Multidimensional NMR Spectroscopy.PDF

3 Multidimensional NMR Spectroscopy

Models used for the description of NMR experiments

1. energy level diagram: only for polarisations, not for time-dependent phenomena

2. classical treatment (B LOCH EQUATIONS ): only for isolated spins (no J coupling!)

3. vektor diagram: pictorial representation of the classical approach (same limitations)

4. quantum mechanical treatment (density matrix): rather complicated; however, using

appropriate simplifications and definitions – the product operators – a fairly easy and correct

description of most experiments is possible

3.1. B LOCH Equations

The behaviour of isolated spins can be described by classical differential equations:

d M /dt = g M (t) x B (t) - R {M(t) -M 0 }

[3-1]

with M 0 being the B OLTZMANN equilibrium magnetization and R the relaxation matrix:

x y z

1/T 2 0 0

0 1/T 2 0

0 0 1/T 1

The external magnetic field consists of the static field B 0 and the oscillating r.f. field B rf :

R =

B(t) = B 0 + B rf

[3-2]

B rf = B 1 cos(wt + f)e x

[3-3]

The time-dependent behaviour of the magnetization vector corresponds to rotations in space (plus

relaxation), with the B x and B y components derived from r.f. pulses and B z from the static field:

dM z /dt = gB x M y - gB y M x -(M z -M 0 )/T 1

[3-4]

dM x /dt = gB y M z - gB z M y - M x /T 2

[3-5]

dM y /dt = gB z M x - gB x M z - M y /T 2

[3-6]

Product operators

To include coupling a special quantum mechanical treatment has to be chosen for description. An

operator, called the spin density matrix r(t), completely describes the state of a large ensemble of

spins. All observable (and non-observable) physical values can be extracted by multiplying the

density matrix with their appropriate operator and then calculating the trace of the resulting matrix.

The time-dependent evolution of the system is calculated by unitary transformations (corresponding

to "rotations") of the density matrix operator with the proper Hamiltonian H (including r.f. pulses,

chemical shift evolution, J coupling etc.):

r(t') = exp{i H t} r(t) exp{-i H t}

(for calculations these exponential operators have to be expanded into a Taylor series).

The density operator can be written als linear combination of a set of basis operators. Two specific

bases turn out to be useful for NMR experiments:

- the real Cartesian product operators I x , I y and I z (useful for description of observable

magnetization and effects of r.f. pulses, J coupling and chemical shift) and

- the complex single-element basis set I + , I - , I a and I b (raising / lowering operators, useful for

coherence order selection / phase cycling / gradient selection).

Cartesian Product operators

Lit. O.W. Sørensen et al. (1983), Prog. NMR. Spectr. 16 , 163-192

Single spin operators

correspond to magnetization of single spins, analogous to the classical macroscopic magnetization

M x , M y , M z .

I x , I y

(in-phase coherence, observable )

I z

( z polarisation, not observable)

Two-spin operators

2I 1x I 2 z , 2I 1y I 2 z , 2I 1z I 2 x , 2I 1z I 2y

(antiphase coherence, not observable)

2I 1 z I 2z

(longitudinal two-spin order, not observable)

2I 1x I 2x , 2I 1y I 2x , 2I 1x I 2y , 2I 1y I 2y

(multiquantum coherence, not observable)

Sums and differences of product operators

2 I 1x I 2x + 2 I 1y I 2y = I 1 + I 2 - + I 1 - I 2 +

zero-quantum coherence

2 I 1y I 2x - 2 I 1x I 2y = I 1 + I 2 - - I 1 - I 2 +

(not observable)

2 I 1x I 2x - 2 I 1y I 2y = I 1 + I 2 + + I 1 - I 2 -

double-quantum coherence

2 I 1x I 2y + 2 I 1y I 2x = I 1 + I 2 + - I 1 - I 2 -

(not observable)

The single-element operators I + and I - correspond to a transition from the m z = - 1 / 2 to the m z = + 1 / 2

state and back, resp., hence "raising" and "lowering operator". Products of three and more operators

are also possible.

Only the operators I x and I y represent observable magnetization. However, other terms like antiphase

magnetization 2 I 1x I 2z can evolve into observable terms during the acquisition period.

Pictorial representations of product operators

(cf. the paper in Progr. NMR Spectrosc. by Sørensen et al.)

coherences

I x

1 x

2 z

I y

1 y

2 z

polarisations

I 1z

I 2z

I+I

1 z

2I 1z 2z

2 z

In the energy level diagrams for coherences, the single quantum coherences I x and I y are

symbolically depicted as black and gray arrows. Both arrows in each two-spin scheme (for the

coupling partner being a or b) belong to the same operator; in the vector diagrams these two species

either align (for in-phase coherence) or a 180° out of phase (antiphase coherence). In the NMR

spectrum, these two arrows / transitions correspond to the two lines of the dublet caused by the J

coupling between the two spins. The term 2I 1x I 2z is called antiphase coherence of spin 1 with

respect to spin 2.

For the populations, filled circles represent a population surplus, empty circles a population deficit

(with respect to an even distribution). I 1z and I 2z are polarisations of one sort of spins only, I 1z +I 2z is

the normal B OLTZMANN equilibrium state, and 2 I 1z I 2z is called longitudinal two-spin order (with the

two spins in each molecule preferentially in the same spin state).

Evolution of product operators

Chemical shift

I 1x

I 1x cos(W 1 t) + I 1y sin(W 1 t)

[3-7]

I 1y

W 1 tI 1z

I 1y cos(W 1 t) - I 1x sin(W 1 t)

[3-8]

W 1 tI z

Effect of r.f. pulses

bI y

I 1z

I 1z cosb + I 1x sinb

[3-9]

bI y

I 1x

I 1x cosb - I 1z sinb

[3-10]

bI y

I 1y

The effects of x and z pulses can be determined by cyclic permutation of x , y , and z. All rotations

obey the "right-hand rule", i.e., with the thumb of the right (!) hand pointing in the direction of the

r.f. pulse, the curvature of the four other fingers indicate the direction of the rotation.

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