<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>1</mml:mn><mml:mi/><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>,<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mi/><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>, and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mn>2</mml:mn><mml:mi/><mml:mrow><mml:mmultiscripts><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mprescripts/><mml:mrow/><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mrow/><mml:mrow/></mml:mmultiscripts></mml:mrow></mml:math>States of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="normal">H</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:math>and of He

The ionization energy $J$, including mass-polarization and relativistic corrections but not the Lamb shift correction, was evaluated for the $1^{1}S$ state of the negative-hydrogen ion using determinants up to order $n=444$. We get $J(444)=6083.0943$ ${\mathrm{cm}}^{\ensuremath{-}1}$, and, by extrapolation, $J(\ensuremath{\infty})=6083.0958$ ${\mathrm{cm}}^{\ensuremath{-}1}$. A search for bound states $2^{1}S$ and $2^{3}S$ of ${\mathrm{H}}^{\ensuremath{-}}$ led to negative results. In the case of helium, an upper bound to the nonrelativistic energy eigenvalue $\ensuremath{\nu}$ for the $1^{1}S$ state was evaluated at $n=1078$ to be ${\ensuremath{\nu}}_{+}=198317.866$ ${\mathrm{cm}}^{\ensuremath{-}1}$, as against the previously determined lower bound of ${\ensuremath{\nu}}_{\ensuremath{-}}(1078)=198317.374$ ${\mathrm{cm}}^{\ensuremath{-}1}$. For the $2^{3}S$ state this gap is already completely closed at $n=715$, with ${\ensuremath{\nu}}_{+}(715)=38453.1299$ ${\mathrm{cm}}^{\ensuremath{-}1}$ and ${\ensuremath{\nu}}_{\ensuremath{-}}(715)=38453.1292$ ${\mathrm{cm}}^{\ensuremath{-}1}$. At $n=1078$, $J=38454.827375$ ${\mathrm{cm}}^{\ensuremath{-}1}$, and the electron charge density at the nucleus $D(0)$ comes out 33.18414092, in agreement with previously extrapolated values. This substantiates a disagreement of the order of one part in ${10}^{5}$ between theory and experiment in the hyperfine structure of the $2^{3}S$ state of ${\mathrm{He}}^{3}$ which was established by White, Chow, Drake, and Hughes. With Suh and Zaidi's value for the Lamb shift of -0.109\ifmmode\pm\else\textpm\fi{}0.009 ${\mathrm{cm}}^{\ensuremath{-}1}$, the ionization energy of the $2^{3}S$ state comes out 38454.718\ifmmode\pm\else\textpm\fi{}0.009 ${\mathrm{cm}}^{\ensuremath{-}1}$, as against Herzberg's experimental value of 38454.73\ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. For the $2^{1}S$ state we get $J(615)=32033.318$ ${\mathrm{cm}}^{\ensuremath{-}1}$, which with a Lamb shift of -0.104\ifmmode\pm\else\textpm\fi{}0.014 ${\mathrm{cm}}^{\ensuremath{-}1}$ evaluated by Suh and Zaidi, leads to an ionization energy of 32033.214\ifmmode\pm\else\textpm\fi{}0.014 ${\mathrm{cm}}^{\ensuremath{-}1}$. The experimental value is, according Herzberg, equal to 32033.26\ifmmode\pm\else\textpm\fi{}0.03 ${\mathrm{cm}}^{\ensuremath{-}1}$ or, at worst, \ifmmode\pm\else\textpm\fi{}0.05 ${\mathrm{cm}}^{\ensuremath{-}1}$. A summary is given of the verification to date of the Lamb shift in two-electron atoms.

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