.. _sphx_glr_api_gallery_quantum_helium:

Variational helium atom
-------------------------

The Kellner-Hylleraas effective-nuclear-charge treatment of helium and
its two-electron isoelectronic ions: electron screening from the
variational principle.


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    <div class="sphx-glr-thumbcontainer" tooltip="Helium was the first test of quantum mechanics beyond one electron. Kellner (1927) and Hylleraas (1928-1929) attacked it with the variational method: take both electrons in hydrogen-like 1s orbitals but let the nuclear charge they &quot;see&quot; be an adjustable \zeta. The energy E(\zeta)=(\zeta^2-2Z\zeta+\tfrac{5}{8}\zeta)E_h (~chemistrykit.quantum.systems.helium.helium_like_variational_energy) is lowest at \zeta=Z-5/16=27/16 for helium (~chemistrykit.quantum.systems.helium.optimize_helium_like_effective_charge): each electron shields the other from 5/16 of a nuclear charge. The result, -2.848\,E_h, is within 2% of the exact -2.9037\,E_h, which Hylleraas approached by adding the interelectronic distance to the trial function.">

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  :doc:`/api/gallery/quantum/helium/plot_01_hylleraas_variational_helium`

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   /api/gallery/quantum/helium/plot_01_hylleraas_variational_helium