3. Cluster structure and properties
3.1. Calculation of the cluster potential
Any stable equilibrium configuration of a cluster with N monomers corresponds to a local minimum of the potential in the -dimensional space of the positions of its individual atoms. In a system of many particles there exist numerous possible equilibrium configurations with different geometrical arrangements of the atoms and rather different binding energies. Usually most of the potential minima are rather shallow and do not really correspond to observable equilibrium structures of a cluster, but a few of them are quite deep. The deepest of all of these potential minima corresponds to the ground state configuration of the cluster and its depth to the dissociation energy of the ground state into the free ions. Other deep local minima correspond to stable isomers of the cluster and their depth to their dissociation energies, which are usually comparable to but (by definition) less than that of the ground state.
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 clusters
The 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 N and all strong local minima within a few eV above the ground state, which represent isomeric structures of the N -cluster. For nearly all cluster sizes there exist some isomers with a bond energy per monomer nearly equal to that of the ground state.
Table 4. Calculated properties of MgO-clusters: Potential , bond energy per monomer , the three principal moments of inertia I and the symmetry number
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 n, R is the Rydberg constant, and and are the charges of the ion and the electron in units of the elementary charge, respectively. The effective charge of the Mg core is found from (30) and the ionisation energy of 15.035 eV required to remove the second 3s-electron from the free ion Mg to be . We apply Eq. (30) to calculate the ionisation energy of a fractionally charged Mg cation with an effective charge of the electron located partially on the cation with given by (20). Then we calculate the ionisation energy as
The energy of formation of a cluster of size N from the free monomers, finally, is obtained from the energy required to form the free ions from the fractionally charged particles bound in the cluster and the energy gain by recombining the free ions into the monomers as
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 N the average bond energy per monomer is much smaller than the bond energy of 6.79 eV of a monomer in the crystal lattice of the solid MgO. It increases with cluster size N and for the clusters with 15 and 16 monomers it is already only one eV below its bulk value. This behaviour is just what one would expect, that is that the bond energy of the monomers in the clusters increases more or less monotonically with increasing cluster size, but remains always less than in the solid.
Fig. 2 shows the energy
required to dissociate the most stable cluster formed from N monomers into a cluster consisting of monomers and into a free MgO molecule in the gas phase. The dependence of this dissociation energy on N shows clearly that clusters of size , 4, 6, 9, 12 and 15 show enhanced stability compared to clusters of size (magic numbers). This fits well to the findings from mass spectrometry experiments (Ziemann and Castleman 1991, Saunders 1988 , 1989) and the results from Hartree-Fock calculations (Rezio et al. 1993).
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