COUPLING CONSTANT
(Communication between nuclei via bonding electrons)
H
Hoe does the spectrum of
H
C
F
Chemical shift
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C
Cl
look like ?
Chemical shift
WHY ?
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The magnetic dipole moment m of spin-half (1/2) nuclei may have
two orientations in a magnetic field
This will be discussed later when introducing the quantum mechanical concept)
B0
”up”
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”down”
The spin-orientation of a neighboring nucleus affects
the magnetic field seen by the other nucleus
Bseen = B0+
HB 
HA
(I)
C = C
Bseen = B0-
HB
HA

(II)
C = C
What will be the number of HA molecules in I and II ?
NA(I)  NA(II)
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Boltzmann distribution
Exercise 3.0A
How will the 1H-NMR spectrum of the following molecular
fragments look like ? Discuss
b)
a)
-CH-Cd)
c)
-CH2 -CH
f)
CH3-CH-
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CH3-C=C-CH3
CH3-CH2
LECTURE IV
MOLECULAR STRUCTURE AND MR ( A QM-APPROACH)
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RETROSPECT
SPIN-FUNCTION; |I,m>
I; Spin quantum number (integer or half-integer)
m; -I, -I+1,…I-1, I
ORIENTATION OF A SINGLE SPIN IN AN EXTERNAL MAGNETIC FIELD.
CORRESPONDING SPIN-FUNCTIONS
I = 1/2
SPINNFUNKSJON
|1/2, -1/2> =  ()
I= 1 SPINNFUNKSJON
|1,-1>
|1,0>
|1/2, 1/2> =  ()
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|1,1>
ONE-SPIN SYSTEM (A-SYSTEM)
(I = ½, 1H, 13C, 31P, …..
NUCLEAR SPIN - MAGNETIC FIELD INTERACTION; ZEEMANN ENERGY)
EZ = -mA.BA
(Classical)
 Z  BA  IˆZ( A)
(Quantum mechanical, Hamiltonian)
NUMBER OF SPINS; 2 (=21), SPIN FUNCTIONS; |f1> = |> and f2 =|>
( A)
 Z f1   Z   B A IˆZ 
( A)
 Z f2   Z   B A IˆZ 
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 (BA / 2)   E1 
 (BA / 2)   E2 
SPINNFUNKSJON
m
m = + 1
()
-1/2
E1  B A / 2
()
1/2
ENERGI
E2  B A / 2
ˆ Zeeman

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TRANSITION

 
  m  B  Iˆz  B  Iˆz  Iˆz
SPIN UP – SPIN DOWN, ……… SPIN…..………. !
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(ONE-SPIN SYSTEM)
H
Cl
(TWO-SPIN SYSTEM)
H
C=C
Cl
H
C=C
R
Cl
R
?
1H-NMR
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ØKENDE MAGNETFELT
OBSERVATION
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INTERACTION BETWEEN NUCLEI
H
m1
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H
m2
INDIRECT DIPOL-DIPOL COUPLING
J AB (Hz )
EJ = -k.mA.mA
(Klassisk)
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 J  k  Iˆ A  Iˆ B  k () 2 Iˆ A  Iˆ B
(kvantemekanisk)
INTRODUCE SHIFT OPERATOR
Iˆ ( IˆX  iIˆY )

 J  J AB Iˆ A  Iˆ B ( Hz )  J AB IˆXA IˆXB  IˆYA IˆYB  IˆZA IˆZB

(Exercise 3.1)


  H Z   J   A IˆZA   B IˆZB  ( J AB / 2)  IˆA IˆB.  IˆA IˆB.  J AB IˆZA IˆZB
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Select the following N trial spin-functions FN( 2N = 22)
F1(A)(B)
F2(A)(B)
F3(A)(B)
F4(A)(B)
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mT
1
0
0
-1
From basic quantum mechanics (QM)
1/ 2
Iˆ I , m  I ( I  1)  m(m  1) I , m  1
(Exersice 3.1)
Iˆ   0
Iˆ   
Iˆ   
Iˆ   0
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What is Ĥ|fi>?
(Exercise 3.2)
Hˆ f1   A / 2  B / 2  J / 4 f1
Hˆ f2   A / 2   B / 2  J / 4 f2  J / 2 f3
Hˆ f3  J / 2 f2   A / 2  B / 2  J / 4 f3
Hˆ f4   A / 2   B / 2  J / 2 f4
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(Exercise 3.3)
What functions are eigenfunctions ?
(Exercise 3.4)
Determine the total z-component of the spin operator Fz
(= IzA + IzB). What can you conclude from this result?
How can we determine the two remaining energies E2 and E3 ?
(Exercise 3.5)
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Since E2 and E3 are now eigenvalues of the Hamiltonian, we may write:
2  (   ) 2
C  J AB
A B
(Exercise 3.6)
mT
E4 = 1/2 .(A + B)+JAB/4
E3 = C/2 -JAB/4
E2 = - C/2-JAB/4
E1 = -1/2 (A + B)+JAB/4
.
Spin-function
-1

0
cosq. sinq.
0
-sinq. cosq.
1
TILLATTE OVERGANGER
(M = +1)

Show that the actual spin-functions are orthonormal. What is q?
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How can we determine the signal intensity Iij ,
From basic QM
I i j 
 (
)
 i IˆA  I B  j d

2
(Exercise
3.7)
TRANSITON
34
24
13
12
RESONANCE FREQUENCY
(Hz)
V/2 - C/2 + J/2
V/2 + C/2 + J/2
V/2 +C/2 - J/2
V/2- C/2 – J/2
V   A  B
2
C  J AB
 ( A   B ) 2
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SIGNAL
INTENSITY
1-sin2q
1+sin2q
1+sin2q
1-sin2q
PROCEDURE TO DERIVE AN NMR SPECTRUM
1. CONSTRUCT AN ENERGY OPERATOR, HAMILTONIAN (Ĥ)
2. DETERMINE THE NUMBER (N) OF SPIN-FUNCTIONS ψN
(N = 2M, M IS THE NUMBER OF SPINS
3. DESIGN SOME INITIAL SPIN-FUNCTIONS fi ( i = 1,..N)
4. DETERMINE A BASIS SET OF EIGENFUNCTIONS i
(= aijfj) SUCH THAT ̂|i>= Ei|i>
5. Ei DEFINES THE EIGENVALUES AND REPRESENT ENERGIES
6. CONSTRUCT AN ENERGY DIAGRAM AND DERIVE ALLOWED
TRANSITION BETWEEN SPIN-STATES AND INTENSITIES
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ENERGY
LEVEL
SPINNFUNKSJON
EN
N
m TN
E2
2
m T2
E1
1
m T1
ENERGI
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m Ti
RESONANSFREKVENSER
h=EFOR mT = + 1
7A. WHAT IS THE INTENSITY (Iij) OF A RESONANCE
BETWEEN LEVEL i AND j (Ei - Ej with m = mi-mj = + 1)


 N



I i  j     i   Iˆ(q)  j  d 


 q 1





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2
Simulation of AB-spectra
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