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17
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84927888643
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Further details will be published elsewhere.
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18
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84927888642
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For N>= 7 some physical considerations should be made. Here we find graphs (e.g., the pentagonal bipyramid) which do not satisfy the previous conditions but which are physically / acceptable as structures since the conditions are only slightly violated. Therefore, we have also considered these graphs as possible cluster structures. The magnetic properties of the Hubbard model on arbitrary graphs will be discussed elsewhere.
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19
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84927888641
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Taking into account all possible structures involves a large computational effort. For instance, for N=7 and N=8 there are, respectively, 853 and 11,117 non equivalent graphs. In fact, it is the exploding number of possible geometrical configurations that limits our present study to N<= 8.
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22
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84927888640
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Throughout this work the term ``exact'' should be understood as exact within the accuracy varepsilon simeq 10-10 when it refers to energies.
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23
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84927888639
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Physically, varying U/t can be viewed as changing the interatomic distance (e.g., t propto Rij-5) or changing the spatial extension of the valence wave function (i.e., the element). Different ν/N can be related either to simple metal clusters in different ionic states (ν=N,N pm 1) or, more indirectly, to different elements in a TM series. These analogies provide our study with a more universal character.
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25
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84927888638
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and references therein.
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26
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84927888637
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We do not obtain the optimal structures reported in citekou for Lisub 4}sup + and Lisub 5}sup +. The optimal Hubbard structure corresponds in these cases to the second best ab initio / structure. Notice, however, that often the energy differences are small and in some cases different ab initio / approximations yield somewhat different geometries (Ref. citekou).
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27
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84927888636
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The Hubbard model is in accordance with nature in a further interesting aspect. While the number of possible lattice configurations or graphs ng increases in a very explosive way with the number of atoms N, the number of structures no which are optimal for some value of the parameters U/t and ν/N remains a handful. For example, for N=7 ; (8), ng=853 ; (11117) while no = 18 ; (23). As the cluster size increases, some growth patterns start to dominate and the same or very similar structures cover large regions of the phase or structural diagram (see Figs. 2 and 3). The situation tends to what one observes in the macroscopic limit (solid state) as we go through the periodic table of the elements (see Ref. citefoot10).
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