Near-degenerate stereomorphs of the doubly-chiral hcp-{213?1} surface

The surfaces of hcp crystals can show a variety of structural features and classes of symmetry that differ markedly from those of simpler fcc or bcc crystals. The hcp-{213?1} surface, for example, can occur in four distinct stereomorphs, interconverted by a combination of mirror operations (linking degenerate enantiomorphically related surfaces) and/or the removal of the outermost atomic layer (linking non-degenerate diamorphically related surfaces). The strict pattern of degeneracy amongst these stereomorphs is analogous to that found for molecules with two chiral centres, and hence it is possible to view this system as doubly-chiral. Simple nearest-neighbour bond-counting arguments, however, suggest that for {213?1} even the diamorphically related cases should be near-degenerate, despite the fact that they differ in having either a notably short or notably long interlayer spacing between the outermost layers (ideal spacing ratio 1:5). In the present work, this counterintuitive result is confirmed at the level of density functional theory, both for the ideal and relaxed {213?1} surfaces of Co, Ru and Re.