Many-electron atoms

The helium atom: two electrons

The preceding sections have been devoted to hydrogen-like species containing one electron, the energy of which depends only on n (equation 1.1); the atomic spectra of such species contain only a few lines associated with changes in the value of n (Figure 1).

                    (1.1) 

It is only for such species that the Schrodinger equation has been solved exactly. The next simplest atom is He (Z = 2), and for its two electrons, three electrostatic interactions must be considered:

* attraction between electron (1) and the nucleus;

* attraction between electron (2) and the nucleus;

* repulsion between electrons (1) and (2).

The net interaction will determine the energy of the system.

In the ground state of the He atom, two electrons with m_s = ‏1/2 and -1/2 occupy the 1s atomic orbital, i.e. the electronic configuration is 1s². For all atoms except hydrogen- like species, orbitals of the same principal quantum number but differing Ɩ are not degenerate. If one of the 1s² electrons is promoted to an orbital with n = 2, the energy of the system depends upon whether the electron goes into a 2s or 2p atomic orbital, because each situation gives rise to different electrostatic interactions involving the two electrons and the nucleus. However, there is no energy distinction among the three different 2p atomic orbitals. If promotion is to an orbital with n = 3, different amounts of energy are needed depending upon whether 3s, 3p or 3d orbitals are involved, although there is no energy difference among the three 3p atomic orbitals, or among the five 3d atomic orbitals. The emission spectrum of He arises as the electrons fall back to lower energy states or to the ground state and it follows that the spectrum contains more lines than that of atomic H.

  In terms of obtaining wave functions and energies for the atomic orbitals of He, it has not been possible to solve the Schrodinger equation exactly and only approximate solutions are available. For atoms containing more than two electrons, it is even more difficult to obtain accurate solutions to the wave equation.

  In a multi-electron atom, orbitals with the same value of n but different values of Ɩ are not degenerate.

Fig. 1 Some of the transitions that make up the Lyman and Balmer series in the emission spectrum of atomic hydrogen.
 

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