Tutorial Session - Exercises
Problem 1: Dipolar Interaction in Water Moleclues
The goal of this exercise is to calculate the dipolar 1 H spectrum of the protons in an
isolated water molecule which does not move.
1
H
104.45◦
16
1
O
H
0.9584 Å
Figure 1: Molecular geometry of the water molecule.
γH
γT
µ0
~
=
=
=
=
26.75 · 107 rad · s−1 T−1
28.53 · 107 rad · s−1 T−1
4π · 10−7 Vs/Am
1.05 · 10−34 Js
The homonuclear dipolar coupling Hamiltonian is given by
2
~
µ0 γH
3 cos2 Θ − 1
1 + −
− +
ˆ
ˆ
Ĥ =
−
2I1z I2z −
I I + Ik In
.
3
2
2 k n
| 4πr
{z }
(1)
dipolar coupling constant
1. Start with the single-spin operators in matrix representation and determine the
matrix representation of the dipolar Hamiltonain. Calculate the Eigenvalues and
allowed transitions. Also calculate the dipolar coupling constant.
2. Which orientation would a single isolated water molecule have to have with respect
to an external field to show a maximum dipolar splitting? How big is the maximum
splitting?
3. How does the spectrum of a powder-like sample containing water molecules in arbitrary orientations look like?
4. Which would be the longest 1 H-1 H distance that could be detected by this experiment assuming a minimum spectral resolution of 100 Hz?
5. How large is the dipolar coupling if one of the protons is replaced by a tritium?
What does the corresponing spectrum look like with respect to the homonuclear
case?
6. Which interactions are present in the liquid state? What would the corresponding
spectrum look like?
Problem 2: Dipolar Coupling and the Spherical Tensor Notation
1. The laboratory-frame Hamiltonian can be expressed via the sum of scalar products
between a spherical spatial- and a spin-tensor operator as
Ĥ =
l
XX
l
(−1)q Al,q Tl,−q
q=−l
Starting from this expression, calculate the dipolar Hamiltonian (Hint: see additional expressions for the spherical tensors at the end of this exercise sheet). Show
that the result is equivalent to the “dipolar alphabet” representation given as
(k,n)
ĤD
µ0 γk γn ~ Â + B̂ + Ĉ + D̂ + Ê + F̂
=−
3
4π rkn
(2)
with
3 cos2 Θ − 1
 = 2Iˆkz Iˆnz
2
1 ˆ+ ˆ− ˆ− ˆ+ 3 cos2 Θ − 1
B̂ = − Ik In + Ik In
2
2
3 sin Θ cos Θe−iϕ
Ĉ = Iˆk+ Iˆnz + Iˆkz Iˆn+
2
3 sin Θ cos Θeiϕ
D̂ = Iˆk− Iˆnz + Iˆkz Iˆn−
2
1 ˆ+ ˆ+ 3 sin2 Θe−2iϕ
Ê =
I I
2 k n
2
2
1 ˆ− ˆ− 3 sin Θe2iϕ
F̂ =
I I
.
2 k n
2
(3)
Assign to the components A, B, C, D, E, F the corresponding rank and order of the
spin and spatial tensor.
2. Taking into account that only the q = 0 components are invariant under z-rotation,
give a reason why all components except A and B are neglected in the secular
approximation.
(k,n)
3. Calculate T̂2,+2 = 12 Iˆk+ Iˆn+ explicitly in matrix form.
2
Problem 3: Single Crystals and MAS
The aim of this problem is to calculate the resonance frequency of a 13 C spin in the carbonyl group of an Alanine single crystal. The components of the chemical-shift tensor in
the principle axis system are σ̂xx = 239 ppm, σ̂yy = 184 ppm and σ̂zz = 106 ppm. The
orientation of the crystal is such that the vectors of the principle components lie along the
x, y and z axis in the laboratory frame. The static magnetic field is along the laboratory
~ e~z )
z-axis (B||
1. Calculate the isotropic chemical shift σiso .
2. Simplify general Hamiltonian describing the interaction of the chemical-shift tensor
and the magnetic field, which is given as,
~ ~
Ĥ = γ Iˆσ̂ B
(4)
and simplify it according to the high-field approximation.
3. The crystal is rotated from its original orientation by an angle
√ γ around a rotation
◦
−1
axis that has the polar coordiates φ = 0 and θ = cos (1/ 3) = 54.7356◦ . The
matrix describing this rotation is given as
R(γ) =

1
(2 + cos γ)
3

 √1 sin γ
√ 3
2
(1 − cos γ)
3
− √13 sin γ
cos γ
q
2
sin γ
3
√

2
(1 − cos γ)
3 q

− 23 sin γ 

1
(1
3
(5)
+ 2 cos γ)
0
0
(γ) = σ̂iso + σ̂1 cos(γ) +
(γ) and express the result in the form σ̂zz
Calculate σ̂zz
σ̂2 cos(2γ). What does σ̂iso correspond to? (Keep in mind that σ̂ 0 (γ) = R(γ)σ̂R(−γ))
0
(γ) it as a function of γ. What happens if we rotate the sample continuously
4. Plot σ̂zz
around the previously chosen axis (i.e. if γ(t) = 2πωr t where ωr is the rotation
frequency)?
3
Problem 4 (Optional): The Tensor Product
The tensor product of two spherical tensors Ak and Bk0 of rank k and k 0 , respectively, can
be expressed with irreducible tensors IK that follow the relation
0
0
IKQ (k, k ) =
k
k
X
X
q=−k q 0 =−k0
hkk 0 qq 0 |KQi Akq Bk0 q0
k−k0 +Q
= (−1)
√
k
k0 X
X
k k0 K
2K + 1
Akq Bk0 q0
q q0 Q
0
(6)
k
k0 X
X
k k0 K
2K + 1
q q0 Q
0
(7)
q=−k q=−k
with
0
k−k0 +Q
0
hkk qq |KQi Akq Bk0 q0 = (−1)
√
q=−k q=−k
as the so-called Clebsch-Gordan coefficients and
(k + k 0 ) ≥ K ≥ |k − k 0 |
Q = q + q0
.
(8)
1. Name all components JKQ of the product A1 ⊗ B1 . How many components would
there be for A2 ⊗ B2 ?
2. As an example, verify that
1
J00 = √ (A1,−1 B1,1 + A1,1 B1,−1 − A1,0 B1,0 )
3
1
J20 = √ (2A1,0 B1,0 + A1,1 B1,−1 + A1,−1 B1,1 )
6
4
(9)
(10)
5
36. Clebsch-Gordan coefficients
1
36. CLEBSCH-GORDAN COEFFICIENTS, SPHERICAL HARMONICS,
AND d FUNCTIONS
p
Note: A square-root sign is to be understood over every coefficient, e.g., for −8/15 read − 8/15.
Y11
Y20
Y21
Y22
r
3
cos θ
4π
r
3
=−
sin θ eiφ
8π
r ³
5 3
1´
cos2 θ −
=
4π 2
2
r
15
sin θ cos θ eiφ
=−
8π
r
1 15
sin2 θ e2iφ
=
4 2π
Y10 =
Yℓ−m = (−1)m Yℓm∗
d ℓm,0 =
j
′
j
r
hj1 j2 m1 m2 |j1 j2 JM i
4π
Y m e−imφ
2ℓ + 1 ℓ
j
d m′ ,m = (−1)m−m d m,m′ = d −m,−m′
= (−1)J−j1 −j2 hj2 j1 m2 m1 |j2 j1 JM i
d 10,0 = cos θ
1/2
d 1/2,1/2 = cos
θ
2
1/2
d 1/2,−1/2 = − sin
1 + cos θ
θ
cos
2
2
√ 1 + cos θ
θ
3/2
sin
d 3/2,1/2 = − 3
2
2
√ 1 − cos θ
θ
3/2
d 3/2,−1/2 = 3
cos
2
2
1 − cos θ
θ
3/2
d 3/2,−3/2 = −
sin
2
2
3 cos θ − 1
θ
3/2
d 1/2,1/2 =
cos
2
2
θ
3 cos θ + 1
3/2
sin
d 1/2,−1/2 = −
2
2
1 + cos θ
2
sin θ
1
d 1,0 = − √
2
1 − cos θ
1
d 1,−1 =
2
d 11,1 =
θ
2
3/2
d 3/2,3/2 =
d 22,2 =
³ 1 + cos θ ´2
2
1 + cos θ
sin θ
=−
2
√
6
d 22,0 =
sin2 θ
4
1 − cos θ
d 22,−1 = −
sin θ
2
³ 1 − cos θ ´2
d 22,−2 =
2
d 22,1
1 + cos θ
(2 cos θ − 1)
2
r
3
d 21,0 = −
sin θ cos θ
2
1 − cos θ
(2 cos θ + 1)
d 21,−1 =
2
d 21,1 =
d 20,0 =
³3
2
cos2 θ −
1´
2
Figure 36.1: The sign convention is that of Wigner (Group Theory, Academic Press, New York, 1959), also used by Condon and Shortley (The
Theory of Atomic Spectra, Cambridge Univ. Press, New York, 1953), Rose (Elementary Theory of Angular Momentum, Wiley, New York, 1957),
and Cohen (Tables of the Clebsch-Gordan Coefficients, North American Rockwell Science Center, Thousand Oaks, Calif., 1974).