The DFT method is used to analyze the singlet and triplet PES cross-sections for a number of reaction pathways of the interaction between a hydrogen molecule and a palladium tetramer. Stationary points characterizing stable singlet and triplet complexes and transition states are determined. Predissociated Pd4H2 complexes with binding energies of 5.0-6.4 kcal/mol are formed without an activation barrier at any initial orientation of reactants in the ground triplet state. The dissociated triplet complexes with adsorption energy of 4.7-8.4 kcal/mol are separated from the predissociated structures by the barrier of 11 kcal/mol. The dissociated singlet structures with hydrogen atoms located in two 3-fold positions and in the bridge positions on nonintersecting Pd-Pd bonds have the same binding energy of 22.1 kcal/mol and correspond to the ground state of the Pd4H2 system. Several more local minima corresponding to dissociated and nondissociated H-H bonds are found on the singlet PES. In contrast to the low-index bulk palladium surfaces, no spontaneous pathway for hydrogen dissociation on the cluster is found. Activated H2 dissociation on the palladium tetramer includes the triplet-singlet transition induced by the spin-orbit interaction as a key step. Numerical estimation of the matrix spin-orbit coupling element and the corresponding transmission coefficient κ is performed within nonadiabatic theory for two reaction pathways, which are expected to have the maximal reaction probability. The estimated κ values of ∼0.1 and ∼0.4 and corresponding activation barriers of ∼25 and ∼20 kcal/mol are found for the reaction pathways leading to the ground-state three-coordinated dissociated complex and to the saddle point heading toward the ground-state bridge structure, respectively.