AMX, ABX and ABC patterns, and various related spin systems are very common in organic molecules. Below some of the structural types which give these patterns.

It is worth examining ABX patterns in some detail because, in the progression from first-order NMR patterns to incomprehensible jungles of peaks, they represent the last stopping point where a complete analysis is still possible, and where insights into the problems that arise in the analysis of more complex systems can be achieved. Specifically, ABX patterns are the simplest systems which show the phenomenon sometimes referred to as "virtual coupling" and they are the simplest systems in which both the magnitude and the sign of J coupling constants is significant. Furthermore, there are several pathological forms of ABX patterns which are sufficiently nonintuitive that the unwary spectroscopist can misassign coupling constants and even structures.

**Development of an ABX Pattern**. Consider the stick diagram below which represents an ABX pattern in which we sequentially turn on first the A-X and then the B-X coupling:

One of the two lines in the A-pattern arises from those molecules with the spin of the X-nucleus aligned against the field ( ) and the other from those which have the X-spin aligned with the field ( ). Similarly for the B-pattern. Note, however, that the line assignments of the pattern with both JAX and JBX nonzero will be different depending on the relative sign of JAX and JBX, as illustrated in the Figure. Up to this point the line positions are identical.

The key to understanding ABX patterns is to realize that the nuclei with X = a and those with X = b are actually on different moleculaes, and cannot interact with each other. Thus, when we finally turn on JAB, it will be the X = a line of A and the X = a line of B that will couple to form an AB-quartet, similarly the two X=b lines will form a second AB-quartet.

**First Order "AMX" Type Solution.** Many ABX patterns are sufficiently close to AMX (i.e., AB>>JAB) that a first-order solution has a good chance of being correct. We identify the distorted doublet of doublets (JAB, JAX) which make up the A portion, as well as the dd (JAB, JBX) for B, and begin the analysis by first removing the JAX and JBX couplings, respectively. This leaves us with an AB pattern, which we can solve in the usual way.

**Correct Analysis of ABX Patterns**. In order to correctly analyze an AB pattern of arbitrary complexity we have to reverse the order of extraction of coupling constants compared to the AMX solution above. We have to first solve for JAB, and then for JAX and JBX. Proceed in the following order:

1. Identify the two ab quartets. These can usually be recognized by the characteristic line separations and "leaning". We will use the notation ab+ and ab- for the two quartets (+ identifies the one with the larger ab). Check to make sure that Jab+ = Jab-, and that the ab quartet with the taller middle lines has the shorter outer lines. An unambiguous choice of ab subquartets cannot always be made.

2. Solve the two ab quartets. Treat the ab subquartets as normal AB patterns, and obtain the four "chemical shifts", a+, b+ and a-, b-.

3. Identify the correct solution. At this point in the analysis we encounter an ambiguity. We know that each of the ab quartets consists of two a and two b lines, but we do not know which half is a and which is b. There are thus two solutions to all ABX patterns which have two ab quartets. The two solutions are obtained by pairing up one each of a d+ and a d- line. The analysis is completed as below:

**Distinguishing Between Solutions 1 and 2**. Which solution is the correct one? This determination can be made according to several criteria:

1. Magnitude of the couplings. Sometimes one of the solutions gives unreasonable couplings. In the example above, if we are dealing with proton-proton couplings Solution 2 looks dubious because one the couplings is larger than usually observed for JHH. In the example JBX for solution 2 is 27.6 Hz, which is not impossible for a proton spectrum, but rather unlikely.

2. Signs of coupling constants. Sometimes the sign of the coupling constants is definitive. If the structure fragment is known, the signs can sometimes be predicted, and may rule out one solution. For example, all vicinal 3JHCCH couplings are positive, geminal 2JHCH couplings at sp3 carbons are usually negative. A common structure fragment which gives ABX patterns is CHX-CHAHB. Here both JAX and JBX must have the same sign. On the other hand, if the pattern is CHA-CHBHX then the signs must be different. Note, however, that if you misidentified the ab subquartets, then the signs of the coupling constants you calculated may be wrong.

3. Analysis of the X-Part. It is important to note that all lines have identical positions in both Solutions 1 and 2. The intensities of the AB part are also identical for both solutions. However, the intensities of the lines in the X-part are always different, and this is the most reliable and general way to identify the correct solution.

For the majority of ABX patterns encountered in organic molecules, Solution 1 is correct. Solution 2 spectra are found when A and B are close in chemical shift and the magnitude of JAX and JBX are very different or they have different signs.

**Analysis of the X-Part of ABX Patterns**. It is a common practice to treat the X-part as a doublet of doublets (which it often closely resembles). However, the couplings obtained are only approximate (the sum of JAX and JBX will be correct, but the individual values incorrect), and will be completely wrong if we are dealing with a Solution 2 pattern.

The X-part consists of 6 lines, of which only four are usually visible. The two additional lines are often weak, but can be seen in Solution 2 patterns for which A and B are close together, and the X-part is consequently significantly distorted. The lines are numbered as follows: the two most intense are 9 and 12, they are separated by JAX + JBX. The inner pair of the remaining lines are 10 and 11, their separation is 2D+ - 2D-. The outer of lines (often invisible) are 14 and 15, separated by 2D+ + 2D-. Line 13 has intensity of xero. These line assignments are not always straightforward: sometimes lines 10 and 11 are on top of each other, resulting in a triplet-like pattern, sometimes 10 and 11 are very close to 9 and 12, to give a doublet.

To carry out an intensity calculation we define lines 9 and 12 to have intensity 1 (i9 = i12 = 1.0), and proceed as outlined below:

A simpler way to distinguish the two solutions is to enter the numeric data into an ABX simulation using WINDNMR, and compare the appearance of the two X part with the experimental spectrum.