Consider the two-electron configuration of a helium-like atom
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Atomic, Molecular & Optical Physics · Many-electron atoms and the periodic table
Deriving the Spectrum, Ch Helium: exchange and the variational method
Problem
Consider the two-electron configuration 1s3d of a helium-like atom, built from two orthonormal one-electron orbitals u1s(r) and u3d(r), each an eigenfunction of the same one-electron Hamiltonian with energies E1s and E3d.
((a)) Construct the two independent normalised spatial combinations of these orbitals for two electrons, symmetric and antisymmetric under exchange of the electron labels.
((b)) Pair each spatial combination with the appropriate spin function(s) so that every resulting two-electron state is totally antisymmetric under exchange, and count the states obtained.
((c)) Now replace the 3d orbital by a second copy of the 1s orbital, so that the configuration becomes 1s2. Show that the antisymmetric spatial combination vanishes identically in this case, and state which of the states found in part (b) survive.