The elastic dipole tensor is a fundamental quantity relating the elastic field and atomic structure of a point defect. We review three methods in the literature to calculate the dipole tensor and apply them to hydrogen in
α
-zirconium using density functional theory (DFT). The results are compared with the dipole tensor deduced from earlier experimental measurements of the
λ
tensor for hydrogen in
α
-zirconium. There are significant errors with all three methods. We show that calculation of the
λ
tensor, in combination with experimentally measured elastic constants and lattice parameters, yields dipole tensor components that differ from experimental values by only 10%–20%. There is evidence to suggest that current state-of-the-art DFT calculations underestimate bonding between hydrogen and
α
-zirconium.