## 3. Cluster structure and properties## 3.1. Calculation of the cluster potentialAny stable equilibrium configuration of a cluster with The basic problem in determining the structure and bond energy of a cluster then consists in finding the absolute minimum of the cluster potential and all other deep local potential minima with comparable depth to that of the absolute minimum. This corresponds to the notoriously difficult problem of optimising a nonlinear quality function. In the present calculation the task of determining the minima of the potential was solved by applying the evolution strategy of Rechenberg (1973 , 1989) which is described in the appendix. We determined by this method the minimum of the potential for all clusters of size up to . Further we determined all local energy minima, within a few eV away from the ground state, if such exist, which correspond to the stable isomers. For each equilibrium configuration, the vibrational frequencies for the cluster can be calculated in the approximation of small vibrations from the eigenvalues of the force matrix. The results will be published elsewhere (Koehler et al. 1997). ## 3.2. Results for MgO clustersThe results of a calculation for the potential and structure of MgO
clusters are shown in Table 4 and Fig. 3, respectively. The
calculations were done for clusters consisting of up to 16 monomers.
For each cluster size many runs of the optimizing procedure with
different initial configurations have been carried out to be sure that
we have found the true energy minimum representing the ground state
configuration of a cluster of size
The bond energy per monomer is shown in Fig. 1. This bond energy is determined by considering that the ionisation energy and electron affinity of the cations and anions in the cluster are different from that in the monomer. The electron affinity of the fractionally charged anions in the clusters is determined by linear interpolation according to between its values for the two limit cases and (given in Table 2). The ionization energy for the fractionally charged Mg-cations in the clusters is calculated according to the effective charge and from the standard formula for the ionisation of hydrogen like ions
is the ionisation energy from a state with
principal quantum number The energy of formation of a cluster of
size is the energy of formation of the monomer from the free ions Mg and O as given by (29). The resulting bond energy per monomer in
the cluster is given in Table 4 and is shown in Fig. 1. For
small cluster sizes Fig. 2 shows the energy required to dissociate the most stable cluster formed from
The structures of the magic clusters are stacked hexagonal rings for , see Fig. 3, and only for the cluster with shows the cubic structure of the solid. For the structure is a rhomboid. There is a slight asymmetry in the bond angles between O-Mg-O and Mg-O-Mg which results from the different polarizabilities. The energy of the induced dipole moments strengthens the bond energy which is responsible for a tendency of the ions to arrange in such a way that the induced dipole moments are as large as possible, with the restriction, however, that equally charged ions should not come too close. This favours smaller bond angles at the particle with the higher polarizability, i.e. at the oxygen ion. Only for two clusters a plane structure is found: for the dimer and for the ring with . The linear double chains with , ..., 6 are warped. The big clusters , , , , and 16 are cage molecules, their structure is probably determined by the high polarizability of the oxygen.
© European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |