The two-electron ion (nuclear charge

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

The two-electron ion O6+\mathrm{O}^{6+} (nuclear charge Z=8Z = 8, two electrons) is treated with the same three approximations used for helium, with Eh=27.211 eVE_{\rm h} = 27.211\,\mathrm{eV}. ((a)) Calculate the independent-electron ground-state energy E0=−Z2EhE_0 = -Z^2 E_{\rm h}. ((b)) Calculate the first-order repulsion energy 58Z Eh\tfrac58 Z\,E_{\rm h}, the corrected ground-state energy E1=E0+58Z EhE_1 = E_0 + \tfrac58 Z\,E_{\rm h}, and the first ionisation energy this predicts (the remaining ion is hydrogenic with nuclear charge ZZ). ((c)) Using the trial function e−ζ(r1+r2)e^{-\zeta(r_1+r_2)}, state the optimal effective charge ζ=Z−516\zeta = Z - \tfrac{5}{16} and the resulting variational ground-state energy −ζ2Eh-\zeta^2 E_{\rm h}.

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