The one-parameter trial function gives

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 one-parameter trial function e−ζ(r1+r2)e^{-\zeta(r_1+r_2)} gives, for a two-electron ion of nuclear charge ZZ, the upper bound Evar(Z)=−(Z−516)2EhE_{\rm var}(Z) = -\left(Z-\tfrac{5}{16}\right)^2E_{\rm h}. The ion is bound if its true ground-state energy lies below the energy of the one-electron ion that remains when one electron leaves to rest at infinity, −12Z2Eh-\tfrac12Z^2E_{\rm h}. Take Eh=27.211 eVE_{\rm h} = 27.211\,\mathrm{eV}. ((a)) For H−\mathrm{H}^- (Z=1Z=1), compare EvarE_{\rm var} with this threshold and say exactly what the comparison proves. ((b)) Treat ZZ as a continuous parameter. Find the smallest nuclear charge ZcZ_c above which this trial function proves the two-electron ion bound, exactly and to four figures. ((c)) H−\mathrm{H}^- is measured to be bound. What does that say about the true critical charge, and about this trial function between Z=1Z = 1 and ZcZ_c?

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