**5-HMR-14 A**_{2}X_{2} and AA'XX' Patterns

© Copyright Hans J. Reich 2017

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University of Wisconsinn

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In **A**_{2}X_{2} and **A**_{2}B_{2} patterns the two **A** nuclei and the two **X** (B) nuclei are magnetically equivalent: they have the same chemical shift by symmetry, and each **A** proton is coupled equally to the two **X** (or **B**) protons. True **A**_{2}X_{2} patterns are quite rare. Both the **A** and **X** protons are identical triplets.

An example of an **A**_{2}X_{2} patterns is shown below. Note that the methyne protons signals are a little broader than the CH_{2}, presumably there is a small long-range coupling to the MeO protons.

More complicated patterns are seen when the chemical shift difference approaches or is smaller than the *J*_{AB} coupling. However, both **A**_{2}B_{2} and **AA'BB'** always give centrosymmetric patters (**A**_{2} part mirror image of the **B**_{2} part).

**AA'XX' Spectra**

**AA'XX'** and **AA'BB'** spectra are much more common. Here each **A** proton is coupled differently to the **B** and **B'** protons (or **X**, **X'** nuclei). Some molecules with such patterns are:

Such molecules give inherently second-order multiplets. Only if the *J*_{AB} coupling is identical to the *J*_{AB'} coupling by accident does the system become **A**_{2}B_{2} or **A**_{2}X_{2}, and a first order pattern is seen (if ν_{AB} is large enough).

**AA'XX' **spectra consist of two identical half spectra, one for **AA'** and one for **XX'**, each a maximum of 10 lines, each symmetrical about its midpoint, ν_{A} and ν_{X}, respectively. See example B below. The appearance of the spectrum is defined by four coupling constants: *J*_{AA'}, *J*_{XX'}, *J*_{AX} and *J*_{AX'}. The spectrum is sensitive to the relative signs of *J*_{AX} and *J*_{AX'}, but not to the relative signs of *J*_{AA'} and *J*_{XX'}. The relationship between these, and the directly measurable values K, L, M, and N are given below and in the graphic.

Each half-spectrum consists of a 1:1 doublet with a separation of N (intensity 50% of the half spectrum), and two **ab** quartets, each with "normal" intensity ratios and ν_{ab} = |L|. One has apparent couplings (*J*_{ab}) of |K| and the other of |M|, as indicated. Unfortunately, K and M cannot be distinguished, the relative signs of *J*_{AA'} and *J*_{XX'} are not known, nor is it known which number obtained is *J*_{AA'} and which is *J*_{XX'}. It is also not known which coupling is *J*_{AX} and which is *J*_{AX'}, but the relative signs of *J*_{AX} and *J*_{AX'} can be determined: if |N| is larger than |L|, signs are the same. Thus the ^{19}F and ^{1}H spectra of 1,1-difluoroethylene (B) are identical, so it is not possible to distinguish which coupling is ^{2}*J*_{FF} and which is ^{2}*J*_{HH}, nor can one tell which is the *cis* *J*_{HF} and which is *trans* *J*_{HF}. This would have to be done using information about such couplings obtained from compounds where the assignments are not ambiguous.

**Solving an AA'XX' Pattern**. If all 10 lines are visible, and can be assigned to the large doublet and the two **ab** quartets, the process is straighforward, as shown for the solution of the ^{19}F NMR spectrum of 1,1-difluoroethylene below:

1. Determine N from the doublet separation (35.3 Hz).

2. Measure K (41.2 and 41.4 Hz) and M (31.7, 32.0 Hz) from the appropriate line separation ("J" of the two **ab** quartets).

3. Calculate L - it is the "δ_{ab}" of each of the **ab** quartets. For the K quartet we get: SQRT[(276.2-181.3)(235.0-222.7)] = 33.8 Hz, for the M quartet: SQRT[(268.1-189.8)(236.4-221.8)] = 34.2 Hz

4. Calculate *J*_{AA'} and *J*_{XX'} by summing and subtracting K and M: *J*_{AA'} = (K+M)/2 = (41.3+31.8)/2 = 36.5 Hz; *J*_{XX'} = (K-M)/2 = (41.3-31.8)/2 = 4.7 Hz. Because we do not know which **ab** quartet is K, and which M, we do not know the relative signs of J_{AA'} and J_{XX'}, nor do we know which coupling is which.

5. Calculate *J*_{AX} and *J*_{AX'} by summing and subtracting L and N: *J*_{AX} = (N+L)/2 = (35.3+34.0)/2 = 34.7 Hz, *J*_{AX'} = (N-L)/2 = (35.3-34.0)/2 = 0.7 Hz. Again, we do not know which coupling is which, but the relative signs can be determined: if |N| is larger than |L|, the signs are the same, as in this case.

**Special Cases of AA'XX' patterns:** Unfortunately a large fraction of **AA'XX'** patterns are missing lines, which means that some or all of the coupling constants may be indeterminate. Below are summarized several common (and some less common) situations where a reduced number of lines is seen.

In the situation where *J*_{AX} = *J*_{AX'} (i.e. L = 0, **A**_{2}X_{2}) the spectrum collapses to a triplet. In other words, the effective "chemical shift" of each of the **ab** quartets is zero, and thus each gives a single line at ν_{A}. This is more or less the situation with many acyclic compounds of the X-CH_{2}-CH_{2}-Y type, provided that X and Y are not too large, but cause very different chemical shifts. See example **C**.

In the situation where *J*_{AA'} ≈ *J*_{XX'} (which is often approximately the case with X-CH_{2}-CH_{2}-Y and p-disubstituted benzenes) the M **ab** quartet collapses to two lines since M = 0. See example A.

In cases where *J*_{XX'} is zero, both **ab** quartets will have the same *J*_{ab} (M = K) and will be identical, leaving only 6 lines. This is nearly the case for situations like symmetrical o-disubstituted benzenes or 1,4-disubstituted butadienes, where *J*_{AA'} is a 3-bond coupling, and *J*_{XX'} a 5-bond coupling. In these situations L is small (i.e.*J*_{AX} is close to *J*_{AX'}) and the central lines of the K and M quartets will likely be superimposed, whereas the small outer lines may be distinct -- the outer lines are separated by just under twice the value of *J*_{XX'}, the inner lines by just a fraction of *J*_{XX'}.

If the signs of *J*_{AX} and *J*_{AX'} are different the N lines will be relatively close together. This is the case for AA'XX' patterns of the AA' vicinal type, where *J*_{AB }is a geminal coupling, hence negative, and *J*_{AB'} is vicinal, and hence positive. In the limit, if *J*_{AX} = -*J*_{AX'} then the N lines can collapse to a singlet.

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