Understanding Cosy With Product Operators

9.5.1 Analyzing the Homonuclear "Front End"

Most homonuclear 2D (1H-1H) experiments have the same "front end": they start with the sequence 90o — t1 — 90o. Let's follow the net magnetization of a single proton, Ha, with offset and a single J coupling to Hb. The preparation pulse is just a 90° pulse on the X axis, which rotates the Ha z magnetization onto the -/ axis. This magnetization undergoes both chemical shift and J-coupling evolution during the t1 delay. This is a complex motion that involves the two components of Ha magnetization (Hb = a and Hb = ft) separating, spreading until they are opposite each other (antiphase), and then coming back together again (in phase). At the same time, the center between the two components is rotating with angular frequency (the chemical-shift offset, corresponding to the center of the Ha doublet). The two components may rotate into antiphase and back again many times during the ti period, and where they end up at the end of the ti period will vary with ti, consisting in general of some fraction of the magnetization being in-phase and some fraction antiphase. In addition, the chemical shift is encoded into the phase of the magnetization at the end of the t1 period. As we saw in the INEPT experiment, antiphase magnetization can be transferred to the other J-coupled nucleus by simultaneous 90° pulses on both nuclei. This is just what the second 90° pulse in the COSY sequence does, since it is a nonselective pulse that affects both Ha and Hb equally. The portion of the Ha magnetization that is antiphase at the end of the ti delay is transferred to Hb antiphase magnetization and contributes to a crosspeak at F1 = F2 = The portion of the Ha magnetization that is in-phase does not undergo transfer, and contributes to a diagonal peak at F1 = F2 = In product operator notation:

During the evolution (t1) period, the Ha magnetization rotates with angular frequency in the X-y1 plane, and the doublet components (Hb = a and Hb = ft) separate from this center position with angular frequency Jab/2 in Hz, or ^Jab in radians. In contrast to the INEPT experiment, we have no control over these two kinds of evolution and both will happen at the same time. A spin echo will not help because pulses do not distinguish between Ha and Hb: both would receive a 180° pulse and we would have no chemical-shift evolution, only J-coupling evolution. Without chemical-shift evolution we cannot create a second dimension!

This is a complicated motion to describe with vectors, but with product operators it is relatively simple, if you are not afraid of a little algebra and trigonometry. First we consider the chemical-shift evolution, which causes the Ha magnetization to rotate through an angle © = t1 radians after t1 s:

Since counterclockwise rotation leads from the -/ axis to the +X axis, the x component has a plus sign. Now consider the effect of the coupling Jab. The pure X and y' magnetization will rotate into and out of the antiphase condition with angular frequency nJ:

ix ^ IX cos(nJt1) + 2iyib sin(nJi1) iy ^ iy cos(nJti) - 2ixib sin(nJti)

Verify for yourself that this makes sense: at t1 = 0 you have the starting magnetizations. At t1 = 1/(2Jab) you have the pure antiphase magnetization. At t1 = 1/Jab you have the starting in-phase magnetizations with the opposite sign, that is, rotated by 180° in the x'-y' plane.

At t1 = 2/Jab you are back to the starting magnetization. With product operators we do not need to draw vector diagrams, because we treat the vectors as pure in-phase (Iax, tfp and antiphase (2IXlb, 2Iylb) components. Now plug these results in wherever you see IX or Iy as a result of chemical-shift evolution:

a (chemical-shift evolution)

(J-coupling evolution) ->

= -Iy cos(^a t1) cos(nJt1) + 2IXIbcos(^a t1) sin(nJt1)

sin(nJt1)] cos(^a t1) + [IX cos(nJt1) + 2IyIb sin(nJt1)] sin(^at1)

We can abbreviate a bit by using s and c for sin(^at1) and cos(^ai1), respectively, and s' and c' for sin(nJtO and cos(nJt1), respectively:

It still looks pretty messy, but there are only four terms: the X and y' components of Ha inphase magnetization (first and third terms), and the X and y' components of Ha magnetization that is antiphase with respect to the spin state (a or f) of the coupled Hb spin (second and fourth terms). This is a full description of what happens to the Ha magnetization that started on the —y' axis at the beginning of the t1 period, and it can be applied to all homonuclear 2D NMR experiments.

There is only one more thing to do: consider the effect of the mixing portion of the 2D experiment, which is just a 90° pulse applied along the X axis of the rotating frame. Each component of the product operators can be treated separately and rotated just as the vectors rotate under the influence of a pulse. All of the individual components behave as follows:

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