Product operator formalism and

Product-Operator Formalism and Cohere nee-Level Diagrams... 

Appendix 6 Product-Operator Formalism and Coherence-Level Diagrams 323... 

The HSQC experiment cannot be explained readily by using spin gymnastics arguments. To understand the HSQC and other more complex pulse sequences, we should consult other texts to learn the product operator formalism and density matrix theory. 

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. 

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. 

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. 

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