Potential Madelung

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. 

A finite array of charges is built taking into account the symmetry elements of the crystal. The charges of the outermost ions are adjusted in order to provide the correct value of the Madelung potential on each cluster site as well as the electrical neutrality... 

The Madelung potential on the 13 sites of the unit cell of the YBa2Cu307 ceramic is given in Table 1. In order to obtain the Madelung potential values with ten significant figures, 3543 ions were considered. 

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... 

Table 1 Madelung potential on the sites of the YBaf u Oj unite cell obtained with Gupta-Gupta [14] charges, in a.u. (infinite crystal)...

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... 

Isolated cluster Cluster in cristal Madelung potential... 

Note that in part of the literature and in Eq. (9.12), the Madelung potential is defined such that iioiji.unii ccii = — where r is the nearest distance between the cation and the anion. For the rock-salt lattice, ft = p/2. 

The recent band calculations by Brooks et al. show indeed from charge density calculations that in stoichiometric UO2 a purely ionic picture is incorrect the oxygen ion is singly charged, the second electron not being retained by the Madelung potential. Simple considerations based on mechanical properties and spectroscopic evidence had already pointed at these conclusions. 

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