Shape and size of simple cations in aqueous solutions: A theoretical reexamination of the hydrated ion via computer simulations

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