主题：剑桥大学NMR讲义

1 What this course is about
This course is aimed at those who are already familiar with using NMR on a
day-to-day basis, but who wish to deepen their understanding of how NMR
experiments work and the theory behind them. It will be assumed that you are
familiar with the concepts of chemical shifts and couplings, and are used to
interpreting proton and 13C spectra. It will also be assumed that you have at
least come across simple two-dimensional spectra such as COSY and HMQC
and perhaps may have used such spectra in the course of your work. Similarly,
some familiarity with the nuclear Overhauser effect (NOE) will be assumed.

That NMR is a useful for chemists will be taken as self evident.
This course will always use the same approach. We will first start with
something familiar – such as multiplets we commonly see in proton NMR
spectra – and then go deeper into the explanation behind this, introducing
along the way new ideas and new concepts. In this way the new things that
we are learning are always rooted in the familiar, and we should always be
able to see why we are doing something.
In NMR there is no escape from the plain fact that to understand all but
the simplest experiments we need to use quantum mechanics. Luckily for us,
the quantum mechanics we need for NMR is really rather simple, and if we
are prepared to take it on trust, we will find that we can make quantum mechanical
calculations simply by applying a set of rules. Also, the quantum
mechanical tools we will use are quite intuitive and many of the calculations
can be imagined in a very physical way. So, although we will be using quantum
mechanical ideas, we will not be using any heavy-duty theory. It is not
necessary to have studied quantum mechanics at anything more than the most
elementary level.

Inevitably, we will have to use some mathematics in our description of
NMR. However, the level of mathematics we need is quite low and should not
present any problems for a science graduate. Occasionally we will use a few
ideas from calculus, but even then it is not essential to understand this in great
detail.
Course structure
The course is accompanied by a detailed set of handouts, which for convenience
is divided up into “chapters”. You will notice an inconsistency in the
style of these chapters; this comes about because they have been prepared (or
at least the early versions of them) over a number of years for a variety of
purposes. The notes are sufficiently complete that you should not need to take
many extra notes during the lectures.
Each chapter has associated with it some exercises which are intended to
illustrate the course material; unless you do the exercises you will not understand
the material. In addition, there will be some practical exercises which
1–2 What this course is about
involve mainly data processing on a PC. These exercises will give you a feel
for what you can do with NMR data and how what you see relates to the
theory you have studied. Quite a lot of the exercises will involve processing
experimental data.
Chapter 2 considers how we can understand the form of the NMR spectrum
in terms of the underlying nuclear spin energy levels. Although this
approach is more complex than the familiar “successive splitting” method for
constructing multiplets it does help us understand how to think about multiplets
in terms of “active” and “passive” spins. This approach also makes it
possible to understand the form of multiple quantum spectra, which will be
useful to us later on in the course. The chapter closes with a discussion of
strongly coupled spectra and how they can be analysed.
Chapter 3 introduces the vector model of NMR. This model has its limitations,
but it is very useful for understanding how pulses excite NMR signals.
We can also use the vector model to understand the basic, but very important,
NMR experiments such as pulse-acquire, inversion recovery and most
importantly the spin echo.
Chapter 4 is concerned with data processing. The signal we actually
record in an NMR experiment is a function of time, and we have to convert
this to the usual representation (intensity as a function of frequency) using
Fourier transformation. There are quite a lot of useful manipulations that we
can carry out on the data to enhance the sensitivity or resolution, depending
on what we require. These manipulations are described and their limitations
discussed.
Chapter 5 is concerned with how the spectrometer works. It is not necessary
to understand this is great detail, but it does help to have some basic
understanding of what is going on when we “shim the magnet” or “tune the
probe”. In this chapter we also introduce some important ideas about how the
NMR signal is turned into a digital form, and the consequences that this has.
Chapter 6 introduces the product operator formalism for analysing NMR
experiments. This approach is quantum mechanical, in contrast to the semiclassical
approach taken by the vector model. We will see that the formalism
is well adapted to describing pulsed NMR experiments, and that despite its
quantum mechanical rigour it retains a relatively intuitive approach. Using
product operators we can describe important phenomena such as the evolution
of couplings during spin echoes, coherence transfer and the generation of
multiple quantum coherences.
Chapter 7 puts the tools from Chapter 6 to immediate use in analysing
and understanding two-dimensional spectra. Such spectra have proved to be
enormously useful in structure determination, and are responsible for the explosive
growth of NMR over the past 20 years or so. We will concentrate on
the most important types of spectra, such as COSY and HMQC, analysing
these in some detail.
Chapter 8 considers the important topic of relaxation in NMR. We start
out by considering the effects of relaxation, concentrating in particular on
the very important nuclear Overhauser effect. We then go on to consider the
sources of relaxation and how it is related to molecular properties.
1–3
Chapter 9 does not form a part of the course, but is an optional advanced
topic. The chapter is concerned with the two methods used in multiple pulse
NMR to select a particular outcome in an NMR experiment: phase cycling
and field gradient pulses. An understanding of how these work is helpful in
getting to grips with the details of how experiments are actually run.
Texts
There are innumerable books written about NMR. Many of these avoid any
serious attempt to describe how the experiments work, but rather concentrate
on the interpretation of various kinds of spectra. An excellent example of
this kind of book is J. K. M. Sanders and B. K. Hunter Modern NMR
Spectroscopy (OUP).
There are also a number of texts which take a more theory-based approach,
at a number of different levels. Probably the best of the more elementary
books if P. J. Hore Nuclear Magnetic Resonance (OUP).
For a deeper understanding you can do no better that the recently published
M. H. Levitt Spin Dynamics (Wiley).
Acknowledgements
Chapters 2 to 5 have been prepared especially for this course. Chapters 6, 7
and 8 are modified from notes prepared for summer schools held in Mishima
and Sapporo (Japan) in 1998 and 1999; thanks are due to Professor F Inagaki
for the opportunity to present this material.
Chapter 9 was originally prepared (in a somewhat different form) for an
EMBO course held in Turin (Italy) in 1995. It has been modified subsequently
for the courses in Japan mentioned above and for another EMBO course held
in Lucca in 2000. Once again I am grateful to the organizers and sponsors of
these meetings for the opportunity to present this material.
Finally, I wish to express my thanks to Professor AJ Shaka and to the
Department of Chemistry, University of California, Irvine, for the invitation
to give this course. The University of Cambridge is acknowledged for a period
of study leave to enable me to come to UC Irvine.
James Keeler
University of Cambridge, Department of Chemistry
March 2002
James.Keeler@ch.cam.ac.uk
www-keeler.ch.cam.ac.uk

2 NMR and energy levels
E2
E1
hν = E2–E1 ν
energy
levels
spectrum
Fig. 2.1 A line in the spectrum
is associated with a transition
between two energy levels.
The picture that we use to understand most kinds of spectroscopy is that
molecules have a set of energy levels and that the lines we see in spectra
are due to transitions between these energy levels. Such a transition can be
caused by a photon of light whose frequency, ν, is related to the energy gap,
E, between the two levels according to:
E = hν

where h is a universal constant known as Planck’s constant. For the case
shown in Fig. 2.1, E = E2 − E1.
In NMR spectroscopy we tend not to use this approach of thinking about
energy levels and the transitions between them. Rather, we use different rules
for working out the appearance of multiplets and so on. However, it is useful,
especially for understanding more complex experiments, to think about
how the familiar NMR spectra we see are related to energy levels. To start
with we will look at the energy levels of just one spin and them move on
quickly to look at two and three coupled spins. In such spin systems, as they
are known, we will see that in principle there are other transitions, called
multiple quantum transitions, which can take place. Such transitions are not
observed in simpleNMR spectra, but we can detect them indirectly using twodimensional
experiments; there are, as we shall see, important applications of
such multiple quantum transitions.
Finally, we will look at strongly coupled spectra. These are spectra in
which the simple rules used to construct multiplets no longer apply because
the shift differences between the spins have become small compared to the
couplings. The most familiar effect of strong coupling is the “roofing” or
“tilting” of multiplets. We will see how such spectra can be analysed in some
simple cases.
2.1 Frequency

2.1 Frequency and energy: sorting out the units
NMR spectroscopists tend to use some rather unusual units, and so we need
to know about these and how to convert from one to another if we are not to
get into a muddle.
Chemical shifts
It is found to a very good approximation that the frequencies at which NMR
absorptions (lines) occur scale linearly with the magnetic field strength. So,
if the line from TMS comes out on one spectrometer at 400 MHz, doubling
the magnetic field will result in it coming out at 800 MHz. If we wanted
to quote the NMR frequency it would be inconvenient to have to specify the
exact magnetic field strength as well. In addition, the numbers we would haveto quote would not be very memorable. For example, would you like to quote
the shift of the protons in benzene as 400.001234 MHz?

We neatly side-step both of these problems by quoting the chemical shift
relative to an agreed reference compound. For example, in the case of proton
NMR the reference compound is TMS. If the frequency of the line we are
interested in is ν (in Hz) and the frequency of the line from TMS is νTMS
(also in Hz), the chemical shift of the line is computed as:
δ =
ν − νTMS
νTMS
. (2.1)
As all the frequencies scale with the magnetic field, this ratio is independent
of the magnetic field strength. Typically, the chemical shift is rather small
so it is common to multiply the value for δ by 106 and then quote its value
in parts per million, or ppm. With this definition the chemical shift of the
reference compound is 0 ppm.
δppm = 106 ×
ν − νTMS
νTMS
. (2.2)
Sometimes we want to convert from shifts in ppm to frequencies. Suppose
that there are two peaks in the spectrum at shifts δ1 and δ2 in ppm. What is
the frequency separation between the two peaks? It is easy enough to work
out what it is in ppm, it is just (δ2 − δ2). Writing this difference out in terms
of the definition of chemical shift given in Eq. 2.2 we have:
(δ2 − δ1) = 106 ×
ν2 − νTMS
νTMS − 106 ×
ν1 − νTMS
νTMS
= 106 ×
ν2 − ν1
νTMS
.
Multiplying both sides by νTMS now gives us what we want:
(ν2 − ν1) = 10−6 × νTMS × (δ2 − δ1).
It is often sufficiently accurate to replace νTMS with the spectrometer reference
frequency, about which we will explain later.
If we want to change the ppm scale of a spectrum into a frequency scale we
need to decide where zero is going to be. One choice for the zero frequency
point is the line from the reference compound. However, there are plenty of
other possibilities so it is as well to regard the zero point on the frequency
scale as arbitrary.
Angular frequency
Frequencies are most commonly quoted in Hz, which is the same as “per
second” or s−1. Think about a point on the edge of a disc which is rotating
about its centre. If the disc is moving at a constant speed, the point returns
to the same position at regular intervals each time it has competed 360◦ of
rotation. The time taken for the point to return to its original position is called
the period, τ .

2.2 Nuclear spin and spin states 2–3
time
start
one
period
Fig. 2.3 A point at the edge of a
circle which is moving at a
constant speed returns to its
original position after a time
called the period. During each
period the point moves through
2π radians or 360◦.
The frequency, ν, is simply the inverse of the period:
ν =
1
τ
.
For example, if the period is 0.001 s, the frequency is 1/0.001 = 1000 Hz.
There is another way of expressing the frequency, which is in angular
units. Recall that 360◦ is 2π radians. So, if the point completes a rotation in
τ seconds, we can say that it has rotated though 2π radians in τ seconds. The
angular frequency, ω, is given by
ω =
2π
τ
.
The units of this frequency are “radians per second” or rad s−1. ν and ω are
related via
ν =
ω
2π
or ω = 2πν.
We will find that angular frequencies are often the most natural units to use in
NMR calculations. Angular frequencies will be denoted by the symbols ω or
     whereas frequencies in Hz will be denoted ν.

Energies
A photon of frequency ν has energy E given by
E = hν
where h is Planck’s constant. In SI units the frequency is in Hz and h is in
J s−1. If we want to express the frequency in angular units then the relationship
with the energy is
E = h
ω
2π
= ¯hω
where ¯h (pronounced “h bar” or “h cross”) is Planck’s constant divided by
2π.
The point to notice here is that frequency, in either Hz or rad s−1, is directly
proportional to energy. So, there is really nothing wrong with quoting
energies in frequency units. All we have to remember is that there is a factor
of h or ¯h needed to convert to Joules if we need to. It turns out to be much
more convenient to work in frequency units throughout, and so this is what we
will do. So, do not be concerned to see an energy expressed in Hz or rad s−1.

2.2 Nuclear spin and spin states
NMR spectroscopy arises from the fact that nuclei have a property known as
spin; we will not concern ourselves with where this comes from, but just take
it as a fact. Quantum mechanics tells us that this nuclear spin is characterised
by a nuclear spin quantum number, I . For all the nuclei that we are going
2–4 NMR and energy levels
to be concerned with, I = 12
, although other values are possible. A spin-half
nucleus has an interaction with a magnetic field which gives rise to two energy
levels; these are characterised by another quantum number m which quantum
mechanics tells us is restricted to the values −I to I in integer steps. So, in
the case of a spin-half, there are only two values of m, −1
2 and +12
Strictly, α is the low energy state .
for nuclei with a positive
gyromagnetic ratio, more of
which below.
By tradition in NMR the energy level (or state, as it is sometimes called)
with m = 12
is denoted α and is sometimes described as “spin up”. The state
with m = −1
2 is denoted β and is sometimes described as “spin down”. For
the nuclei we are interested in, the α state is the one with the lowest energy.
If we have two spins in our molecule, then each spin can be in the α or
β state, and so there are four possibilities: α1α2, α1β2,β1α2 and β1β2. These
four possibilities correspond to four energy levels. Note that we have added a
subscript 1 or 2 to differentiate the two nuclei, although often we will dispense
with these and simply take it that the first spin state is for spin 1 and the second
for spin 2. So αβ implies α1β2 etc.
We can continue the same process for three or more spins, and as each spin
is added the number of possible combinations increases. So, for example, for
three spins there are 8 combinations leading to 8 energy levels. It should be
noted here that there is only a one-to-one correspondence between these spin
state combinations and the energy levels in the case of weak coupling, which
we will assume from now on. Further details are to be found in section 2.6.