Discrete Bound States: Many-Electron Atoms and Ions

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Much of the physics determined for hydrogen and hydrogen-like ions is relevant to atoms and ions with many electrons. In a multi-electron atom or ion, the quantum state energies and wavefunctions are dominated by the central potential arising from the nuclear charge as is the case for hydrogen and hydrogen-like ions. Consequently, quantum wavefunctions similar in form to those found for hydrogen or hydrogen-like ions are produced. There are some corrections needed to the exact electron energies associated with the nuclear electric field experienced by each electron due to shielding effects by other electrons – and some perturbing energy effects due to electron–electron Coulomb repulsion. The electron wavefunctions have similar orbital or angular shapes as for hydrogen and hydrogen-like ions (see Figures 7.2 and 7.3). The same electron-configuration system (i.e. 1s, 2s, 2p, …) can be used and the same degeneracies are associated with different values of principal quantum number n (degeneracy 2 n 2 ) and orbital angular momentum quantum number l (degeneracy 2×2 l(l +1 ) ) as found for hydrogen and hydrogen-like ions. As multi-electron atoms and ions have quantum states following the ‘rules’ established for hydrogen and hydrogen-like ions, the quantum state energy values are primarily determined by a principal quantum number n with values n =1, 2, 3, … . Angular quantum numbers l with l =0, 1, 2, … n − 1 are found to have a small effect on energy values due to shielding of the nuclear charge. As for hydrogen, the number of individual quantum states at a particular energy involves the different magnetic quantum numbers m with values m = − l ,− l + 1, … , 0, 1 … + l and the two possible spin orientations (relative to the orbital angular momentum). The states which the electrons occupy are called the electronic configuration for the atom or ion. An important feature of multi-electron atoms is established by the Pauli exclusion principle which requires that only one electron can occupy an individual quantum state. In the ground state of an atom or ion, the required number of electrons effectively fills the lowest energy quantum states designated by the n , l , m and s values according to the hydrogen degeneracies.