A 1D ¹H NMR spectrum only tells you where protons resonate (chemical shift δ). COSY (COrrelated SpectroscopY) adds a second frequency axis, δ₂, recorded after a magnetization-transfer pulse sequence. If two protons are scalar (J-)coupled — usually 2–3 bonds apart — magnetization from one hops onto the other during the mixing period, producing an off-diagonal cross-peak at (δ₁, δ₂) and its mirror at (δ₂, δ₁). Uncoupled protons show up only on the diagonal.
Diagonal peak (proton i):
I(δ1,δ2) = exp(-[(δ1-δi)² + (δ2-δi)²] / 2σ²)
Cross-peak pair (i,j), J ≠ 0:
I(δ1,δ2) += exp(-[(δ1-δi)² + (δ2-δj)²] / 2σ²)
+ exp(-[(δ1-δj)² + (δ2-δi)²] / 2σ²)
σ (ppm) ≈ linewidth(Hz) / 300 MHz spectrometer frequency
- Spin system — swap between three real molecules; each has a different coupling network (a straight chain gives two cross-peak pairs, a molecule with an isolated methyl gives none for that peak).
- Peak linewidth — wider lines (more Hz) blur neighbouring peaks together, exactly as field inhomogeneity or fast relaxation does on a real spectrometer.
- Vertical scale — exaggerates peak height so weak cross-peaks stay visible next to tall diagonal peaks.
- Cross-peak toggle — turn J-coupling off to see that only the diagonal (the "boring" 1D-equivalent information) survives.
- Drag to orbit the 3D surface; the mountain-plot rendering is exactly how real COSY software (TopSpin, MestReNova) displays the data.