J. Chem. Phys. 116, 1684 (2002); h ps://doi.o g/10.1063/1.1429642 116, 1684
© 2002 Ame ican Ins i u e o Physics.
Su ace model and exchange-co ela ion
unc ional e ec s on he desc ip ion o
Ci e as: J. Chem. Phys. 116, 1684 (2002); h ps://doi.o g/10.1063/1.1429642
Submi ed: 27 Augus 2001 . Accep ed: 31 Oc obe 2001 . Published Online: 10 Janua y 2002
J. R. B. Gomes, F. Illas, N. C uz He nández, J. F. Sanz, A. Wande , and N. M. Ha ison
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Su ace model and exchange-co ela ion unc ional e ec s
on he desc ip ion o PdÕ
␣
-Al2O3„0001…
J. R. B. Gomes and F. Illasa)
Depa amen de Quı
´mica Fı
´sica i Cen e de Rece ca en Quı
´mica Teo
` ica, Uni e si a de Ba celona,
C/ Ma ı
´i F anque
`s 1, E-08028 Ba celona, Spain
N. C uz He na
´ndez and J. F. Sanz
Depa amen o de Quı
´mica Fı
´sica, Facul ad de Quı
´mica, Uni e sidad de Se illa E-41012 Se illa, Spain
A. Wande
CLRC Da esbu y Labo a o y, Da esbu y, Wa ing on WA4 4AD, Uni ed Kingdom
N. M. Ha ison
Depa men o Chemis y, Impe ial College o Science, Technology and Medicine, London SW7 2AY,
Uni ed Kingdom
共Recei ed 27 Augus 2001; accep ed 31 Oc obe 2001兲
The in e ac ion o Pd wi h he Al- e mina ed
␣
-Al2O3(0001) su ace has been in es iga ed using an
embedded clus e model and pe iodic-supe cell app oaches. Fu he mo e, se e al ea men s o
elec onic exchange and co ela ion wi hin densi y unc ional 共DF兲 heo y ha e been employed
including gene alized g adien app oxima ion 共GGA兲and hyb id exchange unc ionals. In he
pe iodic calcula ions he in luence o pseudopo en ials and basis se s ha e also been in es iga ed by
compa ing GGA esul s ob ained using all elec on basis se and pseudopo en ial plane-wa e
app oaches. Fo a gi en choice o he exchange-co ela ion unc ional and o a ixed subs a e, he
clus e and slab models p edic nea ly he same s uc u al pa ame e s and adso p ion ene gies. All
s uc u al models ep oduce he gene al end o he in e ac ion o Pd wi h he
␣
-Al2O3(0001)
su ace, which is ha he e is a sligh p e e ence o adso p ion abo e su ace si es si ing di ec ly
abo e oxygen a oms ei he om he second o i h laye . Howe e , signi ican ly la ge di e ences
exis when compa ing di e en DF me hods wi hin a gi en su ace model. The clus e and pe iodic
slab models p edic a la ge adso ba e-induced elaxa ion wi h a simila desc ip ion o he
me al-oxide in e ace p o ided a minimum numbe o su ace laye s is included in he op imiza ion
p ocedu e. © 2002 Ame ican Ins i u e o Physics. 关DOI: 10.1063/1.1429642兴
I. INTRODUCTION
Unde s anding he g ow h o me al clus e s and me al
hin ilms on an oxide subs a e is a key issue in a a ie y o
echnological applica ions. The ini ial s eps o he me al
a om deposi ion on he oxide su ace a e esponsible o he
esul ing mo phology o he me al o e laye . This is o con-
side able in e es in many applica ions anging om me al-
ce amic-based gas senso s o mic oelec onic de ices o
oxide-suppo ed ansi ion me al ca alys s.1Likewise, he a -
ini y o a single me al a om owa ds a gi en adso p ion si e
o an oxide su ace s ongly a ec s he p ope ies o he sup-
po ed ca alys and i s su ace chemis y. Alumina is widely
used as a suppo o ul a- hin me al deposi ion2and also o
me al suppo ed ca alys s.1Mos o en,
␥
-alumina is used as
suppo because o i s acid-base p ope ies. Howe e , in
some o he cases, as in he indus ial oxida ion eac ion o
e hylene o p oduce oxy ane, a i ually ine suppo is nec-
essa y and
␣
-alumina becomes an ob ious choice.3
Recen ly, he adso p ion o palladium on he
␣
-Al2O3
su ace has been s udied by means o seconda y ion mass
spec ome y in s a ic mode, SSIMS, and by he mal p o-
g ammed deso p ion, TPD.4F om hese s udies i is p e-
dic ed ha Pd g ows on
␣
-Al2O3 ollowing ei he he
S anski–K as ano , SK, model, comple ion o a monolaye
plus 3D c ys alli e g ow h, o he h ee-dimensional Volme –
Webe , VW, mechanism. I is sugges ed ha a low co e age
Pd–Al bonds a e o med i s and ha Pd–O bonds appea o
be o med in a subsequen s ep. Du ing he adso p ion s age,
a iny nega i ely cha ged palladium is obse ed bu he Pd
a oms emain essen ially neu al du ing deposi ion. Palla-
dium clus e s in he 1–3 Å hickness ange exhibi a wo k
unc ion ha emains a a cons an alue o 6.5 eV on
␣
-Al2O3. The e y ecen expe imen s o Pang e al.5using
noncon ac a omic o ce mic oscopy, NC-AFM, show ad-
so bed Pd clus e s which a e ⬃30–40 Å in diame e and
⬃2–3 Å in heigh abo e he
␣
-Al2O3(0001) su ace. These
au ho s also sugges ha hei obse a ions a e consis en
wi h ei he SK o VW g ow h modes. Simila indings ha e
been p e iously epo ed o expe imen s ha use aluminum
oxide hin ilms g own on a me al suppo such as NiAl6–8
ins ead o well-cha ac e ized single c ys al co undum su -
aces.
The expe imen al and echnological in e es o he
alumina-suppo ed ca alys s has igge ed a numbe o heo-
a兲Au ho o whom co espondence should be add essed. Elec onic mail:
[email p o ec ed]
JOURNAL OF CHEMICAL PHYSICS VOLUME 116, NUMBER 4 22 JANUARY 2002
16840021-9606/2002/116(4)/1684/8/$19.00 © 2002 Ame ican Ins i u e o Physics
e ical s udies aimed a unde s anding he na u e o he
me al–oxide in e ac ion and p edic ing he ac i e si es o
me al clus e adso p ion. Ve dozzi e al.9 ecen ly epo ed
esul s om a local densi y app oxima ion 共LDA兲s udy o
he adso p ion o Ag and P a oms on he Al- e mina ed
␣
-Al2O3(0001) su ace. These au ho s ound ha a low
co e age, Ag a oms a e p e e en ially adso bed abo e su -
ace aluminum a oms while P is adso bed abo e he ou e -
mos oxygen a oms. Howe e , o a1MLgeome ical co -
e age, he LDA p edic s ha bo h me als ha e a sligh
p e e ence o di ec adso p ion on su ace aluminum a oms.
Zhang and co-wo ke s10 used LDA and he gene alized g a-
dien app oxima ion, GGA, o s udy he he modynamics o
he in e ac ion o Nb also wi h he
␣
-Al2O3(0001) su ace.
Depending on he oxygen pa ial p essu e, hey ound ha
he Nb/
␣
-Al2O3in e ace o ma ion can e e se he o de o
s abili y leading o an O- e mina ed su ace which is no -
mally less s able han he Al- e mina ed su ace. Ne e he-
less, wo ecen heo e ical s udies claim ha e en unde
high oxygen pa ial p essu e he Al- e mina ed su ace is
s able.11,12 One mus no e ha hese esul s con as wi h he
conclusions o se e al wo ks summa ized in he e iew by
Renaud.13 The in e ac ion o Ag wi h he
␣
-Al2O3(0001)
su ace is cu en ly being s udied by Zhuko skii e al.14 us-
ing a pe iodic supe cell Ha ee–Fock app oach including
elec onic co ela ion e ec s by means o a pos e io i co e-
la ion unc ional. Simila ly, he na u e o he in e ac ion be-
ween Pd and he
␣
-Al2O3(0001) has been in es iga ed by
Gomes e al.15 using a clus e model and he cons ained
space o bi al a ia ion me hod. The clus e esul s sugges a
la ge adso ba e-induced elaxa ion ha has been con i med
by he same au ho s h ough pe iodic supe cell GGA calcu-
la ions wi hin a plane wa e basis se . Using he same pe i-
odic supe cell GGA app oach He na
´ndez and Sanz ha e ana-
lyzed he in e ac ion mechanism o Cu, Ag, and Au wi h he
␣
-Al2O3(0001) su ace.16 Lodziana and No sko ha e also
used his la e app oach in ecen esea ch in ol ing di e en
e mina ions and s oichiome ies o he
␣
-Al2O3(0001)
su ace.17 The a ie y o possible su aces 共Al- o
O- e mina ed兲, su aces models 共embedded clus e o pe i-
odic supe cell兲, and exchange-co ela ion app oxima ions
共Ha ee–Fock o a ious Densi y Func ional ea men s兲ne-
cessi a es a sys ema ic in es iga ion o he e ec s o bo h
ma e ial and compu a ional models be o e eliable conclu-
sions can be d awn. The impo ance o es ablishing he in-
luence o hese app oxima ions is also mani es in p e ious
s udies o he in e ac ion o ansi ion me als on he su ace
o simple oxides. The p e ious wo k o Lopez e al.18 on he
in e ac ion o Pd wi h MgO共100兲s ongly sugges s ha in
DFT me hods he in luence o he choice o exchange-
co ela ion unc ional can be e y signi ican . Fo he adso p-
ion o Cu on MgO共100兲 he alidi y o he clus e model has
been demons a ed explici ly. Embedded clus e models o
inc easing size indica e ha , a leas o his simple oxide,
esul s con e ge a he quickly wi h espec o he numbe o
a oms included.19 Clea ly,
␣
-Al2O3(0001) is a much mo e
complica ed subs a e and i is no ob ious ha his sys em
could be p ope ly desc ibed by means o a clus e model.
Es ablishing he alidi y o a local desc ip ion is o g ea
impo ance as i acili a es modeling o he ubiqui ous su -
ace inhomogenei ies and poin de ec s which a e belie ed o
be ac i e si es o me al g ow h.1Fo such p oblems he clus-
e model and a localized desc ip ion a e o en signi ican ly
mo e con enien han la ge scale pe iodic supe cell calcula-
ions. In compa ing a ious models and making con ac wi h
expe imen a comple e ea men o su ace elaxa ion is im-
po an . Fo his he clus e model mus be su icien ly la ge
and pe iodic supe cell models mus use la ge enough supe -
cells and be based on su icien ly hick slabs. In gene al hese
ques ions can only be add essed by sys ema ic s udies o he
a ia ion o pa icula esul s wi h model size. Howe e , one
migh expec ha he in e ac ion o Pd wi h
␣
-Al2O3(0001),
which induces a la ge subs a e elaxa ion,15 will con e ge
mo e slowly wi h espec o model size han Pd on a simple
oxide su ace such as MgO.
In his wo k we p esen a de ailed and sys ema ic s udy
o he in e ac ion o Pd a oms wi h di e en si es o he
elaxed, clean, Al- e mina ed,
␣
-Al2O3(0001) su ace using
a a ie y o s uc u al models. Bo h ini e clus e and pe i-
odic supe cell models ha e been used o ep esen he co e-
sponding in e ace. In addi ion, wo pe iodic supe cell ap-
p oaches ha e been used. In he i s , a slab pe iodic in wo
dimensions bu ini e in he hi d dimension is used, and in
he second app oach such slabs a e also epea ed pe iodically
in he hi d dimension bu a e sepa a ed by a acuum gap
su icien ly la ge o make slab–slab in e ac ions negligible.
This la e app oach is pa icula ly con enien when a plane
wa e basis se is employed and i is e y widely used in solid
s a e physics, especially o desc ibe me als and me allic su -
aces. In all cases, he su ace elaxa ion induced by he
me al deposi ion has been explici ly aken in o accoun . The
a icle is o ganized as ollows: In Sec. II he di e en su ace
models a e desc ibed. In Sec. III we discuss he a ious com-
pu a ional app oaches. In Sec. IV we p esen he whole se o
esul s ob ained wi h he di e en compu a ional echniques
wi hin he clus e model o pe iodic desc ip ion o he
Pd/Al2O3sys em o a ixed subs a e. In Sec. V we desc ibe
he adso ba e-induced su ace elaxa ion, and compa ison o
expe imen al esul s, whe e a ailable, is epo ed in Sec. VI.
Finally, ou conclusions a e p esen ed in Sec. VII.
II. SURFACE MODELS OF THE Al-TERMINATED
␣
-Al2O3„0001…SURFACE
In he p esen wo k, he
␣
-Al2O3su ace was ep e-
sen ed by wo ex eme ideal models ha a e also ep esen a-
i e o wo di e en ways o hinking abou su ace phenom-
ena. In he i s , one ocuses in he local p ope ies such as
he in e ac ion o an adso ba e wi h he su ace a e y low
co e age. He e, he su ace is modeled by a ini e numbe o
a oms ha a e quan um mechanically ea ed. This clus e
model is embedded in an en i onmen which accoun s o
sho and long ange in e ac ions wi h he emainde o he
c ys al. In he second case one imagines a pe ec , de ec -
ee, su ace in ini e and pe iodic in wo dimensions. The
esul ing model is slab pe iodic in wo dimensions and o
limi ed hickness in he hi d. As discussed abo e, he slab
may hen be subjec o acuum bounda y condi ions in he
1685J. Chem. Phys., Vol. 116, No. 4, 22 Janua y 2002 Densi y unc ional calcula ions o Pd/
␣
-Al2O3(0001)
hi d di ec ion o , o compu a ional con enience, epea ed
pe iodically in he hi d di ec ion wi h slabs sepa a ed by
acuum gaps.
The clus e model used o ep esen he alumina su ace
con ains 2395 cen e s ha a e di ided in h ee di e en e-
gions as schema ically illus a ed in Fig. 1. The i s egion
con ains 23 a oms, 8 Al and 15 O, which a e explici ly
ea ed by a p ope quan um mechanical me hod. The second
shell includes 18 o al ion po en ials,20 TIPs, ha a e cus om-
a ily added o a oid spu ious pola iza ion o he ou e clus e
oxygen a oms.18,19,21,22 In his case he TIP consis s simply o
a pseudopo en ial mimicking an Al3⫹ca ion, no basis unc-
ions a e added o he TIP. Finally, he hi d egion con ains
2354 poin cha ges wi h alues o ⫹3
兩
e
兩
and ⫺2
兩
e
兩
o ca -
ions and anions, espec i ely. The a ay o poin cha ges in
he hi d egion p o ides an adequa e ep esen a ion o he
elec os a ic po en ial in he clus e . Su ace elaxa ion was
explici ly in oduced in his model; he e ical sepa a ion
be ween he ou e mos su ace laye s was aken om p e i-
ous wo k on he econs uc ion o he clean and Al-
e mina ed
␣
-Al2O3su ace calcula ed by pe iodic all-
elec on DFT.23–25 Fu he de ails abou he se up o his
clus e model can be ound elsewhe e.15 Fo he abo e-
desc ibed clus e model, i e di e en si es o he elaxed
␣
-Al2O3(0001) su ace ha e been conside ed o Pd adso p-
ion. These si es a e labeled as Al1,O
2,Al
3,Al
4and O5,
Al1deno es a su ace si e di ec ly abo e an Al ca ion o he
ou e mos a omic laye whe eas O2deno es a su ace si e
loca ed di ec ly abo e an oxygen anion in he second laye .
The no a ion o he es o si es ollows he same logic, c .
Fig. 1. Because o he e y la ge inwa ds elaxa ion o he Al
ou e mos laye in he Al- e mina ed o he clean
␣
-Al2O3(0001) su ace, he Al1and O2can be ega ded as
op si es while Al3,Al
4and O5a e hollow si es.
The pe iodic supe cell calcula ions we e all pe o med
o a 1
3ML Pd co e age, i.e., one Pd a om pe (AlO3Al)nuni
cell, 3*nbeing he o al numbe o a omic laye s in he slab.
In he wo-dimensional pe iodic calcula ions a nine-laye
slab, wi h he same cell pa ame e s used in ou p e ious
wo k,25 was used o ep esen he alumina su ace. In he
h ee-dimensional pe iodic calcula ions, he same uni cell is
used in o de o acili a e compa ison o he models. In his
case a acuum wid h o 10 Å was in oduced be ween he
epea ed slabs. The esul ing uni cell is a hombic p ism
belonging o he hexagonal sys em. No ice ha his s uc u al
model is no iden ical o ha used in ou p e ious h ee-
dimensional pe iodic calcula ions and, hence, esul s in he
ables a e sligh ly di e en om hose epo ed in Re . 15.
III. COMPUTATIONAL MODELS
Fo he clus e calcula ions all elec ons o Al and O
a oms in he local egion we e explici ly ea ed by means o
6-31G and 6-31⫹G*s anda d Gaussian basis se s, espec-
i ely. The la e includes a di use and a pola iza ion unc-
ion o dsymme y. Fo he palladium a om, he 28 inne
elec ons we e included in a ela i is ic e ec i e co e po en-
ial, ECP,26 and he co esponding basis se employed o de-
sc ibe he 4s24p64d10 alence elec ons was o double-
␨
quali y. These ECP and basis se s a e usually e e ed o as
LANL2 and LANL2DZ, espec i ely. The TIPs o Al3⫹
we e also desc ibed by means o he Hay and Wad e ec i e
co e po en ials.26 Wi h his clus e model wo di e en se s
o calcula ions we e ca ied ou o in es iga e he
Pd/
␣
-Al2O3in e ace. In he i s se o calcula ions only he
pe pendicula dis ance o he palladium a om o each o hese
adso p ion si es was op imized and in he second se o cal-
cula ions a pa ial geome y op imiza ion in ol ing he sub-
s a e a oms nea o he me al ada om was ca ied ou o Pd
in e ac ing wi h he Al1and O2adso p ion si es. All clus e
calcula ions we e pe o med wi h he GAUSSIAN 98
package.27
In o de o s udy he e ec o he le el o heo y in he
s udy o he Pd/Al2O3in e ace, se e al exchange-
co ela ion unc ionals we e used. The densi y unc ional
me hods used in he clus e calcula ions a e he GGA p o-
posed by Pe dew and Wang28,29 and wo di e en hyb id
schemes ha ollow he ideas o Becke.30 The o me is cho-
sen because i is he de aul app oach in many band s uc u e
compu e codes whe eas he hyb id app oaches a e inc eas-
ingly used in molecula quan um chemis y. The hyb id
me hods used a e B3LYP and B3PW91. These a e e y simi-
la o he BLYP and BPW91 pu e DFT me hods which use
he Becke’s nonlocal exchange unc ional31 and ei he he
co ela ion unc ional gi en by Lee, Yang, and Pa ,32 based
on he o iginal wo k o Colle and Sal e i on he co ela ion
ac o ,33,34 o he co ela ion unc ional p oposed by Pe dew
and Wang.28,29 Howe e , in he B3LYP and B3PW91 hyb id
me hods h ee pa ame e s o he exchange unc ional a e i
o ep oduce expe imen al he mochemical da a. The op i-
mum mixing being ound o ⬃20% Fock exchange in he
FIG. 1. Schema ic ep esen a ion o he Al26O15 , TIPS included, clus e
model used o ep esen he a ious adso p ion si es on he Al2O3(0001)
su ace. La ge sphe es ep esen anions and small ligh sphe es ep esen he
ca ions ha we e ea ed all-elec on du ing he compu a ion p ocedu e.
Al3⫹TIPS a e ep esen ed as small da k sphe es. Fi h laye aluminum
a oms a e no shown because hey a e loca ed exac ly below he i s laye
aluminum a oms.
1686 J. Chem. Phys., Vol. 116, No. 4, 22 Janua y 2002 Gomes
e al.
exchange unc ional. I is impo an o no e ha he B3LYP
hyb id unc ional has ound wide applicabili y in molecula
calcula ion and ha in pa icula i is able o ep oduce he
he mochemis y o molecules con aining ansi ion me al a -
oms al hough no ansi ion me al compounds we e included
in he da a se used in i s cons uc ion.35–37 All adso p ion
ene gies calcula ed by means o clus e models ha e been
co ec ed o possible basis se supe posi ion e o s using he
s anda d coun e poise me hod.
The pe iodic calcula ions we e pe o med by means o
he CRYSTAL p og am,38 o he pu e wo-dimensional pe i-
odic slab calcula ions, and VASP 4.4.3 code39–41 o he h ee-
dimensional pe iodic calcula ions. He e, i is impo an o
poin ou ha hese calcula ions use di e en basis se s and
ea men s o he co e elec ons. In he CRYSTAL code he
Bloch unc ions a e expanded in e ms o Gaussian ype o -
bi als, GTO, ha ha e an iden ical o m o hose widely used
in molecula quan um chemical calcula ions. In he VASP cal-
cula ions he Bloch unc ions a e expanded in a plane wa e
basis wi h a disc e e se o K- ec o s selec ed o ha e kine ic
ene gy less han an ene gy cu o . Bo h basis se s a e aimed
o expand he same Hilbe space and he same esul s mus
be ob ained wi h ei he basis, p o ided i is ex ended enough
and he same exchange-co ela ion unc ional is used. Re-
sul s in he nex sec ion will p o ide s ong e idence ha his
is indeed he case.
In he CRYSTAL calcula ions all elec ons o heAl, O and
Pd a oms a e explici ly conside ed. The GTO basis se s used
in he p esen s udy a e simila bu somewha mo e ex ended
han hose use in he p e ious s udies on he elaxa ion o he
Al- e mina ed su ace.24,25 Hence, using Pople’s a he s an-
da d no a ion, he O and Al basis se s we e o 8-511G and
8-711G quali y, espec i ely. The Pd basis se was a 9-76311-
631G quali y, i.e., 9 s unc ions plus 5 sp and 3 dshells
cons uc ed om 7⫹6⫹3⫹1⫹1 and 6⫹3⫹1 con ac ion
schemes, espec i ely.42 An auxilia y Gaussian basis se con-
aining 20 e en- empe ed s-symme y unc ions was used o
i he exchange co ela ion po en ial while he ene gy unc-
ional was in eg a ed explici ly on an a om cen e ed g id.
Finally, he plane wa e GGA pe iodic calcula ions ha e been
ca ied ou using ul aso pseudopo en ials o ep esen he
co e elec ons.43 The cu o ene gy o he plane wa es was
337 eV and he Monkho s –Pack se o ou kpoin s was
used.
In o de o ha e a clea -cu compa ison be ween clus e
and pe iodic calcula ions, he la e we e ca ied ou o a
elaxed bu ixed subs a e and o a su ace which was al-
lowed o elax in esponse o he p esence o Pd. In he i s
case only he dis ance o Pd pe pendicula o he su ace
abo e each possible, high symme y, ac i e si e has been
conside ed. Adso ba e-induced su ace elaxa ion was s ud-
ied by allowing he e ical elaxa ion o he wo ou e mos
a omic laye s in he CRYSTAL and VASP calcula ions, espec-
i ely. In he la e case, calcula ions we e epea ed by allow-
ing he six ou e mos subs a e laye s o elax. Howe e , no
signi ican di e ences be ween he wo inal op imized s uc-
u es we e ound. This is impo an because in he geome y
op imiza ions wi h he wo- and h ee-dimensional pe iodic
slab models he e is a Pd o e laye abo e each o he wo
Al- e mina ed su aces o he slab. The eason o his is o
exploi he poin g oup symme y in he CRYSTAL calcula-
ions and a he same ime use he same s uc u al model in
he h ee-dimensional pe iodic calcula ions. No ice ha his
implies a symme ic elaxa ion o he wo ou e mos a omic
laye s in each o he wo su aces o he slab. The simila i y
be ween he op imized s uc u es including jus one Pd o e -
laye wi h he six ou e mos a omic laye s wi h no up–down
symme y and hose esul ing om he symme ic elaxa ion
o he wo ou e mos su aces indica es ha he la e does
no in oduce any a i ac in he model. Finally, we no e ha
o a p ope compa ison be ween clus e and pe iodic ap-
p oaches i is necessa y no only o model he same physical
su ace bu also o use he same app oach o densi y unc-
ional heo y. Hence, CRYSTAL and VASP calcula ions ha e
been ca ied ou by means o he GGA model and CRYSTAL
calcula ions we e indeed ca ied ou o he B3LYP and
B3PW91. The la e is exac ly he same o bo h clus e and
pe iodic calcula ions, howe e he LDA pa o he co ela-
ion unc ional used in B3LYP clus e and pe iodic calcula-
ions is sligh ly di e en . The implemen a ion o he LDA
co ela ion in GAUSSIAN uses he unc ional de ined in Eq.
共3兲o he pape by Vosko e al.44 whe eas CRYSTAL 共and
mos o he plane wa e band s uc u e codes兲use he unc-
ional de ined in Eq. 共5兲o Re . 44 共see also he discussion o
his poin in he pape s by Ma in and Illas兲.45,46
IV. Pd ADSORPTION ON A FIXED
␣
-Al2O3„0001…
SURFACE
The i s poin o conside conce ns he in luence o he
su ace model on s uc u al p ope ies and he ene ge ics o
adso p ion. To his end we conside he GGA esul s ob ained
o he h ee di e en s uc u al models o he elaxed Al-
e mina ed
␣
-Al2O3su ace which is kep ixed du ing he
in e ac ion wi h Pd. The use o a ixed subs a e and o he
same exchange co ela ion unc ional isola es he e ec s o
he su ace model. F om he da a epo ed in Table I i is
TABLE I. E ec o he subs a e model used o s udy he in e ac ion o a Pd
a om wi h i e di e en su ace si es on he
␣
-Al2O3(0001), calcula ed a
he GGA le el o heo y. The oxide subs a e is ixed and only he Pd o
su ace dis ance is op imized. The adso p ion ene gies a e calcula ed wi h
espec o he sepa a ed sys ems and include he coun e poise co ec ion.
The 2D and 3D slabs e e o he CRYSTAL and VASP calcula ions, espec-
i ely. The las column collec s he a e age be ween he di e en models
and he s anda d de ia ion.
P ope y
Su ace model
Si e 2D slab Clus e 3D Slab A e age⫾de ia ion
Al12.53 2.49 2.43 2.49⫾0.05
O22.15 2.06 2.06 2.10⫾0.05
d(Pd-Su 兲/Å Al32.06 1.96 1.97 2.00⫾0.05
Al42.06 1.93 2.00 1.97⫾0.08
O52.05 1.96 1.97 1.99⫾0.05
Al10.83 0.70 0.76 0.76⫾0.09
O20.98 0.86 1.02 0.95⫾0.08
⫺Eads /eV Al30.65 0.47 0.66 0.59⫾0.11
Al40.75 0.69 0.78 0.74⫾0.05
O50.91 0.77 0.93 0.87⫾0.09
1687J. Chem. Phys., Vol. 116, No. 4, 22 Janua y 2002 Densi y unc ional calcula ions o Pd/
␣
-Al2O3(0001)

clea ha he ag eemen be ween he clus e and pe iodic
calcula ions is ema kable, especially be ween he clus e and
he plane wa e calcula ions. No e ha hese calcula ions use
ela i is ic pseudopo en ials o desc ibe he co e elec ons.
Hence, he small de ia ion om he esul s o he pu ely
wo-dimensional calcula ions can be asc ibed o he lack o
ea men o ela i is ic e ec s in he a omic co e which a e
neglec ed in hese all-elec on calcula ions. In ac , he mos
impo an ela i is ic e ec is he con ac ion o he alence
shell due o he scala mass- eloci y e m. This is indi ec ly
in oduced in he pseudopo en ial ei he h ough he i o he
pseudo-o bi al o he ela i is ic all-elec on o bi al in he
o malism o Hay and Wad 26 o by a simila p ocedu e
wi hin he ul a-so app oach.47 In o de o e i y ha his is
he o igin o he disc epancy be ween wo-dimensional and
clus e and h ee-dimensional slab calcula ions we ha e e-
pea ed some o he clus e calcula ions bu using he same all
elec on basis se o Pd as in he CRYSTAL slab calcula ions.
The inal dis ances a e always la ge han hose ob ained
om he ela i is ic ECP by 0.01–0.03 Å. Ne e heless, he
h ee di e en models p edic equilib ium dis ances ha a e
in ag eemen wi hin an e o ba o oughly ⫾0.05 Å.
I is possible o claim ha since he adso ba e–su ace
dis ance is a local p ope y he ag eemen be ween di e en
models desc ibed abo e could be an icipa ed om p e ious
expe ience in clus e models o me al, semiconduc o , and
oxide su aces.48 Howe e , he ag eemen be ween he clus-
e and pe iodic calcula ions discussed abo e is no limi ed o
he equilib ium dis ance. The summa y o adso p ion ene -
gies epo ed in Table I shows ha he ag eemen be ween
clus e and pe iodic calcula ions also holds o he ene ge ics
which a e a e y sensi i e and ul ima ely he mos i al es
o he su ace model. Fo he mos a o able si e 共abo e O2兲,
he adhesion ene gy p edic ed by he clus e and he di e en
slab models di e s by 0.13 eV and, in he wo s case, he
di e ence does no exceed 0.17 eV. In absolu e e ms hese
a e qui e small alues al hough hey do ep esen a a ia ion
o some 15% in he o al adso p ion ene gy.
The esul s o his i s pa can be summa ized in a
a he simple way: he use o a clus e o a slab su ace
model leads o alues in he pe pendicula equilib ium dis-
ance o he Pd a om o he co undum su ace ha ag ee o
⫾0.05 Å and adso p ion ene gies ha ag ee o ⫾0.10 eV. In
he o hcoming discussion we will show ha he unce ain y
caused by he choice o one o ano he su ace model is o
he same o de as ha caused by he choice o a gi en
exchange-co ela ion unc ional among hose ha a e be-
lie ed o be o high accu acy. This s a emen is suppo ed by
he esul s summa ized in Table II o a clus e and a wo-
dimensional slab model using GGA and wo di e en hyb id
unc ionals o a oid he di e ences commen ed abo e be-
ween he wo di e en implemen a ions o B3LYP. Resul s
o he plane wa e basis a e no included because a p esen
nonlocal Fock con ibu ion o he exchange in e ac ion can-
no be con enien ly compu ed wi hin his me hodology. Cu -
en ly, plane wa e calcula ions a e limi ed o LDA and GGA
app oaches. Howe e , om quan um chemical applica ions i
is known ha o molecula sys ems he he mochemis y
and ene ge ics o a gi en chemical eac ion a e mo e accu-
a ely desc ibed using hyb id- ype unc ionals.30,35–37 No ice
ha p e ious wo k has clea ly shown ha GGA has a ma ked
endency o o e es ima e he s eng h o he me al–oxide
in e ac ion.18,19 This is he main eason o including hyb id
unc ionals in he p esen sys ema ic s udy. The analysis o
esul s in Table II e eals ha he e ec o he exchange-
co ela ion unc ional is a leas o he same o de as he
e ec o he su ace model. In ac , e en he equilib ium
dis ance ha one migh expec o be less sensi i e o he
e ec o he unc ional chosen e lec s a non-negligible
a ia ion. This e ec o he DF me hod in he dis ance is
accompanied by an e en la ge e ec in he adso p ion en-
e gy. Ne e heless, all me hods p edic he same o de o
s abili y o he di e en ac i e si es al hough one can sa ely
conclude ha he accu acy o he adso p ion ene gy does no
exceed 0.25 eV. This is somewha la ge han he e o in o-
duced by he choice o a su ace model.
V. ADSORBATE-INDUCED RELAXATION EFFECTS
FOR Pd ON
␣
-Al2O3„0001…
Following p e ious wo k,15 possible adso ba e-induced
elaxa ion e ec s o he in e ac ion o Pd abo e he i e
di e en adso p ion si es we e conside ed by allowing some
o subs a e a oms o elax. Howe e , o echnical easons,
somewha di e en op imiza ion p ocedu es we e adop ed
o he di e en su ace models. Fo he clus e model only
he Al1and O2si es which a e su icien ly a away om he
clus e edge we e elaxed. Fo Pd adso p ion abo e hese wo
TABLE II. E ec o he exchange-co ela ion unc ional in he s udy o he
in e ac ion o a Pd a om wi h i e di e en su ace si es on he
␣
-Al2O3(0001), calcula ed using clus e o pu ely wo-dimensional slab
models. The oxide subs a e is ixed and only he Pd o su ace dis ance is
op imized. The adso p ion ene gies a e calcula ed wi h espec o he sepa-
a ed sys ems and include he coun e poise co ec ion.
P ope y Si e GGA B3LYP B3PW91 A e age⫾de ia ion
Clus e Model
Al12.49 2.54 2.51 2.51⫾0.03
O22.06 2.12 2.09 2.09⫾0.03
d(Pd-Su 兲/Å Al31.96 1.96 1.90 1.94⫾0.03
Al41.93 1.93 1.89 1.92⫾0.02
O51.96 2.05 2.00 2.00⫾0.05
Al10.70 0.44 0.48 0.54⫾0.14
O20.86 0.51 0.56 0.64⫾0.19
⫺Eads /eV Al30.47 0.12 0.15 0.25⫾0.19
Al40.69 0.30 0.32 0.44⫾0.22
O50.77 0.42 0.49 0.56⫾0.19
Two-dimensional model
Al12.53 2.58 2.54 2.55⫾0.03
O22.15 2.20 2.16 2.17⫾0.03
d(Pd-Su 兲/Å Al32.06 2.19 2.12 2.12⫾0.07
Al42.06 2.15 2.08 2.10⫾0.05
O52.05 2.14 2.07 2.09⫾0.05
Al10.83 0.54 0.54 0.64⫾0.17
O20.98 0.63 0.63 0.75⫾0.20
⫺Eads /eV Al30.65 0.34 0.33 0.44⫾0.18
Al40.75 0.44 0.44 0.54⫾0.18
O50.91 0.55 0.57 0.68⫾0.20
1688 J. Chem. Phys., Vol. 116, No. 4, 22 Janua y 2002 Gomes
e al.
si es we elax he e ical posi ion o he nea es Al1a om
and he O2a oms unde nea h. This choice is no sel -e iden ,
bu ollows om he esul s o pe iodic calcula ions epo ed
in a p e ious wo k.15 Fo he wo-dimensional slab model we
ollowed a simila app oach, he e ical posi ion o he Al1
and O2laye s has been op imized and, inally, six laye s
we e ully elaxed in he epea ed 3D slabs as commen ed in
Sec. III. In his la e case, he op imiza ion p ocedu e was
epea ed by allowing only he wo ou e mos , Al1and O2,
laye s o elax. In his way he wo pe iodic calcula ions
ep esen exac ly he same physical model. The esul s om
he wo op imiza ions wi h six o wo laye s elaxed a e al-
mos indis inguishable, he la e being he ones included in
he ables o allow a mo e adequa e compa ison. Resul s in-
cluding he elaxa ion o he six ou e mos laye s ha e been
epo ed in a p e ious wo k al hough using a sligh ly di e -
en uni cell.15
Following he s a egy o he p e ious sec ion we di ide
he discussion o esul s in wo pa s. The i s one conce ns
GGA esul s on di e en models and he second pa is de-
o ed o in es iga e he e ec o he exchange-co ela ion
unc ional on a gi en su ace model. F om he esul s e-
po ed in Table III i appea s ha all su ace models p edic a
e y la ge adso ba e-induced su ace elaxa ion. He e, he
impo an pa ame e s a e he displacemen o he Al1laye ,
⌬(Al1
i⫺Al1
), and he dis ance o Pd o he Al1su ace laye .
P ecisely, ⌬(Al1
i⫺Al1
) is he s uc u al pa ame e exhibi ing
he la ges a ia ion. The p esence o Pd p o okes he Al1
laye o elax ou wa ds. This is easily unde s ood because he
adso p ion o Pd es o es pa ially he bulk coo dina ion. The
O2laye does also dis o , bu because o he lack o C3
symme y on he O2and O5si es he e is no a uni o m
elaxa ion o his laye . The o e all elaxa ion o he O2laye
a oms is small o he elaxa ion o he O2laye when he
Al2O3su ace is cu . The Al- e mina ed
␣
-alumina su ace
cu om an in ini e bulk Al2O3c ys al exhibi s a e y la ge
elaxa ion bu only he Al1a omic laye is signi ican ly dis-
placed om i s ini ial posi ion and owa ds he bulk.23–25
Thus, i is no su p ising ha he p esence o any adso bed
species abo e he
␣
-Al2O3su ace a ec s essen ially he op
aluminum laye . The quali a i e ag eemen o he a ious
models on he na u e o he elaxa ion is hus o be expec ed.
An impo an poin eme ging om he esul s on Table
III is ha he pe pendicula dis ance o he su ace p edic ed
by he wo- and h ee-dimensional slabs is he same. We
know, howe e , ha o a ixed subs a e he e we e non-
negligible di e ences be ween he wo su ace models, c .
Table I, ha a ise om neglec ing scala ela i is ic e ec s
on he all elec on wo-dimensional slab. The e o e, he e y
close ag eemen be ween he wo models is somewha o u-
i ous. In ac , inspec ion o Table III e eals ha he di e -
ence be ween he wo models emains, bu now i is almos
en i ely ans e ed o ⌬(Al1
i⫺Al1
). Ano he , impo an e -
ec o su ace-induced elaxa ion is he change in he o de
o s abili y o he di e en si es, especially o Al1, ha now
becomes he leas a o ed si e. Ne e heless, he gene al
ends p edic ed by he h ee su ace models a e he same.
The only signi ican di e ence, apa om ha al eady com-
men ed o ⌬(Al1
i⫺Al1
), is he absolu e alue o he adhe-
sion ene gy which, as in he case o he ixed subs a e, di -
e s by a mos 0.3 eV.
The inal poin o discuss conce ns he e ec o he
exchange-co ela ion unc ional o he op imized su ace
models. The wo-dimensional slab model calcula ions, wi hin
he GGA le el o heo y, p oduces sho e Pd o su ace dis-
ances han hose calcula ed by he hyb id me hod B3LYP, c .
Table IV. The di e ences be ween he wo unc ionals a e
la ge han hose ound o he ixed su ace. Simila esul s
a e ound o he clus e model, bu he e he e ec o he
exchange-co ela ion unc ional is less p onounced, he di -
e ences a e o he o de o hose al eady ound o he ixed
su ace. The appea ance o la ge di e ences in he s uc u al
pa ame e s p edic ed by he GGA and B3LYP pe iodic cal-
cula ions mus be a ibu ed o he op imiza ion p ocedu e
ha is based on nume ical g adien s as opposed o he clus e
calcula ion which makes use o analy ical g adien s. Hence,
he di e ence in equilib ium geome ies is no only due o
he e ec o he exchange-co ela ion unc ional, bu also o
he ela i ely la po en ial ene gy o he mo ion o Pd pe -
pendicula o he su ace. Wi h espec o he adhesion en-
e gy, i is wo h poin ing ou ha he ela i e ene ge ics o
he bonding o Pd on he se e al adso p ion si es on he
co undum su ace is he same o bo h compu a ional ap-
p oaches. The e ec o he exchange-co ela ion unc ional
in he inal adhesion ene gy is again o ⬃0.25 eV, he same
ha was al eady ound o he ixed subs a e wi h a gene al
end o GGA o p edic la ge alues o he adhesion ene gy.
VI. COMPARISON TO AVAILABLE EXPERIMENTAL
DATA FOR Pd ON
␣
-Al2O3„0001…
F om he sys ema ic s udy p esen ed in he p e ious sec-
ions one can sa ely conclude ha o low Pd co e ages o in
he ini ial s eps o Pd deposi ion on he alumina su ace, he
TABLE III. E ec o he subs a e model used o s udy he in e ac ion o a
Pd a om wi h i e di e en su ace si es on he
␣
-Al2O3(0001), calcula ed
a he GGA le el o heo y. Some a omic laye s o he oxide subs a e a e
elaxed. The adso p ion ene gies a e calcula ed wi h espec o he sepa a ed
sys ems and include he coun e poise co ec ion.
Adso p ion si e
Su ace model Al1O2Al3Al4O5
2D Slab
d(Pd-su .兲/Å 2.37 1.82 1.50 1.44 1.76
d(Al1⫺O2)/Å 0.30 0.49 0.53 0.59 0.44
⌬(Al1
i⫺Al1
)/Å 0.12 0.40 0.45 0.50 0.40
⫺Eads /eV 1.13 1.75 1.44 1.66 1.65
Clus e
d(Pd-su .兲/Å 2.44 1.69
d(Al1⫺O2)/Å 0.53 0.61
⌬(Al1
i⫺Al1
)/Å 0.29 0.37
⫺Eads /eV 1.29 1.60
3D slab
d(Pd-su .兲/Å 2.39 1.79 1.44 1.42 1.74
d(Al1⫺O2)/Å 0.34 0.42 0.52 0.54 0.43
⌬(Al1
i⫺Al1
)/Å 0.20 0.29 0.42 0.43 0.28
⫺Eads /eV 0.89 1.39 1.04 1.27 1.28
1689J. Chem. Phys., Vol. 116, No. 4, 22 Janua y 2002 Densi y unc ional calcula ions o Pd/
␣
-Al2O3(0001)
Pd a oms will end o adso b abo e he O2si es. This is in
ull ag eemen wi h he sugges ions o Eale and Gille 49
who, om he esul s o se e al expe imen al echniques—
XPS, LEED, and EELS, p oposed ha o e y low co e -
ages, Pd is weakly bonded o he alumina su ace wi h Pd
a oms si ing nea he posi ions ha would be occupied by O
a oms, i.e., nea ly abo e he O2o O5si es. No ice ha a he
GGA le el and o a slab model which ully elaxes he ou -
e mos six laye s he adhesion ene gy a hese wo si es di -
e s by 0.1 eV only. Consequen ly, a oom empe a u e,
which is indeed he one used in he expe imen s o Eale and
Gille , he Pd a oms a e almos ee o di use om O2 o O5
si es. These au ho s also p edic ha he e is a small cha ge
ans e om his ansi ion me al a om o he Al si es o he
␣
-Al2O3(001) su ace. This is in ull ag eemen wi h he
analysis o he elec onic s uc u e p edic ed by all heo e i-
cal models and su aces models and discussed a leng h in a
p e ious s udy.15
VII. CONCLUSIONS
In he p esen wo k a sys ema ic s udy o he e ec s o a
gi en su ace model choice ha e been analyzed oge he
wi h he in luence o he choice o a gi en densi y unc ional
app oach. F om his s udy wo main conclusions eme ge ha
can be judged as posi i e and nega i e, espec i ely. The
posi i e poin is ha all su ace models conside ed, clus e o
pe iodic, lead o he same conclusions. Mo eo e , hese con-
clusions do no depend on he choice o he exchange-
co ela ion chosen o sol e he Kohn–Sham equa ions. The
nega i e conclusion is ha , om a quan i a i e poin o iew,
he accu acy eached by he p esen s a e-o - he-a densi y
unc ional me hods in p edic ing s uc u al geome ies and
adhesion ene gies o me als on oxide su aces is a om ha
eached in molecula quan um chemical calcula ions ei he
based on he use o co ela ed wa e unc ions o o hyb id
DF app oaches.
ACKNOWLEDGMENTS
This esea ch has been suppo ed by he Spanish DGI-
CYT G an Nos. PB98-1216-C02-01 and PB98-1125 and, in
pa , by Gene ali a de Ca alunya G an No. 1999SGR-
00040 and he Jun a de Andalucı
´a, G an No. FQM132.
J.R.B.G. hanks he Fundac¸a
˜
o pa a a Cie
ˆnciaeaTecnologia
o a pos -doc g an 共BPD/22098/99兲, N.C.H. hanks he
Jun a de Andalucı
´a o a p edoc o al g an , and N.M.H. ac-
knowledges inancial suppo o he Eu opean Communi y
o a s ay in Ba celona h ough he Imp o ing Human Po en-
ial Con ac No. HPRI-CT-1999-00071 held by he Cen e
de Supe compu acio
´de Ca alunya, CESCA, and Cen e Eu-
opeu de Pa al.lelisme de Ba celona, CEPBA. Pa o he
compu e ime was p o ided by he CESCA and CEPBA,
h ough g an s om he Fundacio
´Ca alana de la Rece ca
and om he Uni e si a de Ba celona. VASP pe iodic cal-
cula ions we e ca ied ou a he Cen o de In o ma
´ ica Ci-
en ı
´ ica de Andalucı
´a, CICA, Se illa.
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TABLE IV. E ec o he exchange-co ela ion unc ional in he s udy o he
in e ac ion o a Pd a om wi h i e di e en su ace si es on he
␣
-Al2O3(0001), calcula ed using clus e o wo-dimensional models. Some
a omic laye s o he subs a e a e elaxed. The adso p ion ene gies a e cal-
cula ed wi h espec o he sepa a ed sys ems and include he coun e poise
co ec ion.
Me hod
Adso p ion si e
Model Al1O2Al3Al4O5
GGA
d(Pd-su .兲/Å 2.37 1.82 1.50 1.44 1.76
d(Al1⫺O2)/Å 0.30 0.49 0.53 0.59 0.44
⌬(Al1
i⫺Al1
)/Å 0.12 0.40 0.45 0.50 0.40
⫺Eads /eV 1.13 1.75 1.44 1.66 1.65
2D Slab B3LYP
d(Pd-su .兲/Å 2.49 1.88 1.77 1.62 1.84
d(Al1⫺O2)/Å 0.42 0.45 0.45 0.50 0.44
⌬(Al1
i⫺Al1
)/Å 0.31 0.35 0.36 0.40 0.30
⫺Eads /eV 0.89 1.21 0.94 1.12 1.18
GGA
d(Pd-su .兲/Å 2.44 1.69
d(Al1⫺O2)/Å 0.53 0.61
⌬(Al1
i⫺Al1
)/Å 0.29 0.37
⫺Eads /eV 1.29 1.60
Clus e B3LYP
d(Pd-su .兲/Å 2.47 1.74
d(Al1⫺O2)/Å 0.51 0.61
⌬(Al1
i⫺Al1
)/Å 0.25 0.36
⫺Eads /eV) 0.97 1.17
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