## 2. A semiempirical potential model for II-VI compoundsWe shall calculate the structure and the properties of the MgO solid on the one hand and the structure and properties of the molecules MgO and their clusters on the other hand on the basis of a single, semi-empirical potential model, which is defined in this chapter. ## 2.1. Born-Mayer potential for ionic crystalsThe magnesium oxide solid is believed to be a pure ionic compound formed from Mg cations and O anions, respectively. The crystal structure of the solid MgO according to X-ray diffraction studies is of the NaCl lattice type. The bond properties and structures of ionic crystals can well be represented by the classical Born-Mayer (1932) potential model representing the Coulomb interaction between the ions of charge (positive for the alkali cations and negative for the halogen anions in the case of I-VII compounds) at mutual distances and the repulsion due to overlap of the outer electron shell represented by the exponential term (Unsöld 1927). The distance measures the steepness of the repulsive part of the potential and its strength. Usually it is assumed that for a given compound the coefficients and are independent of the interacting particles (but they are different for different substances). This potential and its consequences for the structure and properties of ionic crystals are discussed in detail in standard textbooks on solid state physics. For a given crystal structure the summation for the Coulomb potential term can be done with the result for the contribution of the interaction between point charges to
the total energy. The total energy of a big macroscopic ionic crystal with NaCl structure is Minimizing this with respect to the distance can be determined from X-ray diffraction studies or from the relation between the specific weight of the solid, the masses , of the anions and cations and the equilibrium distance in a lattice of the NaCl type. The total lattice energy (per anion-cation pair) is obtained as This energy may be obtained, for instance, from the well known Born-Haber cycle from the vapourisation energy of the solid and from the ionisation energy and electron affinity of the cation and anion, respectively. The compressibility of the solid is where denotes the volume. This yields the following relation at This together with Eq. (5) and the known equilibrium distance
determines the unknown parameters
## 2.2. The Rittner potential model for ion clustersFor small clusters and molecules the Born-Mayer potential model
needs some modification. The ions do not only bear an electric charge
but they are polarizable by the local electric field and then carry
additionally higher electric multipole moments
where is the induced dipole moment due to the local electric field
of all charges and
induced dipoles at the position of particle
Additionally the van der Waals attraction potential contributes to the potential energy. This usually yields a correction of less than one percent to the total bond energy originating from the electrostatic forces and is neglected in the present calculation. For the diatomic ionic molecules the potential may be written as The second row describes the dipole-dipole interaction and the work required to form the induced dipoles if the two ions approach each other from infinity. The induced dipole moments are Usually for ionic molecules one has polarizabilities of the order
of Å Keeping terms up to the second order from this series results in the following potential for the diatomic molecule If one compares this model with the results of quantum mechanical perturbation calculations, it turns out that the model corresponds to the perturbation theoretical result up to the second order, except for the term which is of third order (Brumer and Karplus 1973). For consistency this term is neglected. Brumer and Karplus call this special potential model the T(runcated)-Rittner-Potential. The potential model depends on the two free parameters ## 2.3. Extension of the T-Rittner potential modelThis T-Rittner potential model has been applied by Zieman and
Castleman (1991) to model small alkaline-earth clusters, but with
unsatisfactory results. It turns out that the parameters - For the solid II-VI compounds the two different ions carry a charge equal to two elementary charges. Causa et al. (1986) give a value of for MgO.
- The effective charge of the atoms in the diatomic molecule on the other hand is much less. Zieman and Castleman (1991) and Recio et al. (1993) found from ab initio calculations an effective charge of the O and Mg in MgO of .
That this difference in the effective charge carried by the ions in the molecule and the solid, respectively, is real can be seen from the fact that the properties of the diatomic molecule can be fit much better with a T-Rittner potential if an effective charge of is used (Zieman and Castleman 1991). On the molecular level, the bonding of the II-VI compounds is not (more or less) purely ionic as it is in the case of I-VII compounds but it has a pronounced covalent contribution. The moderate difference of the electronegativity of such compounds also points to a considerable contribution of covalent bonding (Pauling 1960). This covalent contribution, however, disappears in the solid where the bonding for the alkaline earth compounds is of (nearly) purely ionic character. The potential of covalently bound diatomic molecules usually can be modeled with sufficient accuracy by the well known Morse potential where denotes the equilibrium distance of
the atoms and is the depth of the potential
which equals its dissociation energy reduced by the contribution of
the zero-point energy of the vibrations. is the reduced mass and the vibrational frequency of the molecule. The Morse potential may be written as with which shows a repulsive term similar to the corresponding term in the Born-Mayer potential and an exponential attractive term characteristic of the covalent bonding. The transition between the two types of bonding, the mixed ionic and covalent bonding for the diatomic molecule and the pure ionic bonding in the solid, must occur somewhere in the size region of clusters. Parallel to the change in bonding character the effective charge carried by the ions must increase from its value for the diatomic molecule to for the solid. This suggests merging both potential models, the T-Rittner potential (15) on the one hand and the Morse potential (18) on the other hand, into a combined model to obtain a potential model for the whole size regime between molecules and solids The quantity With this changes from slightly more than
for to
in the limit .
may be used to adapt the cluster size region
where the transition between the two limit cases is assumed to occur.
We choose in our calculation which results in
at a cluster size , i.e.
we essentially use the small cluster limit for
all small clusters which may be of relevance for a possible
condensation process. The term
The potential of a cluster of size where is the induced dipole moment on particle ## 2.4. Fitting the parametersThe three free parameters For the potential (19) we obtain for the dissociation energy of the diatomic molecule with being the equilibrium distance of the two atoms. follows from which yields the equation The vibrational frequency of the molecule follows from the equation of motion in the harmonic oscillator approximation (17) as For the solid on the other hand the potential (22) reduces to the limiting case of the standard Born-Mayer potential (4) where is the equilibrium distance in the lattice which follows from We determined the parameters of our model by fitting three measured quantities of the diatomic molecule and the solid exactly. Other procedures are also possible, for instance least square fittings of more than three measured properties. For the calculation of thermodynamical data for the clusters the ground state energy needs the highest accuracy of all calculated data. Thus we decided to fit our model to the following set of measured properties: - The bond energy of the molecule
- The vibrational frequency of the molecule
- The crystal lattice energy.
Thus, we solved Eqs. (24), (26) and (27) for the parameters
1.) The parameters of the solid are rather uncritical. The lattice energy follows from the Born-Haber cycle as kcal/mol. We used a value for the electron affinity for O of eV for the solid which reproduces the lattice energy of alkaline earth oxides as given in the table in Weast et al. (1988). 2.) The value of the bond energy for the MgO
is rather uncertain. Values between 2.3 eV and 4.3 eV have been
reported. The lowest experimentally derived value is
eV (Operti et al. 1989). Most experimentally
determined values up to 1985 are listed in the JANAF table (Chase et
al. 1985). They favour a value of eV which is
the value for which we are using in our
calculations. A similar value of eV is
favoured in the J. Chem. Ref. Data (Pedley and Marshall 1983).
Theoretical calculations for the MgO molecule have been performed by
Bauschlicher et al. (1982 , 1993) and Cuesta et al. (1991 , 1993). The
results for The bond energy of the MgO molecule with respect to formation from the free ions follows from the dissociation energy of the molecule, the ionisation energy of the Mg atom, and the electron affinity of the oxygen atom as This number is used for calculating formation energies of clusters from monomers. 3.) The polarizability of the free doubly charged negative oxygen
ion O of cm 4.) A comparison of some calculated and measured values for MgO is given in Table 3. The match between measured properties of the diatomic molecule and the solid and that of the corresponding quantities calculated from the potential model is as good as can be expected for such calculations and can be considered satisfactory, though our model for II-VI compounds is not as accurate as the corresponding model for pure ionic I-VII compounds (e.g. Martin 1983). © European Southern Observatory (ESO) 1997 Online publication: June 30, 1998 |