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4. The Nuclear Magnetic Resonance Interactions
4a. The Chemical Shift interaction
The most important interaction for the utilization of NMR in chemistry is the “chemical shift”.It makes it possible to distinguish between chemically inequivalent nuclei. In anexternal magnetic field the surrounding electronic densities of the nuclei generate a field at the nuclear positions that point in most cases in a direction opposing the external magnetic field. This shielding field shifts the Larmor frequency according to:
)1()( 00 zzeL BBB
Thus it turns out that the magnitude of the additional field is proportional to the external field.
The source of the induced field can be understood by considering the example of an electron in an s-rbital. The external magnetic field generates an overall current proportional to this field, and this current generates in turn a field in opposite direction (diamagnetic shift).
The order of magnitude of the shielding field is about 10 -6 times B0 and seems to be anisotropic. The orientation of the molecule in the external fielddetermines the shift. The induced field can thus point in all directions depending on the relative orientation of the molecules but we must consider only its z-component (z//B0).
Electrons in p-orbitals behave of course differently (paramagnetic shift) . “Very schematic”
B0 B0 B0
<e> <e>
B0 B0
Be
65
0
0
0
BB
B
B
B
zzzyzx
yzyyyx
xzxyxx
ez
ey
ex
e
The chemical shift can be represented as a tensor in matrix form. Choosing a coordinate system with B0 pointing in its z-direction:
And because , the only relevant component is 0BBe
0BB zzez
cos
sinsin
sincos
00
00
00
0
0
0
B
B
B
B
B
B
B
ZZ
YY
XX
ez
ey
ex
e
Or: The chemical shift can be represented as a tensor in matrix form. Choosing the Principle Axis System coordinate system on the molecule , with the direction of the field defined by polar angles ():
And because , the only relevant component is in the direction of the main field.
The magnitude of that component is :
0BBe
B0
)cossinsinsincos(
cos
sinsin
sincos
cos
sinsin
sincos22222
0
0
0
0
ZZYYXX
ZZ
YY
XX
B
B
B
B
2
)1cos3(2cossin
2
1
3
2
02
00
ZZZZ
YYXXZZYYXXe BBBB
Isotropic chemical shift “chemical shift anisotropy”
B0
In liquids the anisotropy averages to zero
3ZZYYXX
isoisoPPPP
66
http://orgchem.colorado.edu/hndbksupport/nmrtheory/protonchemshift.html
TMS
TMSspinppm
)(
For samples in CDCl3 solution. The scale is relative to TMS at =0.
TMS gives one line at high field and is inert!
The shift is measured in terms of
67
Carbon Chemical Shift Ranges*
68
http://ascaris.health.ufl.edu/classes/bch6746/2004_notes/lecture4onscreen.ppt
Example of (de)shielding effects in the neighborhood of -systems or double/triple bonds:
“Understanding” the chemical shift values is a subject on its own, and requires a combinationof empirical facts, shielding and deshielding characteristics of functional groups in terms of their relative position, electronegativity, bond strength, -character, and molecular motion.
Today possible quantum mechanical calculations based on orbital structure, or Hartree-Fockand lately DFT, are possible to predict chemical shift values.
Random Coil Carbon and Proton Shifts of Amino Acids
69
4b. Chemical shift in solids
In solids the chemical shift anisotropy (CSA) does not vanish and the spectral linesbroaden in powders:
CSA powder lineshapes:
See: Multidimensional Solid-State NMR and Polymers; K. Schmidt-Rorr and H.W. Spiess Academic Press (1994)
70
Finally, each individual inequivalent nucleus is described by its own spin ensemble,
with its own magnetization vector in its own rotating frame, its own off resonance and
its own two-level spin system.
tiEtiE ecect //)(
)()(
sin)Re(2)Re(2)(
cos)Re(2)Re(2)(
**
*/*/
*/*/
cccctm
tciceceictm
tccecectm
z
tiEtiEy
tiEtiEx
)(tm
2/ E
2/ E
xy
z
The Free Induction Decay :In the rotating frame:
There exists a correlation between the QM description of a two level system
and the rotation of a vector in a Cartesian axis system. The x and y components of
the vector are proportional to two functions of the coefficients of the eigenstates (coherence)
and the z component to the difference in eigenstate probabilities (population).
In the spin-1/2 case the x- and y-components are observables.
RF pulses will change the coefficients of the wavefunction:
1
2
21
1
1
2
21
1
1
2
21
1
1
2
21
1
1
1
1
1
2/1
2/1
2/1
2/1
y
ia
xz
a
y
ia
xz
a
x
ia
yz
a
x
ia
yz
a
zx
z
a
m
eicc
m
c
m
eicc
m
c
m
ecc
m
c
m
ecc
m
c
m
c
m
c
4c. The vector model and the two level system
71
Finally, each individual inequivalent nucleus is described by its own spin ensemble,
with its own magnetization vector in its own rotating frame, its own off resonance and
its own two-level spin system.
)()()( tctct
1)()()()(
))()()()(()(
))()(Im(2)(
)()(Re(2)(
**
**
*
*
tctctctc
tctctctcItm
tctcItm
tctcItm
zz
yy
xx
)(tm
2/ E
2/ E
xy
z
The Free Induction Decay :In the rotating frame:
4c. The vector model and the two level system
72
a spin-1/2 with three independent coefficients that behave like a vector and follow the Bloch equation
))()()()((
))()(Im(
)()(Re(
)(
)(
)(
**
*
*
tctctctc
tctc
tctc
tI
tI
tI
z
y
x
)(tm
2/ E
2/ E
xy
z
2/2/4 XAE
2/2/3 XAE
2/2/2 XAE
2/2/1 XAE 1XA2XA
3XA4XA
Measurable x-y components of a spin system AX
73
X
X
AA
One spin -1/2
Two spins -1/2: “AX”
))()()()((
))()(Im(
))()(Re(
**
*
*
tctctctc
tctc
tctc
I
I
I
jjii
ji
ji
ijz
ijy
ijx
1533161226324243
413121
Coherences population differences
total
AX EEEEEEEE 24133412 ;
NMR on a spin-1/2 can be represented in a schematic way as:
Spin evolution:
yx II
yz
xz
IxI
IyI
)(
)(RF pulses:
4d. The Spin-Spin interaction
The interaction of two spins immediated by their overlapping wavefunctions is the Spin-Spin Interaction or j-coupling.
To describe the interaction we will restrict ourselves here to the “secular” interaction only.This excludes the interaction between two neighboring equivalent spins.
For example:
tItItItI
tItItItI
zyzz
xyxx
11 sin)0()(;cos)0()(
sin)0()(;cos)0()(
A3
Ethanol proton spectrum
X2
CH3CH2O-
XAj 2/
74
)0( xI
Suppose two spins A and X with off resonance values and .
In their rotating frames the energy level diagram looks like:
A X
There are 4 wave functions and thus six possible coherences:
ntcnn
)(4,3,2,1
},,,,,{ *41
*32
*42
*31
*43
*21 cccccccccccc
and there are 6 “fictitious spin-1/2” systems with 18 “vector components” .
2/2/4 XAE
2/2/3 XAE
2/2/2 XAE
2/2/1 XAE 1XA2XA
3XA4XA
},,{},,{},,{
},,{},,{},,{141414242424343434
232323131313121212
zyxzyxzyx
zyxzyxzyx
mmmmmmmmm
mmmmmmmmm
75
A “vector” with 18 components:
12xm 12
ym12xm
12zm
13ym13
xm
13zm
23ym23
xm
23zm
34ym
34zm
24ym24
xm
24zm
14ym14
xm
14zm
ijp
ijp Im
A X
{13;24} {12;34}
0
34xm
231424133412 ,,,, zzzzzz mmmmmm and are dependent
)}(2),(2{
)}(2),(2{34123412
24132413
yyAz
Xyxx
Az
Xx
yyXz
Ayxx
Xz
Ax
mmIImmII
mmIImmII
23
23
14
14
yXx
Ay
Xy
Ax
ZQy
xXy
Ay
Xx
Ax
ZQx
yXx
Ay
Xy
Ax
DQy
xXy
Ay
Xx
Ax
DQx
mIIIII
mIIIII
mIIIII
mIIIII
The other coherences are:
and the double and zero quantum coherences
We can measure only the single quantum coherences:
)}(),({
)}(),({
34123412
24132413
yyXyxx
Xx
yyAyxx
Ax
mmImmI
mmImmI
ZQyI ZQ
xI
ZQzI
DQyI DQ
xI
DQzI
AxI
AyI
AzI
Xz
Ax II X
zAy II
Xz
Az II
XxI
XyI
XzI
Az
Xx II
Az
Xy mm
Az
Xz II
76
4/2/2/4 JE XA
4/2/2/3 JE XA
4/2/2/2 JE XA
4/2/2/1 JE XA 1XA2XA
3XA4XA
The j-coupling shifts the energies as follows:
Making the spectra look like:
and the spin evolution looks like
Xz
AxA
Xz
Ay
AyA
Ax
IIII
jj
II
22
2/2/
For example when spin A at t =0 is in state <Ix>(0): :
tJtItII
tJtItII
tJtItItJtItI
AAx
Xz
Ax
AAx
Xz
Ay
AAx
AyA
Ax
Ax
2/sinsin)(2
2/sincos)(2
2/cossin)(;2/coscos)(
and the A-spin signal is
}])2/sin()2/{sin(})2/cos()2/[{cos(5.0
}2/cossin2/cos{cos)(
tJtJitJtJI
tJtitJtItS
AAAAAx
AAx
A
A X
77
DQyXA
DQx
ZQyXA
ZQx
II
II
)(
)(
Az
XxX
Az
Xy
XyX
Xx
IIII
jj
II
22
2/2/
jJ 2;2
The extension to more coupled spins is straightforward:
A X
2jAX jAX
The number of A-lines in A-Xn is (n+1): the (n+1) multiplet
A and X can be spins of the same type or of different types: 1H-1H or 13C-1H etc.
A – X2
jAX and jAX
A
X
X
X
X
H
jCH
C
2jCHcarbon spectrum Proton spectrum
(The energy level diagrams are evaluated in the rotating frames of all interacting spins)
78
A X13CH2
Vicinal Coupling (3J, H-C-C-H)
28.02cos:18090
28.02cos:900
18030
030
jj
jj
ab
ab
Karplus equation:
79
D
AxI
AyI
Xz
Ay II
Xz
Ax II
0t
X Axis Title
X Axis TitleFFT
t
A – signals:
80
0
j
0
Time evolution of AX spin system
The spectrum of A
detectable
non-detectable