A Simplified Approach to Product Operator Formalism - Journal of

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A Simplified Approach to Product Operator Formalism Christopher E. Spiese Department of Chemistry, Juniata College, Huntingdon, PA 16652; [email protected]

The product operator formalism is a valuable model for describing weakly coupled systems of nuclei observed by nuclear magnetic resonance spectrometry (NMR). Derived from the density matrix, it is applicable to two-dimensional experiments and adequately explains quantum coherences. The product operator formalism, however, retains the fundamental quality of the classical vector model—it is easily visualized, especially with regards to the primary operators (Iz, Ix, and Iy ). The combination of these two aspects makes the product operator formalism a nearly ideal model for NMR spectrometry. Traditionally, the introduction of product operator formalism has been reserved for a graduate-level course in spectroscopy. This article, however, presents the product operator formalism in a mathematical way that is easily understood by undergraduate students. By eliminating the complex derivations and focusing on the formalism as an extension of the vector model, only a working knowledge of vectors and trigonometry is required to utilize the product operator formalism. The Vector Model Before discussing the product operator formalism, it is imperative that some basic concepts of the vector model be presented. Derome (1) presents a good explanation, and it is his text that forms the basis of this explanation of the vector model. In NMR, there is an imagined a set of axes corresponding to the standard three axes, x, y, and z. There are two frames of reference from which these axes can be viewed. The first is a static frame called the laboratory frame. In this frame, both the axes and the observer are fixed in their position. The magnetic moment of the sample precesses around the z axis with a certain angular frequency, ω, which corresponds to the Larmor frequency, assuming that the magnetic moment of the sample is in resonance with the magnetic field of the instrument. This precession complicates the visualization of the magnetization, and so this frame of reference will not be utilized in this article.

The second frame of reference is a set of axes that rotate at exactly the angular frequency ω. Likewise, the observer, too, rotates at this speed. Using this frame of reference eliminates the precession of the magnetic moment of the sample and simplifies the subsequent mathematics. For the rest of this article, it will be assumed that the vectors are viewed in this frame of reference. The magnetic moment of the sample is assumed to be aligned along the positive z axis and, using a hydrogen nucleus as an example, called IHz. As long as there is no perturbing force (i.e., a pulse), this vector will remain oriented as such. When a pulse is applied, however, a torque is created. This torque pushes the vector towards the xy plane, as shown in Figure 1. How far the projection of this vector is pushed into the xy plane is determined by the duration of the pulse. Typically, pulse durations are indicated by an angle. This angle corresponds to the angle between the magnetic moment and the z axis. Notice that after a pulse of B1, the magnetic moment lies between being completely aligned along the z axis and being completely within the xy plane. This vector can be decomposed into its component vectors, one of which is aligned along the z axis, while the other vector lies within the xy plane. The decomposition of the net magnetization after a pulse of angle θ into its component vectors is shown in Figure 2. The vector in the xy plane has a magnitude equal to that of the original vector, M, multiplied by the cosine of the pulse angle. Similarly, the vector aligned along the z axis has a magnitude equal to M multiplied by the sine of the pulse angle. The derivation of these vectors is simple. When a torque is created by the application of a magnetic field, the torque is equal to the cross product of the magnetic moment and the magnetic field of the pulse, B1. The vector resulting from this cross product can be decomposed into the original vec-

Mz

y M

B0

y x z

B1, +x

My

M × B1

x

B1

Figure 1. Effect of a pulse, B1, on the net magnetization of the sample, M, in a magnetic field of B0. The torque created on net magnetization is equal to the cross product of the magnetization and the pulse.

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z Figure 2. Decomposition of net magnetization into component vectors, My and Mz.

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tor (aligned with the z axis) multiplied by the cosine of the pulse angle and a vector in the xy plane multiplied by the sine of the pulse angle. The phase of the component in the xy plane is determined by simply taking the cross product of the magnetic moment and the applied magnetic field. For example, with a magnetic moment aligned along the z axis (IHz) with a π兾6 pulse aligned along the x axis, we get the vectors: π , +x 6

IHz

IHz cos

π 6

− IHy sin

π 6

In this expression, notice that the notation of calling the magnetic moment Ikp, where k is the nucleus and p is the phase, is carried throughout the entire process. In fact, the pulse itself could be written IHx to preserve this notation. Second, this expression does not explicitly state that a cross product of vectors is performed. The cross product is implied, with the pulse being the first vector in the pair. Therefore, one could write the previous expression as:

IHx × IHz = − IHy sinθ This is precisely the component of the resulting vector in the xy plane. Notice that the right hand rule is followed when determining the cross product of vectors, even though many classical NMR descriptions follow a left hand rule. This convention is used by Sørensen et al. (2) and is in keeping with the standard mathematical definition of a cross product of vectors, and therefore will be the convention that will be used in this article. Focusing on the component of the magnetization vector in the xy plane, the process of chemical shift evolution can be described. Because of shielding effects, certain nuclei come into resonance with the magnetic field at slightly different frequencies. As time passes between the end of the pulse and the start of either the acquisition of the signal or the second pulse, these nuclei begin to separate based on the difference in frequency from the Larmor frequency. In terms of

ω0 + ω

y

ω0 − ω x

the vector, the component in the xy plane is again decomposed into two rotating vectors, each rotating at the angular frequency of ω0 ± ω, where ω0 is the Larmor frequency and ω is the resonant frequency of the nucleus. This is shown in Figure 3. It is at this point that the classical vector model fails. Because the vector model uses exclusively single-spin operators, spin coupling cannot possibly be explained adequately. Therefore, a new model is required for systems of nuclei that exhibit weak coupling. This model is the product operator formalism. The Product Operator Formalism The product operator formalism not only encompasses everything contained in the classical vector model but also covers other aspects such as spin coupling and quantum coherences. The former will be discussed in depth; The latter, however, will not be discussed. This is because quantum coherences are not necessary for a basic understanding of the product operator formalism. A good explanation of quantum coherences, as well as a more visual model based on nonclassical vectors, can be found in reference 3. The discussion of the product operator formalism will begin with an explanation of two-spin operators, a concept unknown in the classical vector model. Then, the product operator formalism will be presented in full, from the first pulse until the acquisition. Much of the notation will be identical to that used in the classical vector model.

Two-Spin Operators To describe spin coupling, a new set of vectors is required. Sørensen describes these vectors as magnetizations of one nucleus, antiphase with respect to the other. In terms of notation, these vectors are written in a similar manner as the single-spin operators. These vectors all have the basic form 2IkpIls, where k and l are the coupled nuclei and p and s are the phases of k and l, respectively. The final vector is the unity vector, E兾2. This vector is mathematically equivalent to a scalar value of 1兾2 and occurs whenever a two-spin operator is affected by another vector with the same phase and can generally be ignored. The Formalism As in the vector model, the initial condition of the sample is a single vector aligned along the positive z axis and representing the net magnetization. Remember that this vector is being viewed in a rotating frame of reference. When the first pulse is applied, a pulse with a phase of +x, the vector begins to bend down toward the xy plane. This pulse will last for a duration of π兾2. Note that this pulse angle is measured in radians, which is simply a convention of mathematics. This pulse is written as:

IHz z Figure 3. Separation of net magnetization as decomposition into two vectors rotating with angular velocity of ω0 ± ω.

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(π 2) IHx

IHz cos

π 2

− IHy sin

π 2

Simplification of the two final vectors yields ᎑IHy as the magnetization following the π兾2 pulse. This will be true of all π兾2 pulses—only one vector will be the result.


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After this pulse, as in the vector model, chemical shift evolution is evolved. For the product operator formalism, this is written in a similar manner. The “pulse” is written as the chemical shift operator, 2πδτ Ikz, where k is the nucleus evolving. The δ term is equivalent to the actual shift of the nucleus from the zero point. The chemical shift operator can be thought of as a pulse that forces a rotation around the z axis. As such, chemical shift evolution can be written as: (2πδτ ) IHz

− IHy

(πJτ)2IHz ICz

− IHy cos (πJτ) − 2IHx ICz sin(πJτ)

At this point, either of two processes may happen. The first is acquisition of the signal. This means that any singlespin operators with a phase of either x or y will create a signal that the spectrophotometer will read and interpret as a free-induction decay signal. The second is the application of an additional pulse. If this is the case, the pulse will affect any operator that has a magnetization of the same nucleus as the pulse. This means that a pulse on the hydrogen channel (IHp) will not affect any aspect of the carbons, as in the operator 2IHxICz. This will hold true whenever multiple pulses are applied. If a second pulse is applied, then chemical shift and spin coupling must also be evolved. Chemical shift is evolved just as though it were another pulse. Spin coupling, however, presents a new problem. Spin coupling involves the use of a twospin operator as a pulse. When this operator is combined with another two-spin operator, however, a single-spin operator is often the result. How does this come about? Remember how combinations of operators have been discussed. Throughout this article, these combinations have been described in terms of a cross product of vectors. The combination of a pair of two-spin operators is no different. This combination is written as: (πJτ)2IHz ICz

2IHx ICz

(πJτ)2 Ikz Ilz

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1

H:

π 2

π 2

τ1 x

acq x

where acq is the acquisition of the signal. Using the vectors, we will assume two protons that interact via coupling. The first pulse will push both vectors into the xy plane.

I1z + I2z

(π 2) IHx

−I1y − I2y

Focusing solely on hydrogen 1 (᎑I1y ), chemical shift evolution and coupling are resolved. Note that the same interactions are applicable to hydrogen 2. Only the relevant vector results are shown. − I1y

(2πδτ1) I1z

(π Jτ1)2 I1z I2z

I1x

I1z sin (2 πδ τ1) − I1y cos (2 πδτ1) 2 I1y I2z sin (2 π δ τ1) sin (πJτ1)

After the first delay of τ1, a second π兾2 pulse is applied.

2 I1y I2z sin (2 π δ τ1) sin (πJτ1)

(π 2) IHx 2 I1z I2y sin (2 π δ τ1) sin (πJτ1)

Finally, during the acquisition, coupling, and chemical shift evolution occur once again to yield the final observable vector, [I2xsin(2πδτ1)sin(πJτ1)sin(2πδτ2)sin(πJτ2)]. 2 I1y I2z sin (2 π δ τ1) sin (π Jτ1)

(2 πδτ1) I1z

2 I1z I2y sin (2 π δ τ1) sin (π Jτ1)sin (2πδ τ2)

IHy sin (πJτ) + 2IHx ICz cos(πJτ)

Notice that a single-spin operator is created from the combination. This single-spin operator is the result of the cross product of the second term (ICz ) of the two operators being combined. Remember that when operators of the same phase are combined, the unity operator E兾2 is the result. In this case, the terms of the coupling nucleus have the same phase. Therefore, E兾2 is the answer for that combination. Simplifying all the terms returns the single spin operator, IHy. In general, this means that:

2Ikp Ilz

An example of a pulse sequence will be worked through so that the rules can be seen in application. The COSY sequence correlates hydrogens coupled through spin–spin coupling interactions. COSY can often reveal the entire structure of a molecule without the need for additional spectra. The pulse sequence, as found in Braun et al. (4), is as follows,

IHx cos (2πδτ ) − IHy sin (2πδτ)

This is the point at which the vector model broke down. In that model, no vectors existed that could adequately depict spin coupling. With the two-spin operators in the product operator formalism, spin coupling can now be determined. To do this, we use the spin coupling equivalent of the chemical shift operator. This vector is πJ τ2IkzIlz, where J is the coupling constant. This again can be thought of as a pulse, and as such spin coupling can be written as:

− IHy

Example: COSY

I (z × p ) sin (πJτ) + 2Ikp Ilz cos(πJτ)


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2 I1y I2z sin (2 π δ τ1) sin (π Jτ1)

(π Jτ2)2 I1z I2z

I2x sin (2 πδ τ1) sin (π Jτ1)sin (2πδ τ2) sin (π Jτ2)

This is not to say that [I2xsin(2πdt1)sin(πJt1)sin(2πdt2)sin(πJt2)] is the only observable vector; actually, there is a family of vectors for each nucleus. The distinguishing feature of this vector is that this is the one that creates the cross peaks in the spectrum as it describes the interaction between the two nuclei.


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Discussion The product operator formalism is a valuable tool for students who have an interest in NMR spectroscopy. It adequately explains spin coupling and quantum coherences, which are beyond the scope of the vector model. It is also much less complex and cumbersome than the quantum mechanical spin-density matrix. The product operator formalism as presented in this article does not include any mention of Hamiltonians, eigenfunctions, or any other quantum mechanics because, for the undergraduate, they are simply not needed. The undergraduate often needs a better tool than the vector model, but in most cases has not had any quantum mechanics. Therefore, this method of presenting the product operator formalism gives that tool to the undergraduate without requiring the student to tackle complex quantum mechanical derivations. Conclusion The product operator formalism has been presented with the undergraduate in mind. Quantum mechanics has been avoided, as it tends to complicate a straightforward model of NMR spectroscopy. This method presents the product op-

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erator formalism as an extension of the classical vector model, with which most undergraduate chemistry students are already familiar. Acknowledgments I would like to thank Tom L. Fisher of Juniata College for vision and the encouragement, without which this article would never have taken shape. I would also like to thank Paul Schettler of the same institution for assistance with editing and revising. Literature Cited 1. Derome, A. E. Modern NMR Techniques for Chemistry Research; Pergamon: Oxford, United Kingdom, 1988; Chapter 4. 2. Sørensen, O. W.; Eich, G. W.; Levitt, M. H.; Bodenhausen, G.; Ernst, R. R. Prog. NMR Spec. 1983, 16, 163–192. 3. Donne, D.; Gorenstein, D. A. Pictorial Representation of Product Operator Formalism: Non-classical Vector Diagrams for Multidimensional NMR. http://www.biophysics.org/img/ Donne.D.pdf (accessed Sep 2003). 4. Braun, S.; Kalinowski, H.-O.; Berger, S. 150 and More Basic NMR Experiments; Wiley-VCH: Weinheim, Germany 1998; p 353.


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A Simplified Approach to Product Operator Formalism - Journal of

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