The two matrices on the bottom represent antiphase coherence on the y' axis:

where B is the "destination" state. Thus the spin state Ax = Iax — IPx will move towards the spin state Ayz = 2iyib — 2IbyIaz :

In this weird environment, the individual spins states ix and I? are not important. The "collective spin mode" £x = Iax+Ix is stable and the "collective spin mode" Ax = Iax — i°x moves to Ayz and back to itself in an oscillatory manner. It is these collective spin modes that characterize TOCSY mixing, as opposed to the individual (independent) spin modes we normally deal with. Note that the off-diagonal terms of the Hamiltonian, which are important in strong coupling, are completely dominant in isotropic mixing. Not only is chemical shift missing from the central 2 x 2 region of the Hamiltonian (Av = 0), but it is also gone from the Hu and H44 elements (v = 0). In this case we can look at individual spin states as a linear combination of the collective spin modes:

If we start with for example, in a selective 1D TOCSY, it will evolve in the spin lock to give:

since is stationary (commutes with the Hamiltonian). Substituting the individual operators, we have:

ix ^ 1(Ix + ix) + 2 ax — Ix)cos(2nJT) + 1(2Iy Ib — 2IyIZ )sin(2nJr )

= 2 ix(1 + cos(2nJr)) + 1 ib(1 — cos(2nJr)) + ^(2iy Ib — 2Ib I^) sin(2nJr)

This is the result that we showed without proof in Chapter 9. In-phase Ha coherence on x' is completely converted into in-phase Hb coherence on X after t = 1/(27), with the antiphase term reaching a maximum in the middle of this period at t = 1/(47).

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