We use density-functional theory (DFT) to analyse the interaction of trans- and cis-porphycene with Cu(111) and their interconversion by intramolecular H-transfer. This tautomerisation reaction is characterised by small values for the reaction energy and barrier, on the order of ∼0.1 eV, where the trans configuration is thermodynamically more stable upon adsorption according to the experiments [J. N. Ladenthin et al., ACS Nano 9, 7287-7295 (2015)]. To gain even a qualitatively correct description of this reaction at the DFT level, an accurate treatment of dispersion interactions and a careful choice of the exchange contribution are required in order to predict the subtle energetics. Analysis of the electronic structure shows that adsorption is contributed by a van der Waals (vdW) interaction, mainly responsible for stabilising the polyaromatic fragments, and by a significant charge redistribution localised between Cu and the unsaturated N atoms of the molecule central cavity. We find that different vdW functionals can produce qualitatively different electronic structures, while yielding small trans vs. cis energy differences. Unlike other functionals surveyed here, vdW-DF with PBE exchange satisfactorily reproduces not only the experimental energetics but also the scanning tunneling microscopy images. This gives us confidence that this functional achieves a reliable balance between the two mechanisms contributing to the adsorption of porphycene.