# Product Operators 6.1 A quick review of quantum mechanics

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1 6 Product Operators The vector model, introduced in Chapter 3, is ver useful for describing basic NMR eperiments but unfortunatel is not applicable to coupled spin sstems. When it comes to two-dimensional NMR man of the eperiments are onl of interest in coupled spin sstems, so we reall must have some wa of describing the behaviour of such sstems under multiplepulse eperiments. The tools we need are provided b quantum mechanics, specificall in the form of densit matri theor which is the best wa to formulate quantum mechanics for NMR. However, we do not want to get involved in a great deal of comple quantum mechanics! Luckil, there is a wa 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 eperiments and is well suited to calculating the outcome of modern multiple-pulse eperiments. One particularl appealing feature is the fact that the operators have a clear phsical meaning and that the effects of pulses and delas can be thought of as geometrical rotations, much in the same wa as we did for the vector model in Chapter A quick review of quantum mechanics n this section we will review a few ke concepts before moving on to a description of the product operator formalism. n quantum mechanics, two mathematical objects wavefunctions and operators are of central importance. The wavefunction describes the sstem of interest (such as a spin or an electron) completel; if the wavefunction is known it is possible to calculate all the properties of the sstem. The simplest eample of this that is frequentl 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 sa that the square of the wavefunction gives the probabilit of finding the electron at that point. Wavefunctions are simpl mathematical functions of position, time etc. For eample, the s electron in a hdrogen atom is described b the function ep( ar), where r is the distance from the nucleus and a is a constant. n quantum mechanics, operators represent "observable quantities" such as position, momentum and energ; each observable has an operator associated with it. Operators "operate on" functions to give new functions, hence their name operator function = (new function) An eample of an operator is ( dd); in words this operator sas "differentiate with respect to ". ts effect on the function sin is Chapter 6 "Product Operators" James Keeler, 998, 2002 &

2 d ( sin )= cos d the "new function" is cos. Operators can also be simple functions, so for eample the operator 2 just means "multipl b 2 ". A mass going round a circular path possesses angular moment, represented b a vector which points perpendicular to the plane of rotation. p 6.. Spin operators A mass going round a circular path (an orbit) possesses angular momentum; it turns out that this is a vector quantit which points in a direction perpendicular to the plane of the rotation. The -, - and -components of this vector can be specified, and these are the angular momenta in the -, - and -directions. n 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, and ) are represented b the operators, and Hamiltonians The Hamiltonian, H, is the special name given to the operator for the energ of the sstem. This operator is eceptionall important as its eigenvalues and eigenfunctions are the "energ levels" of the sstem, and it is transitions between these energ levels which are detected in spectroscop. To understand the spectrum, therefore, it is necessar to have a knowledge of the energ levels and this in turn requires a knowledge of the Hamiltonian operator. n NMR, the Hamiltonian is seen as having a more subtle effect than simpl determining the energ levels. This comes about because the Hamiltonian also affects how the spin sstem evolves in time. B altering the Hamiltonian the time evolution of the spins can be manipulated and it is precisel this that lies at the heart of multiple-pulse NMR. The precise mathematical form of the Hamiltonian is found b first writing down an epression for the energ of the sstem using classical mechanics and then "translating" this into quantum mechanical form according to a set of rules. n this chapter the form of the relevant Hamiltonians will simpl be stated rather than derived. n NMR the Hamiltonian changes depending on the eperimental situation. There is one Hamiltonian for the spin or spins in the presence of the applied magnetic field, but this Hamiltonian changes when a radiofrequenc pulse is applied. 6.2 Operators for one spin 6.2. Operators n quantum mechanics operators represent observable quantities, such an energ, angular momentum and magnetiation. For a single spin-half, the - - and -components of the magnetiation are represented b the spin angular momentum operators, and respectivel. Thus at an time the state of the spin sstem, in quantum mechanics the densit operator, σ, can be 6 2

3 represented as a sum of different amounts of these three operators σ()= t at () + bt () + ct () The amounts of the three operators will var with time during pulses and delas. This epression of the densit operator as a combination of the spin angular momentum operators is eactl analogous to specifing the three components of a magnetiation vector. At equilibrium the densit operator is proportional to (there is onl - magnetiation present). The constant of proportionalit is usuall unimportant, so it is usual to write σ eq = 6..2 Hamiltonians for pulses and delas n order to work out how the densit 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 dela), H free, is H free = Ω n the vector model free precession involves a rotation at frequenc Ω about the -ais; in the quantum mechanical picture the Hamiltonian involves the - angular momentum operator, there is a direct correspondence. The Hamiltonian for a pulse about the -ais, H pulse, is Hpulse, = ω and for a pulse about the -ais it is Hpulse, = ω Again there is a clear connection to the vector model where pulses result in rotations about the - or -aes Equation of motion The densit operator at time t, σ(t), is computed from that at time 0, σ(0), using the following relationship σ()= t ep( iht) σ( 0) ep( iht) where H is the relevant hamiltonian. f H and σ are epressed in terms of the angular momentum operators if turns out that this equation can be solved easil with the aid of a few rules. Suppose that an -pulse, of duration t p, is applied to equilibrium magnetiation. n this situation H = ω and σ(0) = ; the equation to be solved is σ( tp)= ep( iωtp) ep( iωtp ) Such equations involving angular momentum operators are common in quantum mechanics and the solution to them are alread all know. The identit required here to solve this equation is ( ) ( ) ep iβ ep iβ cosβ sin β [6.] This is interpreted as a rotation of b an angle β about the -ais. B 6 3

4 putting β= ω t p this identit can be used to solve Eqn. [6.] σ( tp)= cosωtp sinωtp The result is eactl as epected from the vector model: a pulse about the - ais rotates -magnetiation towards the -ais, with a sinusoidal dependence on the flip angle, β Standard rotations Given that there are onl three operators, there are a limited number of identities of the tpe of Eqn. [6.]. The all have the same form ( ) ( ) ep iθa {old operator} ep iθa cosθ {old operator} + sinθ {new operator} where {old operator}, {new operator} and a are determined from the three possible angular momentum operators according to the following diagrams; the label in the centre indicates which ais the rotation is about 6 4 Angle of rotation = Ωt for offsets and ω t p for pulses First eample: find the result of rotating the operator b θ about the -ais, that is ( ) ( ) ep iθ ep iθ For rotations about the middle diagram is required. The diagram shows that (the "old operator") is rotated to (the "new operator"). The required identit is therefore ( ) ( ) + ep iθ ep iθ cosθ sinθ Second eample: find the result of ( ){ } ( ) ep iθ ep iθ This is a rotation about, so diagram is required. The diagram shows that (the "old operator") is rotated to (the "new operator"). The required identit is therefore ep iθ ep iθ cosθ sin θ ( ) { } ( ) { }+ { } cosθ sinθ Finall, note that a rotation of an operator about its own ais has no effect e.g. a rotation of about leaves unaltered Shorthand notation To save writing, the arrow notation is often used. n this, the term Ht is written over an arrow which connects the old and new densit operators. So, for eample, the following

5 is written σ( tp)= ep iωtp σ 0 ep iωtp ( ) ( ) ( ) ωt p σ( 0) σ t ( p) For the case where σ(0) = p cos ω t sinω t ω t 6..6 Eample calculation: spin echo p p a b e f 90 delaτ 80 delaτ acquire ( ) ( ) At a the densit operator is. The transformation from a to b is free precession, for which the Hamiltonian is Ω ; the dela τ therefore corresponds to a rotation about the -ais at frequenc Ω. n the short-hand notation this is Ωτ σ b ( ) To solve this diagram above is needed with the angle = Ωτ; the "new operator" is Ωτ cosωτ + sin Ωτ n words this sas that the magnetiation precesses from towards +. The pulse about has the Hamiltonian ω ; the pulse therefore corresponds to a rotation about for a time t p such that the angle, ω t p, is π radians. n the shorthand notation ω cosωτ + sin Ωτ p t σ() e [6.2] Each term on the left is dealt with separatel. The first term is a rotation of about ; the relevant diagram is thus ω t p cosωτ cosωτcosω t cosωτsinω t p p However, the flip angle of the pulse, ω t p, is π so the second term on the right is ero and the first term just changes sign (cos π = ); overall the result is π cosωτ cosωτ The second term on the left of Eqn. [6.2] is eas to handle as it is unaffected b a rotation about. Overall, the effect of the 80 pulse is then π cosωτ + sin Ωτ cosωτ + sin Ωτ [6.3] As was shown using the vector model, the -component just changes sign. The net stage is the evolution of the offset for time τ. Again, each term on the right of Eqn. [6.3] is considered separatel Ωτ cosωτ cosωτcosωτ sin ΩτcosΩτ Ωτ sin Ωτ cosωτsin Ωτ + sin Ωτsin Ωτ Collecting together the terms in and the final result is cosωτcosωτ + sin Ωτsin Ωτ cosωτsin Ωτ sin ΩτcosΩτ ( ) + ( ) 6 5

6 The bracket multipling is ero and the bracket multipling is = because of the identit cos 2 θ + sin 2 θ =. Thus the overall result of the spin echo sequence can be summarised 90 ( ) 80 τ ( ) τ n words, the outcome is independent of the offset, Ω, and the dela τ, even though there is evolution during the delas. The offset is said to be refocused b the spin echo. This is eactl the result we found in section 3.8. n general the sequence τ 80 () τ [6.4] refocuses an evolution due to offsets; this is a ver useful feature which is much used in multiple-pulse NMR eperiments. One further point is that as far as the offset is concerned the spin echo sequence of Eqn. [6.4] is just equivalent to 80 (). J 2 J 2 Ω Ω 2 The spectrum from two coupled spins, with offsets Ω and Ω 2 (rad s ) and mutual coupling J 2 (H). 6.2 Operators for two spins The product operator approach comes into its own when coupled spin sstems are considered; such sstems cannot be treated b the vector model. However, product operators provide a clean and simple description of the important phenomena of coherence transfer and multiple quantum coherence. 6.. Product operators for two spins For a single spin the three operators needed for a complete description are, and. For two spins, three such operators are needed for each spin; an additional subscript, or 2, indicates which spin the refer to. spin : spin 2 : represents -magnetiation of spin, and 2 likewise for spin 2. represents -magnetiation on spin. As spin and 2 are coupled, the spectrum consists of two doublets and the operator can be further identified with the two lines of the spin- doublet. n the language of product operators is said to represent in-phase magnetiation of spin ; the description in-phase means that the two lines of the spin doublet have the same sign and lineshape. Following on in the same wa 2 represents in-phase magnetiation on spin 2. and 2 also represent in-phase magnetiation on spins and 2, respectivel, but this magnetiation is aligned along and so will give rise to a different lineshape. Arbitraril, an absorption mode lineshape will be assigned to magnetiation aligned along and a dispersion mode lineshape to magnetiation along. 6 6

7 absorption dispersion Ω Ω 2 2 Ω 2 There are four additional operators which represent anti-phase magnetiation: 2 2, 2 2, 2 2, 2 2 (the factors of 2 are needed for normaliation purposes). The operator 2 2 is described as magnetiation on spin which is anti-phase with respect to the coupling to spin 2. Ω Ω Ω 2 The absorption and dispersion lineshapes. The absorption lineshape is a maimum on resonance, whereas the dispersion goes through ero at this point. The "cartoon" forms of the lineshapes are shown in the lower part of the diagram. Ω 2 Note that the two lines of the spin- 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. - or -operators) terms and correspond to multiple-quantum coherences; the are not observable multiple quantum : Ω Finall there is the term 2 2 which is also not observable and corresponds to a particular kind of non-equilibrium population distribution. α Ω β spin state of spin 2 The two lines of the spin- doublet can be associated with different spin states of spin Evolution under offsets and pulses The operators for two spins evolve under offsets and pulses in the same wa as do those for a single spin. The rotations have to be applied separatel to each spin and it must be remembered that rotations of spin do not affect spin 2, and vice versa. For eample, consider evolving under the offset of spin and spin 2. The relevant Hamiltonian is H = Ω + Ω free 22 where Ω and Ω 2 are the offsets of spin and spin 2 respectivel. Evolution under this Hamiltonian can be considered b appling the two terms sequentiall (the order is immaterial) Hfreet Ωt+ Ω2t2 Ωt Ω2t2 The first "arrow" is a rotation about Ωt Ω2t2 cosω t + sin Ω t 6 7

8 The second arrow leaves the intermediate state unaltered as spin-2 operators have not effect on spin- operators. Overall, therefore Ωt+ Ω2t2 cosω t + sin Ω t A second eample is the term 2 2 evolving under a 90 pulse about the -ais applied to both spins. The relevant Hamiltonian is H = ω + ω 2 The evolution can be separated into two successive rotations ωt ωt2 2 2 The first arrow affects onl the spin- operators; a 90 rotation of about gives (remembering that ω t = π/2 for a 90 pulse) ω t cosω t2 sinω t π 2 ω t 2 The second arrow onl affects the spin 2 operators; a 90 rotation of about takes it to π The overall result is that anti-phase magnetiation of spin has been transferred into anti-phase magnetiation of spin 2. Such a process is called coherence transfer and is eceptionall important in multiple-pulse NMR 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: H = 2 J πj2 2 where J 2 is the coupling in H. Evolution under coupling causes the interconversion of in-phase and antiphase magnetiation according to the following diagrams V π 2 V π 2 angle = πjt For eample, in-phase magnetiation along becomes anti-phase along according to the diagram V 2πJ t 2 2 cosπj t + sinπj t note that the angle is πj 2 t i.e. half the angle for the other rotations,. Anti-phase magnetiation along becomes in-phase magnetiation along ; using diagram V: 2πJ2t 2 2 cosπj t2 + sinπj t

9 The diagrams appl equall well to spin-2; for eample 2πJ2t 2 2 cosπj t2 + sinπj t Complete interconversion of in-phase and anti-phase magnetiation requires a dela such that πj2t = π 2 i.e. a dela of /(2J 2 ). A dela of /J 2 causes in-phase magnetiation to change its sign: 2πJ2t 2 t= 2J2 2πJ2t 2 t= J Spin echoes t was shown in section that the offset is refocused in a spin echo. n this section it will be shown that the evolution of the scalar coupling is not necessaril refocused Spin echoes in homonuclear spin sstem n this kind of spin echo the 80 pulse affects both spins i.e. it is a nonselective pulse: τ 80 (, to spin and spin 2) τ At the start of the sequence it will be assumed that onl in-phase - magnetiation on spin is present:. n fact the starting state is not important to the overall effect of the spin echo, so this choice is arbitrar. t was shown in section that the spin echo applied to one spin refocuses the offset; this conclusion is not altered b the presence of a coupling so the offset will be ignored in the present calculation. This greatl simplifies things. For the first dela τ onl the effect of evolution under coupling need be considered therefore: 2πJ2τ 2 cosπj τ + sinπj τ The 80 pulse affects both spins, and this can be calculated b appling the 80 rotation to each in succession π π2 cosπj τ + sinπj τ where it has alread been written in that ω t p = π, for a 80 pulse. The 80 rotation about for spin has no effect on the operator and 2, and it simpl reverses the sign of the operator π π2 cosπj τ + sinπj τ2 cosπj τ sinπj τ The 80 rotation about for spin 2 has no effect on the operators and, but simpl reverses the sign of the operator 2. The final result is thus π cosπj τ + sinπj τ2 cosπj τ sinπj τ π2 cosπj τ + sinπj τ Nothing has happened; the 80 pulse has left the operators unaffected! So, for the purposes of the calculation it is permissible to ignore the 80 pulse and simpl allow the coupling to evolve for 2τ. The final result can therefore just be written down: 2 6 9

10 τ 80 ( ) τ cos2πj τ + sin 2πJ τ From this it is eas to see that complete conversion to anti-phase magnetiation requires 2πJ 2 τ = π/2 i.e. τ = /(4 J 2 ). The calculation is not quite as simple if the initial state is chosen as, but the final result is just the same the coupling evolves for 2τ: τ 80 ( ) τ cos 2πJ τ + sin 2πJ τ 2 n fact, the general result is that the sequence τ 80 (, to spin and spin 2) τ is equivalent to the sequence 2τ 80 (, to spin and spin 2) in which the offset is ignored and coupling is allowed to act for time 2τ nterconverting in-phase and anti-phase states So far, spin echoes have been demonstrated as being useful for generating anti-phase terms, independent of offsets. For eample, the sequence 90 () /(4J 2 ) 80 () /(4J 2 ) generates pure anti-phase magnetiation. Equall useful is the sequence /(4J 2 ) 80 () /(4J 2 ) which will convert pure anti-phase magnetiation, such as 2 2 into inphase magnetiation,. a spin spin 2 b spin spin 2 c spin spin 2 τ τ τ Three different spin echo sequences that can be applied to heteronuclear spin sstems. The open rectangles represent 80 pulses. τ τ τ 6..3 Spin echoes in heteronuclear spin sstems f spin and spin 2 are different nuclear species, such as 3 C and H, it is possible to choose to appl the 80 pulse to either or both spins; the outcome of the sequence depends on the pattern of 80 pulses. Sequence a has alread been analsed: the result is that the offset is refocused but that the coupling evolves for time 2τ. Sequence b still refocuses the offset of spin, but it turns out that the coupling is also refocused. Sequence c refocuses the coupling but leaves the evolution of the offset unaffected. Sequence b t will be assumed that the offset is refocused, and attention will therefore be restricted to the effect of the coupling 2πJ2τ 2 cosπj τ + sinπj τ 2 The 80 () pulse is onl applied to spin π cosπj τ + sinπj τ 2 cosπj τ sinπj τ 2 [6.5] The two terms on the right each evolve under the coupling during the second dela: 6 0

11 cosπj 2πJ2τ 2 τ cosπj τcosπj τ + sinπj τcosπj τ πJ2τ 2 sinπj2τ 22 cosπj2τsinπj2τ 22 + sinπj2τ sinπj2τ Collecting the terms together and noting that cos 2 θ + sin 2 θ = the final result is just. n words, the effect of the coupling has been refocused. Sequence c As there is no 80 pulse applied to spin, the offset of spin is not refocused, but continues to evolve for time 2τ. The evolution of the coupling is eas to calculate: 2πJ2τ 2 cosπj τ + sinπj τ 2 This time the 80 () pulse is applied to spin π cosπj τ + sinπj τ 2 2 cosπj τ sinπj τ The results is eactl as for sequence b (Eqn. [6.5]), so the final result is the same i.e. the coupling is refocused. Summar n heteronuclear sstems 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 pla a ke role in multiple pulse NMR eperiments. 6.4 Multiple quantum terms 6.4. Coherence order n NMR the directl observable quantit is the transverse magnetiation, which in product operators is represented b terms such as and 2 2. Such terms are eamples of single quantum coherences, or more generall coherences with order, p, = ±. Other product operators can also be classified according to coherence order e.g. 2 2 has p = 0 and 2 2 has both p = 0 (ero-quantum coherence) and ±2 (double quantum coherence). Onl single quantum coherences are observable. n heteronuclear sstems it is sometimes useful to classif operators according to their coherence orders with respect to each spin. So, for eample, 2 2 has p = 0 for spin and p = ± for spin Raising and lowering operators The classification of operators according to coherence order is best carried out be re-epressing the Cartesian operators and in terms of the raising and lowering operators, + and, respectivel. These are defined as follows = + i = i [6.6] + where i is the square root of. + has coherence order + and has coherence order ; coherence order is a signed quantit. 6

12 Using the definitions of Eqn. [6.6] and can be epressed in terms of the raising and lowering operators ( ) = ( ) = i + [6.7] from which it is seen that and are both mitures of coherences with p = + and. The operator product 2 2 can be epressed in terms of the raising and lowering operators in the following wa (note that separate operators are used for each spin: ± and 2± ) 2 = 2 2 ( 2 + ) ( ) [6.8] = 2 ( )+ 2 ( ) The first term on the right of Eqn. [6.8] has p = (++) = 2 and the second term has p = ( ) = 2; both are double quantum coherences. The third and fourth terms both have p = (+ ) = 0 and are ero quantum coherences. The value of p can be found simpl b noting the number of raising and lowering operators in the product. The pure double quantum part of 2 2 is, from Eqn. [6.8], double quantum part[ 2 2 ]= 2 ( ) [6.9] The raising and lowering operators on the right of Eqn. [6.9] can be reepressed in terms of the Cartesian operators: 2 ( )= 2 ( + i) ( 2 + i2)+ ( i) ( 2 i2) [ ] [ ] = So, the pure double quantum part of 2 2 is 2 ( ); b a similar method the pure ero quantum part can be shown to be 2 ( 22 22). Some further useful relationships are given in section 6.9 J 3 J 2 Ω α α β β spin 2 α β α β spin 3 The doublet of doublets from spin coupled to two other spins. The spin states of the coupled spins are also indicated. 6.5 Three spins The product operator formalism can be etended to three or more spins. No reall new features arise, but some of the ke ideas will be highlighted in this section. The description will assume that spin is coupled to spins 2 and 3 with coupling constants J 2 and J 3 ; in the diagrams it will be assumed that J 2 > J Tpes of operators represents in-phase magnetiation on spin ; 2 2 represents magnetiation anti-phase with respect to the coupling to spin 2 and 2 3 represents magnetiation anti-phase with respect to the coupling to spin represents magnetiation which is doubl anti-phase with respect to the couplings to both spins 2 and

13 As in the case of two spins, the presence of more than one transverse operator in the product represents multiple quantum coherence. For eample, 2 2 is a miture of double- and ero-quantum coherence between spins and 2. The product is the same miture, but antiphase with respect to the coupling to spin 3. Products such as contain, amongst other things, triple-quantum coherences Evolution Evolution under offsets and pulses is simpl a matter of appling sequentiall the relevant rotations for each spin, remembering that rotations of spin do not affect operators of spins 2 and 3. For eample, the term 2 2 evolves under the offset in the following wa: Ωt Ω2t2 Ω3t3 2 cosω t 2 + sin Ω t The first arrow, representing evolution under the offset of spin, affects onl the spin operator. The second arrow has no effect as the spin 2 operator 2 and this is unaffected b a -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 sstem. For eample, evolution of under the influence of the coupling to spin 3 generates 2 3 2πJ t 3 3 cosπj t + sinπj t Further evolution of the term 2 3 under the influence of the coupling to spin 2 generates a double anti-phase term 2πJ2t 2 2 cosπj t 2 sinπj t n this evolution the spin 3 operator if unaffected as the coupling does not involve this spin. The connection with the evolution of under a coupling can be made more eplicit b writing 2 3 as a "constant" γ 2πJ t γ 2 2 cosπj t γ sinπj t 2 γ which compares directl to πJ t 2 2 cosπj t sinπj t Representations of different tpes of operators. 6.6 Alternative notation n this chapter different spins have been designated with a subscript, 2, 3... Another common notation is to distinguish the spins b using a different letter to represent their operators; commonl and S are used for two of the smbols 22 2S Note that the order in which the operators are written is not important, although it is often convenient (and tid) alwas to write them in the same sequence. n heteronuclear eperiments a notation is sometimes used where the letter represents the nucleus. So, for eample, operators referring to protons are given the letter H, carbon-3 atoms the letter C and nitrogen-5 atoms 6 3

14 the letter N; carbonl carbons are sometimes denoted C'. For eample, 4C H N denotes magnetiation on carbon-3 which is anti-phase with respect to coupling to both proton and nitrogen Conclusion The product operator method as described here onl applies to spin-half nuclei. t can be etended to higher spins, but significant etra compleit is introduced; details can be found in the article b Sørensen et al. (Prog. NMR Spectrosc. 6, 63 (983)). The main difficult with the product operator method is that the more pulses and delas that are introduced the greater becomes the number of operators and the more comple the trigonometrical epressions multipling them. f pulses are either 90 or 80 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 tr to simplif the calculation as much as possible, for eample b recogniing when offsets or couplings are refocused b spin echoes. A number of computer programs are available for machine computation using product operators within programs such as Mathematica. These can be ver labour saving. 6.8 Multiple -quantum coherence 6.8. Multiple-quantum terms n 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, and +, to the product; passive spins contribute onl -operators,. n a sense the spins contributing transverse operators are "involved" in the coherence, while those contributing -operators are simpl spectators. For double- and ero-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 epressed in terms of the Cartesian or raising and lowering operators. double quantum, p =± 2 ( ij) DQ 2 ( 2ij 2ij ) 2 ( i+ j+ + i j ) ( ij) DQ 2 ( 2ij + 2ij ) 2i ( i+ j+ i j ) ero quantum, p = 0 ( ij) ZQ 2 ( 2ij + 2ij ) 2 ( i+ j + i j+ ) ( ij) ZQ 2 ( 2ij 2ij ) 2 i ( i+ j i j+ ) Evolution of multiple -quantum terms Evolution under offsets The double- and ero-quantum operators evolve under offsets in a wa 6 4

15 which is entirel analogous to the evolution of and under free precession ecept that the frequencies of evolution are (Ω i + Ω j ) and (Ω i Ω j ) respectivel: ( ij) Ωiti+ Ωjt j ( ij) ( ij) DQ cos Ω + Ω tdq sin Ω Ω tdq i i j j ij DQ cos Ω + Ω tdq sin Ω Ω tdq ZQ ( ) + ( + ) i j i j ( ij) Ωt + Ω t ( ) ( ij) ( i j) ( i + j) ( ij) Ωiti+ Ωjt j ( ij) ( ij) cos( Ωi Ωj) tzq + sin( Ωi Ωj) tzq ( ij) Ωiti+ Ωjt j ( ij) ( ij) cos Ωi Ωj t sin Ωi Ωj t ( ) ( ) ZQ ZQ ZQ 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 ero-quantum operators evolve under passive couplings in a wa which is entirel analogous to the evolution of and ; the resulting multiple quantum terms can be described as being anti-phase with respect to the effective couplings: ( ij) ( ij) ( ij) DQ cosπj t DQ + cosπj t 2 DQ ij ij DQ cosπj t DQ sinπj t 2 DQ ij ij ZQ cosπj tzq + sinπj t 2 ZQ ZQ DQ,eff DQ,eff k ( ) ( ) ( ij) DQ,eff DQ,eff k ( ) ( ) ( ij) ZQ,eff ZQ,eff k ( ij) ( ij) ( ij) cosπj ZQ,efftZQ sinπjzq,efft 2kZQ J DQ,eff 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. J ZQ,eff is 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 eample in a three-spin sstem the ero-quantum coherence between spins and 2, anti-phase with respect to spin 3, evolves according to ( 2) ( 2) ( 2) 23ZQ cosπjzq,efft 23ZQ sinπjzq,efft ZQ where J = J J ZQ,eff 2 23 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 (Oford Universit Press, 987). 6 5

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