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
For a two-electron ion of nuclear charge Z, the chapter gives the first-order energy E1=(−Z2+85Z)Eh and the variational energy Evar=−(Z−165)2Eh. Take Eh=27.211eV.
((a)) Show that E1−Evar is the same for every Z, and find it in eV.
((b)) The measured ground-state energies are −79.005eV for helium and −198.1eV for Li+. For each, find how far Evar lies above the measured energy, in eV and as a percentage of the measured energy's size.
((c)) Use (a) and (b) to explain why the relative error of both estimates falls as Z grows.