where 𝛾 𝐻 , 𝛾 𝐶 are the gyromagnetic ratios for 1 H and C , 𝐼 ̂𝑧,𝑖 is the z-projection spin operator of the i-th proton, 𝑆 ̂𝑧 is the z-projection spin operator of the C nucleus, 𝐽 𝑙𝑘 is the scalar coupling constant between l-th and k-th spins, 𝑰 ̂𝒍 is the spin operator of the l-th spin, and the summation goes over all magnetic nuclei of the molecule.