Core term

In addition to this term, account must be taken of the decreasing screen in g of then iicleus by th e electron s as the in teratom ic dis-tance becomes very small,. At very small distances the core-core term should approach the classical form. To account for this, an additional term is added to the basic core-core repulsion integral in MlXnO/3 to give ... 

Also shown in Figure 8.15 is the line due to removal of a 2p electron from the sulphur atom in thiophene. Spin-orbit coupling is sufficient to split the resulting core term, as it is called, into 3/2 and 1/2 states, the multiplet being inverted. 

For the multipolar modeling [11, 12] of the X-ray diffraction data the program XD [13] was used. The atomic density contributions are parametrized into a core term, p re, a spherical valence term, p,leiKe, and a set of multipolar functions ... 

Note that, unless otherwise indicated, I am using Ie and (y+)j = (Y+)exp thls review. Furthermore, the simple equations prisented so far do not include terms for the cosphere overlap, the hard core term, and ion-cavity interactions. 

Values with Camahan-Starling hard-core term for Fisher s original treatment see Ref. 198. c Ebeling s original version. d Ebeling theory with an improved hard-core term. 

Fourth, the significance of a proper hard-core term near the critical density remains subject to debate. Near the critical density, the MSA theory is more sensitive to the choice of the hard-core term than is the DH theory [198,199]. At high ion densities the need for an appropriate choice of the hard-core term is unquestionable—for example, to prevent the coexistence curve from reaching states beyond the close packing of the b.c.c. solid phase of the RPM (Pbcc — 1-3 [252]). Figure 9 compares the coexistence curves of the FL and WS models with the simulation results of Orkoulas and Panagiotopoulos [52],... 

Hamiltonian is of course the same as in MO theory, and is composed in this case of two core terms and a bielectronic repulsion ... 

The potential matrix element consists of a hydrogenic term and, if the matrix element is diagonal, a sum of core terms similar to the case of an inert target (7.83). 

The expression (7.91) is the density matrix for two active electrons added to the core term (7.82). 

The above approach leans heavily however upon the supposed dominance of the one-electron core terms (which may well not be justified as far as the f-orbital — ligand interactions are concerned), and also provides only a little help in predicting the energetic ordering of the f-levels in D8h symmetry. Clearly if the one-electron picture is assumed with... 

In this model the hard core term is as before, but the Gumey term is different. Instead of adding or subtracting a constant term for the region between r = r, + rj and r = r, + rj + d, this term is allowed to vary and to depend on the extent to which the co-spheres of the two ions overlap. The new term which is used is taken to be proportional to the volume of the overlapping part of the co-spheres, and again the magnitude will depend on the identities of the ions. 

This involves a modified core term where the hard core or hard sphere repulsion is replaced by a non-hard sphere short range repulsion which is proportional to This replaces the Debye-Hiickel core term. The Gurney term remains proportional to the volume of overlap of the co-spheres. 

The constant of proportionality in the core term is taken as known, being that which will give the correct value for the sum of the crystallographic radii. 

The potential energy expressions for each of these terms, the core terms, all the Gurney terms and the cavity terms, can all be written down, and each will have a specific dependence on the distance between the pair of ions considered. The final total potential energy is equal to the sum of each term which is taken to be incorporated in the new model, and this total potential energy will have an explicit dependence on distance. 

The expressions for any will derive from the quantity 0, the total potential energy of aU the interactions for the electrolyte solution. Contributions to 0 can be as simple or as complex as is wished, e.g. hard core terms or other short range repulsions, ion—dipole interactions, Gurney terms, cavity terms, ion—solvent interactions, solvent—solvent interactions and others where the molecular structure of the solvent is considered. All these can be included, or selections can be made, and all the resultant g compared. In these cases, the total potential energy, 0, is then taken to be a sum of pair-wise terms for interactions between the ions concerned ... 

Now, by assuming that the core density of the system remains unchanged at any distance during the interaction, and that there is practically no overlap between the core densities of all the atoms that form part of the molecules that interact with each other, then one can show that if Ng(( N, the sum of the terms in ( Eg fg[pf ] - E(.(,re[Pi]) associated with the first two terms in the right hand side of Eq. (27) is approximately equal to - AVjsjn. If it is further assumed that the core terms difference related with the second functional derivatives of the kinetic and the exchange-correlation energies cancel each other, one finds, from Eq. (32), if the chemical potential remains constant, that... 

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