Central-Field Approximation for Many-Electron Atoms

Questions:

(a) Briefly describe the central-field approximation for many-electron atoms (in around 200 words). Your answer should explain why this approximation predicts that atomic orbitals with the same pair of values of n  and l, but different values of  m, are degenerate. In the Coulomb model of a hydrogen atom, states with the same value of  n  but different values of l have the same energy. Explain why many-electron atoms do not exhibit this degeneracy. Give a qualitative argument indicating why atomic orbitals in a many-electron atom have energies that increase as l increases for any fixed value of  n .

 (b) A titanium atom has an excited-state configuration

              1s ² 2s ² 2p ⁶ 3s² 3p⁶ 4s ² 3d ¹ 4p ¹

When residual electron–electron interactions are taken into account, this configuration splits into a number of atomic terms. List these atomic terms, labelled by their L and S quantum numbers, and also give each term an appropriate spectroscopic symbol. Outline your reasoning.

(c) Specify the degree of degeneracy of each atomic term in part (b) and confirm that the total number of states associated with all these atomic terms agrees with the number expected from the and s quantum numbers of the two valence electrons in the given configuration.

(d) When the spin–orbit interaction is taken into account in the LS-coupling scheme, the atomic term with L = 1 and S = 1 splits into a number of atomic levels. Label these atomic levels by their L, S and J quantum numbers, and give each level an appropriate spectroscopic symbol. Outline your reasoning.

Solution:

(a) Central-field approximation for many-electron atoms:

By increasing the number of electrons in the atom, the contribution of electron-electron repulsion terms in the Hamiltonian becomes more significant (not treatable with perturbation theory), and the separation of variables (that is the main property of the independent-particle model) becomes impossible with the exact Hamiltonian of the many-electron atoms.By applying the central-field approximation, the exact Hamiltonian can be replaced by an effective Hamiltonian that is the sum over effective Hamiltonians of electrons, i.e.,

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