Many Electron Atoms

When an atom has more than one electron, the electronic structure of the atom cannot be simply described as the electron-electron repulsion makes solving Schrodinger’s equation very hard. Instead, an approximation is made that the wavefunction of a two electron atom or ion is obtained by assigning each electron to one of the atomic orbitals of hydrogen. This is known as the orbital approximation.

For a two electron species, such as He or H^–, both electrons can occupy the 1s atomic orbital, and this gives rise to the ground state electronic configuration 1s². The ground electronic state of the species is that where the electrons occupy the orbitals with the lowest possible energies. States of two electron systems with electronic configurations 1s¹2s¹, 1s¹2p¹, 1s¹3s¹, etc., are known as excited states.

When an electron is added or removed from a species, the difference in energy difference between the two states of the species gives the electron affinity and the ionization energy. For example:

I₁(He) = [energy of He⁺] – [energy of He] = 24.58 eV

A(H) = [energy of H] – [energy of H^–] = 0.75 eV

I₁ is the first ionization energy of He, and A is the electron affinity of H, and also the ionization energy of H^–.

These energy differences can be interpreted by associating an orbital energy, e, with each occupied orbital, and using this in the same way as the energy of a hydrogenic system. For the ionization energy, I = -ε_i, where i is the label of the orbital from which the electron is lost. This assignment of the ionization energy is known as Koopman‘s theorem [note that the ionization energy is positive as the orbital energies are all negative].

In absorption or emission spectra, the energy changes corresponding to the observed transitions are given by:

ΔE = ε_i – ε_j

From the values of the transition energies, electron affinities and ionization energies, the energies of the atomic orbitals in different species may be calculated. The energies of the 1s orbitals, in eV, for a range of species are shown in the table.

The large differences between the one-electron and the two-electron species shows the importance of the electrostatic repulsion between the electrons in contributing to the energies of the orbitals. This repulsion opposes the attraction of the electrons towards the nucleus, and gives a positive contribution to the orbital energy, and hence a negative contribution to the ionization energy or electron affinity. This effect is known as electron shielding, or electron screening.