Radial orbital extensions

FIGURE 1. The calculated sizes of the valence ns and np orbitals of group 14 elements. Adapted from Reference 5 

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Of great importance is the radial orbital extension difference, Ar = rp — rs, (Table 1 and Figure 1). Due to the orbital behavior described above, Ar for carbon is only —0.2 pm. However, Ar increases successively in a zig-zag fashion (caused by the d-block... 

The extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

Pseudorotation.—LCAO-MO calculations on the hypothetical molecule PHs confirm (a) that the trigonal bipyramid (t.b.p.) is the most stable configuration, (6) that the apical orbitals of the t.b.p, are more electropositive than the radial orbitals, (c) that there would be an equilibrium of t.b.p. structures, but those with the most electronegative groups in apical orientations will be more stable and predominate extensively over other t.b.p. configurations, and (d) that the lowest fundamental frequency of a t.b.p. corresponds to the equatorial in-plane bending motion as shown in (37), the force constants being nearly four times smaller than those of the... 

This might be compatible with the electrostatic model in that the radial extensions of Ad and 5d orbitals are greater than that of 3d but then the diffuseness of these orbitals increases along the series in Eq. (6.10) and that would tend to decrease the Zioct values. 

Metal clusters in metal oxide systems have not been well-characterized or abundantly investigated up to the present time. Only isolated examples of metal-metal bonded units in oxide lattices have appeared from time to time. It will be the thesis of this presentation to show that highly unusual structures determined by strong metal-metal bonding will be found in ternary and quaternary metal oxide systems, and that opportunities abound for creative work on the synthesis, theory and structure-property relationships of such compounds. Because of the well-known correlation of d-electron population and d-orbital radial extension with metal-metal bond formation,... 

Recent ab initio calculations delineate the remarkable thermodynamic destabilization of lead(IV) compounds by electronegative substituents182,183. Based on population analyses of the molecular wave functions it was proposed that electronegative substituents increase the charge of the metal and increase the difference in the radial extensions of the 6s and 6p orbitals. By increasing the differences in the radial extensions of the s and p orbitals, 6th-row relativistic effects also contribute to a destabilization of the higher valence state. 

Fig. 20. Schematic representation of the s, p, d and f partial contributions to the total energy of electrons in the conduction band of a light actinide metal. The different R s denote the radial extension of the different contributing orbitals. R (f-included) and R 2n-f refer to the equilibrium volumes when the 5 f electrons are itinerant and when they are non-binding (from Ref. 77)...

The Fe-Fe interactions vary sensitively with the ratio of the radial extension of the n-bonding orbitals to the Fe-Fe separation. This ratio is significantly greater for the minority-spin electron at an Fe " ion than for the majority-spin electrons because the minority-spin electron of a high-spin 3 d configuration is more weakly bound to the iron nucleus (by some 3 eV for octahedral-site iron in rutile) than are the 3 d majority-spin electrons of the same symmetry. Therefore Fe-Fe interactions are strongest for the couple. 

Figure 19 illustrates how a disporportionation reaction can be stabilized. The M-O interaction energy (J) as a function of the atomic separation r for the cation-d-orbital contribution has the conventional Morse form with a minimum at r and an inflection at tc. The M-O equilibrium separation ro > r is determined by the sum of all the contributions to the electronic potential, which come primarily from s and p contributions of larger radial extension. A displacement of anions towards one set of cations and away from the others gives a change in energy... 

The gamma functions Ak(p) and Bj(pt) may be obtained by the use of recursion formulas an extensive tabulation is due to Flodmark (141). In the case of Slater orbitals of principal quantum number 4 or 6, application of Slater s rules leads to nonintegral powers of r in the radial wave function consequently, changing to spheroidal coordinates introduces A and B functions of nonintegral k values, that is, incomplete gamma functions. These functions can, however, be computed (56, 57) and the overlap... 

Since the unpaired 4f electrons in lanthanide ions occupy orbitals of extremely low radial extension, the Fermi contact term is small for lanthanide shift reagents, and... 

The Ln(III) cations of the series Ce-Lu possess the extended Xe-core electronic configuration [Xe] 4/1 (n = 1-14), a symbol which perfectly pictures the limited radial extension of the f-orbitals The 4f shell is embedded in the interior of the ion, well shielded by the 5s2 and 5p6 orbitals [63], A plot of the radial charge densities for the 4f, 5s, 5p and 6s electrons for Gd+ visually explains why Ln(III) cations are commonly thought as a tripositively-charged closed shell inert-gas electron cloutf (Fig. 1) [63]. 

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