Theo y o Exci ed S a es o Fini e Sys ems in Coulomb Ex e nal Po en ial
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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)
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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
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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)
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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. The p ojec is co- inanced by he Eu opean Union and he
Eu opean Social Fund. G an OTKA No. K 100590 is also g a e ully acknowledged.
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