Atomic structure polyelectronic atoms

It is useful to represent the polyelectronic wave function of a compound by a valence bond (VB) structure that represents the bonding between the atoms. Frequently, a single VB structure suffices, sometimes it is necessary to use several. We assume for simplicity that a single VB stiucture provides a faithful representation. A common way to write down a VB structure is by the spin-paired determinant, that ensures the compliance with Pauli s principle (It is assumed that there are 2n paired electrons in the system)... 

The hydrogen atom and one-electron ions are the simplest systems in the sense that, having only one electron, there are no inter-electron repulsions. However, this unique property leads to degeneracies, or near-degeneracies, which are absent in all other atoms and ions. The result is that the spectrum of the hydrogen atom, although very simple in its coarse structure (Figure 1.1) is more unusual in its fine structure than those of polyelectronic atoms. For this reason we shall defer a discussion of its spectrum to the next section. 

The above rule can readily be extended to other polyelectronic systems, like the tt system of benzene (6), or to molecules bearing lone pairs as in formamide (7). In this latter case, calling n, c, and o, respectively, the tt atomic orbitals of nitrogen, carbon, and oxygen, the VB wave function describing the neutral covalent structure is given by Equation 3.10 ... 

The model we have just described so greatly oversimplifies the structure of polyelectronic atoms that, although it produces some qualitatively useful ideas about polyelectronic atoms, it is not satisfactory for the description of quantitative atomic properties. To get an accurate description of polyelectronic atoms, we must take into account the electron-electron interactions in a much more detailed manner than simply assuming that they reduce the nuclear charge. 

Spin-orbit coupling is an addition to the Schrodinger equation but it is a natural feature in Dirac s theory which associates relativity theory with quantum mechanics. There are, however, other relativistic effects in the electronic structure of polyelectronic atoms which can be related to changes in the electron mass with velocity (for a review on relativistic effects in structural chemistry, see ref. 62). 

The problem of the structure of the hydrogen atom is the most important problem in the field of atomic and molecular structure, not only because the theoretical treatment of this atom is simpler than that of other atoms and of molecules, but also because it forms the basis for the discussion of more complex atomic systems. The wave-mechanical treatment of polyelectronic atoms and of molecules is usually closely related in procedure to that of the hydrogen atom, often being based on the use of hydrogenlike or closely related wave functions. Moreover, almost without exception the applications of qualitative and semiquantitative wave-mechanical arguments to chemistry involve the functions which occur in the treatment of the hydrogen atom. 

Up to now the discussion has been limited to the simplest possible case, namely, that of the hydrogen atom — the only case for which an exact solution to the Schrodinger equation exists. The solution for a polyelectronic atom is similar to that of the hydrogen atom except that the former are inexact and are much more difficult to obtain. Fortunately, the basic shapes of the orbitals do not change, the concept of quantum numbers remains useful, and, with some modifications, the hydrogen-like orbitals can account for the electronic structure of atoms having many electrons. 

If the electron is in the Is state, the hydrogen atom is in its lowest state of energy. In a polyelectronic atom such as carbon (six electrons) or sodium (eleven electrons) it would not seem unreasonable if all the electrons were in the Is level, thereby giving the atom the lowest possible energy. We might denote such a structure for carbon by the symbol Is and for sodium, ls . This result is wrong, but from what has been said so far there is no apparent reason why it should be wrong. The reason lies in an independent and fundamental postulate of the quantum mechanics, the Pauli exclusion principle no two electrons... 

From the structure of polyelectronic atoms, we would expect that electrons would fill the orbitals according to the following order Is < 2s < 2p < 3s < 3p < 3d < 4s < 4p < 4d < 4f and so on. In fact, data has shown that the energies of the orbitals do not increase progressively according in this order. The order of filling of the orbitals is dependent on the shielding effects (Z ) and the electronic penetration of the orbitals. 

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