Madelung calculation

We thank S. M. Zahurak, A. V. Makhija, and R. C. Haddon for supplyii purified Caoi Needels fin- providing a program for Madelung calculations, and R. C. Haddon, K. Raghavachari, R. E. Wal-stedt, and M. Schluter for r uable suggestions. 

It is sometimes desirable to include the effect of the rest of the system, outside of the QM and MM regions. One way to do this is using periodic boundary conditions, as is done in liquid-state simulations. Some researchers have defined a potential that is intended to reproduce the effect of the bulk solvent. This solvent potential may be defined just for this type of calculation, or it may be a continuum solvation model as described in the next chapter. For solids, a set of point charges, called a Madelung potential, is often used. 

Derive the first four terms of the series to calculate the MADELUNG constant for CsCl (Fig. 7.1). 

Details of the calculation of Madelung constants for all of the common types of crystals are beyond the scope of this book. When the arrangement of ions differs from that present in NaCl, the number of ions surrounding the ion chosen as a starting point and the distances between them may be difficult to determine. They will most certainly be much more difficult to represent as a simple factor of the basic distance between a cation and an anion. Therefore, each arrangement of ions (crystal type) will have a different value for the Madelung constant. The values for several common types of crystals are shown in Table 7.3. 

In our case there are six different sites in the considered cluster. So, to apply the method described previously (system of equations (1)), we need seven shells to reproduce the Madelung potential on these six sites. There are various possibilities to define seven shells by considering various grouping of the 218 charges located on the exterior surface of the complete array. As an example consider the following definition of shells which we used in our calculation SI contains 32 yttrium ions located on the upper and lower faces of the array S2 contains 24 yttrium ions on the lateral faces S3... 

In Table 2 we present the shells which correspond to our best choice as well as the values of the scaling factors which result from the solution of the system of equations (1). The Madelung potential values on each cluster site calculated with the finite adjusted array are given in Table 3. When these values are compared to that of the Table... 

For ionic crystals > = 1, and the A are known (Madelung constants). For van der Waals crystals m — 6 (though small terms in and exist) but in view of the difficulties of calculation we obtain B from the observed heat of vapourisation (from A. 3). The repulsion exponent n varies from about 6 for LiF to 12 for Csl, for gases (Lennard-Jones) a value of about 12 seems the best. We assume a constant value of 11 throughout. 

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