Product-operator formalism

As shown for the simple example in Fig. 2.2 explicit density matrix calculation can be cumbersome and this approach is often not recommended for complex pulse sequences, particularly if large data matrices of multi-spin systems or multi-pulse sequences must be evaluated. Consequently different operator formalisms [2.15 - 2.19] using Cartesian, spherical, shift, polarization and tensor operators, based on different coordinate systems or basic functions, have been developed where each formalism is suitable for a particular type of problem. The criteria used to select the appropriate formalism depend on the spin system being described  

With respect to the pulse sequences the suitable formalism must derive  

Taking into account all the relevant criteria spin-1/2 nuclei in the liquid phase can generally be described using CARTESIAN, spherical and shift product operators as shown in Table 2.3. The spherical operators are not shown because they can be easily derived from the shift operators, see Table 2.5. 

In Table 2.3 the denotes the preferred formalism to analyse the particular spin system attribute. Irrespective of the chosen formalism, any formalism can be modified to describe any attribute of spin system or pulse sequence. 

The derivation of the product operator formalism from the density matrix is relatively straightforward. Starting with the density matrix of an arbitrary defined spin system, the density matrix is expanded into a linear combination of orthogonal matrices, the so-called product operators which specify an orthogonal coherence component 

---

Sprensen, O. W., Eich, G. W., Levitt, M. H., Bodenhausen, G., Ernst, R. R., Product Operator Formalism for the Description of NMR Pulse Experiments, Prog. Nucl. Magn. Reson. Spectrosc., 1983,16, 163... 

The evolution of magnetization during NMR experiments can be followed by means of the so-called product operators formalism. This approach has the advantage of being simple, and of being pictorially representable. 

This is pretty complicated, but the advantage is that we can keep track of everything of importance. Any pulse sequence can, in principle, be examined to see what effect it will have on the sample magnetization and what observable signals will remain at the end. Product operator formalism represents the full quantum-mechanical phenomenon of NMR, so that any type of experiment including mysterious things like multiple-quantum coherences (MQCs) can be represented correctly. 

Sprensen OW, Eich GW, Levitt MH, Bodenhausen G, Ernst RR. Product operator formalism for the description of NMR pulse experiments. Prog. NMR Spectrosc. 1983 16 163-192. 

We can look at this more precisely using the product operator formalism, even though it is more important to focus on the conceptual picture rather than the math. For a resonance with Larmor frequency vG, we have during the first gradient... 

The density matrix representation is actually simpler than the product operator formalism for dealing with zero and multiple quantum coherences. Note that the type of multiple quantum coherence can be read from the lower left elements of the antidiagonal. ... 

POMA A complete mathematica implementation of the NMR product-operator formalism—Guntert et al. JMR 101A, 103-105 (1993). POMA is a flexible implementation of the product operator formalism for spin-1/2 nuclei written for Mathematica. It provides analytical results for the time evolution of weakly coupled spin systems under the influence of free precession, selective and non-selective pulses, and phase cycling. As part of Mathematica, it requires a license, but the source code is free. Mathematica also provides a framework for visualizing and storing results. 

A complete understanding of the processes involved in 2D NMR requires a more powerful theoretical underpinning than used in most of the book, so Chapter 11 is devoted to an introduction to the density matrix and product operator formalisms. These methods are not familiar to many chemists, but they are simple outgrowths of ordinary quantum mechanics. We examine the basic ideas and apply this theory in Chapters 11 and 12 to describe some of the most frequently used ID and 2D NMR experiments. 

In Chapter 11 we shall also introduce the product operator formalism, in which the basic ideas of the density matrix are expressed in a simpler algebraic form that resembles the spin operators characteristic of the steady-state quantum mechanical approach. Although there are some limitations in this method, it is the general approach used to describe modern multidimensional NMR experiments. 

The faithful representation of the shape of lines broadened greatly by dipolar and, especially, quadrupolar interactions often requires special experimental techniques. Because the FID lasts for only a very short time, a significant portion may be distorted as the spectrometer recovers from the short, powerful rf pulse. We saw in Section 2.9 that in liquids a 90°, t, 180° pulse sequence essentially recreates the FID in a spin echo, which is removed by 2r from the pulse. As we saw, such a pulse sequence refocuses the dephasing that results from magnetic field inhomogeneity but it does not refocus dephasing from natural relaxation processes such as dipolar interactions. However, a somewhat different pulse sequence can be used to create an echo in a solid—a dipolar echo or a quadrupolar echo—and this method is widely employed in obtaining solid state line shapes (for example, that in Fig. 7.10).The formation of these echoes cannot readily be explained in terms of the vector picture, but we use the formation of a dipolar echo as an example of the use of the product operator formalism in Section 11.6. 

Chapter 11 Density Matrix and Product Operator Formalisms... 

As we shall see, it is very helpful to be able to compute the behavior of the spin system from the sort of matrix multiplications that we have already carried out. On the other hand, it is often possible to simplify the algebraic expressions by using the corresponding spin operators. In fact, this is the concept of the product operator formalism that we discuss later. Note that from Eqs. 11.35 and 11.36, the (redefined) density matrix at equilibrium can be written in operator form as... 

The product operator formalism is normally applied only to weakly coupled spin systems, where independent operators for I and S are meaningful. That means that it is permissible to treat evolution under chemical shifts separately from evolution under spin coupling. It also means that a nonselective pulse can be treated as successive selective pulses affecting only one type of spin. To simplify the notation and to facilitate the handling of the transformation of each product operator, such separations are almost always made. 

These two detailed examples should illustrate the application of the product operator formalism. It is clear that experiments with several pulses and especially with several evolution periods can provide lengthy expressions. In Chapter 12 we indicate the way in which this formalism can be used to explain several 2D NMR experiments but without providing detailed computations. 

The product operator formalism is normally based on Cartesian coordinates because that simplifies most of the calculations. However, these operators obscure the coherence order p. For example, we found (Eq. 11.80) that products such as IXSX represent both zero and double quantum coherence. The raising and lowering operators I+ and I are more descriptive in that (as we saw in Eq. 2.8) these operators connect states differing by 1 in quantum number, or coherence order. We can associate I+ and I with p = +1 and — 1, respectively, and (as indicated in Eq. 11.79) the coherences that we normally deal with, Ix and Jy, each include both p = 1. Coherences differing in sign contain partially redundant information, but both are needed to obtain properly phased 2D spectra, in much the same way that both real and imaginary parts of a Fourier transform are needed for phasing. 

Now that we have available the more powerful theoretical approaches of the density matrix and product operator formalisms to augment the still very useful vector picture, we can examine the mechanisms of some common NMR methods in more detail. In this chapter we discuss some one-dimensional techniques but concentrate on 2D experiments. We see that some of the 2D experiments can be extended to three or four dimensions to provide additional correlations and to spread out the crowded 2D peaks that sometimes arise from large molecules. 

COSY was the first 2D experiment attempted and in many ways serves as the prototype, as we discussed in Chapter 10. Now that we can apply product operator formalism, let s return to a further consideration of the mechanism for COSY and look at some of the factors that make it a more complex technique than was apparent in our initial treatment. 

---