  The structure and stability of small hydrogen clusters adsorbed on graphene
is studied by means of Density Functional Theory (DFT) calculations. Clusters
containing up to six H atoms are investigated systematically -- the clusters
having either all H atoms on one side of the graphene sheet
(\textit{cis}-clusters) or having the H atoms on both sides in an alternating
manner (\textit{trans}-cluster). The most stable cis-clusters found have H
atoms in ortho- and para-positions with respect to each other (two H's on
neighboring or diagonally opposite carbon positions within one carbon hexagon)
while the most stable trans-clusters found have H atoms in
ortho-trans-positions with respect to each other (two H's on neighboring carbon
positions, but on opposite sides of the graphene). Very stable trans-clusters
with 13-22 H atoms were identified by optimizing the number of H atoms in
ortho-trans-positions and thereby the number of closed, H-covered carbon
hexagons. For the cis-clusters, the associative H$_2$ desorption was
investigated. Generally, the desorption with the lowest activation energy
proceeds via para-cis-dimer states, i.e.\ involving somewhere in the H clusters
two H atoms that are positioned on opposite sites within one carbon hexagon.
H$_2$ desorption from clusters lacking such H pairs is calculated to occur via
hydrogen diffusion causing the formation of para-cis-dimer states. Studying the
diffusion events showed a strong dependence of the diffusion energy barriers on
the reaction energies and a general odd-even dependence on the number of H
atoms in the cis-clusters.
