Understanding COSY with Product Operators

5 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 90° — t — 90°. Let s follow the net magnetization of a single proton, Ha, with 

During the evolution (fi) period, the Ha magnetization rotates with angular frequency I2a in the x -y1 plane, and the doublet components (Hb = a and Hb = f ) separate from this center position with angular frequency /ab/2 in Hz, or jrJab 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 I lb both would receive a 180° pulse and we would have no chemical-shift evolution, only /-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 = il-J 1 radians after t 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 /ab. The pure xf and y magnetization will rotate into and out of the antiphase condition with angular frequency 1rJ  

---

In this chapter, we will introduce a new level of theoretical tools—the density matrix— and show by a bit of matrix algebra what the product operators actually represent. The qualitative picture of population changes in the NOE will be made more exact, the precise basis of cross-relaxation will be revealed, and a new phenomenon of cross-relaxation— chemical exchange—will be introduced. With these expanded tools, it will be possible to understand the 2D NOESY (nuclear Overhauser and exchange spectroscopy) and DQF-COSY experiments in detail. 

---