The d and f Orbitals

The d and f orbitals have more complex shapes there are five equivalent d orbitals and seven equivalent f orbitals for each principal quantum number, each orbital containing a maximum of 2 electrons with opposed spins. 

Krypton is found to be an extremely unreactive element indicating that it has a stable electronic configuration despite the fact that the n = 4 quantum level can accommodate 24 more electrons in the d and f orbitals. 

Relativistic effects are more pronounced for the actinides because of their higher nuclear charge. As a result, the s and p orbitals screen the charge of the nucleus better and the d and f orbitals expand, and are destabilized 2,3). The shielding of the 5/ orbitals by filled outer s and p orbitals is thus not as effective, and actinide ions form more covalent bonds and are found in higher oxidation states, at least at the beginning of the 5/ series. 

In contrast, the valence d and f orbitals in heavy atoms are expanded and destabilized by the relativistic effects. This is because the contraction of the s orbitals increases the shielding effect, which gives rise to a smaller effective nuclear charge for the d and f electrons. This is known as the indirect relativistic orbital expansion and destabilization. In addition, if a filled d or f subshell lies just inside a valence orbital, that orbital will experience a larger effective nuclear charge which will lead to orbital contraction and stabilization. This is because the d and f orbitals have been expanded and their shielding effect accordingly lowered. 

The second quantum number describes the shape of the orbital as s, p, d, f or g. These shapes do not describe the electron s path but rather are mathematical models showing the probability of the electron s location. The s and p orbital shapes are shown in Figure 8.9, but descriptions of the d and f orbitals are reserved for more advanced texts. 

There are five different d orbitals followed by seven different f orbitals. Filling of the d and f orbitals are very complicated. The 3 d orbitals are... 

So far, in this chapter, the band model of the solid has been ignored. This is because magnetism is associated with the d and f orbitals. These orbitals are not broadened greatly by interactions with the surroundings and even in a solid remain rather narrow. The resulting situation is quite well described in terms of localised electrons placed in d or f orbitals on a particular atom. However, some aspects of the magnetic properties of solids can be explained only by band theory concepts. 

Numbers with absolute value equal to unity multiplying the wave functions are without any importance in quantum mechanics, thus the imaginary unit can be left out in the second equation. The resulting products, 0(0) - 0(angular functions for AO have a specific directional dependence. For example, p = sin 0 cos

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Experimental evidence for electronic shells is foimd in the plot of cluster abundance vs. nuclearity and in the variation of the ionization energies of clusters (see Fig. 1.12). Electronic shell effects dominate the properties of alkali metal clusters. They are also broadly apphcable to p-block metals. The properties of transition and nobel metal nanoparticles, however, are influenced more by the formation of geometric shells. In fact, a transition from shells of electrons to shells of atoms is seen in the case of A1 [29,53]. It appears that the abundance of available oxidation states and the directional nature of the d- and f-orbitals (and to a limited extent, of the p-orbital) play a role in determining the shell that governs the property of a particular cluster. 

The periodic table is arranged more or less by chemical reactivity, using the number of electrons in the outermost shell of the element and the energy of those outermost (valence) electrons. In effect, elements are arranged according to their valence orbitals. The periodic table currently lists 109 elements. The first attempt to categorize elements in this manner was by Dmitri Ivanovich Mendeleev (Russia 1834-1907), in the nineteenth century. The first row of elements (H, He) have only the spherical s-orbitals, but the second row (Li, Be, B, C, N, O, F, He) has the Is-orbital and the 2s- and 2p-orbitals are in the outermost shell. The third row introduces 3s- and 3p-orbitals, and d-orbitals appear in the fourth row. Each shell will have one s-, three p-, five d-, and seven f-orbitals (1, 2, 3, 4), and the d- and f-orbitals accept more electrons or give up more electrons than a p-orbital. Indeed, elements with d- and f-orbit-als are characterized by multiple valences. This stands in sharp contrast to... 

Equation (4.8) gives an estimate of the mass of the mercury Is electron as about 23% greater than its rest mass. Since the radius of the Is orbital is inversely proportional to the mass of the electron, the radius of the orbital is reduced by about 23% compared to that of the non-relativis-tic radius. This s orbital contraction affects the radii of all the other orbitals in the atom up to, and including, the outermost orbitals. The s orbitals contract, the p orbitals also contract, but the more diffuse d and f orbitals can become more diffuse as electrons in the contracted s and p orbitals offer a greater degree of shielding to any electrons in the d and f orbitals. 

Field (1982) has combined the well-known molecular-orbital pictures of s and p orbitals with the ligand-field theoretical treatment of the d and f orbitals. This approach has provided very simple and yet promising interpretation for lanthanide oxides. Lanthanide oxides such as CeO, PrO, etc. are highly ionic. The conventional MO theory for the AB diatomic molecule by itself ignores the A B or A B ionic character of such compounds. Although indirectly from MO theory, this can be rationalized as well but the amount of effort is usually more significant using the LCAO-MO approach. 

Atomic orbital (Section 1.1) A three-dimensional representation of the solution of Schrodinger s equation describing the motion of an electron in the vicinity of a nucleus. Atomic orbitals have different shapes, which are determined by quantum numbers. The t orbitals are spherically symmetric,/) orbitals roughly dumbbell shaped, and the d and f orbitals are even more complicated. 

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