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