Spin system notation

270 Nuclear Magnetic Resonance Spectroscopy Part Three Spin-Spin Coupling 

FIGURE 5.35 Afirst-order AX system A V large, and n + 1 Rule applies. 

Each chemically distinct nucleus is assigned a letter and a numerical subscript is used to indicate the number of such nuclei. If the chemical shift difference between two sets of nuclei is large compared with the coupling constant between them (3i - S2 Jn), letters that are well apart in the alphabet are used A, X, M. Such systems are first order and give rise to simple multiplets in the NMR spectra. On the other hand, if the chemical shift difference is of the same order of magnitude as the coupling constant between the two nuclei ( 1 - 2 J12), then consecutive letters are used A, B, C,. . . , X, Y, Z. The latter systems give rise to second-order spectra with complex multiple patterns. 

Some examples to illustrate the use of this spin system notation to distinguish between first- and second-order systems and to explain the concept of magnetic inequivalence will now be discussed. Because is the only spin-1/2 nucleus with 100% natural abundance that forms a wide variety of inorganic ring systems, most of the examples are taken from phosphorus chemistry (for other examples, see Chapter 11). 

Both isomers lack symmetry elements and, consequently, the two three-coordinate phosphorus atoms are inequivalent. The chemical environments of the two P(III) centres are similar in both isomers, but the large value of /ab=263 Hz indicates a P-P bond as found in 3.11b.  

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FIGURE 7.40 A first-order AX system Av large, and n + Rule applies. 

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In this equation, 01 is the ifeqiieney of the RF irradiation, oiq is the Lannor ifeqiieney of the spin, is the spm-spm relaxation time andM is the z magnetization of the spin system. The notation ean be simplified somewhat by defining a eomplex magnetization, AY, as in equation (B2.4.3). 

Before writing an equation for the motion of a spin system, we introduce a convenient notation system, the so-called superoperator formalism. It was used for the first time in a description of NMR phenomena by Banwell and Primas, and later developed by Binsch. (12— 15)... 

With an understanding of simple first-order spin systems and of the Pople notations, we can consider the following example. 

The concept of chemical shift equivalence is central to NMR spectrometry. Chemical-shift equivalent (isochronous) nuclei comprise a set within a spin system (Pople notation, Section 3.5). 

If two protons in the same set (i.e., chemical-shift-equivalent protons in the same multiplet) couple equally to every other proton in the spin system, they are also magnetically equivalent, and the usual Pople notations apply A2, B2, X2 etc. However, if two protons in a set are not magnetically equivalent, the following notations apply AA, BB, XX, etc. To rephrase Two chemical-shift-equivalent protons are magnetically equivalent if they are symmetrically disposed with respect to each proton in the spin system. Obviously magnetic equivalence presupposes chemical-shift equivalence. In other words, do not test for magnetic equivalence unless the two protons in question are chemical-shift equivalent. 

For each compound given below (a-o), describe all spin systems (using Pople notation where appropriate), chemically shift equivalent protons, magnetic equivalent protons, enantiotopic protons, and diastereotopic protons. 

A short example will illustrate the usefulness (and complexity) of density matrix notation. Consider the INEPT sequence for a13C-1 H (S-I) spin system. The pulse sequence is simply... 

Simulated "pulse sequences" are built line-by-line similarly to operator assignations. Typical operations on a spin system s density matrix, such as the application of a pulse or evolution over time f, can be applied explicitly in an operator by operator manner (Table 1, lines 8-9) or through the use of built-in grouped operators (Table 1, lines 8-9 alternative coding). Either way, the user has complete control to structure the object-oriented code in a manner which reads almost as naturally as the original physics notation itself. Two examples of spectral simulation pulse sequences coding are shown in the next section. 

Spin systems (collections of interacting nuclei) are often labeled (Pople notation) by assigning a letter from the alphabet to each set of magnetically equivalent nuclei. Nuclei that are close (but not identical) in chemical shift are given letters that are close in the alphabet (e.g., AB). Two nuclei that are chemically equivalent but not magnetically equivalent are assigned the same letter but one letter is primed (e.g., AA ). 

Use Pople notation to identify the types of spin systems that give rise to the spectra shown in Figures 8.20a and b (review problem 8.7). 

Use Pople notation to identify the type of hydro-gen/phosphorus spin system in the structure below ... 

As we saw in Chapter 6, in a spin system consisting of N nuclei, each of spin Y2, there are 2N spin product functions, labeled 2, etc. (or alternatively in the ket notation, l>, 2>, etc.). Each of the 2N wave functions nuclear spins in the molecule can always be described as a linear combination... 

For a macroscopic sample, consisting of a large number of chemically identical molecules, the spin product functions are the same for all individual molecules, but the expectation value of the operator must be averaged over all molecules (spin systems) in the ensemble. With the notation that has been introduced, this average may be obtained by considering the probability of occurrence of each pure state, as expressed in terms of the basis functions ... 

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