Theory of excited states of finite systems in Coulomb external potential

Theo y o Exci ed S a es o Fini e Sys ems in Coulomb Ex e nal Po en ial This a icle has been downloaded om IOPscience. Please sc oll down o see he ull ex a icle. 2013 J. Phys.: Con . Se . 410 012155 (h p://iopscience.iop.o g/1742-6596/410/1/012155) Download de ails: IP Add ess: 193.6.181.137 The a icle was downloaded on 12/02/2013 a 11:56 Please no e ha e ms and condi ions apply. View he able o con en s o his issue, o go o he jou nal homepage o mo e Home Sea ch Collec ions Jou nals Abou Con ac us My IOPscience Theo y o Exci ed S a es o Fini e Sys ems in Coulomb Ex e nal Po en ial ´ A. Nagy Depa men o Theo e ical Physics, Uni e si y o Deb ecen, Deb ecen, Hunga y E-mail: [email p o ec ed] Abs ac . Recen ly a heo y o exci ed s a es o Coulomb sys ems (P. W. Aye s, M. Le y and ´ A. Nagy, Phys. Re . A 85, 042518 (2012)) has been been pu o wa d. The alk will p esen and de elop his new heo y. I will be shown ha he Coulomb densi y de e mines he Hamil onian and he deg ee o exci a ion. The de ini ion o a single, uni e sal unc ional which is enough o desc ibe Coulomb sys ems is p esen ed. The a ailabili y o he heo y is discussed. 1. In oduc ion Nowadays exci a ion ene gies a e equen ly calcula ed wi h ime-dependen densi y unc ional heo y(See, e. g. [1, 2, 3]), hough he e a e ime-independen heo ies, oo. The i s igo ious heo y was he subspace heo y o Theophilou [4] which was enla ged o an ensemble heo y by G oss e al [5]. These app oaches ha e, howe e , he disad an age ha hei applica ion o highly exci ed s a es is a he complica ed and hey canno be used in case o co e exci a ions. The e also exis heo ies o a single exci ed s a e. The Le y-Nagy heo y [6], o example, is a bi unc ional heo y, ha is, he exci ed-s a e ene gy is a unc ional no only o he exci ed-s a e densi y, bu also o he ex e nal po en ial (o he g ound-s a e densi y). The Nagy heo y o a single exci ed s a e [7, 8], on he o he hand, is based on Ka o’s heo em and alid o Coulomb ex e nal po en ial. Recen ly, he la e has been gene alized [9]. The alk will p esen and de elop his new heo y. 2. Cusp condi ion o exci ed s a es The g ound-s a e elec on densi y is su icien in p inciple o de e mine all molecula p ope ies. This can be simply unde s ood ollowing B igh Wilson’s [10] a gumen : A well-known heo em o quan um mechanics, Ka o’s heo em [11] leads o [12] Zβ=−1 2n( ) ∂¯n( ) ∂  =Rβ ,(1) whe e he pa ial de i a i es a e aken a he nuclei βand ¯n( ) is he angula a e age o he densi y. So he cusps o he densi y ell us whe e he nuclei a e (Rβ) and wha he a omic numbe s Zβa e. On he o he hand, he in eg al o he densi y gi es us he numbe o elec ons: N=Zn( )d .(2) IC-MSQUARE 2012: In e na ional Con e ence on Ma hema ical Modelling in Physical Sciences IOP Publishing Jou nal o Physics: Con e ence Se ies 410 (2013) 012155 doi:10.1088/1742-6596/410/1/012155 Published unde licence by IOP Publishing L d 1 Ka o’s heo em is alid no only o he g ound s a e bu also o he exci ed s a es. So, i he densi y nko he k- h elec on s a es is known he Hamil onian ˆ His also in p inciple known and i s eigen alue p oblem ˆ HΨi= ( ˆ T+ˆ V+ˆ Vee)Ψi=EiΨi(3) can be sol ed, whe e ˆ T= N X j=1 (−1 2∇2 j),ˆ Vee = N−1 X i=1 N X j=i+1 1 | i− j|,ˆ V= N X i=1 M X β=1 −ZJ/| i−Rβ|,(4) a e he kine ic ene gy, he elec on-elec on and he elec on-nucleon ope a o s, espec i ely. The e a e ce ain special cases, howe e , whe e Eq. (1) does no de e mine he a omic numbe . The simples example is he 2po bi al o he hyd ogen a om. In his case he sphe ical a e age o he de i a i e o he densi y is ze o and he alue o he densi y n2p( ) = c 2e−Z (5) is also ze o a he nucleus. I means ha in his case Ka o’s heo em does no gi e he a omic numbe . Simila cases occu in hose highly exci ed a oms, ions o molecules, in which he e a e no s-elec ons. The e exis , howe e , a mo e gene al cusp exp ession o he densi y, ha can be applied e en in hese special cases. The co esponding ela ions o he wa e unc ions we e de i ed by Pack and B own [13]. (Fu he wo ks conce ning he cusp o he densi y a e Re . [14, 15, 16].) The cusp ela ions o highly exci ed s a es o he densi y we e also de i ed [17, 18]: ∂¯ηl( ) ∂  =Rβ =−2Z l+ 1ηl(Rβ),(6) whe e ηl( ) = n( ) 2l(7) and lis he smalles in ege o which ηlis no ze o a he nucleus. In he example men ioned abo e Eq. (7) leads o η2p( ) = n2p 2=ce−Z (8) and he new cusp ela ion has he o m: −2Zη2p(0) = 2η0 2p(0) .(9) So we can again eadily ob ain he a omic numbe om he elec on densi y. (Fu he use ul cusp ela ions can be ound in [19].) The a gumen abo e p o es Theo em 1: The densi y o a Coulomb sys em de e mines he ex e nal po en ial . 3. Theo y o exci ed s a es in Coulomb ex e nal po en ial Theo em 2: The exci ed s a e densi y nkde e mines no only he Hamil onian bu he eigen alue Ek, oo. To p o e his heo em, suppose ha we ha e wo wa e unc ion Ψkand Ψ0 k o which he densi y IC-MSQUARE 2012: In e na ional Con e ence on Ma hema ical Modelling in Physical Sciences IOP Publishing Jou nal o Physics: Con e ence Se ies 410 (2013) 012155 doi:10.1088/1742-6596/410/1/012155 2 is he same: nk=n0 k. I is well known, ha he asymp o ic beha iou o he densi y is go e ned by he ioniza ion ene gy I[20, 21]. Consequen ly, Ik=Ek−E(N−1) 1,(10) whe e E(N−1) 1is he g ound-s a e ene gy o he sys em a e an elec on is emo ed. Simila ly, we can w i e ha I0 k=E0 k−E(N−1) 1. As he densi y is he same nk=n0 k, so is he ioniza ion ene gy Ik=I0 k. The e o e Ek=E0 k. So, o a non-degene a e s a e he densi y uniquely de e mines bo h he ene gy and he wa e unc ion. Eq. (10) is alid only i ε=E(N−1) 1−Ek>0,(11) Fo highly exci ed s a es εmigh be nega i e. A mo e gene al exp ession alid o any exci ed s a e can be ound in [21]: Wi h he nomencla u e o Wigne [22] he eigen unc ions o he Hamil onian ˆ HCoul can be classi ied by he i educible ep esen a ions D(l), l = 0,1, ..., [N/2]. I he eigen unc ion belongs o he i educible ep esen a ion D(l)o he symme ic g oup SN, i can be ionized in o N−1-elec on s a es which ans o m acco ding o he i educible ep esen a ions D(l)o D(l−1) o he symme ic g oup SN−1. Deno ing by E(N−1) 1,l he lowes eigen alue o he Hamil onian HN−1and de ining ˜ E(N−1) 1=min E(N−1) 1,l , E(N−1) 1,l−1,(12) he decay o he wa e unc ion is de e mined by εgen =˜ E(N−1) 1−Ek.(13) The p esen heo y is alid p o ided his quan i y is posi i e, ha is, o a bound s a e. De ine now he unc ional F. F[n] = E−Zn( ) ( )d (14) is alid o any s a iona y s a e o any Coulomb sys em. As he e is no known way o decide (wi hou cons uc ing he ex e nal po en ial and sol ing he Sch ¨odinge equa ion), whe he a gi en densi y is Coulombic i would be be e o de ine a unc ional F o all elec on densi ies. Fi s , Fis de ined as a unc ional o a ial densi y nand a Coulomb densi y nCoul co esponding o he he k h s a e o some Coulomb Hamil onian F[n, nCoul] = min Ψ→n {hΨ|ΨCoul j[nCoul]i=0}k−1 j=1 hΨ|ˆ T+ˆ Vee|Ψi.(15) The minimiza ion is done wi h he cons ain s ha each Ψ yield n( ) and is simul aneously o ogonal o he i s k−1 s a es o he Coulomb sys em speci ied by nCoul( ). Then a uni e sal unc ional F[n] can be cons uc ed as ollows. Assume ha he e exis s a unique Coulomb densi y ha is closes o he (non-Coulomb) densi y n. (The bes measu e o “closes ” is no de ailed he e.) I he e a e se e al Coulomb densi ies om he same ’dis ance’ om n, he one yielding o he smalles F(Eq. (15)) is selec ed: FCoul [n] = min nCoul F[n, nCoul]; ||nCoul −n|| ≤ . (16) IC-MSQUARE 2012: In e na ional Con e ence on Ma hema ical Modelling in Physical Sciences IOP Publishing Jou nal o Physics: Con e ence Se ies 410 (2013) 012155 doi:10.1088/1742-6596/410/1/012155 3 is supposed o be la ge enough o ensu e he exis ence o a leas one s a iona y s a e Coulomb densi y in he dis ance smalle han . Wi h min deno ing he smalles possible alue o , FCoul[n] = FCoul min [n].(17) This p ocedu e should be done in he “ icini y” o all Coulomb densi ies nk( ). In his way he unc ional FCoul is de ined o any densi y. Supposing ha his unc ional is unc ionally di e en iable we a e led o he he Eule equa ion Coul([n], ) = −δFCoul[n] δn (18) up o a cons an . The heo y p esen ed he e is o non-dege a e s a es. The gene aliza ion o dege a e s a es will be published elsewhe e. An ex ension o he Kohn-Sham scheme is unde p og ess. Acknowledgmen s The wo k is also suppo ed by he TAMOP 4.2.1/B-09/1/KONV-2010-0007 and he TAMOP 4.2.2/B-10/1-2010-0024 p ojec s. 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