Interpreting ¹H NMR Signal Splitting: The (n + 1) Rule

The (n + 1) rule states that a proton coupled to n equivalent neighboring protons yields a multiplet signal split into n + 1 peaks. For example, a proton with one neighbor produces a doublet with two peaks, while a proton with two equivalent neighbors produces a triplet with three peaks. This rule provides a straightforward way to predict the number of peaks in a split NMR signal.

Peak intensities in multiplets follow a predictable pattern determined by Pascal's triangle, an array where each entry is the sum of the entries to its left and right in the row above it. For a doublet, peaks have equal 1:1 intensity. A triplet shows 1:2:1 intensity, with the central peak twice as intense. A quartet displays 1:3:3:1 intensity ratios, allowing chemists to predict relative peak heights.

When proton A couples to two equivalent protons, each can align with or against the external magnetic field, creating three possible spin state combinations. Two combinations produce the same energy, resulting in a central peak twice as intense as the outer peaks. This 1:2:1 triplet pattern reflects the three distinct magnetic environments experienced by proton A.

An AX proton spin system consists of two coupled protons: A and X. Proton A can sense the two spin states of proton X, resulting in a doublet NMR signal with two peaks of equal 1:1 intensity. This is the simplest coupling scenario and serves as the foundation for understanding more complex multiplet patterns in spin-spin coupling.

In an AX3 spin system, proton A couples to three equivalent neighbors, creating four possible spin state combinations: all aligned with the field, two with and one against, one with and two against, or all against. This produces a quartet with 1:3:3:1 intensity ratios. The AX2 system produces only three combinations, yielding a triplet with 1:2:1 intensities.

Pascal's triangle predicts the relative intensities of peaks in multiplet signals by showing how spin state combinations distribute across energy levels. Each row corresponds to a different number of equivalent neighbors, allowing chemists to anticipate peak height ratios without calculating individual probabilities. This tool simplifies interpretation of complex NMR splitting patterns.

Count the number of peaks in a multiplet signal and subtract one to find the number of equivalent neighboring protons. A doublet (2 peaks) indicates one neighbor, a triplet (3 peaks) indicates two neighbors, and a quartet (4 peaks) indicates three neighbors. This direct application of the (n + 1) rule allows rapid identification of molecular connectivity from splitting patterns.