  Icosahedral Au clusters with three and four shells of atoms are found to
deviate significantly from the commonly assumed Mackay structures. By
introducing additional atoms in the surface shell and creating a vacancy in the
center of the cluster, the calculated energy per atom can be lowered
significantly, according to several different descriptions of the interatomic
interaction. Analogous icosahedral structures with five and six shells of atoms
are generated using the same structural motifs and are similarly found to be
more stable than Mackay icosahedra. The lowest energy per atom is obtained with
clusters containing 310, 564, 928 and 1426 atoms, as compared with the commonly
assumed magic numbers of 309, 561, 923 and 1415. Some of the vertices in the
optimized clusters have a hexagonal ring of atoms, rather than a pentagon, with
the vertex atom missing. An inner shell atom in some cases moves outwards by
more than an \AA{}ngstr\"om into the surface shell at the vertex site. This
feature, as well as the wide distribution of nearest-neighbor distances in the
surface layer, can strongly influence the catalytic properties of icosahedral
clusters. The structural optimization is initially carried out using the GOUST
method with atomic forces estimated with the EMT empirical potential function,
but the atomic coordinates are then refined by minimization using electron
density functional theory (DFT) or Gaussian approximation potential (GAP). A
single energy barrier is found to separate the Mackay icosahedron from a lower
energy structure where a string of atoms moves outwards in a concerted manner
from the center so as to create a central vacancy while placing an additional
atom in the surface shell.
