1 NMR and ESR Spectroscopy K. R. Shamasundar and M. Nooijen University of Waterloo Introduction Nuclear Magnetic Resonance (NMR) spectroscopy and Electron Spin Resonance (ESR) spectroscopy are two widely used spectroscopic techniques to infer structure and properties of complex molecules (even bio-molecules such as proteins). Both these methods use angular momentum, .i.e., pure “spin” or “total angular momentum” of the relevant particles to extract molecular structural information. In ESR spectroscopy, molecules in a state containing unpaired electrons, .i.e., with non- zero spin-angular momentum (molecules in non-singlet states, 0 S ≠ ) are placed in constant magnetic field. The ESR spectrum resulting from transitions between molecular states with different S M components contains chemically relevant information. For example, the ESR spectrum of unpaired electrons in transition metal complexes contains information on how ligands are arranged around the metal ion. Fine structure results from the interaction between electrons and nuclear spin and this can serve as a fingerprint for molecular, nuclear and electron spin density. Likewise, in NMR spectroscopy, a composite system of nuclei in molecules with non- zero nuclear spins ( 0 I ≠ ) is placed in a constant magnetic field. Similarly, the NMR spectrum resulting from transitions between different states of nuclear spin system contains a wealth of information regarding chemical environment of the nuclei, and this is used to extract structural information of molecules. Despite their wide-spread use, the underlying principles of both these spectroscopic techniques are fairly simple. In fact, the machinery needed for a quantum mechanical description of the basic phenomena of these methods mainly involves the angular momentum theory we have discussed in the class. Additionally, we would require basic first-order (and some times second-order) perturbation theory, and variational method and related secular problem. The Hamiltonians used are simple enough to enable manual solution of first-order perturbation theory
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1
NMR and ESR Spectroscopy
K. R. Shamasundar and M. Nooijen
University of Waterloo Introduction
Nuclear Magnetic Resonance (NMR) spectroscopy and Electron Spin Resonance (ESR)
spectroscopy are two widely used spectroscopic techniques to infer structure and properties of
complex molecules (even bio-molecules such as proteins). Both these methods use angular
momentum, .i.e., pure “spin” or “total angular momentum” of the relevant particles to extract
molecular structural information.
In ESR spectroscopy, molecules in a state containing unpaired electrons, .i.e., with non-
zero spin-angular momentum (molecules in non-singlet states, 0S ≠ ) are placed in constant
magnetic field. The ESR spectrum resulting from transitions between molecular states with
different SM components contains chemically relevant information. For example, the ESR
spectrum of unpaired electrons in transition metal complexes contains information on how
ligands are arranged around the metal ion. Fine structure results from the interaction between
electrons and nuclear spin and this can serve as a fingerprint for molecular, nuclear and electron
spin density.
Likewise, in NMR spectroscopy, a composite system of nuclei in molecules with non-
zero nuclear spins ( 0I ≠ ) is placed in a constant magnetic field. Similarly, the NMR spectrum
resulting from transitions between different states of nuclear spin system contains a wealth of
information regarding chemical environment of the nuclei, and this is used to extract structural
information of molecules.
Despite their wide-spread use, the underlying principles of both these spectroscopic
techniques are fairly simple. In fact, the machinery needed for a quantum mechanical description
of the basic phenomena of these methods mainly involves the angular momentum theory we
have discussed in the class. Additionally, we would require basic first-order (and some times
second-order) perturbation theory, and variational method and related secular problem. The
Hamiltonians used are simple enough to enable manual solution of first-order perturbation theory
2
and the secular equations. Finally, use of simple symmetry arguments are helpful to reduce the
problem to smaller problems.
Magnetic moment of charged particles with Intrinsic Spin.
Both NMR and ESR involve appropriate (nucleus or electrons) charged particles with
non-zero intrinsic spin. The spin of individual particles is fixed and typically 1/2 or 1 (in units of
h ), although higher integral values are possible (and complicate the spectrum somewhat). A
charged particle possessing angular momentum is related to a classical charged particle rotating
around its axis. This gives raise to a magnetic (dipole) moment µr .
2
g Jqm
µ β
β
=
=
rr
(1)
Where, q and m are charge and mass of the particle, and Jr
is the total angular
momentum (spin or orbital or their combination) for the particle. The quantity β is known as
“magneton”, and g is known as the “g-factor”. The g-factor is unitless, and is typically of the
order of 1-10. Both these quantities are characteristics of the particle.
For electrons, β is negative (since electron charge q is negative) and is usually referred
to as “Bohr-magneton”, and is denoted by eβ . Usually, the negative sign in β is explicitly
written into Eqn (1), and it is rewritten as.
for electrons
2
e e e
ee
g Jem
µ β
β
= −=
rr
(1-e)
Therefore, for electrons the magnetic moment is in reverse direction of total angular
momentum Jr
. Using relativistic theory of electron, the g-factor can be shown to be 2 for a free
electron. For electrons in chemical environment, g-factor will depend on the molecular state.
Such electrons have non-zero spin angular momentum ( S ) as well as orbital angular momentum
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( L ), giving raise to a total angular momentum ( J ). For this case, the g-factor is given by Lande
factor as,
( 1) ( 1) ( 1)12 ( 1)e
J J L L S SgJ J
+ − + + += +
+ (2)
For ground and low-lying states of molecules, usually the orbital angular momentum ( L )
is zero or small 0L ≈ . Therefore, total angular momentum is almost equal to the spin-angular
momentum ( J S≈ ). In such a case, the g-factor is close to 2. For molecules involving unpaired
electrons in d-orbitals of transition metals, the g-factor can be significantly different from 2.
For nuclei, β is positive (because nuclear charge is positive), and is usually referred to as
“nuclear-magneton”, and is denoted by Nβ . For nuclei, Eqn (1) can be rewritten as,
for nuclei
2
N N N
NN
g Jqm
µ β
β
==
rr
(1-N)
Therefore, magnetic moment is the same direction as Jr
. The g-factor for nuclei depends
on constituent particles (protons and neutrons), and is not easy to calculate. For protons
(Hydrogen atom nucleus), g-factor is 5.58. For practical purposes, g-factor for a particular type
of nucleus is a fixed value, and is experimentally determined.
In literature, h appearing in Jr
is often absorbed into definition of magneton β and is
expressed in units of Joules per Tesla (J/T). In this case, magnetic moment is expressed in units
of “magnetons”. The magnitude of β determines the energy gap between the states whose
transitions are recorded, and sets the energy range (in the electromagnetic spectrum) where the
spectroscopy is observable. For electrons, the Bohr-magneton 2e
e
em
β =h is 9.2*10-24 J/T. For
magnetic field strengths used in practice (1-10 Tesla), this leads to transitions of few cm-1 and
falls in the micro-wave (one to a few hundreds of GHz) region. Since proton (and other nuclei)
are about 1000 times heavier than electrons, the nuclear-magneton for proton
( )2N p
p
eprotonm
β β= =h is about 1000 times smaller than Bohr-magneton. As a result, this leads
to transitions observable in radio-wave region (one to a few hundred MHz).
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Unlike electron which has spin 1/2, the spin of nucleus depends on individual spin of
nucleons (protons and neutrons). Again, for all practical purposes, spin of a particular type of
nucleus can be considered as a fixed value (integer or half-integer) determined theoretically or
experimentally. Most common nucleus used in NMR, the proton, has spin 1/2.
For our purposes, and for the purpose of chemists who use ESR and NMR, we will
consider β and g-factor as being provided to us. The product of β (without the h ) and g-factor
is known as gyro-magnetic ratio and is denoted byγ .
g
J
γ β
µ γ
=
=rr (3)
Again for nuclei, gyro-magnetic ratio is positive and for electrons it is negative. In
literature, h appearing in Jr
is not absorbed into gyro-magnetic ratio. As we shall see, the
magnitude of gyro-magnetic ratio (measured in per second per Tesla) determines the amount of
splitting of energy levels. While β sets the energy scale for the spectroscopy, γ determines the
sensitivity of the given spin-system within this scale. Within the energy scale set by β , different
spin-systems with different gyro-magnetic ratio will resonate at significantly different
frequencies.
Furthermore, the higher the value of gyro-magnetic ratio, the smaller the external
magnetic field required for achieving resonance transitions and the easier it is to observe them.
For example, the gyro-magnetic Hydrogen nucleus (proton) is about one-order more than that of 13C nucleus. Therefore, it is easier to observe proton nuclear resonances compared to 13C. It also
implies that a group of hydrogen nuclei in a given chemical environment resonate at different
frequencies than a group of 13C nuclei in the same chemical environment! This means that a
NMR of a group of nuclei of a given type in a given environment can be safely observed
disregarding groups of other nuclei in the same environment, if these groups of nuclei have
different gyro-magnetic ratios.
Interaction of the Magnetic moment with External Magnetic Field.
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When placed in external magnetic field, a magnetic dipole µr will interact with the
applied magnetic field Br
. The interaction potential energy of is given by (refer to MS Problem
13-49),
E Bµ= − •rr (4)
As can be seen, the interaction energy will be minimum when µr is in the same direction
as Br
. Therefore, a magnetic dipole will tend to align parallel to the magnetic field. Usually, the
direction of applied magnetic field is taken to be along the z-axis, and hence the energy is given
by,
0z z zE B Bµ µ= − = − (5)
The quantum mechanical Hamiltonian for this interaction is obtained by expressing zµ in
above expression by ˆz zJµ γ= obtained from Eq (3).
0 0ˆ ˆ
zH B Jγ= − (6)
The eigenstates of the are just the angular momentum eigenfunctions corresponding to a
total angular momentum quantum number. In absence of external magnetic field, these (2 1)J +
states of the spin-system with different axial quantum numbers ( J− to J+ ) are all degenerate. In
presence of magnetic field, this degeneracy completely breaks (known as Zeeman effect) and
energy levels split. Considering spin-1/2 systems (consisting of α and β levels), the magnitude
of the energy level splitting is given by,
0E Bγ∆ = h (7)
Exercises:
1. Prove Eq (7) for a single spin 1/2 system.
2. Derive the energy level splitting for a single spin system with total angular
momentum quantum number J .
3. For the system above, consider the selection rule that only transitions with 1JM∆ = ±
will be allowed. How many transitions will be observed? Does Eq (7) require any
modification?
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Note that the magnitude of the energy splitting depends on applied field strength.
Depending on the sign of gyro-magnetic ratio, either α state or β state will be lower in energy.
For electrons, β state will be lower, and for nucleus α state will be lower.
When electro-magnetic radiation of appropriate angular frequency (given by 0Bω γ= ) is
applied perpendicular to the direction of the constant magnetic field, .i.e., in x-y plane, the
system undergoes transition. This can be detected and recorded as a spectrum.
In the following, we will mainly consider the case of NMR spectroscopy. We will mainly
study quantum mechanical treatment of NMR spectroscopy using a many-particle Hamiltonian
obtained by generalization of Eq (6) to a system consisting of a group of Hydrogen nuclei (spin
1/2) in a given chemical environment (as in a molecule or a crystal). We will first study the
effect of chemical environment immediate to a given nuclei on its resonance positions, and
define chemical shift. We will then consider the finer effects of spin-spin interactions between
these nuclei in different chemical environments, and show that it leads to multiplet structure in
NMR spectra. The material presented here may be read in conjunction with Chapter 14 of
McQuarrie and Simon.
It should be kept in mind that techniques presented here can also be used to study hyper-
fine effects in ESR spectroscopy.
Magnetic Shielding Effect and Chemical Shift:
According to Eq (7), all Hydrogen nuclei in a molecule absorb at the “same” frequency
given by,
0( )
bare spectrometer
E E EB β α α βω γ ω− ∆ →
= = = =h h
(8)
Here the subscript bare is included because this is the same frequency at which a free (or
bare) proton will absorb at the same magnetic field. This is the frequency at which the
spectrometer is said to be operating.
If all Hydrogen nuclei in a molecule were to absorb at exactly the same frequency as
above, then NMR would not be more any more useful than to test for the presence of Hydrogen
atoms in a molecule. In reality, different Hydrogen nuclei in different immediate chemical
environments in the molecule absorb at slightly different frequencies around the frequency given
7
by Eq (8). This difference is what makes NMR spectroscopy a very useful tool for extracting
structural information about the molecule.
The reason for this difference is that Hydrogen nuclei in a molecule are surrounded by
clouds of electrons. This cloud of electrons is locally different around different nuclei. When the
molecule is placed in external magnetic field, these electron clouds interact with the magnetic
field. As a result of this interaction, a small additional magnetic field elecB develops at the
nucleus. For most molecules, this generated field is in opposite direction to the applied field, and
its magnitude is proportional to the applied field.
0elecB Bσ= − (9)
Here, σ is the proportionality constant (which is unit-less). The negative sign in Eq.(9)
shows that elecB is opposite to applied field. The constant σ is a measure of effective shielding
of the nucleus offered by electrons, and is known as shielding constant. Typically the generated
field is about 5 orders of magnitude smaller than applied field, .i.e., σ is usually of the order of
10-5. The local field at the nucleus localB and corresponding change in Eq (6) is therefore given
by,
0 0 0
0 0
(1 )ˆ ˆ ˆ(1 )local
local z z
B B B B
H B I B I
σ σ
γ γ σ
= − = −
= − = − − (10)
In the above equation, nuclear-spin denoted by zI replaces the general angular
momentum ˆzJ used in earlier equations. As opposed to Eq (8), the resonant frequency including
shielding effects is given by,
0 (1 )
bare
Bω γ σω σω= −
∆ = − (11)
The shift in frequency ω∆ depends on the applied magnetic field (higher magnetic fields
will give better resolutions). It is usually very small and is of the order of a few hundred Hz
depending on the applied field. Please refer to Figure. 1 where this splitting into α and β , and
the transition between them, has been illustrated by two levels of the first proton
The value of shielding constant for a given nucleus depends on the type of local
electronic environment the nucleus is in. Therefore, two protons in different chemical
environments experience different elecB , and absorb at slightly different frequencies. Observing
8
this frequency difference between different protons in a molecule, we can elucidate molecular
structure.
The frequency difference depends (or is proportional to) applied magnetic field or
operating frequency of the spectrometer, comparing NMR spectra from different spectrometers
becomes difficult. To resolve this, a number called as chemical shift Hδ is assigned to each
proton in molecule as follows.
( ) 610H TMSH
spectrometer
ω ωδ
ω−
= × (12)
Here, TMSω is the resonance frequency of the protons of reference molecule known as
tetra methyl silane (TMS) Si(CH3)4, and spectrometerω is the operating frequency (which is related to
the applied field) of the spectrometer. The TMS is a relatively unreactive, and its 12 equivalent
protons are highly shielded, and give a clear signal even if it is present in small traces.
Chemical shift is related to the difference in absorption frequencies of proton in a given
molecule and the 12 equivalent protons in TMS molecule. It measures how much a proton in a
given molecule is shielded compared to protons in TMS. It is a dimensionless quantity, and is
expressed in parts per millions (ppm). Its value is indicative of the kind and amount of electron
cloud around the proton. Higher chemical shift for a proton usually implies depletion of electrons
around that proton. Bare proton has highest chemical shift. Different protons in a molecule have
different chemical shifts, and their difference will be independent of spectrometer characteristics.
Non-interacting two proton system
Consider two protons (labelled by 1 and 2) in two different chemical environments with
shielding constants 1σ and 2σ . If the protons are not interacting, the total 2-particle Hamiltonian
The final normalized solution can be written in terms of X and J as follows,
1
2 22
1c Jc XJ X
= =
+ c
Similarly, the eigenfunction 3ϕ (or equivalently 1 2,d d ) corresponding to energy 3E , is
found by solving the corresponding eigenvalue equation.
2 31
23 3
1 14 2 01 12 4
E J J ddJ E J
− − =
− −
h h
h h
E
E
By substituting the value of 3E we found out, we can solve these equations. Final
normalized solution is as follows.
1
2 22
1d Xd JJ X
= = −+
d
You can verify that the vectors c and d vectors are orthonormal, as they should be. We
have completely solved the secular equations, and we now have the complete energies and
eigenstates for the general case of two interacting non-equivalent protons.
In previous section, we have seen that first-order perturbation theory did not predict the
correct spectrum for the case of two interacting equivalent proton system. Now that we have
obtained the exact solutions using the full variational method, let us substitute 1 2σ σ σ= = in the
these expressions and obtain energies and eigenstates.
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1 0
2
3
4 0
1(1 )4
1434
1(1 )4
B J
J
J
B J
γ σ
γ σ
= − − +
= +
= −
= + − +
h h
h
h
h h
E
E
E
E
Note that the second and third states which were degenerate for the zeroth-order
Hamiltonian are no longer degenerate. Let us also see what are the eigenstates of two interacting
equivalent protons. Substituting 1 2σ σ σ= = in the expressions for eigenstates we obtained from
solving the Secular problem, we get,
( ) ( )22 2 20 1 2 0 1 2X B B J X Jγ σ σ γ σ σ= − + − + ⇒ =
1
2 22
11 112
c Jc XJ X
= = ⇒ =
+ c c
1
2 22
11 112
d Xd JJ X
= = ⇒ = − −+
d d
Therefore, the four eigenstates of two-equivalent protons are,
1 1 (1) (2)ϕ ψ α α= =
( ) ( )2 2 31 1 (1) (2) (1) (2)2 2
ϕ ψ ψ β α α β= + = +
( ) ( )3 2 31 1 (1) (2) (1) (2)2 2
ϕ ψ ψ β α α β= − = −
4 4 (1) (2)ϕ ψ β β= =
You will recognize that these states are precisely the normalized eigenstates of total spin
operator for two spin-1/2 particles. We constructed these functions in the class when we coupled
spin angular momentum of two spin-1/2 particles. We also proved in the class that
1 2 4, ,ϕ ϕ ϕ are eigenfunctions of 2,ˆ totalI with same eigenvalue 1I = and they form a triplet,
and 3ϕ is eigenfunction of 2,ˆ totalI with eigenvalue 0I = , that is it is a singlet. The factor 12
is
the normalization factor.
30
In an earlier section, we considered non-interacting two proton system. We showed that
the Hamiltonian 0H for this system commutes with ˆtotalzI , and that eigenfunctions of ˆtotal
zI are
also eigenfunctions of 0H . We further proved that 0H does not in general commute with 2,ˆ totalI ,
but it does so only if two protons are equivalent. In this case, we can find eigenfunctions of 0H
which would be simultaneous eigenfunctions of 2,ˆ totalI operator as well. Therefore, we could
construct a set of functions, which are simultaneous eigenfunctions of 0H , ˆtotalzI and 2,ˆ totalI .
The appearance of 2,ˆ totalI and ˆtotalzI eigenfunctions 1 2 3 4, , ,ϕ ϕ ϕ ϕ as eigenfunctions
even for the case of complete interacting Hamiltonian H , shows that H and 2,ˆ totalI also
commute. This is true in general. It is possible to prove in general that Hamiltonian for a system
of interacting equivalent interacting protons commutes with 2,ˆ totalI and ˆtotalzI . Using this, it is
possible to also prove that eigenstates of equivalent proton Hamiltonian are same as the
simultaneous eigenfunctions of 2,ˆ totalI and ˆtotalzI operators. We will return to this in the following
section.
Let us use the same selection rule 1IM∆ = ± as we used earlier, and find out the allowed
transitions and calculate the exact spectrum. We get,
1 2 1 0
1 3 2 0
2 4 3 0
3 4 4 0
with (1 )
with (1 )
with (1 )
with (1 )
B
B J
B
B J
ϕ ϕ ω γ σ
ϕ ϕ ω γ σ
ϕ ϕ ω γ σ
ϕ ϕ ω γ σ
→ = −
→ = − −
→ = −
→ = − +
The selection rule 1IM∆ = ± seems to predict three lines each separated by amount J .
However, since we have calculated the energies by full variation principle, they can not be
incorrect. Therefore, the selection rule 1IM∆ = ± must be inappropriate for the special
equivalent proton case. In the next section, we will work out appropriate selection rules for both
equivalent and non-equivalent cases. Using those selection rules we will then show that, for the
case of equivalent protons, the transitions 1 3ϕ ϕ→ and 3 4 ϕ ϕ→ are actually forbidden. In
this special case, the only allowed transitions are 1 2ϕ ϕ→ and 2 4ϕ ϕ→ allowed, and as
31
can be seen from above, they have the same frequency. This leads to only one line in the NMR
spectrum of equivalent protons.
NMR Selection Rules
Selection rules are widely used in spectroscopy to find out allowed and forbidden
transitions when exposed to electro-magnetic radiation. Transitions can take place in general
between any two quantum states of a system. Since the quantum states of a system are
characterized by quantum numbers, the selection rules can be formulated in terms of allowed
changes in quantum numbers between initial and final states. Note that, although generally not
explicitly stated, selection rules are usually meant for a particular mechanism of transition.
Common and important mechanisms are the electric and magnetic dipole transitions, electric
quadrupole transitions and so on.
In practice, transitions only take place from those states of the system where it is likely to
be found in. This depends on the nature of quantum states (electronic, vibrational, rotational
etc.,), as well as the temperature. Since the energy gaps of between various states of nuclear spin
systems used in NMR (and even ESR) are very small compared to average thermal energy at
experimental temperatures, all these states are almost equally occupied, and transitions can take
place between any of them.
To obtain NMR selection rules, let us note that transitions are induced by a small
oscillating magnetic field 1( ) cos( )t tω= 01B B , applied in addition to the strong constant magnetic
field 0 ˆB z along z-axis which splits the energy levels of NMR spin system. Here, 01B is the vector
specifying direction and amplitude of the oscillating magnetic field. As shown in Problem 14-37
of McQuarrie and Simon, NMR spin system with total magnetic moment ˆˆ totalIµ γ= interacts
with this oscillating magnetic field as,
( )ˆ ˆ( ) cos( )totalH t I tγ ω= − ⋅ 01B
Accordingly, a transition from system initially in eigenstate iΨ to the final state jΨ
can happen only if the integral is non-zero.
ˆ 0totalj iIΨ ⋅ Ψ ≠0
1B
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As discussed in Problem 14-37, it can be shown that, if the constant magnetic field is in
z-direction, the component of 01B in along z-direction does not induce any transition. And
therefore, the transition is possible if the following holds.
ˆ 0totalj x iIΨ Ψ ≠ and/or ˆ 0total
j y iIΨ Ψ ≠
Using angular momentum relations, you can show that this is equivalent to,
ˆ 0totalj iI+Ψ Ψ ≠ and/or ˆ 0total
j iI−Ψ Ψ ≠
To find selection rules, note that proton system eigenstates iΨ and jΨ are in general
eigenfunctions of ˆtotalzI . Let us assume that iΨ and jΨ are characterized by quantum
numbers IM and JM . Since ˆtotalI+ increases the ˆtotalzI quantum number by one, the final state
jΨ must be eigenfunction of ˆtotalzI with eigenvalue 1IM + , if the above integrals were not to
vanish. Similarly, using properties of ˆtotalI− we can argue that jΨ must be eigenfunction of
ˆtotalzI with eigenvalue 1IM − . Therefore, transitions can take place only between two states
which differ by one in their ˆtotalzI eigenvalues. Therefore, for NMR, the selection rule in general
is 1IM∆ = ± .
However, we know from angular momentum theory that ˆtotalI+ and ˆtotalI− operators can not
change the 2,ˆ totalI eigenvalue of a function. Therefore, if iΨ and jΨ happen to be eigenstates
of 2,ˆ totalI , then an additional selection rule 0I∆ = must also be considered. We have seen that in
general cases, our NMR Hamiltonian does not commute with 2,ˆ totalI . There is no spherical
symmetry in the spin space, and hence I is not a good quantum number in general, and it is not
defined. Therefore, 0I∆ = selection rule does not apply in general.
However, for the case of equivalent protons, Hamiltonian commutes with 2,ˆ totalI , and I
becomes a good quantum number. The eigenstates of equivalent proton system are the
eigenstates of total angular momentum. Therefore, additional 0I∆ = selection rule applies in
this case.
33
This is the reason why, in equivalent two proton system, the transitions 1 3ϕ ϕ→ and
3 4 ϕ ϕ→ are forbidden, as they involve change of total spin quantum number .i.e., triplet and
singlet states. Therefore, there is only one line for group of equivalent protons.
Exercises:
Deduce NMR selection rules for equivalent and non-equivalent cases from the condition of non-
zero values for the above integrals. Use angular momentum theory to express totalxL and total
yL in
terms of totalL+ and totalL− .
A revisit to the interacting equivalent two-proton case.
In this section, we will revisit the case of interacting equivalent two proton case, and
reconsider it from the point of view of angular momentum theory. The techniques used would
apply in a similar way for ESR spectroscopy and spin-orbit coupling in atoms. The interacting
two equivalent proton Hamiltonian can be written as,
( )0 0 1 0 2 1 2
0 1 2
0 1 2
1ˆ ˆ ˆ ˆ ˆ ˆ ˆ(1 ) (1) (1 ) (2)
1ˆ ˆ ˆ ˆ (1 ) (1) (2)
1ˆ ˆ ˆ (1 )
z z
z z
totalz
H H V B I B I J I I
B I I J I I
B I J I I
γ σ γ σ
γ σ
γ σ
= + = − − − − + ⋅
= − − + + ⋅
= − − + ⋅
h
h
h
Earlier, we proved that the spin-spin interaction V commutes with ˆtotalzI . We will now
prove that V commutes with 2,ˆ totalI as well, and hence the whole Hamiltonian commutes with 2,ˆ totalI . To prove this, we will try to rewrite V which will give insight into structure of this
Hamiltonian.
( ) ( )2,1 2 1 2
1 1 2 2 1 2 2 1
ˆ ˆ ˆ ˆ ˆ
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
totalI I I I I
I I I I I I I I
= + ⋅ +
= ⋅ + ⋅ + ⋅ + ⋅
Since angular momentum operators of different particles commute, we have,
1 2 2 1ˆ ˆ ˆ ˆI I I I⋅ = ⋅
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Using this, we get, 2,
1 1 2 2 1 2
2 21 2 1 2
ˆ ˆ ˆ ˆ ˆ ˆ ˆ2ˆ ˆ ˆ ˆ 2
totalI I I I I I I
I I I I
= ⋅ + ⋅ + ⋅
= + + ⋅
Therefore, we can rewrite 1 2ˆ ˆI I⋅ as follows.
( )2, 2 21 2 1 2
1ˆ ˆ ˆ ˆ ˆ2
totalI I I I I ⋅ = − +
Using the above relation to rewrite V as,
( )2, 2 21 2
1 1ˆ ˆ ˆ ˆ 2 2
totalV J I J I I= − +h h
Now, using this form consider commutation of V with 2,ˆ totalI . The first term can be
easily seen to commute. From angular momentum theory, we know each term in ( )2 21 2ˆ ˆI I+
commutes with 2,ˆ totalI . Let us prove this. The second term involves commutators such as 2, 2
1ˆ ˆ,totalI I (consider first particle as example). We will expand 2,ˆ totalI , by using the following
formula. 2, 2,ˆ ˆ ˆ ˆ ˆtotal total total total total
z zI I I I I− += + + h
For the proof, we need to just consider terms in 2,ˆ totalI involving first-particle only,
because all others commute with 21I . Collecting all terms in 2,ˆ ˆ ˆ ˆtotal total total total
z zI I I I− + + + h which
contain the co-ordinates of first particle, we get, 2, 2 2
1 1 1 1 1 1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, ,total total total total total
z z z z zI I I I I I I I I I I I− + − + = + + + + h
Using 21 1ˆ ˆ, 0zI I = and 2
1ˆ ˆ, 0total
zI I = , we get
2, 2 2 21 1 1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , ,total total totalI I I I I I I I− + − + = +
Using 21 1ˆ ˆ, 0I I±
= we get,
2, 2 2 21 1 1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ, , ,total total totalI I I I I I I I− + − + = +
Using 2 21 1 1
ˆ ˆ ˆ ˆ, , 0totalI I I I± ± = = , we prove that
2, 21
ˆ ˆ, 0totalI I =
35
Therefore, the Hamiltonian for equivalent protons commutes with 2,ˆ totalI as well. Note
that in this process we have also proved that Hamiltonian commutes with 21I and 2
2I as well.
Using this new form for spin-spin coupling, the Hamiltonian can be rewritten as follows,
( )2, 2 20 1 2
1 1ˆ ˆ ˆ ˆ ˆ(1 ) 2 2
total totalzH B I J I J I Iγ σ= − − + − +
h h
Let us now summarize the full symmetries of this Hamiltonian.
ˆ ˆ, 0totalzH I =
2,ˆ ˆ, 0totalH I =
21
ˆ ˆ, 0H I =
22
ˆ ˆ, 0H I =
Therefore, the Hamiltonian has full spherical symmetry. Actually, you can see that