The Two-Spin System

As we would expect, Eq. 11.47 describes a magnetization that moves in a circle at angular frequency o phased so as to lie along — y at t = 0. 

We could continue to manipulate this one-spin system with additional pulses and evolution times, but instead we turn to a system of more interest. 

Consider now the two-spin system, in which chemical shifts and scalar coupling come into play. In Chapter 6 we discussed the two-spin system in detail, both the weakly coupled AX system and the general case, AB. To illustrate the application of the density matrix, we concentrate first on the AX system and then indicate briefly how the results would be altered for AB. To simplify the notation, we call the nuclei I and S, rather than A and X, and use the common notation in which the spin operators and their components are designated, for example, Ix and Sx, rather than the more cumbersome 4(A) or /. Although the I-S notation is usually applied to heteronuclear spin systems, we use it here to include homonuclear systems (e.g., H-H) as well. 

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In general, a relatively direct and straightforward means of analysis may be performed in the case of slow exchange on the chemical shift time-scale by combining the relaxation matrices of the free and bound state with the kinetic matrix to describe the effect of exchange [12]. For the two spin systems described above the expanded relaxation matrix R can be written as ... 

Figure 4 shows the results of a simulation of CT-VPP-REDOR evolution curves calculated for the two different S-I4. five-spin systems and the S-I two-spin system from Figure 2. Again the distances were chosen so that the resulting second moments as calculated employing Equation (12) are identical for the five-spin and the two-spin systems. As expected, the AS/So values at fpp/TR=0.5 match the data points of the REDOR curves (cf. Figures 2 and 4).

Thus, with the assumption of rapid isotropic bond reorientations, the Hamiltonian describing the two-spin system under high resolution conditions (Hhh) consists of three terms, Hz, Hg and H, ... 

H2 and H6. But where do we go from here The Hd/e peak connects to Hb and to Hc, but we cannot tell which is H3 and which is H5 because we lost the specific trails of the two spin systems when they crossed at Hd/e. 

To understand any coherence other than SQC, we need a new and more general definition of coherence. Coherence arises from the quantum mechanical mixing or overlap of spin states ( superposition ). In the two spin system (I, S = ll, 13C) we have four spin states (aa, up, pa, and PP), which are all stable states of defined energy. Let s talk about a single - C pair (one molecule). It is possible for this pair to be in any one of the four energy states, but it is also possible for the pair to be in a mixture or overlap or superposition of two states. This is one of the fundamental tenets of quantum mechanics Sometimes you cannot be sure which energy state a particle is in. Let s say that this particular pair is in a mixture of states aa and pp ... 

Let us start by considering a molecule with two coupled nuclei (A and B) of the same isotope (e.g., H). There are three independent variables that describe the system completely the chemical shifts (8 or 5v) of A and B and their homonuclear coupling constant 7. The exact appearance of the NMR spectrum for this system, that is, the position and intensity of each line, can be calculated from the values of these three variables (and the operating frequency of the instrument if 8 values are used). The general solution for the two-spin system is a four-line spectrum, with each line having the position and intensity listed below ... 

Solution (a) From either Figure 9.14b or the general equations for the two-spin system, we can see that the separation between lines 2 and 3 is 2C - 7. But in this case we have arbitrarily set this separation equal to 7. Therefore, 2C - 7 = 7, or C = 7. Using this relation, we can calculate the position and intensity of each line ... 

Equation 6.33 is completely general. For the two-spin system, it results in the transitions we identified in Fig. 6.2, while the double quantum transition between and 4, and the zero quantum transition between 02 and 03 are forbidden. Note that this statement is true for this treatment, which employs stationary state wave functions and time-dependent perturbations, but as we shall see in Chapter 11, it is easy with suitable pulse sequences to elicit information on zero quantum and quantum double processes. For our present purposes in the remainder of this chapter we accept the validity of Eq. 6.33. 

For the two-spin system the only symmetry operation is the interchange of the two nuclei, and the correct linear combinations, and could be constructed by inspection. When three or more symmetrically equivalent nuclei are present, the symmetry operations consist of various permutations of the nuclei. The correct symmetrized functions can be determined systematically only by application of results from group theory. We shall not present the details of this procedure. 

Consider the two-spin system, with energy levels shown in Fig. 7.1. We can derive a simple and useful expression for a pair of spins by expanding the scalar product of Eq. 7.2 in terms of the x, y, and 2 spin components and using spherical polar coordinates (r, 0, and ) for the spatial coordinates. This gives the expression... 

In Table 11.1 we sketch the form of the density matrix for the two-spin system to show the significance of the elements. Px —P4 refer to the populations of the four states, I and S represent single quantum I and S transitions, and Z refers to zero quantum transitions and D to double quantum transitions. We saw in Eq. 11.9 that an off-diagonal element pm is nonzero only if there is a phase coherence between states m and n, and in Eq. 10.19 we saw that pmn, evolves with a frequency determined by the difference in energies Em — En. Thus, these off-diagonal elements represent not only transitions, but single quantum, double quantum, and zero quantum coherences, which evolve in free precession at approximate frequencies of v, vs, vt + vs, and vt — vs. In Eq. 11.53 we see that p(r) has... 

We have seen that the density matrix can be applied in principle to any spin system, but even for the two-spin system the algebra often becomes very tedious. There are computer programs that permit larger spin systems and more complex experiments to be treated by density matrix procedures. However, it is often... 

Let s now look at the density matrix as it evolves for the pulse sequences that we outlined in Section 11.5. We have illustrated INEPT and related pulse sequences by the two-spin system in which I = H and S = 13C.The equilibrium density matrix can be considered as the sum of two matrices ... 

In compounds where both the donor and the acceptor molecules (or ions) are paramagnetic, two situations may arise (1) if the interchain interactions are negligible, the ESR spectrum is the superposition of the ESR spectra of each individual species or (2) if they are weak but there is significant fast spin exchange between the two spin systems, a single spectrum is observed with a g value that is averaged by the contributions to the spin susceptibilities of each chain [12] ... 

Appendix 3 Quantum Mechanical Treatment of the Two-Spin System 303... 

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