N.7 Shield­ing ap­prox­i­ma­tion lim­i­ta­tions

In the he­lium atom, if you drop the shield­ing ap­prox­i­ma­tion for the re­main­ing elec­tron in the ion­ized state, as com­mon sense would sug­gest, the ion­iza­tion en­ergy would be­come neg­a­tive! This il­lus­trates the dan­gers of mix­ing mod­els at ran­dom. This prob­lem might also be why the dis­cus­sion in [25] is based on the zero shield­ing ap­prox­i­ma­tion, rather than the full shield­ing ap­prox­i­ma­tion used here.

But zero shield­ing does make the base en­ergy lev­els of the crit­i­cal outer elec­trons of heavy atoms very large, pro­por­tional to the square of the atomic num­ber. And that might then sug­gest the ques­tion: if the en­ergy lev­els ex­plode like that, why doesn't the ion­iza­tion en­ergy or the elec­troneg­a­tiv­ity? And it makes the ex­pla­na­tion why he­lium would not want an­other elec­tron more dif­fi­cult. Full shield­ing puts you in the ob­vi­ously more de­sir­able start­ing po­si­tion of the ad­di­tional elec­tron not be­ing at­tracted, and the al­ready present elec­trons be­ing shielded from the nu­cleus by the new elec­tron. And how about the size of the atoms im­plod­ing in zero shield­ing?

Over­all, this book prefers the full shield­ing ap­proach. Zero shield­ing would pre­dict the he­lium ion­iza­tion en­ergy to be 54.4 eV, which re­ally seems worse than 13.6 eV when com­pared to the ex­act value of 24.6 eV. On the other hand, zero shield­ing does give a fair ap­prox­i­ma­tion of the ac­tual to­tal en­ergy of the atom; 109 eV in­stead of an ex­act value of 79. Full shield­ing pro­duces a poor value of 27 eV for the to­tal en­ergy; the to­tal en­ergy is pro­por­tional to the square of the ef­fec­tive nu­cleus strength, so a lack of full shield­ing will in­crease the to­tal en­ergy very strongly. But also im­por­tantly, full shield­ing avoids the reader’s dis­trac­tion of hav­ing to rescale the wave func­tions to ac­count for the nonunit nu­clear strength.

If even­tu­ally X-ray spec­tra need to be cov­ered in this book, a de­scrip­tion of hot rel­a­tivis­tic in­ner elec­trons would pre­sum­ably fix any prob­lem well.