Repulsion correction

Close-packed spheres occupy 74.04% of a total volume, hence the hard-sphere radius of I" in these 2 1 salts in 2.03 A. Correction for the electrostatic attraction alone would give a monovalent iodide radius of about 2.24, an opposite repulsion-correction for the different co-ordination number would reduce this to about 2.10 A for the monovalent sodium-chloride type (see Appendix). Such values are consistent with our earlier estimates, but incompatible with the electron-density minimum value (4) of 1.94 A. 

Gritsenko O, Pemal K, Baerends EJ (2005) An improved density matrix functional by physically motivated repulsive corrections, J Chem Phys, 122 204102... 

The electronic bands of an infinite crystal can cross as a function of some parameter (pressure, concentration etc.). If one treats the e /r,2 term of the electron repulsion correctly, one sees that the crossing transition of the two bands is a first-order phase transition, between the metallic and insulating states. This transition was predicted by Mott in 1946 and has carried his name ever since. In fact, the original Mott criterion does not predict such a transition for Hg, but the criterion was derived for monovalent atoms. For divalent mercury it should not be applicable. Also the semiempirical Herzfeld criterion, which was very successful in predicting the insulator to metal transition in compressed xenon, predicts bulk Hg to be non-metallic. All this seems to imply that Hg is a rather special case. 

One notes the strong dependence of AEsteric as a function of the type of zeolite and also the large repulsive correction AEsteric for the zeolite with the narrow one-dimensional pore (ferrierite). We illustrate in the next section how the values of AEsteric can be used to assist kinetic analysis. 

These considerations are frequently overlooked in studies that claim, on numerical grounds, that 5 is an overcorrection, or simply unhelpful. For example, in an SCF study on FH. OH2 Kocjan et al observed that correction of the STO-2G result at the calculated removed all attraction. Their conclusion that overcorrects is debatable, however, since was probably so short that a repulsive corrected A should have been anticipated. 

Dealing with electrons we know that the dominant interaction between them is the Coulomb repulsion corrected, because electrons are fermions, by interactions induced by their spin. The spin-orbit interaction is already included in the one-electron Dirac Hamiltonian but the two-electron interaction should also include interactions classically known as spin-other-orbit, spin-spin etc... Furthermore a relativistic theory should incorporate the fact that the speed of light being finite there is no instantaneous interaction between particles. The most common way of deriving an effective Hamiltonian for a many electron system is to start from the Furry [11] bound interaction picture. A more detailed discussion is given in chapter 8 emd we just concentrate on some practical considerations. 

In case of large overlapping or mutually penetrating cores a core-core repulsion correction (CCRC) to the point charge repulsion model in Eq. 27 is needed. A similar core-nucleus repulsion correction (CNRC) has to be applied for the interaction between nuclei of atoms treated without ECP and centers with large-core ECPs. A Bom-Mayer type ansatz proved to be quite successful to model the pairwise repulsive correction [206,207]... 

Figure 16. Core-core repulsion correction in Hgj for two-valence electron Hg ECPs. Errors per atom due to a superposition of pairwise eorrections in larger highly-symmetric Hg clusters (n=3 equilateral triangle n=4 tetrahedron n=6 octahedron n=13 icosahedron) are also shown. The underlying core-core repulsion corrections were determined from small-core (20-valence electron) PP frozen-core calculations on the Hg + core systems. The interatomic distances in Hg2 and the bulk are indicated by vertical lines.

Again, note that the self-consistency requirement lies in the fact that each of the Xia and Y,a coefficients involves all the others because of the occurrence of the electron-repulsion correction G x + y) and because x, y involve all the coefficients. 

The diagonal terms in the Cl matrix are all of a similar form being the energy of a set of doubly occupied MOs. They are most conveniently expressed as the energy of o minus the energy of the removed MO plus the energy of the added MO plus the repulsion corrections ... 

In fact, if we develop a theory for the effect of a time-dependent perturbation oscillating at a single frequency we can always use a sum of such terms to give the Fourier synthesis of an arbitrary time-dependent perturbation. Let us try to develop such a theory for the self-consistent (i.e. including changes in electron repulsion) corrections to the energies and orbitals of a single-determinant wavefunction. 

A number of efforts have been made to calculate ionization-potential sums from thermochemical data and appropriate Born-Haber cycles. When an isostructural set of compounds is used, and covalence/repulsion corrections are made from a systematic lanthanide-actinide comparison, such sums can be quite reliable, as has been repeatedly demonstrated for the trivalent lanthanides [88]. For example, Morss [89] was able to estimate the sum of the first three ionization energies (/i +I2 + I3) for Pu as... 

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