Atomic structures of magic ZnSe clusters from first principles calculation*

We report the atomic and electronic structures of {\it magic} (ZnSe)$_n$ ($n$ = 13, 33, and 34) clusters, employing first principles technique based on a pseudopotential approach. These sizes are important as laser ablated plumes of ZnSe have clusters with ($n$) = 6, 13, 19, 23, \& 33 ZnSe molecular units in high abundance suggesting their high stablity and magic behavior. Earlier we had predicted the atomic structures of these clusters to be filled cage structures with a Se centered 3-D structure for $n$ = 13 and a cage/core structure for $n$ = 33 \& 34. In the later two cases, a core of Zn$_5$Se$_5$ and Zn$_6$Se$_6$, respectively, is enclosed by a Zn$_{28}$Se$_{28}$ cage to form a 3-D structure. In contrast to ZnSe clusters, ZnO clusters in this size range have empty cage structures. Therefore, we have performed further calculations using both, GGA-PBE and hybrid HSE06 type of exchange-correlation functionals that suggest that our conclusion for the size $n$ = 13 remains unchanged, but for larger clusters of sizes $n$ = 33 \& 34, hollow cage stuctures made up of 4- and 6-membered rings of ZnSe, are energetically more favourable than the filled cage structures. We shall discuss the trends in the electronic structure, binding enery, and HOMO-LUMO gap, as we vary the ZnSe size.