on O or N have highly variable chemical shifts
sensitive to environment, esp. hydrogen bonding.
•The -system of alkenes, aromatic compounds and
carbonyls deshield attached protons.
32
NMR peak positions vary with magnetic field
recall
 =  B0 / 2
HO-CH2-CH3
TMS
Low B field
H
H
H
Chemical shift (ppm) 0
TMS
High B field (better separation of peaks)
H
H
Chemical shift (ppm)
H
33
0
Equivalent H-atoms are degenerate
However, the energy levels may be split by adjacent H-atoms
(this is where much of the chemical information comes from)
H is influenced by H
H is influenced by H and H
H is influenced by H
HO-CH2-CH3
These peaks will split
TMS
H
H
H
Chemical shift (ppm)
0
34
Splittings are described by Pascal’s triangle
1
1
Coupled to 1 H
Coupled to 2 H
Coupled to 3 H
1
1
1
2
3
1
3
1
Coupling is significant
Splitting is detected
HO-CH2-CH3
Coupling is only to protons
attached to adjacent nuclei
Coupling is weak
Splitting is not detected except
at very high field (we’ll ignore)
1
1
Coupled to 1 H
H
H
Coupled to 2 H
Coupled to 3 H
1
1
1
2
3
1
3
1
1
Coupling is significant
1
Coupled to 1 H
H
H
Coupled to 2 H
Coupled to 3 H
1
1
1
2
3
 Area
HO-CH2-CH3
1
3

Coupling is weak because this
H-O proton exchanges with
local environment
1
1
H
TMS
2
H
3
H
Chemical shift (ppm)
0
1H
NMR Spectrum of Ethanol
HO-CH2-CH3
Area = 1
Area = 3
TMS
Area = 2
Chemical shift (PPM)
increasing B experienced by nucleus;
increasing  at fixed B.
38
NMR Spectrum of Ethanol
HO-CH2-CH3
downfield
deshielding
greater Beff
TMS
upfield
shielding
smaller Beff
Chemical shift (PPM)
increasing B experienced by nucleus;
increasing  at fixed B.
39
why isn’t the OH proton also a triplet?
when traces of water are present, the OH proton exchanges with water
CH3CH2OH + H2O  CH3CH2O- + H3O+
This proton exchange disrupts the spin-spin coupling between the OH and CH2 protons,
so only a single OH proton is observed.
At low temperatures and with pure ethanol, the OH signal is a triplet and the CH2 signal
is a(n) ???
40
Interpretation of NMR Spectra
•Nuclei in different chemical environments resonate
at different frequencies (chemical shift). Chemical
shift charts can be used to assign peaks to groups
on the basis of their resonance frequencies.
•Signal intensities (from integrated peak areas) are
usually directly proportional to the number of
protons in each distinct chemical environment.
•Interaction of nearby nuclei produce a splitting
pattern (spin-spin coupling) that can be correlated
with the structure of the molecule.
41
Chemical shifts
The magnetic field, B, experienced at the nucleus of
an atom, is not identical to the applied field, B0, due
to shielding effects of the surrounding electrons:
B = B0 (1 - s) or NMR =  B0 (1 - s) / 2
where s is a (dimensionless) shielding constant.
The electronic environment of the nucleus depends
on chemical bonding, hence NMR is sensitive to the
detailed molecular structure
42
CH3-CH2-CH3
What do we expect to see?
Two types of protons 6:2 intensity ratio
1
1
H
Coupled to 2 H
H
1
1
1
2
3
1
3
1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Coupled to 6 H
43
 peak
CH3-CH2-CH3
 peak
3
H
1
H
44
o
How about this molecule?
o
First, identify all of the protons
45
o
o
H
o
H H
H
H
H
o
H
H
H
H
H
H
Now, identify which protons are unique
46
H
H
o
o
H
H
H
H
d (2)
(2) f
o
e
(1) g
H
H
o
H
(2) e
o
H H
o
H
H
H
H
H
Next, identify splittings
using Pascal’s triangle
c (2)
b (2)
a (3)
f
These protons are not going to be split
their chemical shift is dominated by the ring current
47
Split by 0 protons
(2) e
unsplit
d (2)
(2) f
o
o
c (2)
Split by 2 protons
Split by 5 protons
e
(1) g
b (2)
a (3) Split by 2 protons
f
1
1
1
1
1
2
3
1
3
1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Next, what are relative integrated
peak intensities?
48
Split by 0 protons
(2) e
unsplit
d (2)
(2) f
o
o
c (2)
Split by 2 protons
Split by 5 protons
b (2)
a (3) Split by 2 protons
e
(1) g
f
1
1
1
1
1
2
3
1
3
1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
∫Intensity = 5
2
2
2
3
Next, what are relative chemical shifts of these peaks?
49
(2) e
d (2)
(2) f
o
c (2)
b (2)
a (3)
e
(1) g
o
f
d
a,b
e-f
50
c
Split by 0 protons
(2) e
unsplit
d (2)
(2) f
o
o
c (2)
Split by 2 protons
Split by 5 protons
b (2)
a (3) Split by 2 protons
e
(1) g
f
1
1
1
1
Most highly shielded
(largest + chemical shift)
1
2
3
Least shielded
(smallest + chemical shift)
1
3
1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
∫Intensity = 5
2
2
2
3
Order might be shifted
So identify by intensity
51
o
Answer
52