Point charge approximation

HgBa2Ca iCu 02 +2 (n = 1, 2, 3) EEG tensor at the copper, barium, and mercury sites, by Cu( Zn), Ba( Cs), and Hg ( Au) Mossbauer emission spectroscopy. Comparison with point-charge approximation and Cu NMR data showed that the holes originating from defects are localized primarily in the sublattice of the oxygen lying in the copper plane (for HgBa2Ca2Cu30g, in the plane of the Cu(2) atoms)... 

The simplest version of the Stem theory consists in eliminating the point-charge approximation of the diffuse-layer theory. This is done in exactly the same way [Fig. 6.66(a)] as in the theory of ion-ion interactions (see Chapter 3) the ion centers are taken as not coming closer titan a certain distance a from the electrode. 

Global hardness. The partial charge concept helps the chemist to visualize within a molecule or a network, how the electronic density changes as a function of the spatial location of the various atomic constituents. Another useful information for chemical reactivity would be to visualize where the electronic chemical potential variation should be the largest or the lowest. Such a parameter is called a frontier index and may be defined in density-functional theory as f = dQHOMO/LUMO N (16). Point-charge approximation of this relation shows that each atom of a chemical compound should have a frontier index fj such that (17) ... 

Several methods have been used for the evaluation of the two-electron coulomb integrals. There is, first of all, the point-charge approximation, as proposed by Pople (1953), or the similar approximation of uniformly charged spheres of Parr (1952) and Pariser and Parr (1952 a). 

In a modified point charge approximation, proposed by Mataga and Nishimoto (1957), the two-electron integrals are evaluated from the expression... 

Fig. 35. The polarizational (P), charge transfer (CT), and overall (P + CT) charge reorganization patterns for the four chemisorption arrangements of Figs. 33 and 34, obtained from the CSA calculations using the point charge approximation to dir. The arrow in the CT components indices the CSA predicted direction of the charge flow. In each row different scale factors are applied in the P, CT, and (P + CT) panels.

It has been shown (Section 3.3.8) that the mean thickness of the ionic cloud depends on the concentration. As the concentration of a 1 1 electrolyte increases from 0.001 N to 0.01 iVto 0.1N, k decreases from about 10 to 3 to about 1 nm. This means that the relative dimensions of the ion cloud and of the ion change with concentration. Whereas the radius of the cloud is 100 times the radius of an ion at 0.001 N, it is only about 10 times the dimensions of an ion at 0.1 N. Obviously, under these latter circumstances, an ion cannot be considered a geometrical point charge in comparison with a dimension only 10 times its size (Fig. 3.29). The more concentrated the solution (i.e., the smaller the size k" of the ion cloud Section 3.3.8), the less valid is the point-charge approximation. If therefore one wants the theory to be applicable to 0.1 N solutions or to solutions of even higher concentration, the finite size of the ions must be introduced into the mathematical formulation. 

If one compares Eq. (3.II9) of the fmite-ion-size model with Eq. (3.90) of the point-charge approximation, it is clear that the only difference between the two expressions is that the former contains a term 1/(1 + ku) in the denominator. Now, one of the tests of a more general version of a theory is the correspondence principle, i.e., the general version of a theory must reduce to the approximate version under the conditions of applicability of the latter. Does Eq. (3.119) from the finite-ion-size model reduce to Eq. (3.90) from the point-charge model ... 

As to the direction of the dipole moments in the ground state of the molecules, two kinds of experimental measurements are available. Stewart has calculated the X-ray dipole moment of uracil within the point-charge approximation using the atomic charges for a standard STO L-shell. The calculated dipole moment is 4.0 1.3 D and its direction of 71° + 12° from N-l-C-4 toward N-3 atom. These values obtained within a point-charge model have an estimated standard error of about 30%. Nevertheless, the X-ray dipole moment is of reasonable magnitude and in close agreement with the value of 4.1 D obtained by solution measurement of uracil in dioxan (Table XX). 

A model related to EEM has been proposed by Rappe and Goddard and is called the charge equilibration method (QEq). The main difference between EEM and QEq lies in the treatment of the electrostatic interactions. In EEM these interactions are calculated using the point charge approximation assuming a 1/r dependence for the interaction energy, whereas in QEq the interactions are calculated between ns Slater orbitals (n = 1, 2,. . . ) ... 

Typically, electrostatic interactions are represented by some form of Coulomb s law-type function. In this type of calculation, the atoms in the molecule are assigned a charge distribution consisting of discrete point charge approximations centered on each atom. Other variations of these simple functions include exponential-type approximations (12) and variable dielectric approximations (3,11). The point charge approximation is often determined by ab-initio or semi-empirical molecular orbital calculations. The user may select the particular functional representation used in the calculation as well as the parametric form of the function. The variable dielectric function is represented by an approximation to the rigorously derived function (3,11) for computational efficiency. 

Similarly, the point-charge approximation is the leading term in the interaction between nuclei and cores... 

In semiempirical calculations, yfJlj 10.8 eV is commonly used for two electrons occupying a p-orbital on the same carbon atom p and the integrals yfJJ, for pffv are usually estimated by a point charge approximation, y e2l(rllv + a), where a = < 2/ =133 pm, that is, y, 1440/(fy1 /pm + 133) eV. 

If there are net charges present, as in proteins, the point charge approximation involves no extra calculation, but the dipole-dipole method requires that charge-charge and charge-dipole interactions also be carried out. The results are similar either way, but the point charge calculation can be carried out more quickly, and this method is usually used for large molecule calculations. 

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