J. Chem. Phys. 113, 6082 (2000); h ps://doi.o g/10.1063/1.1290003 113, 6082
© 2000 Ame ican Ins i u e o Physics.
Ab ini io g oup model po en ials including
elec on co ela ion e ec s
Ci e as: J. Chem. Phys. 113, 6082 (2000); h ps://doi.o g/10.1063/1.1290003
Submi ed: 03 Feb ua y 2000 . Accep ed: 07 July 2000 . Published Online: 02 Oc obe 2000
No ge C uz He nández, and Ja ie Fdez. Sanz
Ab ini io
g oup model po en ials including elec on co ela ion e ec s
No ge C uz He na
´ndez and Ja ie Fdez. Sanza)
Depa amen o de Quı
´mica Fı
´sica, Facul ad de Quı
´mica, E-41012, Se illa, Spain
共Recei ed 3 Feb ua y 2000; accep ed 7 July 2000兲
A me hod o de e mina ion o ab ini io g oup model po en ials, wi h he in en ion o desc ibing he
e ec s o a whole molecule o a chemical g oup wi hin a densi y unc ional heo y amewo k, is
epo ed. The one-elec on pa o he Kohn–Sham equa ions is modi ied by inco po a ion o a
Coulomb ope a o , which accoun s o he classical elec on in e ac ion a ising om he g oup.
Exchange and co ela ion e ec s a e in oduced by a sui able modi ica ion o he
exchange-co ela ion unc ionals. The s ong o hogonali y condi ion, usually equi ed by he heo y
o sepa abili y o many elec on sys ems, is w i en in e ms o i s o de educed densi y ma ices.
In o de o check he me hod a g oup model po en ial o H2O共en i onmen 兲was ob ained and
employed in he calcula ion o HF¯H2O and H2O¯H2O complexes using se e al unc ionals.
Equilib ium in e g oup dis ances and binding ene gies a e compa ed wi h all-elec on calcula ions.
©2000 Ame ican Ins i u e o Physics. 关S0021-9606共00兲30337-3兴
I. INTRODUCTION
A winning app oxima ion in quan um chemis y, espe-
cially help ul in educing he huge compu a ional e o usu-
ally in ol ed, is he classi ica ion o elec ons o a gi en
sys em as ac i e o inac i e.1–4 The inac i e elec ons may
be assumed o beha e as spec a o s and he e o e can be
ozen. This idea has been success ully exploi ed o desc ibe
he beha io o a oms o ions o model en i onmen e ec s
in clus e embedded calcula ions,5,6,7 as well as o he de e -
mina ion o e ec i e po en ials o agmen s ha could be
employed as sa u a o g oups. In his sense, e ec i e po en-
ials o NH3,8SiH3,9COOH,10 and cyclopen adienyl
ligand11 ha e been epo ed.
On he o he hand, some e o s ha e been de o ed o
eplace a comple e molecule, o chemical g oup, in calcula-
ions in which he en i onmen is desc ibed h ough an em-
bedding ype echnique. Along his line, in 1993 Mejı
´as and
Sanz12 epo ed on he de e mina ion o ab ini io g oup
model po en ials 共GMPs兲 o he HF molecule, oge he wi h
some applica ions. La e , F ank e al.13 epo ed on he ea-
sibili y o ep esen ing spec a o molecules h ough e ec i e
po en ials gene a ed by a semiempi ical NDDO calcula ion.
Qui e ecen ly, Day e al.14 came up wi h a me hod o ob ain
wha hey called an ‘‘e ec i e agmen po en ial’’ which
makes use o a dis ibu ed mul ipole expansion, and which
allowed hem o eplace wa e molecules by an e ec i e po-
en ial, wi h excellen esul s.15 Mo e ecen ly we ha e de-
eloped polycen e compac model po en ials in which he
sho - ange con ibu ion was expanded as a spec al ep e-
sen a ion. Also, we epo ed an algo i hm o sol e he p ob-
lem associa ed wi h o a ion o he model po en ial.16
A s aigh o wa d way o imp o e g oup model po en-
ials would in ol e he inco po a ion o he elec on co ela-
ion e ec s be ween ac i e and ozen elec ons. On a gen-
e al basis, his con ibu ion could in p inciple be
inco po a ed in o he densi y unc ional heo y 共DFT兲共Re s.
17, 18兲 amewo k h ough a modi ica ion o he exchange-
co ela ion e m. This opic has been conside ed o a omic
co es in se e al wo ks19,20,21 and, in his con ex , one o he
mos widely pseudopo en ials used in solid-s a e calcula ions
is he so-called ‘‘ul aso pseudopo en ial’’ 共UPP兲o
Vande bil .22
As o he ep esen a ion beyond a omic co es, Aba en-
ko e al.23 epo ed a me hod in which he wa e unc ion o
a small clus e o Li2Mg was compu ed by inco po a ing he
exchange-co ela ion in e ac ion wi h a neighbo ing g oup in
an i e a i e way. Mo e ecen ly Ca e e al.24 p esen ed a
new embedding echnique ha combines pe iodic-DFT
me hods and explici elec on-co ela ion p ocedu es.
G ound-s a e p ope ies o chemical g oups ha e also
been s udied by means o he ‘‘ eeze-and- haw’’ cycle o
Kohn–Sham equa ions wi h cons ained elec on densi y
共KSCED兲as p oposed by Wesolowski and Webe .25 In his
app oach, he elec on densi y o each agmen is kep o-
zen while he in e ac ion ene gy is calcula ed using e ms
de i ed om DFT, and, in addi ion o he exchange co ela-
ion unc ional, he analy ical o m o he app oxima e ki-
ne ic ene gy unc ional is needed. Examples like he wa e
dime , HF–HF, HCl–HCl o HCN–HF may be ound as
applica ions o he s udy o he hyd ogen bond.26
The main goal o his pape is o model he e ec s o a
chemical g oup, o he en i onmen in clus e embedded cal-
cula ion, bu now wi hin he one-pa icle Kohn–Sham sel -
consis en ield 共SCF-KS兲 amewo k. The s a ing poin is
he elec onic sepa abili y p inciple p oposed by McWeeny1
and Huzinaga2–4 bu in ol ing, in his case, he i s -o de
educed densi y ma ices o he sepa able sys ems. Using he
Kohn–Sham o bi als such condi ions lead o an ope a o
simila o he Vande bil 22 and Phillips–Kleinman27 pseudo-
po en ials. The Coulomb in e ac ion be ween elec ons o a
gi en g oup was handled by i ing he elec onic densi y o a
se o auxilia y Gaussian unc ions,28 acco ding o one o he
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 113, NUMBER 15 15 OCTOBER 2000
60820021-9606/2000/113(15)/6082/6/$17.00 © 2000 Ame ican Ins i u e o Physics
s a egies o Dunlap e al.29 The a icle is a anged as ol-
lows. The heo e ical ame is b ie ly o e iewed in Sec. II.
A new sepa abili y condi ion in e ms o g oup densi y ma-
ices is p oposed and he way o inco po a e he exchange-
co ela ion e ec coming om he en i onmen is p esen ed.
In Sec. III some es calcula ions conce ning he H2O model
po en ial a e epo ed. Finally he main conclusions a e sum-
ma ized in Sec. IV.
II. THEORETICAL ASPECTS
In he densi y unc ional heo y, he SCF Kohn–Sham
共SCF-KS兲equa ions17 ha e o be sol ed,
兵
⫺1
2ⵜ2⫹Vc共 兲⫹
␮
xc共
␳
兲
其
␺
i
KS共 兲⫽␧i
␺
I
KS共 兲.共1兲
The e m ⫺1
2ⵜ2is he kine ic ene gy ope a o , Vc( )
ep esen s he elec os a ic 共o Coulomb兲po en ial, and
␮
xc(
␳
) in oduces he exchange and co ela ion con ibu ion,
which, in gene al, may be w i en as
␮
xc共
␳
兲⫽
⳵
Exc共
␳
兲
⳵␳
,共2兲
whe e Exc(
␳
) is he so-called exchange and co ela ion unc-
ional. In local densi y app oxima ion 共LDA兲o in he gen-
e alized g adien app oxima ion 共GGA兲, i is gene ally w i -
en as
Exc共
␳
兲⫽
冕
xc共
␳
,ⵜ
␳
兲d3 .共3兲
The o al elec on densi y
␳
in Eq. 共1兲is ela ed o he
i s -o de densi y ma ix o an Npa icles sys em in he
o m
␳
共 兲⫽
冕
⌫共x,x⬘兲d
␴
,共4兲
whe e he spin–o bi al no a ion x⫽(
␴
) is used and he
i s -o de educed densi y ma ix is de ined as30
⌫共x,x⬘兲⫽N
冕冕
¯
冕
⌽*共x,x1,x2,...,xN⫺1兲
⫻⌽共x⬘,x1,x2,...,xN⫺1兲dx1dx2¯dxN⫺1,
共5兲
⌽(x1,x2,..,xN) being he elec onic wa e unc ion.
The necessa y and su icien condi ion o he educed
densi y ma ix o be N- ep esen able18 can be exp essed as
⌫
ˆ⫽兺
ini
兩
␸
i
典具
␸
i
兩
共6兲
wi h
␸
ibeing he spin–o bi als and 0⭐ni⭐1. Choosing ni
⫽0,1 in Eq. 共5兲, he o al elec on densi y may be spanned on
a se o spinless single-pa icle o bi als, each one occupied
wi h wo elec ons,
␳
共 兲⫽兺
occ
兩
␺
i共 兲
兩
2.共7兲
Le us now assume a sys em cons i u ed by wo mol-
ecules o chemical g oups, Aand B, weakly in e ac ing, wi h
⌽Aand ⌽Bbeing he wa e unc ions. Acco ding o he
building block p inciple o McWeeny and Huzinaga, he o-
al elec onic wa e unc ion ⌽can be w i en as
⌽⫽MA
ˆ共⌽A⌽B兲,共8兲
whe e Mis a no maliza ion ac o and A
ˆis he in e g oup
an isyme ize ope a o . Fo ma hema ical con enience in he
de i a ion o ab ini io model po en ials, he g oup wa e
unc ions a e selec ed so as o be s ongly o hogonal,1i.e.,
冕
⌽A
*共x,i,j,...兲⌽B共x,k,l,...兲dx⫽0. 共9兲
In o de o keep he same condi ion in e ms o elec on
densi y. Eq. 共9兲is mul iplied by
⌽A共2,i,j,...兲⌽B
*共3,k,l,...兲
and in eg a ed wi h espec o i,j,..,k,l,.... Bea ing in mind
he o m o he educed densi y ma ix, Eq. 共9兲becomes
冕
dx⌫B共3,x兲⌫A共x,2兲⫽0. 共10兲
Since in Eq. 共9兲, i is possible o w i e Ain place o B
and Bplace o A, hen he s ong o hogonali y condi ion 共9兲
in e ms o educed densi y ma ices may be w i en as
⌫
ˆA⌫
ˆB⫽⌫
ˆB⌫
ˆA⫽0. 共11兲
I his condi ion is ul illed, he p oblem o inding he local
p ope ies o he g oup o in e es 共clus e , A o B兲,in he
p esence o a spec a o g oup 共en i onmen , B o A兲can be
se up using he DFT o malism.
Acco ding o he de ini ion o he densi y ma ix gi en in
Eq. 共6兲, he s ong o hogonali y condi ion 共11兲is sa is ied i
具
␺
i,clus
KS
兩
␺
j,en
KS
典
⫽0. 共12兲
The o al educed densi y ma ix in a sepa able sys em may
be w i en as
⌫
ˆall⫽⌫
ˆclus⫹⌫
ˆen ,共13兲
and he elec on densi y,
␳
共 兲⫽
␳
clus共 兲⫹
␳
en 共 兲,共14兲
whe e he elec on densi ies
␳
clus( ) and
␳
en ( ) a e no mal-
ized o he numbe o elec ons o he clus e (Nclus) and he
en i onmen (Nen ), espec i ely.
Unde hese condi ions, he clus e ene gy in he p es-
ence o he en i onmen 共 he e ec i e ene gy Eclus
e )is
Eclus
e 共
␳
clus兲
⫽E0共
␳
clus ,nclus兲
⫹Ein 共
␳
clus⇔
␳
en ,
␳
clus⇔nen ,nclus⇔
␳
en ,nclus⇔nen 兲.
共15兲
The i s e m on he igh -hand side o Eq. 共15兲is he iso-
la ed clus e ene gy, excluding all componen s depending on
he en i onmen . The second e m is he in e ac ion o elec-
on densi y in clus e
␳
clus , wi h elec ons and nuclei o he
en i onmen
␳
en and nen , espec i ely. This las e m also
includes he in e ac ion be ween he clus e nuclei, nclus , and
he en i onmen nega i e and posi i e cha ges, which can be
6083J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 G oup model po en ials
added a he end o he elec onic ene gy calcula ion. No ice
ha ollowing Eq. 共7兲 he clus e elec on densi y
␳
clus can be
w i en as
␳
clus共 兲⫽兺
occ
兩
⌿i,clus
KS 共 兲
兩
2.共16兲
Applica ion o he a ia ional p inciple o Eq. 共15兲, o-
ge he wi h he condi ion 共12兲leads o he se o SCF-KS-
like equa ions
兵
⫺1
2ⵜ2⫹Vc
clus⫹Ve 共 兲⫹
␮
xc
e 共
␳
clus ,
␳
en 兲
其
␺
,clus
KS 共 兲
⫽␧i
␺
i,clus
KS 共 兲.共17兲
In his equa ion he hi d and ou h e ms ep esen he in-
e ac ion be ween he clus e and he en i onmen , ep e-
sen ed h ough a g oup model po en ial 共KS-GMP兲. Analyz-
ing hese con ibu ions, one o he cen al poin s in he
p esen wo k, he hi d e m expands as
Ve 共 兲⫽⫺ 兺
i
nucl.
en . zi
兩
⫺Ri
兩
⫹
冕
␳
en 共 ⬘兲
兩
⫺ ⬘
兩
d3 ⬘⫺P
ˆen ,共18兲
whe e he summa ion ep esen s he in e ac ion be ween he
clus e elec ons and he en i onmen nuclei, while he in e-
g al accoun s o he elec on–elec on in e ac ion be ween
he wo g oups. In he p esen wo k, such an in e ac ion has
been simula ed by i ing he cha ge densi y o he spec a o
g oup o a se o coe icien and Gaussian unc ions. The
algo i hm used has been one o hose p oposed by Dunlap
e al.29 whose main poin is o ind he se o coe icien s ha
minimizes he unc ional
D⫽
冕
⌬
␳
共1兲⌬
␳
共2兲
12 d 1d 2共19a兲
wi h
⌬
␳
共 兲⫽
␳
en
exac 共 兲⫺
␳
en 共 兲.共19b兲
The se o Gaussian unc ions used he e was aken om Re .
28. Finally P
ˆen is a weigh ed p ojec ion ope a o o e he
occupied o bi als o he spec a o g oup ha allows o e-
s ic ion o he a ia ional space o he clus e space e lec -
ing he condi ion gi en by Eq. 共11兲,
P
ˆen ⫽
␣
兺
occ
兩
␺
occ,en
KS
典
␧occ,en
KS
具
␺
occ,en
KS
兩
,共20兲
whe e he sum uns o e he occupied molecula o bi als o
he en i onmen wi h eigen alues ␧occ,en
KS . The p ojec o ac-
o
␣
is se o 2.31
Conce ning he ou h e m o Eq. 共17兲, he ope a o
␮
xc
e (
␳
clus ,
␳
en ) a ises om he exchange-co ela ion compo-
nen o he in e ac ion be ween he wo g oups. I can be
w i en as
␮
xc
e ⫽
⳵
Exc,clus
e 共
␳
clus兲
⳵␳
clus
⫽
␮
xc共
␳
clus⫹
␳
en 兲,共21兲
whe e Exc,clus
e (
␳
clus) is he e ec i e exchange-co ela ion en-
e gy o he clus e
Exc,clus
e 共
␳
clus兲⫽Exc共
␳
clus⫹
␳
en 兲⫺Exc共
␳
en 兲共22兲
I a ac ion o he exac Ha ee–Fock exchange is used,
o in a simple Ha ee–Fock calcula ion, he exchange ope a-
o can be simula ed using a nondiagonal spec al
ep esen a ion.5
Equa ions 共21兲and 共22兲in oduce he exchange and
elec on co ela ion con ibu ion be ween clus e and he su -
ounding sys em in a DFT amewo k. These wo equa ions
a e simila o hose p oposed by Vande bil 22 in he UPP
o mula ion in o de o include he co ela ion be ween co e
and alence elec ons, al hough in ou case he de e mina ion
o he Ve ( ) does no need he use o a cu o adius. To
build Ve ( ) we only need o know an app oxima e densi y
and he occupied o bi als o he su ounding sys em, wha -
e e he unc ional used in he clus e g oup is. Mo eo e , in
con as wi h he KSCED,25,26 in ou me hod i is no neces-
sa y o use an app oxima ed kine ic ene gy unc ional, and
he inclusion o he weigh ed p ojec ion ope a o P
ˆen con-
e s a nonlocal componen o Ve ( ) as in he UPP app oach.
The e alua ion o Eq. 共22兲and he pa o he SCF-KS
equa ion ela ed o Eq. 共21兲was sol ed using he g id o
poin s gene a ed by he clus e sys em as i i we e isola ed.32
These algo i hms we e implemen ed in he HONDO
p og am.33 The in eg als in he a omic basis we e e alua ed
using he King, Dupuis, and Rys34,35 and Gauss–He mi e
quad a u es. Fo in eg als conce ning Eqs. 共19兲and 共20兲, he
ecu si e o mulas o Oba a and Saika we e used.36
III. NUMERICAL EXAMPLES
We epo in his sec ion a ew nume ical applica ions o
he p ocedu e de eloped abo e. Among o he aspec s, he
sui abili y o he KS-GMPs o simula e a wa e molecule in
he compu a ion o he 共H2O兲2dime and he HF–H2O com-
plex is explo ed. Mo e ex ensi e examples in which he
me hod is es ed in su ace chemis y calcula ions will be
epo ed in a o hcoming wo k. Fo he p esen we ha e
cen e ed ou a en ion on he use o a wide a ie y o unc-
ionals commonly used in DFT calcula ions. Fo he ex-
change we ha e selec ed ha o Sla e ,37 共SLT兲as well as
ha p oposed by Becke,38 共B88兲. Fo he co ela ion pa we
employed he unc ional o Lee–Yang–Pa 39 共LYP兲, which
added o SLT o B88 gi es wo di e en exchange-
co ela ion unc ionals: SLYP and BLYP. The widely used
hyb id h ee pa ame e unc ional B3LYP has also been
conside ed.40
To se up he p ocedu e, a i s single calcula ion o he
isola ed wa e molecule was ca ied ou using each o hese
unc ionals and om he KS molecula o bi als he KS-GMP
was de e mined. These KS-GMPs we e hen inco po a ed
in o he DFT calcula ions o he ac i e g oup (H2Oo HF兲
and he in e ac ion ene gy was es ima ed a se e al dis ances.
The esul ing p o iles we e compa ed wi h hose a ising om
e e ence DFT all elec on 共AE兲calcula ions ca ied ou on
he supe molecules HF¯H2O and H2O¯H2O. All he cal-
cula ions we e pe o med using he s anda d TZP basis se .
Resul s o he HF¯H2O and H2O¯H2O complexes a e
epo ed in Tables I and II, espec i ely. S a ing wi h he
HF¯H2O sys em we can see ha he equilib ium in e mo-
lecula dis ance es ima ed wi h he KS-GMP ag ees eason-
6084 J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 N. C. He na
´ndez and J. F. Sanz
ably wi h hose ob ained om AE calcula ions. Wi h he ex-
cep ion o SLT based unc ionals, he KS-GMP dis ances a e
ound o be la ge han he AE ones, he e o s anging be-
ween 0.9% and 6.9%. The KS-GMP binding ene gies com-
pu ed a he op imized in e molecula dis ances appea o be
signi ican ly lowe han he AE ones; howe e , i should be
no ed ha he AE binding ene gies a e o e es ima ed be-
cause o he basis se supe posi ion e o 共BSSE兲. Since he
KS-GMP a e ee o BSSE, we co ec ed he AE ene gies
using he well-known coun e poise me hod o Boys and
Be naldi.41 These co ec ed ene gies a e also epo ed in he
able unde Eco en y. As can be seen, he ag eemen is
clea ly imp o ed, wi h de ia ions be ween 3% and 20%. The
obse ed gene al end is an unde es ima ion o he binding
ene gies e lec ing he ac ha he model po en ial is ozen
and, he e o e, no epola iza ion o he elec on dis ibu ion
is allowed. Also, because no basis se is p esen in he GMP,
he elec on densi y and, he e o e, he in e g oup egion a e
no p ope ly desc ibed. An addi ional app oxima ion in o-
duced in he KS-GMP calcula ions a ises om he way o
compu ing he ene gy and in eg als ela ed o Eq. 共21, 22兲,
whe e only he g id gene a ed by he clus e is used. Finally,
in Fig. 1, he p o ile ene gies ob ained om KS-GMP and
AE calcula ions a e plo ed. I shows ha beyond he nume i-
cal easonable ag eemen , he shape o hese cu es appea s
o be qui e sa is ac o y, wi h smoo h beha io and a sui able
desc ip ion o he equilib ium egion.
Resul s ob ained o he H2O¯H2O dime a e epo ed
in Table II and Fig. 2. Bo h in e g oup and binding ene gies
a e ound o ollow he ends al eady men ioned abo e. The
equilib ium dis ances appea o be o e es ima ed 共10%–
14%兲again wi h he excep ion o he SLT unc ionals, while
he binding ene gies a e unde es ima ed wi h espec o he
BSSE co ec ed AE ones. The ene gy p o iles also show a
sui able beha io , and as in he HF¯H2O case, he cu a-
u es a e ound o be lowe acco ding o he unde es ima ion
o he g oup–g oup in e ac ion.
Finally, we will commen on one o he di e ences be-
ween ou KS-GMP and he KSCED me hod epo ed in Re .
25. This di e ence conce ns he use o he p ojec ion ope a-
o de ined in Eq. 共20兲, which is no p esen in he KSCED
o malism. In o de o show he impo ance o his e m, a se
o calcula ions o he HF¯H2O sys em was ca ied ou bu
now se ing he p ojec ion ac o a
␣
⫽0 in Eq. 共20兲. The
p o ile ene gies hus ob ained a e epo ed in Fig. 3. As can
be seen, he absence o he p ojec o gi es ise o a com-
ple ely a ac i e sys em wha e e he unc ional is. This e-
sul shows he necessi y o he p esence o he p ojec ion
ope a o in ou ep esen a ion. The p esence o such a p o-
jec o comes di ec ly om he s ong o hogonali y condi ion
as exp essed in Eq. 共11兲in e ms o he i s -o de densi y
ma ices, and aking in o accoun he N- ep esen abili y con-
di ion leads o Eq. 共12兲.
IV. CONCLUSIONS
In his wo k a me hod o ob ain polycen e g oup model
po en ials wi hin a DFT amewo k is epo ed. The main
idea unde lying his app oach is o ca y ou he calcula ion
o an ac i e subsys em 共clus e 兲 aking in o accoun he in-
luence o he su ounding g oups 共en i onmen 兲 h ough an
ab ini io model po en ial, including elec on co ela ion e -
ec s in he clus e –en i onmen in e ac ion. Wi h his pu -
FIG. 1. Binding ene gy p o iles o he H2O–HF complex om all-elec on
and KS-GMP calcula ions.
TABLE I. Op imized in e g oup dis ances 共Å兲and binding ene gies 共kcal/
mol兲ob ained o he H2O–HF sys em om all elec on and 共KS-GMP兲
calcula ions using SLT, B88, SLYP, BLYP, and B3LYP unc ionals. De ia-
ions in pe cen a e be ween pa en heses.
DFT
unc ional
dBinding ene gy
All elec on KS-GMP All elec on KS-GMP Eco
SLT 1.690 1.573 共6.9兲⫺13.2 ⫺11.2 共3.4兲⫺11.6
B88 1.936 1.954 共0.9兲⫺6.6 ⫺4.7 共14.5兲⫺5.5
SLYP 1.594 1.540 共3.4兲⫺17.5 ⫺15.1 共3.8兲⫺15.7
BLYP 1.786 1.822 共2兲⫺9.6 ⫺6.8 共18兲⫺8.3
B3LYP 1.763 1.849 共4.9兲⫺10.1 ⫺7.2 共20兲⫺9.0
TABLE II. Op imized in e g oup dis ances 共Å兲and binding ene gies
共kcal/mol兲ob ained o he H2O–H2O sys em om all elec on and
共KS-GMP兲calcula ions using SLT, B88, SLYP, BLYP, and B3LYP unc-
ionals. De ia ions in pe cen a e be ween pa en heses.
DFT
unc ional
dBinding ene gy
All elec on KS-GMP All elec on KS-GMP Eco
SLT 1.839 1.785 共2.9兲⫺8.4 ⫺6.5 共6.1兲⫺6.9
B88 2.200 2.576 共14.6兲⫺3.2 ⫺1.9 共24.9兲⫺2.5
SLYP 1.719 1.708 共0.6兲⫺11.8 ⫺8.9 共15.3兲⫺10.5
BLYP 1.976 2.192 共10.9兲⫺5.3 ⫺3.5 共22.2兲⫺4.5
B3LYP 1.953 2.182 共11.7兲⫺5.7 ⫺3.8 共22.4兲⫺4.9
6085J. Chem. Phys., Vol. 113, No. 15, 15 Oc obe 2000 G oup model po en ials

pose in mind, wo main modi ica ions a e in oduced in o he
KS equa ions. The i s one accoun s o he classical
elec on–elec on in e ac ion be ween he clus e and he en-
i onmen . Such an in e ac ion in ol es a new e m in he
one-elec on pa and is ep esen ed h ough a Coulomb op-
e a o buil up by i ing he elec on densi y o he en i on-
men g oup o a se o Gaussian unc ions. The second modi-
ica ion in ol es an adequa e co ec ion o he exchange-
co ela ion ope a o , depending on he unc ional used. The
exchange-co ela ion e ec s a e in oduced in o he KS
equa ions by means o a g id o poin s de ined in he clus e
sys em. In o de o p ese e he s ong o hogonali y condi-
ion be ween g oup wa e unc ions, as equi ed by he build-
ing block p inciple, an equi alen es ic ion is p oposed us-
ing he densi y ma ix o malism. This allows o he
de ini ion o a p ojec ion ope a o simila o ha used in he
g oup model po en ial implemen a ion a he Ha ee–Fock
le el.16
The whole p ocedu e was implemen ed in an all-pu pose
p og am 共HONDO兲,33 pe mi ing us o compu e he clus e
elec on densi y wi h se e al exchange-co ela ion unc ion-
als 共SLT, B88, SLYP, BLYP, B3LYP兲wi hin he DFT
amewo k. In o de o es he me hod a g oup model po en-
ial o H2O was de e mined and used in he calcula ions o
HF¯H2O and H2O¯H2O complexes. Compa ed o all-
elec on DFT calcula ions, ou esul s show a sa is ac o y
beha io o he wa e model po en ials.
ACKNOWLEDGMENTS
This wo k was inancially suppo ed by he DGES and
by he Jun a de Andalucı
´a共Spain, P ojec s Nos. PB98-1125
and FQM132兲. We a e g a e ul o D . A. Ma
´ quez o his
help wi h he HONDO implemen a ion.
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FIG. 2. Binding ene gy p o iles o he H2O–H2O dime om all elec on
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