Astron. Astrophys. 320, 553-567 (1997)

2. A semiempirical potential model for II-VI compounds

We 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 crystals

The 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. M is the Madelung constant which equals for the NaCl lattice structure and N is the number of particles. The repulsive potential rapidly drops with increasing mutual distance of the particles. In a solid with NaCl lattice structure each ion is surrounded by 6 neighbours of the opposite charge at distance r, 12 ions of equal charge at distance , 8 neighbours of opposite charge at distance and so on. For calculating the contribution of the repulsive potential to the total energy it suffices to consider the interaction with nearest neighbours only. The total repulsive energy in this approximation is

The total energy of a big macroscopic ionic crystal with NaCl structure is

Minimizing this with respect to the distance r between nearest neighbours yields for the equilibrium distance in the ionic lattice the equation

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 A and in the repulsive part of the Born-Mayer potential (4). An independent check of the validity of the model is the comparison of calculated lattice energies from (7) and experimental values from the Born-Haber cycle. The accuracy this model is satisfactory for the alkali halogenides (I-VII compounds) (compare the compilation of data in Weast et al. (1988), for instance) and it is applicable with sufficient accuracy to II-VI compounds like MgO (cf. Table 1).

Table 1. Calculated and experimental lattice energies of some oxides.

2.2. The Rittner potential model for ion clusters

For 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 ¹. The induced electric multipole moments mutually interact and contribute to the total energy. For the highly symmetric, infinitely extended ionic crystals of cubic symmetry the local electric fields vanish at the equilibrium position of the ions, such that there is no net induced polarization of the ions and no contribution of this to the potential. For small particles of finite size there is no such high symmetry as viewed from the positions of the individual particles and therefore, there exists a local electric field which polarizes the ion. Rittner (1951) introduced such additional terms in order to discuss the properties of diatomic alkali-halide molecules. The dominating contributions to the potential energy are the monopole-induced dipole interaction, the induced-dipole-induced-dipole interaction and the work required to form the induced dipole moment. It has been shown (O'Konski 1955) that these contributions can be rearranged into the form

where

is the induced dipole moment due to the local electric field of all charges and induced dipoles at the position of particle j. is the polarizability of the j -th particle. This potential was first applied by Rittner (1951) to calculate the properties of diatomic alkali halides and by O'Konski and Higuchi (1955) and by Berkowitz (1958) for calculating the properties of the dimers. Martin (1983) applied this model to the calculation of NaCl clusters. Brumer and Karplus (1973) derived potential energy terms from second order perturbation theory and identified these terms with the Rittner potential model.

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 ų, ų and distances r of the order of Å. Then and one may develop the rhs. of (13) into a series

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 A and which can be determined from experimentally determined properties of the molecule, for instance from its bond length and bond energy. For alkali halide compounds, usually the results for the molecule and the solid agree satisfactorily (cf. Martin 1983) and the potential model then can be applied to calculate cluster properties for the whole size regime between the molecule and the macroscopic solid.

2.3. Extension of the T-Rittner potential model

This 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 A and as determined from the solid and the molecule, respectively, do not agree to within reasonable limits. The reason for the discrepancy can be traced back to the charge carried by the ions:

• 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. a is a free parameter, which usually is fitted by requiring

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 B describes the covalent attraction. It vanishes in the solid state since there the bonding character is purely ionic, but for small clusters it is nonzero. Hence B depends on the cluster size N (N =number of monomers forming the cluster). The same holds for the effective charge on the ions which changes from to . Since presently there is no evidence to indicate how both quantities vary with cluster size we decided to use the following simple interpolation formula

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 B is interpolated in the calculations as follows

B is assumed to contribute only to the interaction potential between the cations and anions since it seems plausible that only these can share to some extent their electrons resulting in a covalent bonding.

The potential of a cluster of size N now is

where

is the induced dipole moment on particle j.

2.4. Fitting the parameters

The three free parameters A, and of the potential model (22) can be determined by fitting these coefficients such that some of the observed quantities of the diatomic molecule and the bulk solid are reproduced. A comparison of additional measured quantities for both substances with predictions for its value from the potential model serve as a test of the reliability and accuracy of the potential model.

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)

The crystal lattice energy is

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 A, B and . The resulting values of the potential parameters for MgO using the data from Table 2 are shown in Table 3. They need some comments:

Table 2. Properties of MgO. The formation energy of the solid is that from free atoms. is the compressibility

Table 3. Fit of the potential model for MgO

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 ab initio calculations tend to be lower than experimental values but it is by no means certain that these lower values are more realistic.

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 cm³ (Landolt-Börnstein 1981) does not allow for a reasonable potential fit. Usually within a compound the polarizability of a particle is less than that for the free particle and we therefore used in our calculations the polarizability of cm³ which is obtained from the Clausius-Mosotti relation for the solid (Fowler and Madden 1985).

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
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