The H molecular ion: Low-lying states

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• 作者：Horacio Olivares-Piló ; np>ap>p> ; p>p>horop@nucleares.unam.mx" class="auth_mail" title="E-mail the corresponding authorp> ; Alexander V. Turbinerp>bp>p> ; p>p>turbiner@nucleares.unam.mx" class="auth_mail" title="E-mail the corresponding authorp>
• 关键词：H2+ ion ; Approximate solutions ; Electronic structure ; Radiative transitions ; Potential curves
• 刊名：Annals of Physics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：373
• 期：Complete
• 页码：581-608
• 全文大小：816 K

文摘

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in Turbiner and Olivares-Pilon (2011). The ten low-lying eigenstates of H₂p>+p> of the quantum numbers pan id="mmlsi179" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si179.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=484e9ca8f3e909443765f8fbd664d47e" title="Click to view the MathML source">(n,m,Λ,&plusmn;)pan>pan class="mathContainer hidden">pan class="mathCode">$(n,m,\Lambda ,\&<font\; color="red">p</font>lusmn;)$pan>pan>pan>  with pan id="mmlsi234" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si234.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=9354eb9ac1243d2f9ec29ba1a37c12f4" title="Click to view the MathML source">n=m=0pan>pan class="mathContainer hidden">pan class="mathCode">$n=m=0$pan>pan>pan> at pan id="mmlsi235" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si235.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=3a1504a4af0b39752d4735891565ee95" title="Click to view the MathML source">Λ=0,1,2pan>pan class="mathContainer hidden">pan class="mathCode">$\Lambda =0,1,2$pan>pan>pan>, with pan id="mmlsi236" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si236.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=aba10e82fcba51a56eab32ee57424d28" title="Click to view the MathML source">n=1pan>pan class="mathContainer hidden">pan class="mathCode">$n=1$pan>pan>pan>, pan id="mmlsi237" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si237.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=c0a82810c20529bf54969cd65ebd0283" title="Click to view the MathML source">m=0pan>pan class="mathContainer hidden">pan class="mathCode">$m=0$pan>pan>pan> and pan id="mmlsi238" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si238.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=bcacb393d48a814f38ddfa6ea5d7926e" title="Click to view the MathML source">n=0pan>pan class="mathContainer hidden">pan class="mathCode">$n=0$pan>pan>pan>, pan id="mmlsi239" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si239.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=80aad7763c0ec78d1dad7af3d15e2b25" title="Click to view the MathML source">m=1pan>pan class="mathContainer hidden">pan class="mathCode">$m=1$pan>pan>pan> at pan id="mmlsi240" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si240.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=4f3344a7dc9837274d7ca48e08aeff41" title="Click to view the MathML source">Λ=0pan>pan class="mathContainer hidden">pan class="mathCode">$\Lambda =0$pan>pan>pan> of both parities are explored for all interproton distances pan id="mmlsi2" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si2.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=f20a41326ffd0e39c6d05cfd7f320bc8" title="Click to view the MathML source">Rpan>pan class="mathContainer hidden">pan class="mathCode">$R$pan>pan>pan>. For all these states this approximation provides the relative accuracy pan id="mmlsi242" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si242.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=1d5fee2dc83b4684c8b39ff9d1651dba" title="Click to view the MathML source">≲10p>−5p>pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> (not less than 5 s.d.) locally, for any real coordinate pan id="mmlsi243" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si243.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=99ca386002276eea7a021107c9336291" title="Click to view the MathML source">xpan>pan class="mathContainer hidden">pan class="mathCode">$x$pan>pan>pan> in eigenfunctions, when for total energy pan id="mmlsi84" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si84.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=1e4ef06bb3d4237d5c832a2c433b8e70" title="Click to view the MathML source">E(R)pan>pan class="mathContainer hidden">pan class="mathCode">$E\left(R\right)$pan>pan>pan> it gives 10–11 s.d. for pan id="mmlsi245" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si245.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=33583cbf5cd29187629756e0b75cdffa" title="Click to view the MathML source">R∈[0,50]pan>pan class="mathContainer hidden">pan class="mathCode">$R\in [0,50]$pan>pan>pan> a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si5.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=b19a0ec54066f0223cc78c490c2283ba" title="Click to view the MathML source">E1pan>pan class="mathContainer hidden">pan class="mathCode">$E1$pan>pan>pan> is calculated with not less than 6 s.d. A dramatic dip in the pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si5.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=b19a0ec54066f0223cc78c490c2283ba" title="Click to view the MathML source">E1pan>pan class="mathContainer hidden">pan class="mathCode">$E1$pan>pan>pan> oscillator strength pan id="mmlsi162" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si162.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=6577e77710f30c00a3977b28b7dfdede" title="Click to view the MathML source">f_{1sσg−3pσu}pan>pan class="mathContainer hidden">pan class="mathCode">$f_{1s{\sigma }_g-3\mathrm{<font\; color="red">p</font>}{\sigma }_u}$pan>pan>pan> at pan id="mmlsi249" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si249.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=f685db7c6fa4b95be2f99412bed2a821" title="Click to view the MathML source">R∼R_eqpan>pan class="mathContainer hidden">pan class="mathCode">$R\sim R_{eq}$pan>pan>pan> is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6 s.d. in oscillator strength. For two lowest states pan id="mmlsi250" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si250.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=7717e6b94bd9904319af75f3fbee2e49" title="Click to view the MathML source">(0,0,0,&plusmn;)pan>pan class="mathContainer hidden">pan class="mathCode">$(0,0,0,\&<font\; color="red">p</font>lusmn;)$pan>pan>pan> (or, equivalently, pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si3.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=38b7c7b110e3a399e5fa1d5e2111212b" title="Click to view the MathML source">1sσ_gpan>pan class="mathContainer hidden">pan class="mathCode">$1s{\sigma }_g$pan>pan>pan> and pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si4.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=716b8ed5fceb1095ee576151fe7e0852" title="Click to view the MathML source">2pσ_upan>pan class="mathContainer hidden">pan class="mathCode">$2\mathrm{<font\; color="red">p</font>}{\sigma }_u$pan>pan>pan> states) the potential curves are checked and confirmed in the Lagrange mesh method within 12 s.d. Based on them the Energy Gap between pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si3.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=38b7c7b110e3a399e5fa1d5e2111212b" title="Click to view the MathML source">1sσ_gpan>pan class="mathContainer hidden">pan class="mathCode">$1s{\sigma }_g$pan>pan>pan> and pan id="mmlsi4" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si4.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=716b8ed5fceb1095ee576151fe7e0852" title="Click to view the MathML source">2pσ_upan>pan class="mathContainer hidden">pan class="mathCode">$2\mathrm{<font\; color="red">p</font>}{\sigma }_u$pan>pan>pan> potential curves is approximated with modified Pade pan id="mmlsi255" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si255.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=c05e14f81345b2c40a8b6175add24bde" title="Click to view the MathML source">Rep>−Rp>[Pade(8/7)](R)pan>pan class="mathContainer hidden">pan class="mathCode">pan>pan>pan> with not less than 4–5 figures at pan id="mmlsi256" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si256.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=ab7772a93ae3ae8d9b96401921de9217" title="Click to view the MathML source">R∈[0,40]pan>pan class="mathContainer hidden">pan class="mathCode">$R\in [0,40]$pan>pan>pan> a.u. Sum of potential curves pan id="mmlsi257" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si257.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=e0335b90e5562de43e815113d65ff29e" title="Click to view the MathML source">E_1sσg+E_2pσupan>pan class="mathContainer hidden">pan class="mathCode">$E_{1s{\sigma }_g}+E_{2\mathrm{<font\; color="red">p</font>}{\sigma }_u}$pan>pan>pan> is approximated by Pade pan id="mmlsi258" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si258.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=370200606da4cfec022bcf6fc96bc9b2" title="Click to view the MathML source">1/R[Pade(5/8)](R)pan>pan class="mathContainer hidden">pan class="mathCode">$1/R\left[Pade(5/8)\right]\left(R\right)$pan>pan>pan> in pan id="mmlsi256" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0003491616301191&_mathId=si256.gif&_user=111111111&_pii=S0003491616301191&_rdoc=1&_issn=00034916&md5=ab7772a93ae3ae8d9b96401921de9217" title="Click to view the MathML source">R∈[0,40]pan>pan class="mathContainer hidden">pan class="mathCode">$R\in [0,40]$pan>pan>pan> a.u. with not less than 3–4 figures.