AB spin system

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

For example, the observed transitions of an AB spin system have a Liouville matrix given m equation (B2.4.35). The coupling constant is J, and it is assumed that ciig = = -5/2, so that 5 is the frequency... 

However, all the nuclei have spin. An example of this occurs in the intemiolecular exchange of an AB spin system, as in equation (B2.4.4I). 

Figure B2.4.5. Simulated lineshapes for an intennolecular exchange reaction in which the bond joining two strongly coupled nuclei breaks and re-fomis at a series of rates, given beside tlie lineshape. In slow exchange, the typical spectrum of an AB spin system is shown. In the limit of fast exchange, the spectrum consists of two lines at tlie two chemical shifts and all the coupling has disappeared.

Fio. 3. Schematic representation of energy levels, populations and resultant patterns of polarization in the n.m.r. s]3eotrum of an AB spin system. (Relative population of the energy levels is indicated by the thickness of the bars.)... 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

Fig. 4. Simulations of the mutual exchange in an AB spin system, along with the components of each lineshape. All the spectra in this figure are the sum of four lines.

Let us now look at what happens when the chemical shift difference between the coupled nuclei is small and the simple patterns become distorted, i.e. non-first-order spectra. The simplest example of this is an AB spin system. Remembering that an AX system will give a pair of doublets, we will now look at a system in which the chemical shift difference is relatively small, so that the spin system is now described as... 

Fig. 9.1 Part of the H NMR spectrum of C60H4 at 500MHz in a 2 1 solution of C6D6/CS2 showing a typical AB spin-system (Alemany et al. 1997a)...

The mode of attachment of the [M(CO)s] fragment to the 1,2,4-selenadiphosphole ring was established as being via the phosphoms P-2 atom bonded directly to selenium. The 31P NMR spectrum of both compounds 19 and 34 shows a pattern of lines typical for an [AB] spin system with satellites due to further coupling to the 77Se nucleus. Additional... 

Declare a spin system AB spin system spin system AB(2)... 

However, we must also consider the minor isotopomers containing two Hg atoms, [ Hg- Hg-Hg]2+ and [ Hg-Hg- Hg]2+. The latter is not detected because it contains two equivalent nuclei whose resonances will be coincident with those of [ Hg-Hg-Hg]2+. In contrast, the nonequivalent spins in the remaining isotopomer, [ Hg- Hg-Hg]2+, are expected to couple strongly with each other. The direct Hg- Hg coupling (139 600 1000 Hz ) is large compared to the difference in chemical shifts ( 45 000 Hz), so an AB spin system is formed. The lines at —1400 and -1550 8 are the intense inner transitions of the AB spin system, while the two outer transitions are well outside the spectral width and are not observed. This is believed to be the first observation of direct Hg- Hg coupling. 

As with the net E and A effects, the multiplet effects can also be rationalized on the basis of nonequilibrium spin state populations. Recall the AB spin system we discussed in... 

Section 9.1. Consider Figure 11.6, which is derived from Figure 9.2. Suppose a chemical reaction generated a product containing an AB spin system (with J > 0), where the polarization increased the populations of the parallel spin states (Figure 11.6a). Lines 1 and 2 would exhibit an A effect, while lines 3 and 4 would exhibit an E effect, resulting in El A multiplet effects in both doublets. Alternatively, if the antiparallel states were favored (Figure 11.6c), an A/E multiplet effect would be observed in both doublets. 

Figure 11.6. Way in which spin state populations give rise to multiplet effects in an AB spin system.

For the AB spin system we have seen that there are four allowed transitions 0i- From the expressions that we derived for energy levels we can easily determine the frequencies of these four transitions, and from the expressions for the wave functions and transition probabilities, we can compute the relative intensities of the four lines. 

Frequencies and intensities of the iines in the two-dimensional 7-spectrum of an AB spin system when both spins experience the same 180 non-selective inverting pulse (45)... 

FIG. 12. Experimental absolute-value display of the two-dimensional J-spectrum of the AB spin system provided by methyl 5-nitrofuran-2-carboxylate in (CD3)2CO. The four traces on the right are theoretical simulations with Lorentzian line-shapes. From ref. 149. 

Check it 3.2.1.2 illustrates magnetic field effects on a weakly and strongly coupled nuclear spins using a AX and AB spin system. 

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