| The magnetic transition-metal (Mn, Co, Fe) clusters are very attractive because of their unique electronic structures and magnetic properties. Furthermore, it is very interesting that the structures and properties of the magnetic transition-metal (TM) clusters are strongly size-dependent, giving rise to the quantum size effect. The electronic structure and chemical bond of cluster dominate the geometric configuration of the cluster, whereas, the geometric configuration reflects electronic structure and the property of the chemical bond. Therefore, the determination of the ground-state geometric configuration of the cluster is a principal task. However, in theory, the partially filled and local 3d electronic level of the TM cluster induces the complexity of the geometric structure of the cluster. There are often many low-lying structural isomers, whose energies are very close to that of the ground state, with the increasing of the size of the cluster. It is very difficult to find the ground state of the cluster among the many isomers for the more than ten-number cluster. Fortunately, the density functional theory gives a good theoretical project to treat the larger system such as the TM cluster.Both in experiment and in theory, the research for the electronic structure and the growth mechanism of the small magnetic TM cluster is not enough. And in generally, the investigation is focus on the very small clusters. In this paper, we have performed a global minimization of the total energy for MnN, FeN and CoN (N=2-13) clusters using the all-electron density functional theory (DFT) implemented in the DMol package and employing a double-numerical basis with polarized functions (DNP) and the BLYP exchange-correlation functional. The geometric optimization is performed for the abundant low-lying isomers of the clusters. The cluster with high symmetry has heavy degeneracy and the Jahn-Teller effect plays an important role in determining the ground state. The results show:(1) The geometries prefer to planar structures for the very small CoN ( 2≤N≤4) clusters and the growth of cobalt clusters with large cluster size is in an icosahedral pattern. For Fe3 and Fe4, two new ground states are found. The ground-state structure of the former is a C2v geometry and that of the latter is a structure with C s symmetry. For MnN, FeN and CoN (N=2-13) clusters, 6- and 10-atom clusters have more stable structures and are so-called magic clusters.(2) The magnetic moment of the small transition-metal cluster is much larger than that of the corresponding bulk. The magnetic moment per atom of Co13 cluster is very near the magnetic moment of the bulk of cobalt. The magnetic moment per atom for FeN (N=2-13) clusters shows small variations with cluster size and remains in the vicinity of 3 .0μB /atom over this size range.(3) For MnN (N=2-13) clusters, the transition range of the magnetic ordering is in the vicinity of N=4,5,6 with a degeneracy state including FM and AFM ordering. The magnetic ordering is FM ordering for the smallest (N=2,3) clusters and a clear AFM ordering at N=7 and beyond.In order to research the growth mechanism of the magnetic TM carbon mixed cluster, we present a systematic study of the geometries, electronic structures, and magnetic moments of small FeNCM and CoNCM (N=1,2, M=1-8) clusters using all-electron DFT. The influence of the addition with one or two Fe or Co atom(s) to small CN (N=1-8) clusters on the structures of the CN (N=1-8) clusters is discussed. The results show:(1) For the FeCN (N=1-8) clusters, the most stable states are predicted to be linear structures with the Fe atom at one end except in the case of FeC2. For the Fe2CN (N=1-6) clusters, the most stable states show a competition between linear structures and two-dimensional (planar) clusters with significant odd-even alternation. Furthermore, the addition of one Fe atom to small CN (N=1-8) clusters leaves the linear structures of the CN (N=1-8) clusters almost unaltered. The addition of a second Fe atom to small FeCN (N=1-6) clusters, however, significantly influences the linear structures of the FeCN (N=1-6) clusters.(2) For the CoCN (N=1-8) and Co2CN (N=1-6) clusters, the most stable states are predicted to be linear structures except in the case of CoC2 and CoC7. The ground states of the mixed clusters with one Fe atom are the C v structures with the Fe atom at one end and the ground states of the mixed clusters with two Fe atoms are the D h structures with the two Fe atoms at the two ends. Furthermore, the addition of one Fe atom and a second Fe atom to small CN (N=1-8) clusters leaves the structures of the CN (N=1-8) clusters almost unaltered and the CN (N=1-8) clusters are still retaining linear structures.(3) In all the cases except in the case of FeC2 and CoC2, no carbon dimer with clear Met-cars characteristics is discovered.
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