Shape and size o simple ca ions in aqueous solu ions: A heo e ical
eexamina ion o he hyd a ed ion ia compu e simula ions
Jose
´M. Ma ı
´nez, Ra ael R. Pappala do, and En ique Sa
´nchez Ma cosa)
Depa amen o de Quı
´mica Fı
´sica, Uni e sidad de Se illa, 41012 Se illa, Spain
~Recei ed 22 June 1998; accep ed 15 Oc obe 1998!
The simples ep esen a ion o monoa omic ca ions in aqueous solu ions by means o a sphe e wi h
a adius chosen on he basis o a well-de ined p ope y ~ ha o he ba e ion o i s hyd a e!is
eexamined conside ing classical molecula dynamics simula ions. Two cha ged sphe e–wa e
in e ac ion po en ials we e employed o mimic he ba e and hyd a ed ca ion in a sample o 512 wa e
molecules. Sho - ange in e ac ions o i alen ca ions we e desc ibed by Lenna d-Jones po en ials
which we e i ed om ab ini io calcula ions. Fi e s a is ically independen uns o 150 ps o each
o he i alen sphe es in wa e we e ca ied ou in he mic ocanonical ensemble. A compa ison o
s uc u al and dynamical p ope ies o hese simple ion models in solu ion wi h hose o a sys em
con aining he C 31hyd a e ~@C ~H2O!6]31) is made o ge insigh in o he size and shape de ini ion
o simple ions in wa e , especially hose ha a e highly cha ged. Ad an ages and sho comings o
using simple sphe ical app oaches a e discussed on he basis o e e ence calcula ions pe o med
wi h a mo e igo ous hyd a ed ion model @J. Phys. Chem. B 102, 3272 ~1998!#. The impo ance o
nonsphe ical shape o he hyd a e o highly cha ged ions is s essed and i is pa adoxically shown
ha when sphe ical shape is e ained, he big sphe e ep esen ing he hyd a e leads o esul s o ionic
solu ion wo se han hose ob ained wi h he small sphe e. A low-cos me hod o gene a e hyd a ed
ion–wa e in e ac ion po en ials aking in o accoun he shape o he ionic agg ega e is
p oposed. © 1999 Ame ican Ins i u e o Physics. @S0021-9606~99!51703-0#
I. INTRODUCTION
Ionic sol a ion o monoa omic ions has long been unde -
s ood in i s ene ge ic, s uc u al and dynamical aspec s on he
basis o h ee landma k classical de elopmen s: he Bo n
heo y, he Debye–Hu
¨ckel, and Debye–Hu
¨ckel–Onsage
heo ies.1They supply a simple and elegan model o ionic
solu ions allowing ui ul applica ions in mul i ude o physi-
cochemical amewo ks whe e ions in solu ion a e in ol ed.
I is gene ally accep ed ha he simplici y o i s o mula ion,
only a ew sys em-dependen pa ame e s a e needed ~e.g.,
ion cha ge, dielec ic pe mi i i y o he sol en and ionic
adius, when applied!, is ce ainly one o he keys o hei
success. The phenomenological ac o s usually added o
hese models in o de o i o expe imen al esul s canno
hide ha hese c ude heo ies e ain some o he main ea-
u es o he ion sol a ion.2–4 Al hough hese heo ies come
om he 1920s, he ecen imp o emen s in quan um and
s a is ical mechanics o condensed medium ha e compelled
di e en au ho s o ge insigh s in o he mic oscopical in e -
p e a ion o he pa ame e s. Thei aim has been o de elop
new concep s o a be e unde s anding in he molecula
basis o hese heo ies.2,5,6 A common ea u e o hese mod-
els is he sphe ical shape adop ed o he ions in solu ion, a
ea u e ha may easily be accep ed o he case o simple
monoa omic ions. To ge ai ag eemen wi h expe imen al
da a, he mos equen ly al e ed pa ame e is he adius, as
shown by La ime e al. in hei ea ly s udy.7Wi hin his line
an impo an ac i i y has appea ed du ing he las en yea s
o sugges a physically meaning ul se o adii o simple
ions.8,9 Nowadays, compu e simula ions a e p o iding a c u-
cial b idge be ween gene al heo ies o solu ion and he mi-
c oscopical le el o he s udied sys em, as hey allow a la ge
numbe o nume ical expe imen s which es he main basis
and ends classically poin ed ou by he pionee heo ies o
elec oly e solu ions.10–14
A chemical concep coming om ea ly s udies o ionic
solu ions was ha o he hyd a ed ion. This concep ecog-
nizes ha some ions, mainly me allic and highly cha ged
ca ions, in aqueous solu ions beha e as mo e complex en i-
ies han expec ed. The agg ega e o med by he ion and a
gi en numbe o sol en molecules su ounding i
@M~H2O!m]n1, was called he hyd a ed ion. I could explain
physicochemical p ope ies whose obse ed alues canno be
easily unde s ood on he basis o simple ba e ions, Mn1.15
Ou g oup has used his concep wi hin he amewo k o
compu e simula ions o ionic solu ions. The implemen a ion
o his old elec ochemical concep wi hin s a is ical simula-
ions has been pe o med by de eloping an ion–wa e in e -
ac ion po en ial, whe e he ion is p esen in i s mos s able
hyd a ed o m, @M~H2O!m]n1. Thus, an ab ini io
@Zn~H2O!6]21-H2O in e molecula po en ial was i s de el-
oped and es ed by Mon e Ca lo ~MC!simula ions o he
Zn21hyd a ion.16 This po en ial has been called HIW ~hy-
d a ed ion-wa e !. Fu he MC and molecula dynamics
~MD!simula ions no only wi h Zn21,17 bu also on he mo e
in ol ed C 31hyd a ion,18,19 ha e been p omising, since
hey ha e simul aneously supplied sa is ac o y esul s o en-
e ge ic, s uc u al, and dynamical p ope ies, wi hou he in-
a!Elec onic mail: [email p o ec ed]
JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 3 15 JANUARY 1999
16690021-9606/99/110(3)/1669/8/$15.00 © 1999 Ame ican Ins i u e o Physics
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
clusion o any kind o empi ical pa ame e s in he p oposed
po en ials. Likewise, se e al au ho s ha e ecen ly used he
hyd a ed ion concep o de elop, in di e en ways, in e mo-
lecula in e ac ion po en ials o me al ca ions.20–25 Thei e-
sul s also poin ou he ad an ages o his app oach o de-
sc ibe a wide se o physicochemical p ope ies o ionic
solu ions.
Ou analy ical po en ial, de eloped o he hyd a ed ion–
wa e in e ac ions, p ecisely desc ibes he shape o his clus-
e . Scheme 1 shows a wo-dimensional map o isoene ge ic
cu es co esponding o he mos a o able in e ac ion be-
ween he @C ~H2O!6]31and a p obe wa e molecule.18,19~a!
The d awn plane is ha con aining he ca ion and ou oxy-
gen a oms. I is seen ha al hough a long dis ances he iso-
lines a e almos ci cula , a sho dis ances hey ollow he
molecula shape o he hyd a e. Scheme 1 compelled us o
unde ake a new s udy examining how changes o he shape
and size o hese me al ions a ec he p ope ies o ionic
solu ions, pa icula ly he less s udied dynamical p ope ies.
To achie e his goal, he esul s o simula ions o wo simple
sphe ical models o a i alen ca ion will be compa ed wi h
hose p e iously ob ained o he C 31hyd a e.19~a!Fo he
i s i alen cha ged sphe e, he adius is chosen such ha
he ba e ion is mimicked, whe eas in he second case he
adius alue co esponds o ha de i ed om a hyd a ed ion
~Scheme 2!. The o me model will be called Q3S ~cha ge
13 small!and he pa icle will ha e a mass equal o ha o
he ch omium a om. The la e model will be called Q3B
~cha ge 13 big!and since i is a c ude ep esen a ion o he
hyd a ed ion, he mass assigned will be ha o he C 31
hexahyd a e. When gene a ing he in e ac ion po en ials o
hese sphe es wi h wa e molecules, special a en ion was
paid o using c i e ia ha did no in oduce signi ican di e -
ences wi h espec o he HIW po en ial o he C 31, apa
om he in insic opological ones. I is wo h poin ing ou
ha he objec i e o hese wo new simple po en ials is no o
imp o e he well- es ed ep esen a ion gi en by he HIW
po en ial,18,19 bu a he o ge insigh s in o basic ac o s con-
olling he in e pa icle o ces in ionic solu ions.
II. METHODOLOGY
A. In e molecula po en ials.
The @C ~H2O!6]31–H2O in e molecula po en ial has al-
eady been desc ibed elsewhe e.18,19~a!The gene al exp es-
sion o his HIW po en ial is
EHIW5(
i
HI si es
(
j
W si es
S
C4
ij
ij
41C6
ij
ij
61C12
ij
ij
12 1qiqj
ij
D
,~1!
whe e indices i,j un o e he si es o he HI and wa e mol-
ecule, espec i ely.
Q3S– and Q3B–wa e in e ac ion po en ials ~Q3XW,
X5S,B!ha e been de ined by means o a Lenna d-Jones
plus a Coulombic e m:
EQ3XW54
e
F
S
s
QO
D
12
2
S
s
QO
D
6
G
1(
j
W si es qQqj
Qj,~2!
whe e qQ513.
s
alues ~1.825 and 3.625 Å o Q3SW and
Q3BW, espec i ely!we e chosen in such a way ha hey
led o minimum alues in he Lenna d-Jones pa o he
Q3SW and Q3BW in e ac ion cu es a he same ion–
oxygen dis ances han hose cha ac e is ic o compu a ions o
C 31in wa e . Thus, o he small sphe e he ab ini io C -OI
dis ance in he hexahyd a e ~2.05 Å!,18 and o he big sphe e
he i s maximum o he gC -O o he p e ious C 31hexahy-
d a e simula ion ~which co esponds o an ion–second–shell
dis ance, C -OII54.06 Å18!, we e he applied c i e ia. Figu e
1 shows he Lenna d-Jones cu es o bo h cha ged so
sphe es.
e
alues ~426.7626 and 52.1092 kJmol21 o Q3SW
and Q3BW, espec i ely!we e ob ained om he Ha ee–
Fock in e ac ion ene gy o a iple cha ge poin and a TIP4P-
geome y wa e molecule a he p e iously men ioned dis-
ances. F om he o al in e ac ion ene gy, he elec os a ic
pa is sub ac ed. Basis se s o he wa e molecule we e he
same as hose used in he de elopmen o he HIW in e ac-
ion po en ial.18 Fo he sake o compa ison, he in e ac ion
ene gy co esponding o he op imized a angemen o a wa-
e molecule a ound he HI and Q3B has been calcula ed, i s
Scheme 1. Isoene gy cu es co esponding o he mos a o able in e ac ion
o a p obe H2O wi h he C 31hexahyd a e.
Scheme 2. Rep esen a ion o he h ee i alen ca ion models in e ac ing
wi h wa e molecules: HI ~ op!, Q3S ~le -hand side!, and Q3B ~bo om!.
1670 J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
alue being 2149.6 and 2160.9 kJmol21, espec i ely. The
simila i y o hese wo quan i ies means ha he Q3BW po-
en ial model has p ima y ion–wa e in e ac ions in he sec-
ond shell egion close o ha o he mo e e ined HIW po-
en ial.
B. MD simula ion de ails
Molecula dynamic simula ions we e pe o med in he
NVE ensemble using pe iodic bounda y condi ions. The sys-
em was o med by he co esponding cha ged sphe e plus
512 H2O which we e desc ibed by he TIP4P model.26 The
ion–wa e in e ac ions we e desc ibed by he wo p e iously
p esen ed po en ials: Q3SW and Q3BW. The basic cell was a
cubic box wi h 24.8 Å pe side. Simula ions we e un wi h
he MOLDY p og am27 ~ e sion 2.10!. New on–Eule equa-
ions o mo ion we e in eg a ed using a modi ied o m o he
Beeman algo i hm28,29 which gua an ees a good s abili y o
molecula sys ems. O ien a ions o sol en molecules, which
we e assumed o be igid, we e desc ibed by he qua e nion
o malism.30 The ime s ep employed was 0.3 s in o de o
a oid ene gy d i .
Coulombic in e ac ions we e compu ed using he Ewald
sum echnique,31 including he cha ged sys em e m.32 Al-
hough his ea men is cos ly, he impo ance o i s use o
ob ain eliable s uc u al and, pa icula ly, dynamical esul s
in he case o ionic aqueous solu ions, has been shown.32–34
A sphe ical molecula cu o was applied o he eal space
pa o he Ewald ene gy as well as he sho - ange po en ial,
which we e ea ed by an implemen a ion o he link cell
me hod.35,36 Co ec ions o he po en ial ene gy and p essu e
a ising om he use o he cu o we e included.
The maliza ion ime was abou 40 ps and empe a u e
was 298 K along he simula ion. Fo each sys em he o al
simula ion ime was 750 ps di ided in o 5 s a is ically inde-
penden uns o 150 ps. To achie e such condi ion, a small
eequilib a ion pe iod was applied be ween subsequen uns
o he simula ed sys em. T ajec o ies and eloci ies we e col-
lec ed e e y 40 ime s eps. Compu a ions we e ca ied ou on
a pa allel HP X-CLASS SPP-2000 Se ies whe e an e icien
pa alleliza ion o he p og am was ob ained.
III. RESULTS
A. S uc u al esul s
P elimina y s uc u al esul s o he cha ged sphe es in
wa e de i ed om simula ions wi h sho e unning ime
ha e al eady been p esen ed elsewhe e.19~a!Analyses do no
change signi ican ly o longe simula ion imes, so ha in
his sec ion we will summa ize he mo e impo an esul s, in
o de o help in unde s anding he dynamical esul s p e-
sen ed below. Figu e 2 shows he RDF ~ adial dis ibu ion
unc ion! o ion–oxygen ~a!and ion-hyd ogen ~b!pai s ob-
ained wi h he wo cha ged sphe es and he hyd a ed ion
model. Acco ding o de ini ions, only he Q3S simula ion
gi es in o ma ion on he i s shell. Maxima o he i s
Q3S–O and Q3S–H peaks a e cen e ed a 2.05 and 2.75 Å,
espec i ely, and in eg a es o 9 wa e molecules. This o e -
es ima ion has been p e iously epo ed in simula ions o
ionic solu ions37 and a ibu ed o la ge many-body e ec s.38
The peaks co esponding o he second hyd a ion shell a e
cen e ed a 4.26 Å ~Q3S–O RDF!, 4.02 Å ~Q3B–O RDF!,
and 4.07 Å ~HI–O RDF!and in eg a es o 18 ~Q3S–O!,25
~Q3B–O!, and 14 ~HI–O!oxygen a oms. Ca ion-hyd ogen
RDF o he solu ion con aining Q3S p esen s a wide peak
co esponding o he second shell cen e ed a 4.8 Å ~ he in-
eg a ion numbe is 44!, he Q3B p esen s a wide double
peak a 4–5 Å which in eg a es o ;55 hyd ogen a oms, and
he HI p esen s a wide peak cen e ed a 4.65 Å ~ he in eg a-
ion numbe is ;32!.
FIG. 1. Lenna d-Jones cu es o wa e –so sphe es in e ac ion ene gies.
FIG. 2. Ion–oxygen ~a!and ion–hyd ogen ~b!RDFs o he di e en simu-
la ions.
1671J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
To isualize he ype o a angemen a ound he cha ged
sphe es, andom, bu ep esen a i e, con igu a ions aken
om he p oduc ion pe iod o Q3S and Q3B simula ions a e
p esen ed in Fig. 3,39 oge he wi h a andom con igu a ion
o he HI simula ion. In he Q3B model @Fig. 3~b!# he hy-
d a ion numbe o he second shell is la gely o e es ima ed
~;25!, wa e molecules do no o ien a e hei dipole momen
ec o s owa d he ca ion, which explains he double peak
shown by he Q3B–H RDF @Fig. 2~b!#. The hyd a ion s uc-
u e adop ed is close o hose o cla h a es o alkali me al
ca ions, as sugges ed by looking a Fig. 3~b!.40 This means
ha he concu ence in Q3BW o an iso opic and damped
~second-shell!in e ac ion po en ial p e en s an ex ensi e
b eaking o he wa e s uc u e a ound Q3B. The absence o
p e e en ial in e ac ion si es in Q3B allows he e en ion o a
pa o he wa e –wa e in e ac ions aking place in he bulk.
On he con a y, Q3S is su ounded by wo qui e well-
de ined hyd a ion shells, whe e he 18 molecules o he sec-
ond shell bound by pai s he 9 i s -shell wa e molecules
@Fig. 3~a!#. This is o say, hough o e es ima ed, speci ic
in e ac ions be ween he i s and second shells de ine a num-
be o p e e en ial binding si es in he second-shell egion
which a e esponsible o he highly ca ion-o ien ed wa e
s uc u e. Howe e , due o he o e es ima ion o he i s -
shell hyd a ion numbe , he s uc u al e o is p opaga ed o
he second shell. The hyd a ed ion app oach supplies a ai
o de ing o sol en s uc u e a he second-shell egion on he
basis ha a co ec hyd a ion numbe o he i s shell is
p e iously imposed by he model @Fig. 3~c!#.
B. Dynamical esul s
T ansla ional sel -di usion coe icien s, D, which de-
sc ibe he mobili y o he cha ged sphe es ha e been calcu-
la ed by he mean-squa e displacemen ~MSD!me hod. The
alues ob ained and ha o he C 31hyd a e p e iously ob-
ained unde simila condi ions19~a!a e:
HI:~0.6860.16!31025cm2s21,
Q3S:~0.7460.15!31025cm2s21,
Q3B:~0.4360.09!31025cm2s21.
Likewise, he compu a ion o hese coe icien s om he e-
loci y au oco ela ion unc ion ~VACF!yields alues o D
which a e he same wi hin he unce ain y deg ee o hese
es ima es. The e o e, om he h ee models employed o ep-
esen he C 31ca ion, he big sphe e is he pa icle wi h a
smalle mobili y, whe eas he small sphe e and he hyd a ed
ion show s a is ically equi alen mobili ies.
Con a y o s uc u al and ene ge ical
in o ma ion,6,12,13,41 he e is sca ce de ailed compa ison on
he dynamical beha io o single ions as a unc ion o he
adius size o sphe es4~a!,11,42 and, in case a hyd a e is ec-
ognized, as a unc ion o he shape. The basic ques ion a he
dynamical le el ha may be p oposed e e s o he in luence
ha hese wo basic ea u es ha e on he ionic mobili y. A
i s sigh , wo opposi e causes may be in oked o help in
p edic ing he possible Dsequence. On he one hand, i ion
size was he dominan ac o , i would lead o displacemen s
o Q3B and HI, which ha e he same mass ~160 amu!bu
di e en shape, mo e g ea ly hinde ed han o he much
smalle Q3S, whose mass is ha o he ca ion ~52 amu!, i.e.,
DQ3B'DHI,DQ3S . On he o he hand, i ion–wa e in e ac-
ions we e he dominan ac o , hen hey would be s onge
o Q3S han o Q3B and HI in a wo old sense: Fi s , he
Q3SW po en ial includes in e ac ions o he ion wi h i s -
shell wa e molecules, and second, his po en ial implici ly
FIG. 3. Random snapsho s aken om simula ions con aining he ions Q3S
~a!, Q3B ~b!, and HI ~c!, whe e he ion en i onmen can be obse ed.
1672 J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
o e es ima es in e ac ions due o many-body e ms, so he
expec ed sequence would be DQ3B'DHI.DQ3S . Howe e ,
he sequence ob ained is nei he o hem, bu a he a com-
p omise among hese wo ac o s and an addi ional one:
wa e –wa e in e ac ions among he molecules o ming he
close en i onmen o Q3S ~ he i s hyd a ion shell!as well
as hei in e ac ion wi h ou e sol a ion shells. In his sense,
Lee and Rasaiah11 ha e ecen ly shown by MD simula ions
how is possible o ep oduce he obse ed maximum in he
mobili ies o alkaline ca ions in wa e as a unc ion o size
and in e ac ion ene gies, Rb1being he ca ion wi h he high-
es mobili y along he se ies Li1–Cs1.
As shown in he s uc u al analysis o Q3S–O and
Q3S–H RDFs, and i is now ound again in he dynamical
esul s, ha Q3S goes h ough solu ion su ounded by a igh
i s shell o sol en molecules. In his sense, he small size
o Q3S mus be unde s ood on he basis ha he pa icle o
which i s mobili y is calcula ed is a sphe e ha ing a iple
cha ge and he mass o he ch omium a om. Howe e , he
s ong in e ac ions wi h i s i s shell o wa e molecules
o ces Q3S o mo e h ough he solu ion wi h such a sol en
cosphe e a ached. This is also he case o he HI, whe e a
p io i a i s shell is imposed by he model, bu in his case 6
ins ead o 9 sol en molecules accompany he i alen ca -
ion. Acco ding o hese hyd a ion numbe s, one could expec
ha Q3S mo ed slowe han HI. Howe e , since he wa e
model employed is no pola izable, in e ac ions be ween he
i s and second hyd a ion shells o he HI simula ion a e
s onge han o he Q3S simula ion. When building he
HIW po en ial,18 he i s -shell wa e molecules a e pola ized
because quan um chemical calcula ions deal wi h
@C ~H2O!6]31. Howe e , in he Q3S simula ion, he 9 sol en
molecules o ming he i s shell bea he cha ge dis ibu ion
o a TIP4P H2O. To check his poin he maximum in e ac-
ion ene gy o a TIP4P wa e molecule wi h he
@Q3S~H2O!9]31agg ega e is 2113 kJ/mol. This alue is
smalle han he co esponding alue o HI ~2149.6 kJ/
mol!. The s uc u al in o ma ion goes in he same way: The
dis ance be ween he ion and he oxygen o he molecule
in e ac ing wi h he agg ega e is 4.16 and 4.11 Å o Q3S
and HI, espec i ely. The e o e, bo h esul s show ha in e -
ac ions be ween he i s and second shells a e s onge in he
case o he HI model and hose should ope a e in hinde ing
he mobili y o he ca ion. In his sense, i may be concluded
ha a subs an ial cancella ion o e ec s akes place in he
DQ3S alue, due o he opposi e beha io ha his magni ude
has on he many-body e ms, leading o a nh59, and he
TIP4P model o nonpola izable wa e .
Le us analyze he esul o he big sphe e, Q3B. The
s uc u al in o ma ion gi en by Q3B–O and Q3B–H RDFs
shows a pic u e ha is a om simple, as de i ed om he
peculia s uc u e adop ed by he second hyd a ion shell @Fig.
3~b!#. The cla h a elike s uc u e ound o he ensemble o
wa e molecules a ound he big sphe e hinde s he mobili y
o he ca ion in such deg ee ha i s DQ3B alue becomes he
smalles one o hose s udied.
An addi ional dynamical magni ude desc ibing he en i-
onmen o he ion is he mean esidence ime ~MRT!o
wa e molecules inside he second hyd a ion shell. Calcula-
ions ha e been ca ied ou applying Impey e al.’s me hod43
in he same way as in he p e ious s udy o he C 31hyd a e
in wa e .19~a!Thus, wo alues o *, 0 and 2 ps, ha e been
used aiming o es ablish a MRT ange o alues ha ac-
coun s easonably o his magni ude. Table 1 collec s he
esul s o he wo cha ged sphe es and o he C 31hyd a e.
Analysis o hese da a indica es ha he pe sis ence o wa e
molecules in he second shell is qui e simila o he simula-
ions o Q3S and HI, whe eas ha o Q3B MRTs a e longe .
This is a consequence o he peculia cla h a e s uc u e
adop ed by wa e molecules a ound he big sphe e, e lec ing
he pe sis ence o a well-de ined wa e en i onmen when
compa ing wi h Q3S and HI solu ions. Likewise, a eexami-
na ion o M–O and M–H RDFs ~Fig. 2!shows ha he la g-
es decay in he second– hi d shell ansi ion egion is ound
o Q3B. This ea u e ag ees wi h he dynamical beha io o
he ion as well as wi h he MRT compu ed o he second-
hyd a ion-shell wa e molecules. The la ge second-shell wa-
e molecules’ MRT alues in he Q3B simula ion, joined o
he p e ious s uc u al in o ma ion allows an addi ional
physical ounda ion o unde s and he smalles mobili y o
he big sphe e. Al hough he dynamical beha io is ob ained
by analyzing one sphe e wi h he mass o he C 31hexahy-
d a e, a signi ican pa o he molecules o ming he second
shell accompany he Q3B du ing i s mo ion. Thus, he ac ual
di using objec becomes la ge han he co esponding ionic
en i ies mo ing h ough he Q3S and HI solu ions.
IV. DISCUSSION
The igh binding o wa e molecules a ound ca ion has
long been in oked, in pa icula o highly cha ged ones, o
jus i y he asc ip ion o an ionic adius much la ge han ha
expec ed om i s elec onic s uc u e. The mos common
ame whe e his idea has been used, assumed a simple ep-
esen a ion o he sol en as a pola izable dielec ic con-
inuum o an ensemble o ha d o so sphe es. The dynami-
cal esul s p esen ed he e show ha he use o a big and
uni o mly cha ged sphe e does no gi e a easonable ep e-
sen a ion when desc ibing he mobili y o a hyd a ed ion in a
sol en desc ibed a he mic oscopical le el.
The s anda d Bo n adii a e in ac pa ame e s which y
he minimiza ion o wo impo an sho comings o he Bo n
equa ion: he con inuum ep esen a ion o he sol en ~in ou
case wa e !and he use o an unique dielec ic pe mi i i y
alue o his pola ized con inuum.7,15 When a disc e e ep-
esen a ion o he sol en is used, he dielec ic esponse o
he high elec ic ield de ined by ions in solu ion is implici
in he mic oscopical desc ip ion o he sol en and in he
TABLE I. Mean esidence ime ~ps!o wa e molecules in he second hy-
d a ion shell o he di e en ypes o i alen ca ion model. ~Fo de ini ion
o *see he ex .!
Ca ion model
*Q3S Q3B HI
0 4.760.6 2363661
2206255663265
1673J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45

in e pa icle o ces de ined in he sys em. The local dielec ic
esponse is o pa icula impo ance in he close en i onmen
o he iple cha ge, gi en ha dielec ic dec emen phenom-
ena appea o a la ge ex en .44
I is wo h commen ing on he dielec ic sc eening asso-
cia ed wi h he h ee di e en models o i alen ca ion em-
ployed in his s udy. Fo he HI model, he i s -shell wa e
molecules ha e been quan um-mechanically desc ibed
wi hin he hyd a e. Thus elec onic and nuclea pola iza ion,
as well as pa ial cha ge ans e e ec s a e collec ed in hese
molecules and a e esponsible o impo an sc eening e ec s
on he es o sol en molecules. Addi ionally, sc eening e -
ec s a e implici in he HIW po en ial gi en ha he i ed
ene gies o hyd a e–wa e in e ac ions come om ab ini io
calcula ions whe e he es wa e molecule is pola ized as a
unc ion o he o ien a ion and dis ance o he hyd a e. Fo
he case o he wo sphe es, he only ype o wa e molecule
p esen is TIP4P, so ha mainly indi ec dielec ic sc eening
e ec s a e p esen . These indi ec e ec s ha e a wo old o i-
gin. The i s one is he ab ini io in e ac ion ene gy which
accoun s o he elec onic pola iza ion o he es wa e mol-
ecule, as p e iously men ioned in he case o he HIW po-
en ial. The second o igin is he a p io i de ini ion o he
op imum ion–wa e dis ance, ha is ex e nal da a e lec ing
in some way di ec in e ac ions among he ion and i s i s
wo hyd a ion shells. Since he impo ance o he local di-
elec ic esponse in he i s hyd a ion shell has long been
ecognized,45 esul s o he Q3B model may be unde s ood,
in pa , on he basis o an insu icien local dielec ic e-
sponse wi hin his model. The big sphe e shows o he solu-
ion an homogeneous su ace ha allows o he close wa e
molecules a ce ain deg ee o eedom o imp o e hei in-
e ac ions wi h neighbo ing wa e molecules mo e easily
han i he i s shell ion showed speci ic in e ac ions si es.
The sol en molecules o ming he i s hyd a ion shell o
Q3S make possible he p esence o such speci ic si es and
hen, he Q3SW po en ial includes implici ly some dielec ic
sc eening e ec s in he i s shell. The e o e, he hyd a ion
model o Q3S is an in e media e si ua ion be ween hose o
HI and Q3B. The ensemble o s uc u al and dynamical e-
sul s ein o ces he p e ious conclusion gi en by Hyun
e al.10 on he c i ical impo ance o noncon inuum beha io
o he s uc u e o he sol en in he i s shell o each ea-
sonable es ima es o he ion sol a ion ene gy.
Ou esul s sugges he in iguing pa adox ha he ial
o using a big sphe e o a highly cha ged ion in a sol en
desc ibed a he molecula le el leads o a desc ip ion o he
solu ion e en wo se han ha de i ed om he use o he
in insic ionic adius ~ba e ion!. Bea ing in mind he p ece-
den discussion, his may be asc ibed o he ac ha he
ep esen a ion o he solu e and he sol en is unbalanced.
The pa adoxical ac is ha i was jus o desc ibe hyd a ion
phenomena o ansi ion me al and highly cha ged ions, such
as C 31,Rh
31
,Al
31
,Cu
21
, e c., whe e he use o he hy-
d a ed ion concep was mo e widesp ead.7,9,15 As a conse-
quence, hese cases we e he pa e n aken o associa e ions
in solu ion wi h big sphe es o e ec i e ~and la ge! adii.
Based on he di e gen esul s ob ained wi h he use o Q3B
and he HI model ~Scheme 1!, in addi ion o he weakness o
he Q3B model ela ed o he poo local dielec ic esponse, a
new ac o seems o ge ele ance: he shape o he agg ega e
in solu ion. Thus, he shape o he hyd a ed ion, as shown in
Fig. 4 o he C 31hexahyd a e, is qui e di e en om a
sphe e. When dynamic p ope ies a e e alua ed he shape
appea s as a c ucial ac o . This is con i med by he esul s
ob ained o he small sphe e model which, hough limi ed
by s ong many-body e ec s, is able o ix igh ly sol en
molecules a ound i and impose a mo e adequa e o de o he
ou e sol en s uc u e. Sho - ange aniso opy implici in he
i s -shell egion is a key poin o a co ec desc ip ion o a
single highly cha ged ca ion. In his sense, i is unde s and-
able ha hese ypes o ca ions a e nei he so simple no
eally sphe ical in aqueous solu ions.
F om his conclusion a s a egy o build a low-cos hy-
d a ed ion po en ial ~LCHIW!may be en isaged: he geom-
e y o he LCHI is he same as he HI; he Non-Coulombic
in e ac ion be ween he i s -shell wa e molecules and hose
o he bulk a e desc ibed by he TIP4P po en ial; elec os a ic
cha ges a e hose o he HIW po en ial and he ch omium ion
is only in ol ed in he Coulombic e m o he po en ial. This
p ocedu e o ob aining he in e ac ion po en ial sa es he
la ge numbe o quan um mechanical compu a ions ~;1200!
which a e needed o build he HIW po en ial,16,18 and he i
o he ab ini io in e ac ion ene gies o a sui able unc ional
o m. In his sense, he e m ‘‘low-cos ’’ may be jus i ied, as
LCHIW only needs he quan um mechanical op imiza ion o
he hyd a ed ion plus a i ing o he e ec i e cha ges on he
clus e a oms in o de o ep oduce he elec os a ic po en ial
gene a ed by he molecula wa e unc ion. RDFs de i ed
om a MD simula ion ~150 ps un unde he same condi ions
as he es o p esen ed simula ions!o such a kind o hy-
d a ed ion plus 512 H2O a e shown in Fig. 5 oge he wi h
he dis ibu ions ob ained o he Q3S model. These esul s
show he co ec end in p edic ing he posi ion o he peaks
o he second hyd a ion shell as well as in he in eg a ion
numbe . The key ea u e o a well-beha ed po en ial is hen
a well shape-adap ed and cha ge-dis ibu ed desc ip ion o
he ion and i s close en i onmen .
Bea ing in mind his discussion, he ollowing wo main
ema ks may be concluded.
FIG. 4. Isodensi y su ace o he C 31hexahyd a e.
1674 J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
~1!The desc ip ion o highly cha ged ions in wa e and
o he pola sol en s based on heo y o solu ions migh be
imp o ed i shapes i ing he sol a ed ion a e conside ed. In
his sense, gene al con inuum sol a ion models such as ha
o Tomasi’s g oup ~PCM model!46 e ains his p ope y.
Likewise, Rick and Be ne in he simple case o he aqueous
sol a ion o wa e also conclude he impo ance o he sol-
u e’s shape when compa ing a con inuum me hod wi h MD
simula ions.47
~2!F om compu e simula ions, p ac ical ways migh be
en isaged o in oduce he aniso opy o sol a ed ions by
means o he inclusion o es ic ions o a gi en numbe o
sol en molecules su ounding he ion. E en hough he
sho - ange po en ial o hese sol en molecules is no an
‘‘ad hoc’’ one, i will imp o e he desc ip ion o he sol a-
ion phenomenon. Me bach e al.20 ha e ecen ly adop ed
his ype o s a egy and ound qui e sa is ac o y esul s. This
low-cos line o p ocedu e may be o pa icula ele ance
when dealing wi h sal e ec s on biomolecules o medium
and la ge sizes.3These ypes o po en ials, on he one hand
gua an ee a ai desc ip ion o in e ac ion ene gies once
elimina ed om he la ges pa o many-body e ms, and on
he o he hand, p e en uncon olled dehyd a ion phenomena
and collapse o biomolecule h ee-dimensional s uc u es.
ACKNOWLEDGMENTS
Spanish DGICYT is hanked o inancial suppo
~PB95-0549!and CICA ~Cen o In o ma
´ ico Cien ı
´ ico de
Andalucı
´a! o gene ous alloca ion o compu e esou ces.
J.M.M. hanks he Minis e io de Educacio
´n y Cul u a o
Spain o a p e-doc o al ellowship.
1~a!M. Bo n, Z. Phys. 1,45~1920!;~b!P. Debye and E. Hu
¨ckel, ibid. 24,
185 ~1923!;~c!L. Onsage , ibid. 28, 277 ~1927!;~d!R. M. Fuoss and L.
Onsage , J. Phys. Chem. 66, 1722 ~1962!.
2B. Roux, H.-A. Yu, and M. Ka plus, J. Phys. Chem. 94, 4683 ~1990!.
3B. Honig and A. Nicholls, Science 268, 1144 ~1995!.
4~a!P. G. Wolynes, Annu. Re . Phys. Chem. 31, 345 ~1980!;~b!J. Hub-
ba d and P. G. Wolynes, in The Chemical Physics o Ion Sol a ion, edi ed
by R. R. Dogonadze, E. Kalman, A. A. Ko nyshe , and J. Uls up
~Else ie , Ams e dam, 1985!, P . C, Chap. 1.
5P. G. Wolynes, J. Chem. Phys. 68, 473 ~1978!.
6J.-K. Hyun and T. Ichiye, J. Phys. Chem. 101, 3596 ~1997!.
7W. M. La ime , K. S. Pi ze , and C. M. Slansky, J. Chem. Phys. 7, 108
~1939!.
8A. A. Rashin and B. Honig, J. Phys. Chem. 89, 5588 ~1985!.
9Y. Ma cus, Chem. Re . 88, 1475 ~1988!.
10J.-K. Hyun, C. S. Babu, and T. Ichiye, J. Phys. Chem. 99, 5187 ~1995!.
11S. H. Lee and J. C. Rasaiah, J. Phys. Chem. 100, 1420 ~1996!.
12~a!G. Humme , L. W. P a , and A. E. Ga cı
´a, J. Phys. Chem. 100, 1206
~1996!;~b!J. Chem. Phys. 107, 9275 ~1997!.
13R. M. Lynden-Bell and J. C. Rassaiah, J. Chem. Phys. 107, 1981 ~1997!.
14F. Figuei ido, G. S. Del Buono, and R. M. Le y, J. Phys. Chem. 101, 5622
~1997!.
15~a!R. R. Robinson and R. H. S okes, Elec oly e Solu ions, 2nd ed. ~Bu -
e wo hs, London, 1959!;~b!J. O.’M. Bock is and A. K. N. Reddy,
Mode n Elec ochemis y ~Plenum, New Yo k, 1973!, Vol. 1.
16R. R. Pappala do and E. Sa
´nchez Ma cos, J. Phys. Chem. 97, 4500 ~1993!.
17~a!E. Sa
´nchez Ma cos, J. M. Ma ı
´nez, and R. R. Pappala do, J. Chem.
Phys. 105, 5968 ~1996!;~b!,108, 1752 ~1998!.
18R. R. Pappala do, J. M. Ma ı
´nez, and E. Sa
´nchez Ma cos, J. Phys. Chem.
100, 11748 ~1996!.
19~a!J. M. Ma ı
´nez, R. R. Pappala do, E. Sa
´nchez Ma cos, K. Re son, S.
Dı
´az-Mo eno, and A. Mun
˜
oz-Pa
´ez, J. Phys. Chem. B 102, 3272 ~1998!;
~b!J. M. Ma ı
´nez, R. R. Pappala do, and E. Sa
´nchez Ma cos, J. Chem.
Phys. 109, 1445 ~1998!.
20A. Bleuzen, F. Foglia, E. Fu e , L. Helm, A. E. Me bach, and J. Webe , J.
Am. Chem. Soc. 118, 12777 ~1996!.
21E. Wasse man, J. R. Rus ad, and S. S. Xan heas, J. Chem. Phys. 106, 9769
~1997!.
22~a!X. Pe iole, D. Allouche, J. P. Daudey, and Y. H. Sanejouand, J. Phys.
Chem. B 101, 5018 ~1997!;~b!X. Pe iole, D. Allouche, A. Ramı
´ ez-Solı
´s,
I. O ega-Blake, J. P. Daudey, and Y. H. Sanejouand, ibid. 102, 8579
~1998!.
23~a!F. Flo is, M. Pe sico, A. Tani, and J. Tomasi, Chem. Phys. Le . 199,
518 ~1992!;~b!Chem. Phys. 195, 207 ~1995!.
24M. N. D. S. Co dei o and J. A. N. F. Gomes, J. Compu . Chem. 14, 629
~1993!.
25A. P. Lyuba se and A. Laaksonen, Phys. Re . E 55, 5689 ~1997!.
26W. L. Jo gensen, R. W. Impey, J. Chand asekha , J. D. Madu a, and M. L.
Klein, J. Chem. Phys. 79, 926 ~1983!.
27K. Re son, MOLDY Use ’s Manual Re . 2.10 MOLDY code can be ob ained
om he CCP5 p og am lib a y o by anonymous p om p.ea h.ox-
.ac.uk.
28D. Beeman, J. Compu . Phys. 20, 130 ~1976!.
29K. Re son, PhysicaB&C131, 256 ~1985!.
30H. Golds ein, Classical Mechanics, 2nd ed. ~Addison-Wesley, Reading,
MA, 1980!.
31M. P. Allen and D. J. Tildesley, Compu e Simula ions o Liquids ~Ox o d
Uni e si y P ess, Ox o d, 1987!.
32J. E. Robe s and J. Schni ke , J. Phys. Chem. 99, 1322 ~1995!.
33J. D. Madu a and B. M. Pe i , Chem. Phys. Le . 150, 105 ~1988!.
34L. Pe e a, U. Essmann, and M. L. Be owi z, J. Chem. Phys. 102, 450
~1995!.
35B. Quen ec and C. B o , J. Compu . Phys. 13, 430 ~1975!.
36D. W. He mann, Compu e Simula ion Me hods, 2nd ed. ~Sp inge , Be lin,
1990!.
37~a!M. M. P obs , E. Spoh , and K. Heinzinge , Chem. Phys. Le . 161,
405 ~1989!;~b!Mol. Simul. 7,43~1991!;~c!A. Gonza
´lez-La on , J. M.
Lluch, A. Oli a, and J. Be a
´n, Chem. Phys. 111, 241 ~1987!;~d!M. N.
D. S. Co dei o, J. A. N. F. Gomes, A. Gonza
´lez-La on , J. M. Lluch, and
J. Be a
´n, ibid. 141, 379 ~1990!.
38~a!L. A. Cu iss, J. W. Halley, J. Hau man, and A. Rahman, J. Chem.
Phys. 86, 2319 ~1987!;~b!N. J. El od and R. J. Saykally, Chem. Re . 94,
1975 ~1994!.
39G. Scha enaa , MOLDEN: A po able Elec on Densi y P og am, QCPE
P og am 619; QCPE Bull. 12,3~1992!.
40~a!A. Selinge and A. W. Cas leman, J ., J. Chem. Phys. 95, 8442 ~1991!;
~b!J. Cioslowski and A. Nanayakka a, In . J. Mod. Phys. B 23&24, 3687
~1992!;~c!J. Lipkowski, Annu. Rep. P og. Chem., Sec . C: Phys. Chem.
92, 307 ~1996!.
41M. I. Be nal-U uchu u and I. O ega-Blake, J. Chem. Phys. 103, 1588
~1995!.
42Th. Kowall, F. Foglia, L. Helm, and A. E. Me bach, J. Am. Chem. Soc.
117, 3790 ~1995!.
FIG. 5. Ion–oxygen and ion–hyd ogen RDFs o he simula ions con aining
he low-cos hyd a ed ion ~LCHI!and he small cha ged sphe e ~Q3S!.
1675J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45
43~a!A. W. Impey, P. A. Madden, and I. R. McDonald, J. Phys. Chem. 87,
5071 ~1983!;~b!A. E. Ga cı
´a and L. S ille , J. Compu . Chem. 14, 1396
~1993!.
44J. M. G. Ba hel, H. K ienke, and W. Kunz, Physical Chemis y o Elec-
oly e Solu ions ~Sp inge , Da ms ad , 1998!, p. 104.
45R. Po el, in Wa e . A Comp ehensi e T ea ise, edi ed by F. F anks ~Ple-
num, New Yo k, 1973!, Vol. 3, Chap. 8.
46~a!S. Mie us, E. Sc occo, and J. Tomasi, Chem. Phys. 55, 117 ~1981!;~b!
S. Mie us and J. Tomasi ibid.,65, 239 ~1982!.
47S. W. Rick and B. J. Be ne, J. Am. Chem. Soc. 116, 3949 ~1994!.
1676 J. Chem. Phys., Vol. 110, No. 3, 15 Janua y 1999 Ma ı
´nez, Pappala do, and Ma cos
Reuse o AIP Publishing con en is subjec o he e ms: h ps://publishing.aip.o g/au ho s/ igh s-and-pe missions. Downloaded o IP: 150.214.182.116 On: Thu, 20 Oc
2016 14:52:45