“Education is what remains after
you forget what you learned.”
Unknown NMMI instructor, ca. 1974.
Who is he?
“Rule #54. Avoid virgins. They’re too clingy.”
--Wedding Crashers (2005)
Spectroscopy of Macromolecules
Danger: this is “what you need to know.” One
could easily spend a whole semester on this alone.
For chemistry in general, spectroscopy is more often
applied than, say, scattering, but…many other courses
teach it better.
Focus on these three:
2. Circular Dichroism/ORD
Reference: VanHolde (newer edition, “Physical Biochemistry” but
the older editions are special, if you can still get them) 3
Spectroscopy ↔ Quantum Mechanics
A Postulate View of Quantum Mechanics
1. A system (e.g. molecule) of n particles (electrons,
nuclei) is described by wavefunctions (q1,q2…q3n,t)
that describe locations (q1,q2,q3) of each particle at
2. The probability of finding the system in the differential
volume element d3V at time t is the squared complex
modulus of the wave functions:
3. Physical observables (dipole moment, energy,
momentum, etc.) are replaced with Hermitian
operators that might typically take some kind of
derivative of the wave function. Hermitian will be
explained later. The operator is represented by P .ˆ
4. The average value of some physical property is
* P d 3 q
This should resemble our now-familiar average of
something is probability times something divided
by a normalizing sum of all somethings. The
wavefunction just sandwiches the something like
bread around meat.
5. The operators for position, momentum, time and energy
are as shown in this table.
Classical Observable Quantum
q position ˆ
t time t
where h 6.63x1034 J s Planck' s cons tan t
Position and time pass through unchanged;
momentum and energy are filtered through the calculus.
6. The wavefunction is determined from
The Hamiltonian operator H follows from a
classical mechanics system worked out by
Hamilton, where the classical operator was
H = K + U, the sum of kinetic and potential
energies. This equation ties the action of the
energy operator to how the wave function
responds. Compare this to Fick’s 2nd law:
Dd2c/dx2 = - dc/dt. Like concentration, a
wavefunction describes where something is.
Why Quantum Mechanics is Hard
Because we don’t study enough classical mechanics.
Debye is said to have seen very little new in Quantum Mechanics; at
least, the math is common to other stuff…if you have studied
enough other stuff…which hardly any of us do!
So far, we have seen that expectation values are similar to
other averages we have computed: sum of probability
times thing, divided by sum of probabilities to normalize.
The wave function “sandwich” is new…and often
associated with a particularly simple matrix mathematics.
We also see that Schrodinger’s equation resembles
Fick’s equation….which in turn resembles the heat
flow equation all engineers learn.
Eigenfunctions may permit us to replace calculus
Sometimes, the operation on the wavefunction can be
replicated simply by multiplying that same wavefunction by a
constant. Examples include some trig and exponential
sin( kx) k sin( kx)
Replacing calculus by simple multiplication? Good deal!
A fair amount of time in quantum mechanics and other
disciplines is spent trying to find functions that will permit
such simplification…and the constants (-k2 in this case) that
make it so.
Basis sets let us create functions by
summation of other functions.
It sometimes works out that a set of eigenfunctions
can be used to represent other functions. We say
the desired function can be expanded in the set of
eigenfunctions. Compare this to writing a vector
in terms of unit vectors.
s s x x sy y sz z
ˆ ˆ ˆ
Eigenfunctions are most useful when they are orthogonal and
complete, meaning that they do not project on to one another
and are sufficient to express arbitrary functions (again, compare
the traditional unit vectors). Hermitian operators satisfy this.
A simple example of an operator:
kinetic energy only
Classically, kinetic energy is Ek=mv2/2 = p2/2m since p =
mv. Check the table of Postulate 5 to get the QM
ˆ ˆ ˆ
x j k
The operator is discussed in the Math Tuneup, as is its “square”, 2, the Laplacian.
Time may not matter.
Often, the time-dependence of (q,t) is particularly boring; it
might just be an oscillation that can be factored out from
the positional, q-dependent part.
(q, t ) (q ) (t ) (q )e i ( E / )t
This case is appropriate for stationary operators—ones with no time
dependence. If you put the above equation into the full, time-dependent
Schrodinger’s equation of Postulate 6, you get (see VanHolde—it is very
easy) the simpler, time-independent Shrodinger’s equation, which is an
Eigenequation with the particularly interesting and useful Eigenvalue, E.
H (q ) E (q )
H-sigh equals E-sigh. Another equation for permanent memory storage.
Van Holde treats all the usual
These notes derive from a modern VanHolde (VanHolde,
Johnson & Ho). The authors march through systems
you probably saw in PChem already, such as:
– Free particle
– Particle in box
– Hydrogen atom
– Approximate solutions (perturbation)
– Small molecules (LCAO)
In time, I hope this presentation will grow to cover some of
those subjects, at least briefly. Meanwhile, I hope the
foregoing made QM seem less weird.
For now, we need a leap of reasonableness.
Leap of Reasonableness
QM is the explanation of things we know about
atoms, going all the way back to Dalton’s Law of
QM explains why it’s CH4, not C1.251H3.785
It’s those stupid waves; together with boundary
conditions (like the electron has to be
somewhere) they give constructive & destructive
interferences—nodes—which makes Chemistry
an integer science.
Think laser cavity, think wave trough, think
booming bass at some positions in a room.
For particle existence to be tied to wave amplitude
is tantamount to saying: there are discrete
energy levels associated with standing waves.
Example: nearly harmonic vibration potential.
Weird quantum-mechanical things:
•There is a zero-point energy.
•Discrete states are separated by near multiples of the zero point energy.
•More energy = more nodes in the wave function = fancier dancing by
•If we consider an energy diagram for electronic energy levels, not vibrational
levels, that fancier dancing corresponds to more complex orbitals—e.g.,
complicated f orbitals instead of simpler s orbitals.
E = h
QM is useful to describe transitions
between atomic or molecular energy states.
• Beer-Lambert Law (see Van Holde)
• Einstein-Planck Coefficients (to be added;
meanwhile, I could weep, these authors make it
• Transition dipole
• Orientation of transition dipole Let’s do these for now
Absorption vs. Scattering
• Earlier, we talked about how light grabs
electrons and shakes them to produce
scattering. We used the analogy of a boat
floating on rough seas; it produces a little
ripple as it bobs up and down.
• Molecules that absorb light do the
grabbing instead. This stronger interaction
is more like the water going over Niagara
(this is a sentence, not a government program)
Van Holde (Ch.8, p 373 et seq.) shows that a
transition from one quantum state to another
(absorption, emission) occurs when:
1) E = h
2) the transition dipole fi is finite and makes a
strong projection on the electric field.
Transition strength E o fi
ˆ i (q)d 3V
E o fi
Think of the initial state as rolling along the
runway. The wings catch some air and—
voila!—transition to vertical acceleration. In this
case, the initial and final vectors (oops,
wavefunctions) are pictured as orthogonal. In
QM transitions, this may or may not be allowed.
The main thing is: the transition operator (wings)
somehow couples one state (horizontal motion)
into another (vertical motion).
Selection rules: will it fly?
Quantum mechanical transitions depend on:
1) how the transition dipole aligns to the
2) obscure rules regarding how the field
being operated on (initial wavefunction)
relates to the new field (final
wavefunction). Some transitions are
“forbidden”—meaning they happen
E o fi It matters
VanHolde Fig. 8.15
UV absorption of crystalline methylthymine
Franck-Condon Principle, Jablonski Diagram
1st excited state
U emission Ground state
Evib = h
But it’s not that simple
Opposing electron spins: singlet (ground state shown)
Aligned electron spins: triplet (excited state shown)
ENERGY-TIMING DIAGRAM FOR ABSORPTION,
FLUORESCENCE & PHOSPHORESCENCE
Internal conversion ~10-12 s Absorption ~10-15 s
crossing ~10-8 s
Phosphorescence, after ~10-8 s delay
instantaneous after delay
of 0.0001 to 100 s
VanHolde Fig 11.2 shows more detail, if you want it.
Fluorescence solvent effects
Example: is that protein aggregating?
Put a hydrophobic probe in (pyrene?) and
see if it “lights up” to indicate aggregation
across the hydrophobic patch.
Example: is that arborol self-assembling?
Same solution to similar problem as above.
Fluorescence effects not always a
tool, sometimes a nuisance.
For FPR, the dye can get quenched and not
This can happen as a result of variables like
pH or salt.
Dye can self-quench if too concentrated.
That is again a tool: calcein leakage test for
Fluorescence Machines Can be Simple
Turner fluorometer, filter
design. This is just
an LS machine with
Remember our timing diagram….Unlike light
scattering, fluorescence is not instantaneous.
It is a little bit more like somehow preparing
radioactive elements: excited molecules
spontaneously decide they have had
enough of life in the fast lane and return to
their ground state. Some tolerate the
excited state longer than others.
Fluorescence Decay: yet another
exponential for us to learn.
at first later
Define: N = number of molecules in the
The number dN decaying back to ground state
is proportional to the number available in the
N The number N is
N (t ) reported by the
ln k (t t o ) proportionate
No number dN that
N (t ) N o e k ( t t o ) 33
Fluorescence decay instruments
are much more complex beasts.
Measure lifetimes to:
probe environmental conditions (viscosity, pH, hydrophobicity)
Infer size, shape of molecules
Infer distance between different parts of molecule.
“Decompose” fluorescence spectra
by components (when spectral features
overlap, time can separate them)
Quantum yield indicates environment.
The minimum rate of decay would be the Einstein A value
for spontaneous emission, which can be calculated from
spectral width (VanHolde Eq. 8.102)
The actual decay rate is higher, due to internal conversion,
intersystem crossing, nonradiative transfer and any
stimulated emission processes from interactions with
Quantum yield is defined as: q = A/k
It is a sensitive indicator of the environment of the dye; q
often increases when a dye binds to a molecule. Free
dye may be quenched (see previous slides).
Another way to think of quantum yield: ratio
of (visible) photons in to photons out.
Some dyes are very efficient light
converters, with quantum yields
approaching 100%. I
Fluorescein (which we often use in
FPR) is so efficient that it is hard emitted
to know how efficient it is. About
90%-100% has been reported. l
High efficiency is a good thing for
FPR: efficient light conversion
means little heat production, less
Fluorescence Resonance Energy Transfer—
There’s nothing to FRET about.
The emitting group does not have to be the same
as the absorbing group.
Donor + Acceptor Donor + Acceptor
Acceptor Acceptor +
Absorbing is easy; transfer grows more
difficult with distance.
Physics problem: Requires 1 (r / Ro ) 6
transition dipole interaction
between donor and Ro = 6 – 45 Å depending
acceptor. Dipole-dipole on the DA chemistry and
solvent. See VanHolde
interactions go like r 6. Table 11.1 for examples.
I Chemistry problem: Donor’s F-
Donor F spectrum must significantly
Acceptor A overlap acceptor’s A-spectrum.
Typical D-A pairs are easily found from the
old Molecular Probes site (now Invitrogen).
(this link gives several D-A pairs & Forster radius.
(methyl ester), perchlorate)
Look up other pairs.
Are donors usually
smaller than acceptors???
D-A is a valuable tool for estimating
inter- or intramolecular distances.
1.0 1 (r / Ro )6
???Does labeling a
0.0 0.5 1.0 1.5 2.0 molecule (twice!)
r/Ro change it? Maybe.
Cyclic AMP (adenosine monophosphate)
Na+ - P O
C-AMP-dependent protein kinase
• 2 regulatory subunits
• 2 catalytic subunits
C-AMP dissociates protein kinase
This is followed by labeling fluorescein to the catalytic
subunit & rhodamine to the regulatory subunit. When
kinase is in compact form, the fluorescein fluorescence
is donated to the rhodamine acceptor. When C-AMP
opens the structure up, fluorescein fluorescence grows. 42
Polarization of fluorescence can be
used to measure a new transport
property, rotational diffusion.
Light scattering is instantaneous, so draw a donut
around the induced dipole and that’s what
radiation pattern you will get.
Fluorescence and, especially, phosphorescence
are slower and the molecule may rotate, taking
the (emission) dipole with it, before emission.
Draw a donut around the dipole at the time of
g is in a molecular frame of
Reference; this is not the
Absorption transition dipole rotation of the molecule but
the different transition dipole of
Emission transition dipole emission and absorption; for
example, the molecule might
distort a bit on absorption, so
emission would be in some
Vertical incident light new direction.
If molecule did not rotate (subscript
You can measure g by
ro (3 cos 2g 1) freezing out the
5 molecule in some
viscous solvent, cool it,
If the molecule does rotate…
Then r will deviate from ro, tending more towards
zero. (If ro was positive, r goes down).
You can use this to estimate rotational diffusion
coefficients, or at least to track how they are
changing due to binding, adhesion or
Problem: it’s the rotation of the fluorphore, not
necessarily the whole molecule.
According to VanHolde, fluorometry is the only
spectroscopic method that senses changes in
molecular weight (e.g., due to aggregation).
What do you think of that assertion?
This is called steady state fluorescence
In this equation, is the
r fluorescence decay time and is a
rotational correlation time
(proportional to the inverse of
rotational diffusion coefficient).
You may recall that Dr = kT/8hR3
from our discussion of Hv DLS.
So this means that 1/Dr or
1 1 represent volume.
Perrin plot: r -1 vs T
How would you use this?
1 1 kT
r ro Vh (T ) r
From intercept, get ro, which is related to g. T/h
Use slope to estimate /V.
If is known from fluorescence decay (or maybe even literature) then obtain V.
Plot not straight?
Could be nonspherical shape, proteins changing with temperature (denaturing),
binding to stuff as a function of temperature, etc.
The well-heeled & talented can do
pulsed, polarized fluorescence…
A recent (and unusual, I think)
application of this is to early
detection of cancer:
r (t ) roe
Van Holde Fig. 11.20
Everything you ever wanted to know about
Fluorescence but were afraid to ask.
Lakowicz: Principles of Fluorescence
A source of fluorphores: probes.invitrogen.com
(worth linking again)
• Chapter 10 stuff from VanHolde.
• Theme of it: molecular birefringence.
Plane-polarized light can be thought of as
two counter-rotating circularly polarized
Circular dichroism, the selective absorption
of left or right-circularly polarized light, is
used to characterize chirality.
If the solvent absorbs heavily in the UV, CD is
probably not possible. Then we use
spectropolarimetry, a.k.a. Optical Rotatory
Have you seen this before?
You could have a whole lifetime in NMR!
The main application is to molecular fingerprinting:
NMR is the primary tool that lets chemists know
they have the right structure.
Increasingly, it competes with X-ray diffraction for
We will do just enough to introduce the variations
that tell us something about polymer dynamics.
This can be internal dynamics of bonds or,
intriguingly, diffusion in complex systems.
Ref: Van Holde, Ch. 12 and articles containing the word, DOSY
Crystallography could not do this.
R. Tycko, Biochemistry 2003, 42 (11), 3151.
NMR is about nuclear spin flips
Spin quantum numbers are multiples of ½. For protons, the choices
are + ½ and –½ . We may just call this + and -.
How do we know? Because we can see energy get absorbed to change the spins, but only when a
magnetic field is applied. The phenomenon is obviously in the quantum domain, but it is reminiscent
of classical spin of a charge, which generates a magnetic dipole.
No energy difference without H. The bigger
H, the bigger E and the faster spin gyrates.
E gH 2mz
Rate of precession where g the "gyromagne tic ratio"
Larmor Frequency: (depends on nucleus)
gH mz is the component of the
gyrating spin along the z axis 60
Nuclear spins in a magnetic field can be compared to
actual spin in a gravitational field.
Play with gyroscope: if we want to deflect its motion,
what is the right stimulus?
NMR = Low-energy spectroscopy
l/nm /MHz Wavenumbers/cm-1 Probes
UV 200 1,500,000,000 50000 Electronic
Visible 500 600,000,000 20000 Electronic
Near Infrared 1000 300,000,000 10000 Vibrational
Infrared 5000 60,000,000 2000 Vibrational
NMR 300000000 1,000 0.033 Nuclear Spin
DLS NA:l <1 NA:l Translation
Today’s biggest NMR is about 900 MHz
The energies associated with nuclear spin flips (from + to -) are much less than
the thermal energy, kT: E+ - E- 0. This is true even in high magnetic fields
( E E ) Unlike vibrational & electronic
1 spectroscopy, most molecules
are not in the ground state. In
N fact, only a tiny excess of
molecules is in the ground state. 62
Net magnetization of lots of spins
( E E ) +1/2 low energy H
The condition e kT
-1/2 high energy
M = net magetization
There is no coherence among spins; they
just rotate in random phase.
If you count very carefully, you’ll see there
are more low-E spins than high-E spins.
Note: there is a good chance the signs are backwards—who cares?
The slight excess of low-E spins sums to produce
a net magnetization (heavy arrow). Because there
are lots of spins, randomly phased, this heavy 63
arrow really sums up parallel to the z-axis.
Old-time continuous wave NMR spectrometers
swept field because sweeping R.F. is hard to do.
Sample in NMR tube ~0.5 mL
e.g. 30 MHz
Big, permanent magnet
4. This stimulates signal Hx detected by
1. Imagine we turn off R.F. receiver coil.
2. Inside the cell, spins precess; 5. Sweep bugger magnet to scan for
random phase, so M points along z. resonance of nuclei in different
3. Apply coherent R.F. (Hy) Now M inverted & environments (scan about 10 ppm for
coherent & it precesses (heavy blue arrow). protons). 64
If NMR were still like this…
It would still be the most important weapon
in the chemist’s arsenal.
Even at low magnetic fields, the
“fingerprinting” of simple
molecules is easily achieved.
Modern Fourier Transform NMR is
Very strong superconducting magnets.
The magnet is not swept, but sometimes spatial gradients in field
“White light” excitation through RF pulses excites all nuclei at
Relaxation experiments not different in basic concept from
Imaging possible; changes in “image” can be related to
What is the Fourier Transform of “Ping”?
A short pulse
corresponds to a
region of nearly
t constant intensity.
Just one example of inverse
relationships. Here’s another: in
optics, if you want to image a tiny
spot in a microscope, you need to
gather scattered light off of it
through a very wide angle
objective (high numerical aperture
or low “f number”). w 67
Actual pulses are short-duration signals
oscillating near the Larmor frequency, o
o t F.T. w
~10sec What would the I(w) spectrum
look like if the number of
o ~ 200 MHz oscillations were 106?
o = # of oscillations = 2000 68
We need a bit more about how it works.
Pulses are delivered along the x-direction.
It’s a coherent pulse, with the energy and duration designed to bring
half the spins in the ground state into the excited state: no net
magnetization along z axis.
It does this with phase coherence, so M rotates in the the x-y plane.
A coil in the x-direction can sense (“acquire”) the oscillating projection
of M onto that direction.
These pulses, and the subsequent acquisition are much faster than
sweeping the field as in a CW instrument. For almost all
applications, FT-NMR has taken over.
x y x y
Kids have to hang arm-to-arm so the last one can ring bell.
Maybe they even hold hands so someone can reach out to do it!
Relax—the nuclei do!
Over time, some of the spins in the high-energy
state fall back to the low energy state. This
enthalpic process happens over a characteristic
time, T1. If no one is on the merry-go-round, the
bell never rings.
Over another time, kids get sick of cooperating.
They lose phase. Although a given group (say,
those wearing red shirts) goes around as often
as they ever did, they can no longer reach out
and touch the bell. This entropic dephasing time
is called T2.
T1 and T2 are the main physical decay terms.
•Spins return to
z T1 z •Enthalpic
x y x y
z T2 z •Entropic
x y x y
FID = Free Induction Decay
Why again do they oscillate?
Why again do they decay?
FID for one nucleus.
-1000 0 1000 2000 3000 4000 5000 6000 7000
FID for two nuclei (very different frequencies).
-1000 0 1000 2000 3000 4000 5000 73 7000
You have to FT your FID.
Frequencies of oscillation give NMR frequencies.
You get strengths at those frequencies, too!
Rapidity of decay controls width of peaks.
An efficient algorithm for doing this was invented
ca. 1950 by Cooley-Tukey and is called Fast
“Window” functions should be applied to limit the
effects of sudden start/stop of acquired dta.
Pulse terminology is easiest in the rotating frame:
as you spin at the Larmor frequency, look up, out or down.
The pulse that was used to bring half the
ground state spins into the excited state,
resulting in coherent oscillation of M in the
x-y plane is called a 90o pulse.
x y x y
A longer and/or more energetic pulse will
take the excess of protons in the ground
state and create a similar excess in the
x y x y
Measuring T1: Inversion Recovery Pulse Sequence*
t FT w
Spins are dunked upside-down, given a time, ,to right themselves by spin-lattice
relaxation. Some spin-spin occurs, too, but doesn’t matter because spins are
refocused with the 90o pulse and measured. The signal depends on the time,.
Imax Waterfall plot: you actually
see something like this for
all of the peaks in the
Imax /e spectrum.
*Just try typing that into Google.
Measuring T2: Spin Echo Pulse Sequence
90 180 Amount of echo goes
down with time due to
Still coherent Some dephasing now
z z z
x y x y x y
A study by 3M used NMR relaxation data to follow
polymer drying during solvent evaporation.
A. ERREDE' and RICHARD A. NEWMARK
Journal of Polymer Science: Part A Polymer Chemistry, Vol. 30,11561161 (1992)
T1, T2 can be used to assess
crystallinity, glass transitions.
T1 >= T2
Fast-moving nuclei (left side of plot)
relax by real motion. Slow nuclei (crystals)
relax by spin dephasing.
The motion time of the nucleus is
called the correlation time, c.
The DOSY Dog Track: NMR is for the dogs.
First, imagine that T2 dephasing is nil.
At the 90 pulse, all the dogs (spins) leave the starting gate
and go along the track however far they go.
Then the “come” command (180 pulse) is issued.
Dogs run back as fast as they ran forward, and bark as
the cross the line: echo!
Thanks to Frank Blum
Wet Dog Track
Suppose now the track is wet, with some lanes being
Dogs come back at the same time!
Unless they cross lanes!
Then, dog that went out on the slow lane might return on a
fast lane and be the first to bark. Dog that went out fast
can return slow and be the last to bark.
The spread in the time from first bark to last bark contains
information about how long it takes dogs to cross tracks
from slow to fast: Diffusion!
PFGNMR = DOSY
Gradient pulses: inhomogeneous magnetic field
= lanes that are faster or slower (spread of
The effect of the gradients is to distribute the echo over a wider time,
which lowers the maximum echo. The echo amplitude depends on a
parameter Q, determined by gradient strength, duration, and timing a
bit similar to DLS or FPR.
PFGNMR-DOSY Pros & Cons
Chemical specificity anyway (DOSY often keeps the identities of the protons):
measure diffusion of everything in a mixture.
Did I mention no labeling?
Works well precisely where DLS doesn’t: small diffusers.
(FPR works for small diffusers—provided the dye doesn’t mess up the
molecule being studied).
Data are very quiet.
Effectively no baseline issues.
Did I mention no labeling?
Slow diffusers: T2 might wipe out your signal before molecules diffuse much.
Does not span the wide range of times and diffusers that DLS or even FPR
Struggles with convection issues.
Diffusion can be over a very short range of space.
Is time-limited, not distance limited.
Software for it still sucks.
DLS and DOSY: one rises, one falls with concentration.
PEO Diffusivity x 1012(m2/S)
Dynamic Light Scattering
PEO MW 106 Da
0 1 2 3 4 5
PEO Concentration (g/L)
R. Cong, et al. Macromolecules, 2003, 36 (1), pp 204–209
Does PFGNMR/DOSY work for rodlike macromolecules?
Ernst von Meerwall--Akron
Figure 7. Diffusion coefficients plotted
against concentration for PSLG 57.5 KDa.
Ernst Von Meerwall &Cornelia Rosu made this result. 86
Superstars use DOSY, too.