Polyelectronic atoms, quantum

Since s = j only, j is not a very useful quantum number for one-electron atoms, unless we are concerned with the fine detail of their spectra, but the analogous quantum number J, in polyelectronic atoms, is very important. 

The treatment of atoms with more than one electron (polyelectronic atoms) requires consideration of the effects of interelectronic repulsion, orbital penetration towards the nucleus, nuclear shielding, and an extra quantum number (the spin quantum number) which specifies the intrinsic energy of the electron in any orbital. The restriction on numbers of atomic orbitals and the number of electrons that they can contain leads to a discussion of the Pauli exclusion principle, Hund s rules and the aufbau principle. All these considerations are necessary to allow the construction of the modern form of the periodic classification of the elements. 

The electron correlation problem occurs with all polyelectronic atoms. To treat these systems using the quantum mechanical model, we must make approximations. The simplest approximation involves treating each electron as if it were moving in a field of charge that is the net result of the nuclear attraction and the average repulsions of all the other electrons. To see how this is done, let s compare the neutral helium atom and the He+ ion ... 

Having defined the four quantum numbers n, /, m and s which describe the electron, we may now return to the problem of the possible electronic states of a polyelectron atom. These are defined by the Pauli exclusion principle, which states that no two electrons in the same atom can possess... 

Limitations on the values of the quantum numbers lead to the familiar aufbau (German, Aujbau, building up) principle, where the buildup of electrons in atoms results from continually increasing the quantum numbers. Any combination of the quantum numbers presented so far correctly describes electron behavior in a hydrogen atom, where there is only one electron. Flowever, interactions between electrons in polyelectronic atoms require that the order of filling of orbitals be specified when more than one electron is in the same atom. In this process, we start with the lowest n, I, and m, values (1, 0, and 0, respectively) and either of the m values (we will arbitrarily use — 5 first). Three mles will then give us the proper order for the remaining electrons as we increase the quantum numbers in the order m , m, I, and n. 

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). 

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... 

One especially important difference between polyelectronic atoms and the hydrogen atom is that for hydrogen all the orbitals in a given principal quantum level have the same energy (they are said to be degenerate). This is not the case for polyelectronic atoms, where we find that for a given principal quantum level the orbitals vary in energy as follows ... 

Any two electrons occupying the same orbital would have a symmetric spatial wave function. Therefore, their spin wave functions must be antisymmetric, as is the case for Equation (4.17). In other words, the Pauli exclusion principle states that no two electrons in the same atom may have all four quantum numbers the same. Each electron in an atom must possess a unique set of quantum numbers. As a result, every hydrogenic orbital in a polyelectronic atom can hold at most two electrons, and then if and only if their electron spins are opposite. Hence, the sets of s, p, d, and f orbitals for a given value of n can hold a maximum of 2, 6, 10, and 14 electrons, as suggested by the blocks of elements shown in Figure 4.12. 

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