Radial distribution functions, valence orbitals

The differences resulting from variations in the radial distribution functions of different valence orbitals become more pronounced for atoms which have nd and nf valence orbitals, because they are significantly more contracted than the (n+l)s and (n+l)p orbitals and therefore behave in a more core like manner [140]. 

Figure 1.2 Radial distribution functions of the valence orbitals in the (a) s-(Mg), (b) p-(Sn), (c) d-(Cr), and (d) f-(Eu) blocks of the periodic table. Black lines correspond to the core density.

In the 3d transition metal series, the electronic state gradually changes with the atomic number, though these elements possess similar properties. Because the valence electronic state of these elements essentially depends upon the nature of the 3d atomic orbital, we depict the radial wave functions, namely the spatial distribution of the 3d atomic orbitals of the elements from Sc to Zn in Fig.5. From the figure. 

The transition metal ions, Cu , Ag and Au", all have a d ( 5) electronic state-configuration, with = 3,4 and 5, respectively. The RCEP used here were generated from Dirac-Fock (DF) all electron (AE) relativistic atomic orbitals, and therefore implicitly include the indirect relativistic effects of the core electron on the valence electrons, which in these metal ion systems are the major radial scaling effect. In these RCEP the s p subshells are included in the valence orbital space together with the d, ( + l)s and ( + l)p atomic orbitals and all must be adequately represented by basis functions. The need for such semi-core or semi-valence electrons to be treated explicitly together with the traditional valence orbitals for the heavier elements has been adequately documented The gaussian function basis set on each metal atom consists of the published 4 P3 distribution which is double-zeta each in the sp and n + l)sp orbital space, and triple-zeta for the nd electrons. 

It is of interest to compare the radial wave functions between 4f and 5f elements. O Figure 18.19 shows relativistic calculations of and Am valence wave functions of the radial part. This figure indicates that the 4f subshell of Eu is core like because of the lower density around the ionic radius of the ion. In the corresponding 5f ion, Am, 5f subshell penetrates the 6s and 6p orbitals. The wider distribution of the 5f orbital allows much participation in bonding for the actinides than for the lanthanides. The 4f orbitals of lanthanide atoms are mainly localized in the core and do not contribute to the chemical bonding. 

The filling of the / shell is a common feature of both lanthanides and actinides. However, there are remarkable differences in the properties of the 4/ and 5/ electrons. The 4/ orbitals of the lanthanides and the 5/ actinide orbitals have the same angular part of the wave function but differ in the radial part. The 5/ orbitals also have a radial node, while the 4/ orbitals do not. The major differences between actinide and lanthanide orbitals depend, then, on the relative energies and spatial distributions of these orbitals. The 5/ orbitals have a greater spatial extension relative to the Is and Ip than the 4/ orbitals have relative to the and 6/t. This allows a small covalent contribution from the 5/ orbitals, whereas no compounds in which 4/ orbitals are used exist. In fact, the 4/ electrons are so highly localized that they do not participate in chemical bonding, whereas the 5d and 6s valence electrons over-... 

It is not difficult to see why the I.a orbital is so hard to reproduce. It is very compacL with an expectation value of r of about 0.3. For comparison, the 2s and 2p expectation values are 1.6 and 1.7. According to (6.5.18), our basis functions with unit exponent have expectation values equal to + and are thus not well suited for describing the core orbital. We conclude that the Litguerrc functions with a fixed exponent are ill suited for describing orbitals with widely different radial distributions and that a large number of such functions would be needed to ensure a uniform description of the core and valence regions of an atomic sy stem. 

Covalent chemical bonds between atoms of the same or a different species rely on the interaction of the outermost — or valence — electrons. Even though one speaks of electrons one should rather think of electron clouds, i.e. of electronic density distributions. The radial and angular distribution of the electron density is described by one electron wave functions — also called atomic orbitals — which are derived as a solution of the quantum mechanical Schrodinger equation ... 

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