The helium-like 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 helium-like ion Li+\mathrm{Li}^+ (nuclear charge Z=3Z = 3, two electrons) can be treated with the same three approximations used for helium. Using Eh=27.211 eVE_{\rm h} = 27.211\,\mathrm{eV}, calculate: ((a)) the independent-electron ground-state energy E0=−Z2EhE_0 = -Z^2E_{\rm h}; ((b)) the energy including the electron-electron repulsion to first order, E=−Z2Eh+58ZEhE = -Z^2E_{\rm h} + \tfrac58 Z E_{\rm h}; ((c)) the variational energy Evar=−(Z−516)2EhE_{\rm var} = -(Z - \tfrac{5}{16})^2E_{\rm h} obtained from the trial function e−ζ(r1+r2)e^{-\zeta(r_1+r_2)} with the optimal effective charge.

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