Gaussian Basis Sets for the Calulation of
Some States of the Lanthanides
P.R. Librelonand F.E. Jorge
Departamento deFsia
UniversidadeFederaldo EspiritoSanto
29060-900Vitoria, ES,Brasil
Reeivedon19Otober,2000
HighlyaurateadaptedGaussianbasissetsareusedtostudythegroundandsomeexitedstates
fortheneutralatomsandalsosomeorresponding6sand4fionizedstatesfromCsthroughLu. Our
totalenergiesareomparedwiththosealulatedwithanumerialHartree-Fokmethod.Themean
error ofourenergyresults isequalto0.74 mhartree. Ouralulations reproduetheexperimental
trendtoinreaseortodereasethe6sand4fionizationpotentialswithinreasingatominumber,
althoughtheyarerespetivelysmallerandlargerthantheexperimentalvalues.
I Introdution
In this last deade lanthanide hemistry and physis
have experiened tremendous growth, for example in
the eld of atalysts [1℄ and high temperature
super-ondutors [2℄. Thus, it would be highly desirable to
eluidate the eletroni struture of lanthanide atoms
at least in theHartree-Fok (HF) approximation. For
these atoms, numerial HF (NHF) alulations [3-5℄
were performedmainlyonthegroundstates.
Inthis work,theadapted Gaussianbasissets
[AG-BSs-onedierentsetofGaussian-typefuntion(GTF)
exponentsfor eah atomi speies for the atoms from
Cs (Z=55) through Lu (Z=71) [6℄ are initially
aug-mented until saturation is ahieved for eah
symme-tryof eahatomandthen, usingthegenerator
oordi-nateHF(GCHF) [7℄method, theyarereoptimizedfor
eah atomi speies. Next the energies for the atoms
Cs-Lu and theirpositiveions arealulated and
om-paredwiththoseobtainedwithaNHF[5℄method. The
ionizationpotentials(IPs)arealsoomputedand
om-paredwiththeorrespondingexperimentalvalues[8,9℄.
II The method
An approah to selet the basis sets arises from the
GCHF method [7℄. In the GCHF method the
one-eletronfuntionsareintegraltransforms,i.e.,
1 (1)= Z i (1;)f i
()d i=1;:::;n; (1)
where
i
are the generator funtions (GTFs in our
ase), f
i
are the unknown weight funtions, and is
the generatoroordinate. Theappliation ofthe
vari-ational priniple to alulate the energy expetation
value built with suh one-eletron funtions leads to
the Gritn-Hill-Wheeler-HF (GHWHF) equations [7℄.
The GHWHF equations are integrated using a
proe-dureknownasintegraldisretization(ID)[10℄.TheID
tehniqueis implemented througha relabelling of the
generatoroordinatespae,i.e.,
=ln
A
; A>1 (2)
where A is a numerially determined saling fator.
In the new generator oordinate spae , an equally
spaedN-pointmeshf
i
gisseleted,andthe
integra-tion range is haraterized by a starting point
min ,
an inrement , and N (number of disretization
points). The highest value (
max
) for the generator
oordinateisgivenby
max =
min
+(N 1): (2)
The hoie of thedisretization pointsdetermines the
exponentsoftheGTFs.
In the last four years, the GCHF [7℄ method was
suessfully tested in the generation of basis sets for
atomiandmoleularsystems[11-16℄.
III III Results and disussion
By employing the GCHF method we have generated
AGBSs for the atomi speies presented in Table 1.
Throughout the alulations we have used the
sal-ing fator A (see Eq. (2)) equal to 6.0, and for all
atomi speies we have sought the best disretization
parameter(
min
and)valuesforeahs,p,dandf
symmetries. All alulations were arriedout using a
modiedversionof theATOMSCF program [17℄,and
for eah atomi speies theoptimization proessis
re-peateduntilthetotalenergyvalueisstabilizedwithin
ten signiant gures. The resulting wave funtions
are available by request through the e-mail address
jorgee.ufes.br.
Table I shows the ground and some exited state
HF total energies (in hartrees)for the neutral atoms
and someationsfrom Cs(Z=55)throughLu(Z=71)
omputedwithourAGBSsandwithaNHF[5℄method.
Ourbasissetsizesarepresentedintheseventholumn.
WereallthattheAGBSsaregeneratedfromthebasis
sets of Ref. [6℄. First, weaugmentedthese basissets
until saturate eah symmetry of eah atom, and
se-ond, usingtheGCHF [7℄method, wereoptimizedeah
AGBSofeahatomispeiesstudiedhere. FromTable
I,weanseethatourtotalenergies,forallatomispeie
ofinterest,areingoodagreementwiththe
orrespond-ing NHF [5℄ valuesand that ourenergy errorsdo not
exeed1.72mhartree. Hereitisimportanttosaythat
thevetorouplingoeÆientsusedinthealulations
of the open-shell ongurations havebeen taken from
the tabulation by Malli and Olive [18℄. These tables
show the vetor oupling oeÆients for the eletron
ongurationss,p n ,sp n ,d n ,sd n ,p m d n ,sp m d n andf n .
TheHFtotalenergiesofthegroundstatesoftheatoms
CeandGdandofsomestatesoftheationsPr + ,Nd + , Pm + , Sm + , Eu + , Tb + ,Dy + ,Ho + , Er + and Tm + are
notalulatedhere,beausetheeletronongurations
of these atomi speies have 5d and 4f and 6sand 4f
openshells,respetively. Theeletronongurationof
Lu +
( 3
H)has5dand4fopenshells,andthusthewave
funtion forthisationisnotgenerated here.
Table II ontainstheIPs(in eV)omputed by
us-ing the Koopmans theorem (" is our orbital energy),
thetotalenergydiereneE =E(X +
) E(X)[X is
the atomi symboland E(X +
)and E(X) are our
to-tal energiesrespetivelyfortheationandtheneutral
atompresentedinTableI℄,andtheexperimentalvalues
(E
expt: )[8,9℄.
From Table II we an see that the dierenes
be-tween our IP's alulated through " (see the fourth
olumn) and through E (see the fth olumn) are
oremworksforthe6sionization. Besidesthis,forthe6s
orbitals,ourIPsalulatedwith thesetwoapproahes
areverysimilartothoseomputedwithaNHFmethod
(seethe sixth olumn). For all lanthanide atoms
pre-sentedinTableII,andfromourresultsfor ",wean
see that the6sorbitals are morediuse thanthe
or-responding 4f orbitals, that is, the 6s IPs are smaller
thanthe4flPs. Fortheseatoms, itisknownthat the
meanvaluesofrforthe6sorbitalsarelargerthanthose
forthe4forbitals,that is,the6seletronsarefarfrom
the nuleus than the 4f eletrons. From La through
Eu,boththealulated " (4.4-4.6eV)and the
ex-perimental (5.4-5.8 eV) 6s IPs are almost onstant.
AfterTb, the "and experimental[8,9℄IPsgradually
inrease. Theexperimental IPsare alwayslargerthan
the " values. Toorretthis disrepany,itis
nees-saryto inlude in the alulationseletronorrelation
eets and relativisti orretions, but this is outside
the sope of this work. Here, it is important to say
that Jorge et al. have developed the generator
oor-dinateDira-Fok(GCDF) [19,20℄method for
losed-shellatoms andasegmentedontrationmethodology
forrelativistiGaussianbasissets[21,22℄. FromTable
II,onlyYb (Z=70)haslosed-shell,thus,fortheother
atomspresentedinthisTable,weannotusetheGCDF
methodtoalulate therelativistiIPs.
Besidesthis,TableIIshowsthatforthelanthanides,
the4fIPsalulatedbyusthrough "andthroughE
giveverydierent results. The ionization of the
ele-trons in theoutermost 6s shell auses small
reorgani-zationonthe wholeeletrondistribution, whereasthe
inner4feletronionizationauseslargerreorganization
eetsbeauseoftheappearaneofaholeintheinner
shell. Thus, for these atoms, itis notappropriated to
usetheKoopmanstheoremtoalulatethe4feletron
ionization. Forall lanthanide atoms, the4fIPs
alu-latedbyus(E)areingoodagreementwiththe
orre-spondingvaluesobtainedwithaNHF[5℄method, and
althoughthealulated4fIPsare1-3eVgreaterthan
theorresponding experimentalvalues [8,9℄,NHF and
our E alulations desribe the experimental trend
well.
IV Conlusions
In this work we have generated AGBSs for the 45
atomispeiespresentedinTableIwiththeGCHF[7℄
method. Thelargestdierenebetweenthetotal
ener-giesalulatedbyusandbyaNHF[5℄methodisequal
to 1.72 mhartree for Lu. Although our 6sand 4f IPs
ulations reprodue theexperimental trendson the6s
and4feletronionizationswell. Forthe4fIPs,ourE
resultsarebetterthanthoseomputedwiththe
Koop-mans theorem, whereasfor 6sIPsthetwoapproahes
givesimilarresults.
Aknowledgements
WeaknowledgethenanialsupportofCNPqand
Dr. L.T. Peixotoforvaluabledisussions.
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