Nuclear magnetic resonance (NMR) signals arise when nuclei absorb a certain radio frequency and are excited from one spin state to another. The exact frequency of electromagnetic radiation that the nucleus absorbs depends on the magnetic environment around the nucleus. This magnetic environment is controlled mostly by the applied field, but is also affected by the magnetic moments of nearby nuclei. Nuclei can be in one of many spin states (Figure 1), giving rise to several possible magnetic environments for the observed nucleus to resonate in. This causes the NMR signal for a nucleus to show up as a multiplet rather than a single peak.

Figure 1: The different spin states of a nucleus (I = ¹/₂) in a magnetic field. These different states increase or decrease the effective magnetic field experienced by a nearby nucleus, allowing for two distinct signals.

When nuclei have a spin of I = ¹/₂ (as with protons), they can have two possible magnetic moments and thus split a single expected NMR signal into two signals. When more than one nucleus affects the magnetic environment of the nucleus being examined, complex multiplets form as each nucleus splits the signal into two additional peaks. If those nuclei are magnetically equivalent to each other, then some of the signals overlap to form peaks with different relative intensities. The multiplet pattern can be predicted by Pascal’s triangle (Figure 2), looking at the n^th row, where n = number of nuclei equivalent to each other but not equivalent to the one being examined. In this case, the number of peaks in the multiplet is equal to n + 1

Figure 2: Pascal’s triangle predicts the number of peaks in a multiplet and their relative intensities.

When there is more than one type of nucleus splitting an NMR signal, then the signal changes from a multiplet to a group of multiplets (Figure 3). This is caused by the different coupling constants associated with different types of nuclei. Each nucleus splits the NMR signal by a different width, so the peaks no longer overlap to form peaks with different relative intensities.

Figure 3: The splitting tree of different types of multiplets.

When nuclei have I > ¹/₂, they have more than two possible magnetic moments and thus split NMR signals into more than two peaks. The number of peaks expected is 2I + 1, corresponding to the number of possible orientations of the magnetic moment. In reality however, some of these peaks may be obscured due to quadrupolar relaxation. As a result, most NMR focuses on I = ¹/₂ nuclei such as ¹H, ¹³C, and ³¹P.

Multiplets are centered around the chemical shift expected for a nucleus had its signal not been split. The total area of a multiplet corresponds to the number of nuclei resonating at the given frequency.

Looking at actual molecules raises questions about which nuclei can cause splitting to occur. First of all, it is important to realize that only nuclei with I ≠ 0 will show up in an NMR spectrum. When I = 0, there is only one possible spin state and obviously the nucleus cannot flip between states. Since the NMR signal is based on the absorption of radio frequency as a nucleus transitions from one spin state to another, I = 0 nuclei do not show up on NMR. In addition, they do not cause splitting of other NMR signals because they only have one possible magnetic moment. This simplifies NMR spectra, in particular of organic and organometallic compounds, greatly, since the majority of carbon atoms are ¹²C, which have I = 0.

For a nucleus to cause splitting, it must be close enough to the nucleus being observed to affect its magnetic environment. The splitting technically occurs through bonds, not through space, so as a general rule, only nuclei separated by three or fewer bonds can split each other. However, even if a nucleus is close enough to another, it may not cause splitting. For splitting to occur, the nuclei must also be non-equivalent. To see how these factors affect real NMR spectra, consider the spectrum for chloroethane (Figure 4)

Figure 4: The NMR spectrum for chloroethane. Adapted from A. M. Castillo, L. Patiny, and J. Wist. J. Magn. Reson., 2010, 209, 123.

Notice that in Figure 4 there are two groups of peaks in the spectrum for chloroethane, a triplet and a quartet. These arise from the two different types of I ≠ 0 nuclei in the molecule, the protons on the methyl and methylene groups. The multiplet corresponding to the CH₃ protons has a relative integration (peak area) of three (one for each proton) and is split by the two methylene protons (n = 2), which results in n + 1 peaks, i.e., 3 which is a triplet. The multiplet corresponding to the CH₂ protons has an integration of two (one for each proton) and is split by the three methyl protons ((n = 3) which results in n + 1 peaks, i.e., 4 which is a quartet. Each group of nuclei splits the other, so in this way, they are coupled.

The difference (in Hz) between the peaks of a mulitplet is called the coupling constant. It is particular to the types of nuclei that give rise to the multiplet, and is independent of the field strength of the NMR instrument used. For this reason, the coupling constant is given in Hz, not ppm. The coupling constant for many common pairs of nuclei are known (Table 1), and this can help when interpreting spectra.

Table 1: Typical coupling constants for various organic structural types.

Structural type	Coupling constant (Hz)
	6 - 8
	5 - 7
	2 - 12
	0.5 - 3
	12 - 15
	12 - 18
	7 - 12
	0.5 - 3
	3 - 11
	2 - 3
	ortho = 6 - 9; meta = 1 - 3; para = 0 - 1

Coupling constants are sometimes written ⁿJ to denote the number of bonds (n) between the coupled nuclei. Alternatively, they are written as J(H-H) or J_HH to indicate the coupling is between two hydrogen atoms. Thus, a coupling constant between a phosphorous atom and a hydrogen would be written as J(P-H) or J_PH. Coupling constants are calculated empirically by measuring the distance between the peaks of a multiplet, and are expressed in Hz.

Example 1

Coupling constants may be calculated from spectra using frequency or chemical shift data. Consider the spectrum of chloroethane shown in Figure 5 and the frequency of the peaks (collected on a 60 MHz spectrometer) given in Table 2.

Figure 5: ¹H NMR spectrum of chloroethane. Peak positions for labeled peaks are given in Table 2.

Table 2: Chemical shift in ppm and Hz for all peaks in the ¹H NMR spectrum of chloroethane. Peak labels are given in Figure 5.

Peak label	δ (ppm)	ν (Hz)
a	3.7805	226.83
b	3.6628	219.77
c	3.5452	212.71
d	3.4275	205.65
e	1.3646	81.88
f	1.2470	74.82
g	1.1293	67.76

To determine the coupling constant for a multiplet (in this case, the quartet in Figure 5), the difference in frequency (ν) between each peak is calculated and the average of this value provides the coupling constant in Hz. For example using the data from Table 2:

Frequency of peak c - frequency of peak d = 212.71 Hz – 205.65 Hz = 7.06 Hz

Frequency of peak b - frequency of peak c = 219.77 Hz – 212.71 Hz = 7.06 Hz

Frequency of peak a - frequency of peak b = 226.83 Hz – 219.77 Hz = 7.06 Hz

Average: 7.06 Hz

∴ J(H-H) = 7.06 Hz

Note:

In this case the difference in frequency between each set of peaks is the same and therefore an average determination is not strictly necessary. In fact for 1^st order spectra they should be the same. However, in some cases the peak picking programs used will result in small variations, and thus it is necessary to take the trouble to calculate a true average.

To determine the coupling constant of the same multiplet using chemical shift data (δ), calculate the difference in ppm between each peak and average the values. Then multiply the chemical shift by the spectrometer field strength (in this case 60 MHz), in order to convert the value from ppm to Hz:

Chemical shift of peak c - chemical shift of peak d = 3.5452 ppm – 3.4275 ppm = 0.1177 ppm

Chemical shift of peak b - chemical shift of peak c = 3.6628 ppm – 3.5452 ppm = 0.1176 ppm

Chemical shift of peak a - chemical shift of peak b = 3.7805 ppm – 3.6628 ppm = 0.1177 ppm

Average: 0.1176 ppm

Average difference in ppm x frequency of the NMR spectrometer = 0.1176 ppm x 60 MHz = 7.056 Hz

∴ J(H-H) = 7.06 Hz

When coupled nuclei have similar chemical shifts (more specifically, when Δν is similar in magnitude to J), second-order coupling or strong coupling can occur. In its most basic form, second-order coupling results in “roofing” (Figure 6). The coupled multiplets point to or lean toward each other, and the effect becomes more noticeable as Δν decreases. The multiplets also become off-centered with second-order coupling. The midpoint between the peaks no longer corresponds exactly to the chemical shift.

Figure 6: Roofing can be seen in the NMR spectrum of chloroethane. Adapted from A. M. Castillo, L. Patiny, and J. Wist, J. Magn. Reson., 2010, 209, 123.

In more drastic cases of strong coupling (when Δν ≈ J), multiplets can merge to create deceptively simple patterns. Or, if more than two spins are involved, entirely new peaks can appear, making it difficult to interpret the spectrum manually. Second-order coupling can often be converted into first-order coupling by using a spectrometer with a higher field strength. This works by altering the Δν (which is dependent on the field strength), while J (which is independent of the field strength) stays the same.