  Herbertsmithite (ZnCu3(OH)6Cl2) is often discussed as the best realization of
the highly frustrated antiferromagnetic kagome lattice known so far. We employ
density functional theory calculations to determine eight exchange coupling
constants of the underlying Heisenberg Hamiltonian. We find the nearest
neighbour coupling J1 to exceed all other couplings by far. However,
next-nearest neighbour kagome layer couplings of 0.019 J1 and interlayer
couplings of up to -0.035 J1 slightly modify the perfect antiferromagnetic
kagome Hamiltonian. Interestingly, the largest interlayer coupling is
ferromagnetic even without Cu impurities in the Zn layer. In addition, we
validate our DFT approach by applying it to kapellasite, a polymorph of
herbertsmithite which is known experimentally to exhibit competing exchange
interactions.
