6 Product Operators*
The vector model, introduced in Chapter 3, is very useful for describing basic NMR experiments but unfortunately is not applicable to coupled spin systems. When it comes to two-dimensional NMR many of the experiments are only of interest in coupled spin systems, so we really must have some way of describing the behaviour of such systems under multiple-pulse experiments.
The tools we need are provided by quantum mechanics, specifically in the form of density matrix theory which is the best way to formulate quantum mechanics for NMR. However, we do not want to get involved in a great deal of complex quantum mechanics! Luckily, there is a way of proceeding which we can use without a deep knowledge of quantum mechanics: this is the product operator formalism.
The product operator formalism is a complete and rigorous quantum mechanical description of NMR experiments and is well suited to calculating the outcome of modern multiple-pulse experiments. One particularly appealing feature is the fact that the operators have a clear physical meaning and that the effects of pulses and delays can be thought of as geometrical rotations, much in the same way as we did for the vector model in Chapter 3.
6.1    A quick review of quantum mechanics
In this section we will review a few key concepts before moving on to a description of the product operator formalism.
In quantum mechanics, two mathematical objects - wavefunctions and operators - are of central importance. The wavefunction describes the system of interest (such as a spin or an electron) completely; if the wavefunction is known it is possible to calculate all the properties of the system. The simplest example of this that is frequently encountered is when considering the wavefunctions which describe electrons in atoms (atomic orbitals) or molecules (molecular orbitals). One often used interpretation of such electronic wavefunctions is to say that the square of the wavefunction gives the probability of finding the electron at that point.
Wavefunctions are simply mathematical functions of position, time etc. For example, the Is electron in a hydrogen atom is described by the function exp(-ar), where r is the distance from the nucleus and a is a constant.
In quantum mechanics, operators represent "observable quantities" such as position, momentum and energy; each observable has an operator associated with it.
Operators "operate on" functions to give new functions, hence their name
operator x function = (new function) An example of an operator is (d/cbc); in words this operator says "differentiate
Chapter 6 "Product Operators" © James Keeler, 1998 & 2002
6-1
with respect to x". Its effect on the function sin x is
d
A mass going round a circular path possesses angular moment, represented by a vector which points perpendicular to the plane of rotation.
dx
(sin x) - cos x
the "new function" is cos x. Operators can also be simple functions, so for example the operator x2 just means "multiply by x2".
6.1.1 Spin operators
A mass going round a circular path (an orbit) possesses angular momentum; it turns out that this is a vector quantity which points in a direction perpendicular to the plane of the rotation. The x-, y- and z-components of this vector can be specified, and these are the angular momenta in the x-, y- and z-directions. In quantum mechanics, there are operators which represent these three components of the angular momentum.
Nuclear spins also have angular momentum associated with them - called spin angular momentum. The three components of this spin angular momentum (along x, y and z) are represented by the operators Ix, I and Iz.
6.1.2 Hamiltonians
The Hamiltonian, H, is the special name given to the operator for the energy of the system. This operator is exceptionally important as its eigenvalues and eigenfunctions are the "energy levels" of the system, and it is transitions between these energy levels which are detected in spectroscopy. To understand the spectrum, therefore, it is necessary to have a knowledge of the energy levels and this in turn requires a knowledge of the Hamiltonian operator.
In NMR, the Hamiltonian is seen as having a more subtle effect than simply determining the energy levels. This comes about because the Hamiltonian also affects how the spin system evolves in time. By altering the Hamiltonian the time evolution of the spins can be manipulated and it is precisely this that lies at the heart of multiple-pulse NMR.
The precise mathematical form of the Hamiltonian is found by first writing down an expression for the energy of the system using classical mechanics and then "translating" this into quantum mechanical form according to a set of rules. In this chapter the form of the relevant Hamiltonians will simply be stated rather than derived.
In NMR the Hamiltonian changes depending on the experimental situation. There is one Hamiltonian for the spin or spins in the presence of the applied magnetic field, but this Hamiltonian changes when a radio-frequency pulse is applied.
6.2    Operators for one spin 6.2.1 Operators
In quantum mechanics operators represent observable quantities, such an energy, angular momentum and magnetization. For a single spin-half, the x- y-
6-2
and z-components of the magnetization are represented by the spin angular momentum operators Ix, I and Iz respectively. Thus at any time the state of the spin system, in quantum mechanics the density operator, a, can be represented as a sum of different amounts of these three operators
a[t) = a[t)Ix+b[t)Iy+c[t)Iz
The amounts of the three operators will vary with time during pulses and delays. This expression of the density operator as a combination of the spin angular momentum operators is exactly analogous to specifying the three components of a magnetization vector.
At equilibrium the density operator is proportional to / (there is only z-magnetization present). The constant of proportionality is usually unimportant, so it is usual to write a = I
eq z
6.1.2 Hamiltonians for pulses and delays
In order to work out how the density operator varies with time we need to know the Hamiltonian (which is also an operator) which is acting during that time.
The free precession Hamiltonian (i.e. that for a delay), //free, is
In the vector model free precession involves a rotation at frequency Q about the z-axis; in the quantum mechanical picture the Hamiltonian involves the z-angular momentum operator, Iz - there is a direct correspondence.
The Hamiltonian for a pulse about the jc-axis, Hpuhe, is and for a pulse about the >>-axis it is
Again there is a clear connection to the vector model where pulses result in rotations about the x- or >>-axes.
6.1.3 Equation of motion
The density operator at time t, ait), is computed from that at time 0, o(0), using the following relationship
a(t) = exp(-iHt) a(0) exp(iHt)
where H is the relevant hamiltonian. If H and a are expressed in terms of the angular momentum operators if turns out that this equation can be solved easily with the aid of a few rules.
Suppose that an jc-pulse, of duration t, is applied to equilibrium magnetization. In this situation H= coJx and o(0) = Iz; the equation to be solved is
aW = exP("^i?p4) h exp(zftVp Ix) Such equations involving angular momentum operators are common in
6-3
quantum mechanics and the solution to them are already all know. The identity required here to solve this equation is
exp(-iK) ^exp(iK) = cos/? Iz -sin(5 Iy [6.1]
This is interpreted as a rotation of / by an angle (5 about the jc-axis. By putting (5= (D\tp this identity can be used to solve Eqn. [6.1]
(j(fp) = cosGVP ^-sinffl^ /,
The result is exactly as expected from the vector model: a pulse about the jc-axis rotates z-magnetization towards the -y-axis, with a sinusoidal dependence on the flip angle, (5.
6.1.4   Standard rotations
Given that there are only three operators, there are a limited number of identities of the type of Eqn. [6.1]. They all have the same form
exp(-i0/fl) {old operator}exp(/0/fl)
= cos6> {old operator} + sin0 {new operator}
where {old operator}, {new operator} and Ia are determined from the three possible angular momentum operators according to the following diagrams; the label in the centre indicates which axis the rotation is about / // ///
Angle of rotation = Qtfor offsets and a^tp for pulses
First example: find the result of rotating the operator / by 0 about the jc-axis, that is
exp(-idlx) Iyexp(i6Ix)
For rotations about x the middle diagram // is required. The diagram shows that / (the "old operator") is rotated to Iz (the "new operator"). The required identity is therefore
exp(-idlx) I exp[i0Ix) = cosO I + sinO Iz
Second example: find the result of
exp(-idly){-lz}exp(idly)
This is a rotation about y, so diagram /// is required. The diagram shows that -/ (the "old operator") is rotated to -Ix (the "new operator"). The required identity is therefore
6-4
cxP(-ieiy){-iz }cxP(ieiy) = cose{-iz }+sme{-ix}
= -cosO Iz -sinO Ix
Finally, note that a rotation of an operator about its own axis has no effect e.g. a rotation of Ix about x leaves Ix unaltered.
6.1.5 Shorthand notation
To save writing, the arrow notation is often used. In this, the term Ht is written over an arrow which connects the old and new density operators. So, for example, the following
aW = exP(-^i?P4) <7(0)exp(/ftvp/x)
is written
For the case where o(0) = Iz
6.1.6 Example calculation: spin echo
90°0)      delay t      1 80°(jc) — delay t acquire
At a the density operator is -/ . The transformation from a to b is free precession, for which the Hamiltonian is Qlz\ the delay t therefore corresponds to a rotation about the z-axis at frequency Q. In the short-hand notation this is
To solve this diagram / above is needed with the angle = Dt, the "new operator"
is 4
y y x
In words this says that the magnetization precesses from -y towards +x.
The pulse about x has the Hamiltonian coJx; the pulse therefore corresponds to a rotation about x for a time t such that the angle, co^, is n radians. In the shorthand notation
- cos flr/y +sin flr^ xr(g) [6.2]
Each term on the left is dealt with separately. The first term is a rotation of y about x; the relevant diagram is thus //
- cos Q.% Iy ——> - cos Q.% cos ft)^ Iy - cos Q.% sin co]tv Iz
However, the flip angle of the pulse, (D\tp, is n so the second term on the right is zero and the first term just changes sign (cos n= -1); overall the result is
- cos Qt I —cos Qt I
The second term on the left of Eqn. [6.2] is easy to handle as it is unaffected by
6-5
a rotation about x. Overall, the effect of the 180° pulse is then
- cos Qt I + sin Qt Ix —cos Qt I + sin Qt Ix [6.3]
As was shown using the vector model, the y-component just changes sign. The next stage is the evolution of the offset for time t. Again, each term on the right of Eqn. [6.3] is considered separately
cos Q.% I —^—> cos Q.% cos Q.% I - sin Q.% cos Q.% I
y y x
sin Qt Ix —Qt!' ) cos Qt sin Qt Ix + sin Qt sin Qt I
Collecting together the terms in Ix and / the final result is
(cos Q.% cos Q.% + sin Q.% sin Q.%) I + (cos Q.% sin Q.% - sin Q.% cos Q.%) Ix
The bracket multiplying Ix is zero and the bracket multiplying / is =1 because of the identity cos2 0 + sin2 0 = 1. Thus the overall result of the spin echo sequence can be summarised
j       90°(^)-t-180°(^)-t- j
z y
In words, the outcome is independent of the offset, 12, and the delay t, even though there is evolution during the delays. The offset is said to be refocused by the spin echo. This is exactly the result we found in section 3.8.
In general the sequence
-t-180°(jc)-t- [6.4]
refocuses any evolution due to offsets; this is a very useful feature which is much used in multiple-pulse NMR experiments.
One further point is that as far as the offset is concerned the spin echo sequence of Eqn. [6.4] is just equivalent to 180°(jc).
6.3    Operators for two spins
The product operator approach comes into its own when coupled spin systems A A A A are considered; such systems cannot be treated by the vector model. However,
£2 product operators provide a clean and simple description of the important
phenomena of coherence transfer and multiple quantum coherence.
The spectrum from two coupled
spins, with offsets Q and £L>    *■ ■* ■*    -n    * ±      e ±
(rad s-1) and mutual coupling   6.1.1   Product operators for two spins
Jl2 *Hz'' For a single spin the three operators needed for a complete description are Ix, I
and /. For two spins, three such operators are needed for each spin; an additional subscript, 1 or 2, indicates which spin they refer to.
sPin 1: hx hy Iiz      sPin 2: hx hy hz
lx represents z-magnetization of spin 1, and /2 likewise for spin 2. Ilx represents jc-magnetization on spin 1. As spin 1 and 2 are coupled, the spectrum consists of two doublets and the operator Ilx can be further identified with the two lines of the spin-1 doublet. In the language of product operators Ilx is said to represent in-phase magnetization of spin 1; the description in-phase
6-6
means that the two lines of the spin 1 doublet have the same sign and lineshape.
Following on in the same way l2x represents in-phase magnetization on spin 2. /j and l2 also represent in-phase magnetization on spins 1 and 2, respectively, but this magnetization is aligned along y and so will give rise to a different lineshape. Arbitrarily, an absorption mode lineshape will be assigned to magnetization aligned along x and a dispersion mode lineshape to magnetization along y.
'1x
•2x
•2y
n0
Vr
■G,
There are four additional operators which represent anti-phase magnetization: 2IlxI2z, 2IlyI2z, 2IlzI2x, 2IlzI2y (the factors of 2 are needed for normalization purposes). The operator 2IlxI2 is described as magnetization on spin 1 which is anti-phase with respect to the coupling to spin 2.
2 V;
y'Zz
2x
a.
2y
a.
Note that the two lines of the spin-1 multiplet are associated with different spin states of spin-2, and that in an anti-phase multiplet these two lines have different signs. Anti-phase terms are thus sensitive to the spin states of the coupled spins.
There are four remaining product operators which contain two transverse (i.e. x- or j-operators) terms and correspond to multiple-quantum coherences; they are not observable
2AA
multiple quantum: 2IlxI2
Finally there is the term 2IlzI2z which is also not observable and corresponds to a particular kind of non-equilibrium population distribution.
6.1.2   Evolution under offsets and pulses
The operators for two spins evolve under offsets and pulses in the same way as do those for a single spin. The rotations have to be applied separately to each spin and it must be remembered that rotations of spin 1 do not affect spin 2, and vice versa.
For example, consider Ilx evolving under the offset of spin 1 and spin 2. The relevant Hamiltonian is
absorption dispersion
The absorption and dispersion lineshapes. The absorption lineshape is a maximum on resonance, whereas the dispersion goes through zero at this point. The "cartoon" forms of the lineshapes are shown in the lower part of the diagram.
spin state of spin 2
The two lines of the spin-1 doublet can be associated with different spin states of spin 2.
H,
free
6-7
where Qx and are the offsets of spin 1 and spin 2 respectively. Evolution under this Hamiltonian can be considered by applying the two terms sequentially (the order is immaterial)
l\x 7
j ClxtlXl+Cl2tl27 l\x 7
The first "arrow" is a rotation about z
hx cos Qxt Ilx + sin Qxt Ily    ^ )
The second arrow leaves the intermediate state unaltered as spin-2 operators have not effect on spin-1 operators. Overall, therefore
hx    n'"M' > cos Qxt Ilx + sin Qxt Ily
A second example is the term 2IlxI2z evolving under a 90° pulse about the y-axis applied to both spins. The relevant Hamiltonian is
The evolution can be separated into two successive rotations
O/    j ml'!ly     \        ml'!2y .
Z,1lx12z 7 7
The first arrow affects only the spin-1 operators; a 90° rotation of Ilx about y gives - Ilz (remembering that (Ox t = nil for a 90° pulse)
■LL\xL2z, 7   ■LL\zL2z, 7
The second arrow only affects the spin 2 operators; a 90° rotation of z about y takes it to x
2i j —n/2!'y s _2/ j —n'2hy > -21 i
■LL\xL2z, 7   ■LL\zL2z, 7 ■LL\zL2x
The overall result is that anti-phase magnetization of spin 1 has been transferred into anti-phase magnetization of spin 2. Such a process is called coherence transfer and is exceptionally important in multiple-pulse NMR.
6.1.3   Evolution under coupling
The new feature which arises when considering two spins is the effect of coupling between them. The Hamiltonian representing this coupling is itself a product of two operators:
Hj=2tU12IuI2z
where Jl2 is the coupling in Hz.
Evolution under coupling causes the interconversion of in-phase and antiphase magnetization according to the following diagrams
6-8
IV V
-yz       zz       yz xz        zz -xz
angle = nJt
For example, in-phase magnetization along x becomes anti-phase along y according to the diagram d
hx   ln]M' >cos rint Ilx + sin nJnt2IlyI2z note that the angle is 7tJnt i.e. half the angle for the other rotations,
Anti-phase magnetization along x becomes in-phase magnetization along y; using diagram V:
2/, A   lK,nthA< > cos nJnt2IlxI2z + sin nJnt Ily
The diagrams apply equally well to spin-2; for example
-2llzI2y   2K,M> )-cosnJ12t2llzI2y+ sinnJntI2x
Complete interconversion of in-phase and anti-phase magnetization requires a delay such that JtJnt - n/2i.e. a delay of l/(2/12). A delay of 1//12 causes in-phase magnetization to change its sign:
T llt]\2tluhz '=V2J\2     \ O J     J J llt]\2tluhz '=VJ\2     v, J
l\x 7jLL\yL2z l2y 7 l2y
6.4    Spin echoes
It was shown in section 6.2.6 that the offset is refocused in a spin echo, section it will be shown that the evolution of the scalar coupling necessarily refocused.
6.4.1   Spin echoes in homonuclear spin system
In this kind of spin echo the 180° pulse affects both spins i.e. it is a non-selective pulse:
— T— 180°(jc, to spin 1 and spin 2) — T —
At the start of the sequence it will be assumed that only in-phase jc-magnetization on spin 1 is present: Ilx. In fact the starting state is not important to the overall effect of the spin echo, so this choice is arbitrary.
It was shown in section 6.2.6 that the spin echo applied to one spin refocuses the offset; this conclusion is not altered by the presence of a coupling so the offset will be ignored in the present calculation. This greatly simplifies things.
For the first delay t only the effect of evolution under coupling need be considered therefore:
hx    27U'2t'"'2' >cos7E/12T Ilx+sm7dl2T 2IlyI2z The 180° pulse affects both spins, and this can be calculated by applying the
6-9
180° rotation to each in succession
cos^/12T Ilx +sin^/12T 2IlyI2z ——
where it has already been written in that G\t = n, for a 180° pulse. The 180° rotation about x for spin 1 has no effect on the operator Ilx and /2, and it simply reverses the sign of the operator Ily
cosnJnTllx + sinnJnT2Il I2z —^-^>cos7tfuTllx -sm7iJuT2IlyI2z —nhx >
The 180° rotation about x for spin 2 has no effect on the operators Ilx and Ily, but simply reverses the sign of the operator /2 . The final result is thus
cos7tJl2Tllx +sin7tJl2T2IlyI2z —%hl > cos nJnTl\x -smnJnT2I\yI2z n'2' > cos xJl2Tllx + sin ^/12T 2IlyI2z
Nothing has happened; the 180° pulse has left the operators unaffected! So, for the purposes of the calculation it is permissible to ignore the 180° pulse and simply allow the coupling to evolve for 2t. The final result can therefore just be written down:
Ilx   t-m°(x)-t )cos27e/12t Ilx+sin2nJnT 2IlyI2z
From this it is easy to see that complete conversion to anti-phase magnetization requires 2nJnx= kI2 i.e. T= 1/(4 Jl2).
The calculation is not quite as simple if the initial state is chosen as Ily, but the final result is just the same - the coupling evolves for 2t.
Ily    t-m°(x)-t )-cos27E/12T/ly +sin27E/12T 2IlxI2z
In fact, the general result is that the sequence
— T— 180°(jc, to spin 1 and spin 2) — T — is equivalent to the sequence
- 2t- 180°(jc, to spin 1 and spin 2) in which the offset is ignored and coupling is allowed to act for time 2t.
6.1.2   Interconverting in-phase and anti-phase states
So far, spin echoes have been demonstrated as being useful for generating antiphase terms, independent of offsets. For example, the sequence
90°(jc) - l/(4/12) - 180°(jc) - 1/(4Jl2) -
generates pure anti-phase magnetization.
Equally useful is the sequence
- 1/(4Jl2) - lS0°(x) - l/(4Jl2) -
which will convert pure anti-phase magnetization, such as 2IlxI2 into in-phase magnetization, A.
6-10
6.1.3   Spin echoes in heteronuclear spin systems
If spin 1 and spin 2 are different nuclear species, such as 13C and 'H, it is possible to choose to apply the 180° pulse to either or both spins; the outcome of the sequence depends on the pattern of 180° pulses.
Sequence a has already been analysed: the result is that the offset is refocused but that the coupling evolves for time 2t. Sequence b still refocuses the offset of spin 1, but it turns out that the coupling is also refocused. Sequence c refocuses the coupling but leaves the evolution of the offset unaffected.
spin 1 -spin 2-
b
spin 1 -spin 2-
JL
JL
JL
Sequenceb
It will be assumed that the offset is refocused, and attention will therefore be restricted to the effect of the coupling
)cos^i2T     +sin7tfi2T 2IlyI2z
The 180°(jc) pulse is only applied to spin 1
cos7t/12t Ilx + sin^/12t 2IlyI2z ■
-*COS7£/12t Ilx
■sin^/12t 2IlyI2z
[6.5]
The two terms on the right each evolve under the coupling during the second delay:
COS7tf12t Ilx-
2?t/|2T/|7/27
COS7Z^12TCOS7Z^12t Ilx + sin TtJX2X cos 7ZJl2T 2IlyI2z
-sin^/12t 2IlyI2z-
2nJn-zIXzI2z
- cos TÜl2T sin TÜnt 2IlyI2z +sin7£/12t sin7£/12t Ilx
Collecting the terms together and noting that cos2 0 + sin2 0 = 1 the final result is just Ilx. In words, the effect of the coupling has been refocused.
Sequencec
As there is no 180° pulse applied to spin 1, the offset of spin 1 is not refocused, but continues to evolve for time 2t. The evolution of the coupling is easy to calculate:
Ilx   2nJ'2tI"'2' )cos7e/12t Ilx +smnJl2T ILL
This time the 180°(jc) pulse is applied to spin 2
COS7£/12t Ilx +Sin7£/12t 2IlyI2z ■
->COS7Z^12t Ilx
\yL2z
■sin^/12t 2IlyI2z
The results is exactly as for sequence b (Eqn. [6.5]), so the final result is the same i.e. the coupling is refocused.
c
spin 1 -spin 2-
JL
Three different spin echo sequences that can be applied to heteronuclear spin systems. The open rectangles represent 180° pulses.
Summary
In heteronuclear systems it is possible to choose whether or not to allow the offset and the coupling to evolve; this gives great freedom in generating and manipulating anti-phase states which play a key role in multiple pulse NMR experiments.
6-11
6.5    Multiple quantum terms
6.5.1 Coherence order
In NMR the directly observable quantity is the transverse magnetization, which in product operators is represented by terms such as Ilx and 2IlzI2y. Such terms are examples of single quantum coherences, or more generally coherences with order, p = ±1. Other product operators can also be classified according to coherence order e.g. 2IlzI2z has p = 0 and 2IlxI2y has both p = 0 (zero-quantum coherence) and ±2 (double quantum coherence). Only single quantum coherences are observable.
In heteronuclear systems it is sometimes useful to classify operators according to their coherence orders with respect to each spin. So, for example, 2IlzI2y has p = 0 for spin 1 and p = ±1 for spin 2.
6.5.2 Raising and lowering operators
The classification of operators according to coherence order is best carried out be re-expressing the Cartesian operators lx and / in terms of the raising and lowering operators, /+ and /_, respectively. These are defined as follows
I+=IX+Uy      I - Ix- ily [6.6]
where i is the square root of -1. /+ has coherence order +1 and /_ has coherence order -1; coherence order is a signed quantity.
Using the definitions of Eqn. [6.6] Ix and / can be expressed in terms of the raising and lowering operators
/,=*(/+ + /_)     /,=*(/+-/_) [6.7]
from which it is seen that lx and / are both mixtures of coherences with p = +1 and -1.
The operator product 2IlxI2x can be expressed in terms of the raising and lowering operators in the following way (note that separate operators are used for each spin: Il± and I2±)
2IlxI2x = 2 x i     +   ) x i (/2+ +12 )
= i(/1+/2+ + /1_/2_) + M/i+/2- + A-/2+)
The first term on the right of Eqn. [6.8] has p = (+1+1) = 2 and the second term has p = (-1-1) = -2; both are double quantum coherences. The third and fourth terms both have p = (+1-1) = 0 and are zero quantum coherences. The value of p can be found simply by noting the number of raising and lowering operators in the product.
The pure double quantum part of 2IlxI2x is, from Eqn. [6.8],
double quantum part[2IlxI2x ] = \ (/1+/2+ + h h ) [6-9]
The raising and lowering operators on the right of Eqn. [6.9] can be re-expressed in terms of the Cartesian operators:
6-12
i (/1+/2+ + hh ) = *[(/„ + uly )(i2x + n2y ) + (ilx- uly )(i2x - u2y)
= }[2IlxI2x+2IlyI2y]
So, the pure double quantum part of 2IlxI2x is j\2IlxI2x + 2IlyI2yj; by a similar method the pure zero quantum part can be shown to be j[2IlxI2x -2IlyI2yj. Some further useful relationships are given in section 6.9
6.6    Three spins
The product operator formalism can be extended to three or more spins. No really new features arise, but some of the key ideas will be highlighted in this section. The description will assume that spin 1 is coupled to spins 2 and 3 with coupling constants Jl2 and /13; in the diagrams it will be assumed that Jl2 > Jl3.
6.6.1 Types of operators
Ilx represents in-phase magnetization on spin 1; 2IlxI2z represents magnetization anti-phase with respect to the coupling to spin 2 and 2IlxI3z represents magnetization anti-phase with respect to the coupling to spin 3. ^I\xhjiz represents magnetization which is doubly anti-phase with respect to the couplings to both spins 2 and 3.
As in the case of two spins, the presence of more than one transverse operator in the product represents multiple quantum coherence. For example, 2IlxI2x is a mixture of double- and zero-quantum coherence between spins 1 and 2. The product 4IlxI2xI3z is the same mixture, but anti-phase with respect to the coupling to spin 3. Products such as 4IlxI2xI3x contain, amongst other things, triple-quantum coherences.
6.6.2 Evolution
Evolution under offsets and pulses is simply a matter of applying sequentially the relevant rotations for each spin, remembering that rotations of spin 1 do not affect operators of spins 2 and 3. For example, the term 2IlxI2 evolves under the offset in the following way:
2/i A '
—> cos X3j? 2/ljc/2? H-sin^f 2IlyI2z
The first arrow, representing evolution under the offset of spin 1, affects only the spin 1 operator Ilx. The second arrow has no effect as the spin 2 operator l2 and this is unaffected by a z-rotation. The third arrow also has no effect as there are no spin 3 operators present.
The evolution under coupling follows the same rules as for a two-spin system. For example, evolution of Ilx under the influence of the coupling to spin 3 generates 2IlyI3z
hx    w'a > cost^V hx + sin^13? 2 V3z
Further evolution of the term 21 ly I3z under the influence of the coupling to spin 2 generates a double anti-phase term
J12
a a P P spin 2 a P    a P spin 3
The doublet of doublets from spin 1 coupled to two other spins. The spin states of the coupled spins are also indicated.
Alii
2/1A
^ ^ i y ^- 4/1x/2z/3z
Representations of different types of operators.
6-13
2IlyI3z   ln]M* >cos7E/12f 2IlyI3z-smnJl3t 4IlxI2zI3z
In this evolution the spin 3 operator is unaffected as the coupling does not involve this spin. The connection with the evolution of Ily under a coupling can be made more explicit by writing 2I3z as a "constant" y
Thy   2K''2""'2' >cos7E/12f ylly - sin 7Ul3t 2yIlxI2z
which compares directly to
Ily )cos7E/12f lly-sh\7Unt 2IlxI2z
6.7 Alternative notation
In this chapter different spins have been designated with a subscript 1, 2, 3 ... Another common notation is to distinguish the spins by using a different letter to represent their operators; commonly / and S are used for two of the symbols
2AA - 2IxSz
Note that the order in which the operators are written is not important, although it is often convenient (and tidy) always to write them in the same sequence.
In heteronuclear experiments a notation is sometimes used where the letter represents the nucleus. So, for example, operators referring to protons are given the letter H, carbon-13 atoms the letter C and nitrogen-15 atoms the letter ./V; carbonyl carbons are sometimes denoted C. For example, ACXHZNZ denotes magnetization on carbon-13 which is anti-phase with respect to coupling to both proton and nitrogen-15.
6.8 Conclusion
The product operator method as described here only applies to spin-half nuclei. It can be extended to higher spins, but significant extra complexity is introduced; details can be found in the article by S0rensen et al. {Prog. NMR Spectrosc. 16, 163 (1983)).
The main difficulty with the product operator method is that the more pulses and delays that are introduced the greater becomes the number of operators and the more complex the trigonometrical expressions multiplying them. If pulses are either 90° or 180° then there is some simplification as such pulses do not increase the number of terms. As will be seen in chapter 7, it is important to try to simplify the calculation as much as possible, for example by recognizing when offsets or couplings are refocused by spin echoes.
A number of computer programs are available for machine computation using product operators within programs such as Mathematica or Maple. These can be very labour saving.
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6.9    Multiple -quantum coherence
6.9.1 Multiple-quantum terms
In the product operator representation of multiple quantum coherences it is usual to distinguish between active and passive spins. Active spins contribute transverse operators, such as Ix, I and /+, to the product; passive spins contribute only z-operators, /. In a sense the spins contributing transverse operators are "involved" in the coherence, while those contributing z-operators are simply spectators.
For double- and zero-quantum coherence in which spins i and j are active it is convenient to define the following set of operators which represent pure multiple quantum states of given order. The operators can be expressed in terms of the Cartesian or raising and lowering operators.
double quantum, p-±2
DQf =\{2IJh +2IiyIjx) -/,_/,_) zero quantum, p-0
= 7 (2IJjx + 2IiyIjy ) = 2 [hJj- + IiJj+ )
ZQf =\{2IiyI]X-21^)^(1^-ItJJ+)
6.9.2 Evolution of multiple -quantum terms Evolution under offsets
The double- and zero-quantum operators evolve under offsets in a way which is entirely analogous to the evolution of lx and / under free precession except that the frequencies of evolution are {Qi + Qp and (Qt - Qp respectively:
DQ(,7) )CQS/o + G.)tDQ{p + sinfa + Q.VDQ^
x y     i j j x y     i j j y
DQ(i/)    Ofi^afl^   )      / Q + Q y DQ(j/) _ sin/ Q +Q\ DQ(y)
zq{P  a^+a'"« > cos(a -n\t zq(/} + sinfa -n\t zQf
x \   1      j j       x v   1      j j y
ZQ(ii) ) cos^_ _ Qj y ZQ(y) _ sin^_ _ Qj yZq(v)
Evolution under couplings
Multiple quantum coherence between spins i and j does not evolve under the influence of the coupling between the two active spins, i and j.
Double- and zero-quantum operators evolve under passive couplings in a way which is entirely analogous to the evolution of lx and Iy; the resulting multiple quantum terms can be described as being anti-phase with respect to the effective couplings:
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DQ?->cos^/DQ,efff DQf +cos^/DQeff? 2/feDQf
DQf->cos^/DQ,efff DQf -sin^/DQeff? 2/feDQ?
ZQ?'}->cos7tfZQ,efffZQf + sinnJZQeSt 2/feZQf
ZQf->cos^/ZQeff?ZQf -sina^f 2/feZQ?
/DQeff is the sum of the couplings between spin i and all other spins plus the sum of the couplings between spin j and all other spins. /ZQeffis the sum of the couplings between spin i and all other spins minus the sum of the couplings between spin j and all other spins.
For example in a three-spin system the zero-quantum coherence between spins 1 and 2, anti-phase with respect to spin 3, evolves according to
2I3zZQ(;2)->cosnJZQettt 2I3zZQ(;2) -smnJZQeSt ZQ(X12)
where /ZQeff = /13 - J23
Further details of multiple-quantum evolution can be found in section 5.3 of Ernst, Bodenhausen and Wokaun Principles of NMR in One and Two Dimensions (Oxford University Press, 1987).
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