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13– Solution State NMR

Experiments: COSY and

Coherence Pathway Selection

As we saw in the previous chapter, in solution state

NMR, we can describe any pulse sequence by using

product operator formalism. In this chapter, we will

continue to use this formalism to understand a

fundamental two-dimensional sequence, namely the

COSY pulse sequence.

13.1 COrrelation SpectroscopY

Although the COSY pulse sequence looks deceptively

simple, it cannot be understood using classical vector

diagrams.

90 90

t1 t2

x x

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So let us use product operator formalism, and let us

begin by assuming that we have a two-spin system,

coupled via the scalar coupling J12. We start with

the equilibrium magnetization

I1z + I2z (13.1)

and apply the first 90o pulse along x:

I1z90Ix

−→ I1zcos(90) − I1ysin(90)

I1z90Ix

−→ −I1y (13.2)

and likewise for the second spin, which we will ignore

for the moment (i.e. we will not write equations for

this one).

Next we have an evolution period t1 during which

chemical shift evolution and evolution due to scalar

coupling occurs, taking the latter one first (order

doesn’t matter) we get:

−I1y2πJ12t1I1z

I2z

−→ −I1y cos(πJ12t1)

+ 2I1xI2zsin(πJ12t1)

−I1ycos(πJ12t1)Ω1t1Iz

−→ −I1y cos(πJ12t1)cos(Ω1t1)

+ I1xcos(πJ12t1)sin(Ω1t1)

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2I1xI2zsin(πJ12t1)Ω1t1Iz

−→ 2I1xI2z sin(πJ12t1)cos(Ω1t1)

+2I1yI2z sin(πJ12t1)sin(Ω1t1)

(13.3)

i.e. we have four terms, two in single-spin product

operators, and two double-spin product operators.

To keep track of the terms, we will make the

following abbreviations:

cos(πJ12t1)cos(Ω1t1) = cJ1cΩ1

cos(πJ12t1)sin(Ω1t1) = cJ1sΩ1

sin(πJ12t1)cos(Ω1t1) = sJ1cΩ1

sin(πJ12t1)sin(Ω1t1) = sJ1sΩ1

(13.4)

where the subscript 1 refers to t1 here and not spin 1.

Now applying the second 90o pulse to the terms in

equation 13.3, we get

− I1ycJ1cΩ1

90Ix

−→ −I1zcJ1cΩ1

I1xcJ1sΩ1

90Ix

−→ I1xcJ1sΩ1

2I1xI2zsJ1cΩ1

90Ix

−→ −2I1xI2ysJ1cΩ1

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2I1yI2zsJ1sΩ1

90Ix

−→ −2I1zI2ysJ1sΩ1

(13.5)

NOTE: The pulse also affects spin 2, which is

why in the third and fourth equation I2z is converted

into I2y.

These equations can be summarized in terms of a

flow chart to show that starting from I1z, we get a

number of different terms:I1z

I1x

I1x

2I1xI2z

2I1xI2z

−I

2I1yI2z

1xI2y−2I −2I

−I

−I

−I1y

1y

1y

1z I 2y1z

The terms in green are not observable, so we do not

need to consider their evolution during t2. Therefore

keeping only the observable terms, we can write out

the resulting evolution due to chemical shift and

scalar coupling:

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for I1x

I1xcJ1sΩ1

Ω1t2Iz

−→ I1xcJ1sΩ1cΩ2 + I1ycJ1sΩ1sΩ2

I1xcJ1sΩ1cΩ2

2πJ12t2I1zI2z

−→ I1xcJ1sΩ1cΩ2cJ2

+2I1yI2zcJ1sΩ1cΩ2sJ2

I1ycJ1sΩ1sΩ2

2πJ12t2I1zI2z

−→ I1ycJ1sΩ1cΩ2cJ2

−2I1xI2zcJ1sΩ1cΩ2sJ2

(13.6)

One can write the equations for the −2I1zI2y term

as well, such that overall the COSY sequence

produces:I1z

I1x

I1x

2I1xI2z

2I1xI2z

I1x

2I1yI2z

I1y

−2I I 2y1z

−2I I 2y1z I1z2I 2x

−I1xI2y−2I

−I

−I

−I1y

1y

1y

1z

−2I −2I I 2y1z I2y

I1x 2I1yI2z I1yI2x I1z2I 2x

1x 2zI

where the terms in red are diagonal peaks (since the

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magnetization belongs to spin 1 during both t1 and

t2, the terms in blue are cross-peaks (since the

magnetization shifts to spin 2), and the other terms

are antiphase unobservable magnetization. Note that

a similar tree can be written down starting from I2z.

Therefore, in summary, the COSY sequence produces

a set of observable and unobservable terms, which

can be represented using product operator vectors

(see following page - note that the rotation

convention in this case is different to the one we

normally use, so all terms will have opposite sign).

The observable terms are modulated by cos and sin

functions, and therefore, form in-phase or anti-phase

doublets in t1 and t2, i.e.

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ref: MLev, p. 397-398.

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In the case of spectral overlap, such peak shapes can

make interpretation very difficult. By using a

double-quantum filtered COSY (DQF-COSY), both

the diagonal and cross-peaks can be made to be

absorptive, thus removing this problem (see

additional handout).

A look at the “tree” of product operators generated

above leads to the idea that we can select for a

branch of this tree by choosing the phase of our

pulses appropriately. This leads us to the idea of

phase cycling and coherence pathway selection.

13.2 Coherences

In the previous chapter, we mentioned that for a

single spin, the density operators and Hamiltonians

are expressed in terms of cartesian base operators

(1/2, Ix, Iy, Iz). We can also write these operators in

terms of a shift operator basis: (1/2, I+, I−, Iz),

where I+ and I− are defined in the usual way.

We can define a parameter p, called coherence order,

which is defined as the number of I+ operators minus

the number of I− operators in a density operator.

Examples:

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Product operator p

I+

1 I+

2 +2

I−1 I+

2 0

I+

1 I2z +1

I+

2 +1

I1zI−

2 -1

Given this definition, we see that the equilibrium

state I1z we had above has a p = 0. For each

experiment, we can follow the experiment by writing

a coherence transfer pathway diagram. For the

COSY sequence above, we can write it as

90 90

t1 t2

01

2

−1−2

x x

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whereas for a DQF-COSY, we have:

90 90

t1

01

2

−1−2

t2

90xx x

How can we rationalize that a 90x pulse changes the

quantum number by ±1? We can do so by

remembering the terms that we derived above,

namely I1z becomes −I1y after the first π/2 pulse. If

we examine these two product operators in matrix

form we can see what is happening:

I1z =1

2

1 0 0 0

0 1 0 0

0 0 −1 0

0 0 0 −1

90Ix

−→ (13.7)

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−I1y =1

2i

0 0 −1 0

0 0 0 −1

1 0 0 0

0 1 0 0

(13.8)

or pictorially,

ββ

βα

αα

αβ βα

αα

αβ

ββ

For a more extensive look at coherence and different

aspects of it, I recommend Malcolm Levitt’s book.

13.3 Phase Cycling

There are two “levels” of phase cycling, A) one

where we change the phase of the pulses with every

scan and sum up the signal, and the other, B) where

we change one phase in the sequence with every (or

every second) t1 increment.

Pulse Phases In the first case, the basic idea is

to change the phase of one or more pulses from scan

to scan:

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90 90

t1 t2

x x, −x

This is like the example given in the previous

chapter, where we varied the phase of certain pulses

and the receiver to add and subtract different

signals. For this sequence, what we have is:

I1z

I1x

I1x

2I1xI2z

2I1xI2z

I1z

I1x

I1x

2I1xI2z

2I1xI2z 2I1y 2z

1x 2y −2I

−I

−I

−I1y

1y

1y

1z I 2y

90x

90−xI 2I

I

1zI

I

I−I

2I1y 2z

1x 2y−2I −2I

−I

−I

−I1y

1y

1y

1z I 2y1z

90x

90x

first scan:

second scan:

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So summing up the operators after two scans, the

green terms cancel out. This is just to illustrate the

concept.

States and TPPI In 2D experiments, the

signal often consists of a combination of absorptive

and dispersive terms.

Often, we would only like to keep the absorptive

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component as this gives us a much nicer peak shape.

To do this we can use States or Time Proportional

Phase Incrementation (TPPI):

t2

t2

t2

t2

t1=2∆

t2 t2

t2

t2

t2

t2

∆t1=

∆t1=

t1=2∆

t1=0

t1=∆

t1=0

x

−x

−y

x

y

States

t1=0

x

y

−x

−y

x

t1=3∆

t1=4∆

TPPI

The general approach on how to use States is

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described in Malcolm Levitt’s book, p. 118-119 and

involves a judicious combination/processing of the

signals:

For more details on coherence pathways, phase

cycling and States/TPPI see the excellent lecture

notes by James Keeler at:

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http://www-keeler.ch.cam.ac.uk/lectures/.