Radial maxima

Differences in spinor energies Ae (in eV) and radial maxima Ar (in bohr) of valence Pj 2 P3/2 spinors for p-block elements from atomic DHF calcula-... 

It is more difficult to obtain a compact basis set representation of the 4f orbitals than any other valence orbitals in the lanthanides. Of course, the 4f AOs are responsible for many of the interesting properties of the lanthanides, such as their highly ionic bonding and sharp f-f UV-visible transitions (as opposed to the broad bands seen for TM complexes in this part of the spec-trum). The 4f orbitals of the lanthanides have radial maxima at r 0.5 A (Figure 3), but still have appreciable tails for r > 2 A. The long-range tails ... 

The highest radial maxima for non-alkali atoms in the same period must decrease uniformly from the alkali values r", depending on the number of electrons at each sublevel. For elements of the />-block, correct values of these radial maxima are predicted as n ... 

From these expressions, it is apparent that we may introduce flexibility in the radial part of the one-electron space not only by using functions with different principal quantum numbers n, but also by using functions with different exponents f . For example, by choosing the exponents in Is STOs, we may hope to span the same space as we do by introducing higher-order STOs since their radial maxima are the same. In Figure 6.5, we compare the Is, 2s and 3s STOs with a fixed exponent equal to 1 with the sequence of Is STOs with exponents 1, and 5. 

Fig. 6.5. The radial distribution functions for Ij, 2s and 3s STOs with exponents = 1 (left) and for li STOs with exponents = 1, 1/2 and 1/3 (right) (atomic units). The notation is ns( ). The radial maxima are the same in the two figures.

For the description of core correlation, functions are needed with radial maxima close to the atomic nuclei. In the core-valence sets cc-pCVXZ [24], the standard valence sets cc-pVXZ are augmented with tight correlating orbitals, with exponents selected to maximize the magnitude of the core correlation energy... 

Fig. 7.14. Relief plots of the negative of the Laplacian distributions for triplet and singlet states of CFj. The lower diagrams are for the plane containing the nuclei, the upper ones for the perpendicular symmetry plane containing the C nucleus, the plane containing the non-bonded charge maxima. There are two non-bonded maxima in the triplet, one in the singlet. The point labelled a is a (3, — 1) critical point in the VSCC of triplet carbon. There is no radial maximum or lip at the point labelled h and its mirror image and the VSCC of singlet carbon exhibits holes at these two points. The maxima present in the VSCCs of the F atoms are not shown as they are larger by a factor of 10 than those on the carbons.

Some intrinsec metal characteristic features are also indicated by EXAFS analysis of these catalysts. By comparing the Fourier transforms of the tartaric acid treated catalysts with those of the standard compounds, a similarity to RuOj is quite evident in the first 4 A of the transforms. This claims for a quite oxidated state of the local Ru-environment in these catalysts. The. shift of the first radial maximum towards larger distances, characteristic of the Ru-Cl bond, suggests a certain Cl contribution to the local environment of Ru, however still dominated by the oxygen of the support... 

FIGURE 17.7 Special 2G-2s orbital optimized using the outer tail of the Sambe 4G-2s with just one additional inner Gaussian optimized to produce the same radial maximum as a Slater-Type 2s fimction. Here the orbital is scaled to 2s =1-5 for the B2s orbital. Note vertical scale is much smaller for the 2s than the Is in Figure 17.6. 

For the lanthanides, the rms radii and the 95% density radius are in shell order, i.e., 4d < 4f << 5s < 5p < 5d <<6s. The radial maximum of the 4f is inside that of the 4d, but otherwise the shell order is observed. What is perhaps not obvious from these plots is that the radial maximum of the 6s is outside the 95% density radius of the 5p shell, whereas the radial maximum of the 5d is inside the 95% density radius of the 5p shell. If the radial maximum is taken as some measure of where the midpoint of a bond would be, this indicates that bonding with the 6s does not incur much repulsion of the ligand orbitals by the outer core (5s and 5p) of the lanthanide, whereas there would be somewhat more repulsion from bonding with the 5d. In any case, the 5d is substantially inside the 6s on all measures of radial extent. 

The radial maximum of the 7s falls at about the same distance as the 95% density radius of the 6p+, and only a little outside that of the 6p and the 6s. As for the 5d in the lanthanides, the radial maximum of the 6d spinors in the actinides is inside the 95% density radius of the outer core spinors, but these spinors extend out almost as far as the 7s spinors at the 95% density radius. Bonding that involves any of the valence spinors is likely to involve a fair... 

The magnitude and shape of such a mean-field potential is shown below [21] in figure B3.1.4 for the two 1 s electrons of a beryllium atom. The Be nucleus is at the origin, and one electron is held fixed 0.13 A from the nucleus, the maximum of the Is orbital s radial probability density. The Coulomb potential experienced by the second electron is then a function of the second electron s position along the v-axis (coimecting the Be nucleus and the first electron) and its distance perpendicular to the v-axis. For simplicity, this second electron... 

The magnitude and "shape" of sueh a mean-field potential is shown below for the Beryllium atom. In this figure, the nueleus is at the origin, and one eleetron is plaeed at a distanee from the nueleus equal to the maximum of the Is orbital s radial probability density (near 0.13 A). The radial eoordinate of the seeond is plotted as the abseissa this seeond eleetron is arbitrarily eonstrained to lie on the line eonneeting the nueleus and the first eleetron (along this direetion, the inter-eleetronie interaetions are largest). On the ordinate, there are two quantities plotted (i) the Self-Consistent Field (SCF) mean-field... 

This factor is reminiscent of the radial distribution function for electron probability in an atom and the Maxwell distribution of molecular velocities in a gas, both of which pass through a maximum for similar reasons. 

Thus, the electric field is radial. Electron density is at a maximum a short distance from the anode. Electrons progress radially toward the anode only as they lose kinetic energy, mainly through inelastic (ionizing) coUisions with molecules (40). 

C An individual pocket of porosity shall not exceed the lesser of Tw/2 or Vh in in its greatest dimension. The total area of porosity projected radially through the weld shall not exceed an area equivalent to 3 times the area of a single maximum pocket allowah le to any square inch (645 mnr) of projected weld area. 

Price (1968) made similar measurements, but placed a m.onolith right over the bed. This eliminated the radial components of the flow velocity after the fluid left the bed. Price also operated at an Rep an order of magnitude higher than Schwartz and Smith and his conclusion was that the maximum is at a half-pellet diameter distance from the wall. Vortmeyer and Schuster (1983) investigated the problem by variational calculation and found a steep maximum near the wall inside the bed. This was considerably steeper than those measured experimentally above the bed. 

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