Reorganization energies for charge transfer reactions in binary mixtures of dipolar hard sphere solvents: A Monte Carlo study

J. Chem. Phys. 110, 473 (1999); h ps://doi.o g/10.1063/1.478108 110, 473 © 1999 Ame ican Ins i u e o Physics. Reo ganiza ion ene gies o cha ge ans e eac ions in bina y mix u es o dipola ha d sphe e sol en s: A Mon e Ca lo s udy Ci e as: J. Chem. Phys. 110, 473 (1999); h ps://doi.o g/10.1063/1.478108 Submi ed: 07 July 1998 . Accep ed: 25 Sep embe 1998 . Published Online: 21 Decembe 1998 C. Denk, M. Mo illo, F. Sánchez-Bu gos, and An onio Sánchez ARTICLES YOU MAY BE INTERESTED IN On he Theo y o Oxida ion-Reduc ion Reac ions In ol ing Elec on T ans e . I The Jou nal o Chemical Physics 24, 966 (1956); h ps://doi.o g/10.1063/1.1742723 On he Theo y o Elec on-T ans e Reac ions. VI. Uni ied T ea men o Homogeneous and Elec ode Reac ions The Jou nal o Chemical Physics 43, 679 (1965); h ps://doi.o g/10.1063/1.1696792 Elec on–elec on and elec on-hole in e ac ions in small semiconduc o c ys alli es: The size dependence o he lowes exci ed elec onic s a e The Jou nal o Chemical Physics 80, 4403 (1984); h ps://doi.o g/10.1063/1.447218 Reo ganiza ion ene gies o cha ge ans e eac ions in bina y mix u es o dipola ha d sphe e sol en s: A Mon e Ca lo s udy C. Denka) and M. Mo illo Uni e sidad de Se illa, Fı ´sica Teo ´ ica, Apa ado 1065, E-41080 Se illa, Spain F. Sa ´nchez-Bu gos and An onio Sa ´nchez Uni e sidad de Se illa, Quı ´mica Fı ´sica, Facul ad de Quı ´mica, C/ P o eso Ga cı ´a Gonza ´lez s/n, E-41012 Se illa, Spain ~Recei ed 7 July 1998; accep ed 25 Sep embe 1998! We s udy he beha io o he eo ganiza ion ene gy o simple cha ge ans e eac ions in mix u es o dipola ha d sphe e luids by Mon e Ca lo simula ion. The s a ic dielec ic cons an s o he sol en s a e also ob ained om he simula ion. They a e used as inpu in he eo ganiza ion ene gy exp essions p o ided by he Ma cus heo y and he mean sphe ical app oxima ion. Thus, a compa ison be ween he alues ob ained om he heo e ical exp essions and ou simula ion esul s is possible. The dependence o he eo ganiza ion ene gy wi h he mix u e composi ion and he in luence o p e e en ial sol a ion e ec s is also discussed. © 1999 Ame ican Ins i u e o Physics. @S0021-9606~99!51401-3# I. INTRODUCTION Elec on ans e ~ET! eac ions in pola sol en s a e im- po an in many biological and chemical p ocesses. Sol en luc ua ions p o ide he ansi ion s a e con igu a ions o he sol en –solu e sys em necessa y o he ET. Ma cus1–4 has gi en a physical pic u e o ET eac ions, desc ibing he mul- idimensional ee ene gy su aces o he p oduc and eac- an s a es as pa abolic su aces in e ms o a sui able eac- ion coo dina e. He ela ed he ac i a ion ee ene gy DG‡ o he eac ion ee ene gy DGand he sol en eo ganiza ion ene gy l. Ma cus also de i ed a simple exp ession o l, based on a mac oscopic ea men o he sol en . The eo - ganiza ion ene gy lis hen gi en in e ms o he s a ic and op ical dielec ic cons an s o he sol en and a geome ical ac o . Al hough he dielec ic con inuum model ~Ma cus o - mula!p o ides an adequa e quali a i e desc ip ion o he e- o ganiza ion ene gy, i s quan i a i e p edic ions a e o en imes a a iance wi h he expe imen al indings.5–7 Molecu- la desc ip ions o he sol en a e, in p inciple, capable o o e coming some o he di icul ies associa ed wi h a mac- oscopic ea men o he sol en . The mean sphe ical ap- p oxima ion ~MSA! ea men o he eo ganiza ion ene gy ecognizes he sol en molecula i y by desc ibing he sol en molecules as ha d sphe es wi h poin dipoles in hei cen e s. Wi hin he MSA, an exp ession o lhas been de eloped ha includes he ha d sphe e adius o he sol en molecules.8,9 The need o a mic oscopic desc ip ion is o pa icula in e es in mix u es o pola sol en s. Fo mos mix u es one obse es a nonideal ~nonlinea !beha io o he sol a ion en- e gy o an ionic o dipola solu e wi h espec o he mola ac ions o he species p esen in he sol en .10 This beha - io is usually e med as p e e en ial sol a ion. Fo ET eac- ions in mix u es o pola sol en s, one migh also expec an in luence o p e e en ial sol a ion e ec s on he sol en e- o ganiza ion ene gy l. Bo h op ical11 and he mal12,13 ex- pe imen al da a o ET eac ions in mix u es o wa e and o ganic cosol en s show a beha io o l ha canno be ex- plained wi h a con inuum model o he sol en . Analy ical app oxima ions o he mic oscopic desc ip ion o he sol a- ion ene gy in pola mix u es ha e been p esen ed in he li e a u e.10,14 In his pape we will in es iga e he beha io o he e- o ganiza ion ene gy lin pola sol en s by means o Mon e Ca lo ~MC!simula ions. A quan i a i e p edic ion o expe i- men al da a12,13 would equi e e y sophis ica ed simula ion echniques. A his poin we a e a he in e es ed in he gen- e al beha io o l. Fo his eason we ha e adop ed a simple sol en model, wi h as ew adjus able pa ame e s as possible. Ou sol en will be modeled ei he as a pu e sol en o as a bina y mix u e. In bo h cases we will conside dipola , non- pola izable ha d sphe e molecules. The solu e consis s o wo cha ged ha d sphe es. We will s udy cha ge sepa a ion and cha ge ecombina ion p ocesses o his sys em. As he sol- en molecules a e conside ed nonpola izable, he op ical di- elec ic cons an will be e op 51 o all sol en s. In o de o compa e he nume ical esul s o lwi h he heo e ical p e- dic ions, we need o e alua e he s a ic dielec ic cons an . Ex ensi e simula ions ha e been ca ied ou in o de o de- e mine e 0 o ou di e en sol en models. The ou line o he pape is as ollows: In Sec. II we desc ibe ou sol en model and e iew he mos impo an simula ion de ails. In Sec. III we discuss he e alua ion o he dielec ic cons an s o he sol en models used in his wo k. In Sec. IV, he applica ion o he MC echnique o he calcula ion o ee ene gy su aces o cha ge ans e eac- ions is summa ized. The esul s a e p esen ed and discussed in Sec. V. A compa ison wi h he heo e ical esul s as ob- a!Elec onic mail: [email p o ec ed] JOURNAL OF CHEMICAL PHYSICS VOLUME 110, NUMBER 1 1 JANUARY 1999 4730021-9606/99/110(1)/473/11/$15.00 © 1999 Ame ican Ins i u e o Physics ained o he con inuum sol en model ~Ma cus!and he mic oscopic desc ip ion ~MSA!is made. Finally, Sec. VI con ains some concluding ema ks. II. MODEL AND MC SIMULATIONS Ou sol en is modeled as a liquid o dipola , nonpola - izable ha d sphe e molecules. Two sol en molecules iand k in e ac ia a long anged dipole–dipole in e ac ion: Vi,k5 m i– m k ik 323~ m i– ik!~ m k– ik! ik 5 ik>2 s~1! and a epulsi e sho anged in e ac ion ~ he ha d sphe e po- en ial, Vi,k5` o ik5 u ik u 5 u i2 k u ,2 s). He e, sis he adius o he ha d sphe es, m iis he dipole momen and iis he posi ion o molecule i. In his wo k we s udy pu e sol en s and bina y mix u es o dipola ha d sphe e luids. The adius o he sol en mol- ecules is aken o be he same o all sol en componen s. A packing ac ion o h 50.417 co esponding o a dense liquid was used in all simula ions. The pola i y o a pu e sol en will be exp essed in e ms o he dimensionless pa ame e y54 p m 2 9kBT,~2! whe e is he densi y o he liquid. A mix u e o wo sol- en s will be composed o wo species, he less pola species L, cha ac e ized by i s pola i y yL, and ano he species H wi h a highe pola i y yH. The composi ion o he mix u es will be desc ibed by he mola ac ions o he mo e pola species H h oughou his wo k. Pe iodic bounda y condi ions wi h he minimum image con en ion15,16 we e applied o a cubic simula ion box o side leng h L. This me hod equi es he use o a cu o o he long anged dipola in e ac ions. In o de o accoun o he con ibu ions beyond he cu o adius c<L/2 we ha e adop ed he gene alized eac ion ield me hod.17 We conside he mo ing bounda y dielec ic implemen a ion, i.e., he sub- sys em inside he cu o sphe e a ound each molecule is hough o be imme sed in a con inuum dielec ic cha ac e - ized by a mac oscopic s a ic dielec ic cons an e RF . A mol- ecule in e ac s di ec ly wi h all molecules inside i s cu o sphe e ia Eq. ~1!and wi h he eac ion ield. The eac ion ield a he cen e o molecule idue o he molecules inside he cu o sphe e su ounding i is p opo ional o he o al dipole momen inside i s cu o sphe e16 m i–Ei 52~ e RF21! 2 e RF11 1 c 3 m i•( ik< c m k5 am i•( ik< c m k,~3! whe e we ha e de ined a sc eening cons an a . Summing up all e ms con ibu ing o he o al po en ial ene gy we ob ain E o 51 2( kÞi, ik< c Vi,k dd 21 2 a ( i m i 2~4! wi h an e ec i e in e ac ion ene gy Vi,k dd 5 m i– m k~12 a ik 3! ik 323~ m i– ik!~ m k– ik! ik 5.~5! The las e m in Eq. ~4!desc ibes he cons an sel - in e ac ion o he molecules wi h hei own eac ion ields. The o m o he e ec i e in e ac ion in Eq. ~5!is e y con- enien o compu a ional pu poses. The inclusion o he e- ac ion ield only equi es wo ex a loa ing poin ope a ions o each e alua ion o he in e ac ion ene gy, which ende s his me hod compu a ionally easible, e en o ela i ely la ge sys ems. I has been shown ha he compu a ionally mo e expensi e Ewald summa ion echnique yields equi a- len esul s o he dielec ic cons an when applied o simila sys ems.18,19 Analogous conclusions ha e been d awn o ee ene gy calcula ions o ionic hyd a ion.20 Sys em con igu a ions we e gene a ed in he ollowing manne : s a ing om a gi en con igu a ion, a molecule was selec ed andomly. I was displaced om i s ini ial posi ion in a cube o side leng h Dxwi h a uni o m dis ibu ion. Now, a o a ional axis (x,yo z) was selec ed a andom and he dipole o ien a ion o he molecule was o a ed by a uni o mly dis ibu ed angle in he ange 2D < <D a ound his axis. The pa ame e s Dxand D we e adjus ed o each an accep ance a io o app oxima ely 30%. The implemen a ion o he simula ion is along he lines o he s anda d Mon e Ca lo echniques.15,16 III. DIELECTRIC CONSTANT The applica ion o he s a is ical mechanical heo y o he dielec ic cons an 21,22 o ini e size simula ion sys ems wi h bounda y condi ions equi es some modi ica ions. This issue has been lucidly add essed by Neumann.18 F om his analysis i ollows ha he s a ic dielec ic cons an e 0in he eac ion ield ~RF!geome y is gi en by e 052 e RF~11 z !11 112 e RF2 z ,~6! wi h z 54 p 3 b ^ M2 & L353ygK.~7! He e ^ M2 & is he con igu a ional a e age o he squa ed o al dipole momen o he sys em and b 51/kBT. We ha e also indica ed he ela ion o z wi h he pola i y yand he Ki k- wood g ac o gK5 ^ M2 & /N m 2. The dielec ic cons an e RF ha cha ac e izes he eac ion ield was de e mined sel - consis en ly in he simula ions, so ha e RF' e 0 o all sol- en s. The sol en molecules we e ini ially p epa ed in an cc s uc u e and he sys em was hen allowed o elax du ing 53106MC con igu a ions. Mean alues we e ob ained om NMC553108subsequen MC con igu a ions, excep whe e o he wise s a ed. In o de o ob ain an es ima e o he s a is- ical e o s, we calcula ed mean alues o z o e blocks o 106con igu a ions. We ca ied ou a ious es s o check he con e gence o he alues o he dielec ic cons an ob ained in ou simula ions. In Fig. 1 we show he unning a e ages o e 0 o wo sol en s wi h y52.18 and y52.90 o wo di e - en alues o he maximum o a ional angle D . In he case o he sol en wi h he highe pola i y, he alues o e 0con- e ge e y slowly. E en o NMC553108con igu a ions we 474 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. s ill ind a de ia ion o D e 051.6 be ween he wo simula- ions ~ heo e ically, he simula ion esul s should be indepen- den o D ). The s a is ical e o s es ima ed using he block a e ages we e D e 050.4 o simula ion ~a!and D e 050.5 o simula ion ~b!. This indica es ha , e en wi h 53108MC con igu a ions, he phase space o his highly pola sys em has no been su icien ly explo ed o ob ain p ecise alues o e 0. The de ia ion o he wo cu es will be used as an es i- ma e o he e o o he dielec ic cons an ~app oxima ely 3%! o sol en s wi h high pola i y. Sol en s wi h lowe mo- lecula dipole momen s show a much as e con e gence o e 0. Fo sol en s wi h a dielec ic cons an e 0<30 we ound 23108MC con igu a ions o be su icien o de e mine e 0 wi h a ela i e e o o app oxima ely 3%. We ha e also s udied he dependency o e 0on he sys- em size. Simula ion ~a!o Fig. 1 was epea ed o a sys em wi h N5864 pa icles; he esul ing alue o e 0is shown in Table I @simula ion ~e!#. The esul s indica e ha e 0is inde- penden o he size o he sys em o N>256 wi hin he e o limi s. As al eady men ioned, he pa ame e e RF was adjus ed sel -consis en ly by epea ing each simula ion, using he e- sul e 0o a simula ion as an inpu pa ame e e RF in he nex simula ion. In all cases, i was su icien o epea each simu- la ion only once, as e 0depends e y weakly on e RF .In simula ion ~ !~see Table I!we ha e epea ed simula ion ~b! wi h e RF558. The esul s a e p ac ically iden ical. The dielec ic cons an e 0was hen de e mined o a pu e sol en o e a wide ange o alues o he molecula pola i y y. A sys em wi h N5256 pa icles was used in hese simula ions and he mean alues o e 0we e ob ained a e 53108MC con igu a ions. Fo he highes pola i y, y 53.0, we inc eased he numbe o MC con igu a ions o NMC5109 o he easons men ioned abo e. The esul s a e shown in Fig. 2. The heo y o liquids p o ides a ious app oxima ions o he s uc u e o he dipola ha d sphe e luid.23 O pa - icula in e es is he so-called MSA, as i p o ides analy ical exp essions o he co ela ion unc ions and he dielec ic cons an .24 We ind i ins uc i e o compa e ou simula ion esul s wi h he MSA heo e ical p edic ions. The pai dis i- bu ion unc ion can be expanded as23 h~1,2!5hS~R!1hD~R!D~1,2!1hD~R!D~1,2!,~8! whe e D~1,2!is he cosine o he angle o med by he dipole o ien a ions o wo molecules and 2D(1,2) m 2/R3is he dipole–dipole in e ac ion as de ined in Eq. ~1!. The MSA p o ides he unc ions hS(R),hD(R) and hD(R) in e ms o he adial dis ibu ion unc ion ~ d !o he Pe cus–Ye ick ~PY!solu ion o ha d sphe es a di e en densi ies. In Fig. 3 we show he adial dis ibu ion unc ion gS(R)5hS(R)11 o a highly pola sol en (y53.00). Sol- en s wi h a lowe pola i y ha e a e y simila d wi h a sligh ly lowe main peak. The MSA esul o gS(R)~also shown in Fig. 3!is jus he PY d o ha d sphe es a densi y and does no depend on he molecula pola i y. Al hough he ag eemen is globally good, he e a e de ia ions be ween he MSA and he simula ions. In he egion close o con ac , he MSA subes ima es he alue o gS(R), which is o p i- ma y impo ance o many he modynamic p ope ies o he liquid. I also p edic s a somewha slowe decay om he peak alue o he i s minimum when compa ed wi h he simula ions. The dielec ic p ope ies o he sol en a e ela ed o hD(R), which desc ibes he angula co ela ion o wo mol- ecules a a gi en dis ance R. The dielec ic cons an e 0is FIG. 1. Running a e ages o e 0 o ~a!,~b!y52.90 and ~c!,~d!y52.18. The sys ems we e simula ed wi h wo di e en alues o he maximum o a ional angle D 5 p /2 ~a!and ~c!and D 51.0 ~b!and ~d!. FIG. 2. The s a ic dielec ic cons an e 0 o a pu e sol en . The e o ba s o he simula ion esul s indica e he es ima ed ela i e e o o 3% ~see main ex !. The solid line is a ou h deg ee in e pola ion polynomial. The do ed line co esponds o he heo e ical esul as ob ained om he MSA. TABLE I. Dielec ic cons an s ob ained o sol en s wi h y52.90 ~a!,~b!,~e!,~ !and y52.18 ~c!,~d! e 0NMC ND e RF ~a!59.68 53108256 p /2 70 ~b!58.05 53108256 1.0 70 ~c!28.29 53108256 p /2 30 ~d!29.05 53108256 1.0 30 ~e!58.41 23108864 p /2 70 ~ !58.78 53108256 1.0 58 475J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. de e mined by he pola i y yand he Ki kwood g ac o gK @see Eq. ~6!#. The Ki kwood g ac o gKcan be ob ained om hD(R) by in eg a ion gK5114 p 3 E 0 `hD~R!R2dR.~9! In Fig. 4 we show hD(R) o a highly pola sol en (y 53.0) as ob ained om ou simula ions and he co espond- ing MSA esul . The MSA does no p o ide a good app oxi- ma ion o hD(R) as i unde es ima es he angula co ela ion. This esul s in an unde es ima ion o gKand hus e 0. This de ia ion is mo e p onounced o sol en s wi h a high pola - i y ~see Fig. 2!. Bu , e en o sol en s wi h he lowes pola - i y unde conside a ion in his wo k, he MSA esul o hD(R) does no ag ee wi h he simula ion da a. The dielec ic cons an as ob ained by he MSA gi es a good app oxima- ion o e 0only o sol en s wi h gK'1. These limi a ions o he MSA a e well known and mo e sophis ica ed heo ies based on he hype ne ed chain ~HNC!app oxima ion p o- ide much be e angula co ela ion unc ions and dielec ic cons an s o dipola ha d sphe e luids. Compa isons o he MSA and o he heo ies can be ound in he li e a u e.23,25–33 Le us now conside he case o bina y sol en mix u es. We s udy wo di e en ypes o mix u es: ~A!mix u es wi h wo componen s o simila pola i ies (yH53.0 and yL 52.18), and ~B!mix u es whose componen s a e o a he di e en pola i y (yH53.0 and yL50.75). Fo each ype o mix u e, he dielec ic p ope ies will only depend on i s mo- la composi ion ~de ined by he mola ac ion o species H, H), i he empe a u e is held ixed. This gi es us he pos- sibili y o s udying wo qui e di e en mix u es, spanning a wide ange o dielec ic cons an s. The simula ion p ocedu e is he same as o pu e sol en s, and he dielec ic cons an is e alua ed by using Eqs. ~6!and ~7!. In Fig. 5 we show he dielec ic cons an s as ob ained o he simula ed composi ions o mix u es o ype ~A!and ~B!. The alues o e 0 o he pu e sol en s ( H50 and H51) we e aken om he simula ions desc ibed abo e ~see Fig. 2!. Fo bo h mix u es we obse e an almos quad a ic depen- dence o he s a ic dielec ic cons an on he mola ac ion H. This beha io can be unde s ood by no icing ha in he case o a bina y mix u e, ^ M2 & /Nis o a good app oxima ion a quad a ic unc ion o he mola ac ion H. Fo e 0 ' e RF , i ollows om Eq. ~6! ha he dielec ic cons an is essen ially a linea unc ion o z , which in u ns is p opo - ional o ^ M2 & /Nas can be seen in Eq. ~7!. IV. REORGANIZATION ENERGIES A. Calcula ion o he eo ganiza ion ene gy om molecula simula ions In Sec. III we ha e analyzed he dielec ic beha io o dipola ha d sphe e sol en s. We will now s udy he si ua ion when a solu e is imme sed in he sol en . Ou aim he e is o s udy he ene ge ics o he mal cha ge ans e eac ions be- ween wo solu e molecules in he p esence o a pola sol- en . We ha e adop ed a simple model o he solu e: The solu e consis s o wo ha d sphe e molecules ~dono and ac- cep o !wi h gi en adii dand asepa a ed by a ixed dis- FIG. 3. The adial dis ibu ion unc ion gS(R)(R5 /2 s) o a sol en wi h y53.00 ~ illed ci cles!. The do ed line ep esen s gS(R) as ob ained om he MSA. FIG. 4. The angula co ela ion unc ion hD(R) o a highly pola sol en (y53.0, illed ci cles!. The do ed line co esponds o hD(R) as p o ided by he MSA. FIG. 5. S a ic dielec ic cons an e 0 o bina y mix u es o dipola ha d sphe es wi h yL52.18 @mix u e ~A!, ci cles#and yL50.75 @mix u e ~B!, squa es#in e ms o he mola ac ion H. In bo h cases yH53.0. The solid lines ep esen quad a ic in e pola ion polynomials. 476 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. ance d. The dono and accep o in he eac an s a e ca y a cha ge qdand qa, espec i ely; a nega i e ne cha ge ans- e s om he dono o he accep o . We will conside he eac ion coo dina e o his p ocess in e ms o a cha ging pa ame e j , so ha qd~ j !5qd1 j e,~10! qa~ j !5qa2 j e.~11! He e, eis he elemen a y cha ge ~ ha we will ake as posi- i e!. The eac an s a e o he solu e is gi en by j 50, while he p oduc s a e is ob ained by se ing j 51. In his wo k we ha e s udied wo ypical ans e eac ions: ~i!a cha ge e- combina ion ~CR!p ocess qd52eand qa51eand ~ii! he in e se p ocess, a cha ge sepa a ion ~CS!p ocess (qd5qa 50). The a es o hese p ocesses a e go e ned by an ac i- a ion ee ene gy law. Ma cus1–4 ob ained o he nonadia- ba ic elec on ans e a e ke 5 k A 4 p l/ b exp~2 b DG‡!,~12! whe e he ac i a ion ee ene gy DG‡is de ined as DG‡5~DG1l!2 4l.~13! He e, k is a ma ix elemen desc ibing he elec onic cou- pling be ween eac an and p oduc s a e, and DGand l deno e he ee-ene gy change o he eac ion and he eo - ganiza ion ene gy, espec i ely. Equa ion ~12!was de i ed unde he assump ion o a classical sol en ha esponds linea ly o a edis ibu ion o cha ges. The p ocedu e used o he e alua ion o eo ganiza ion ene gies in cha ge ans e eac ions om simula ions is well documen ed.34–36 He e we will b ie ly indica e he main poin s. Le H j deno e he sol en –solu e in e ac ion ene gy o a solu e s a e cha ac e ized by he cha ging pa ame e j and a ce ain ixed sol en con igu a ion. The ene gy gap DV5H12H0desc ibes he ene gy di e ence be ween p od- uc s and eac an s o a gi en con igu a ion o he sol en . An elec on ans e will ake place only o sol en con igu- a ions ha ul ill he condi ion DV52Ei~F ank–Condon p inciple!, whe e Eiis he in insic ene gy di e ence be ween he gas phase elec onic s uc u es o he ini ial and inal s a e o he solu e. Le us de ine he andom a i- able D5DV, wi h a p obabili y law gi en by p j (D) 5 ^ d (D2DV) & j , whe e he angula b acke s indica e an equilib ium a e age aken wi h a canonical dis ibu ion de- sc ibing a sys em a empe a u e Tand Hamil onian H j . The a e o he ET p ocess is p opo ional o he p obabili y den- si y p0(2Ei) o he ene gy gap DVha ing a alue 2Ei. This p obabili y dis ibu ion co esponds o a sol en in he - modynamical equilib ium wi h he eac an s a e o he sol- u e, j 50. The main p oblem in simula ions lies in he cons uc ion o his p obabili y densi y as he en i e phase space o he sol en deg ees o eedom has o be explo ed. This is an imp ac ical ask, and one eso s o a ee ene gy pe u ba ion me hod,34 which is based on he ollowing. Fo a gi en alue o he ac ional cha ge pa ame e j ,DVis sampled a ound a alue D j wi h a dis ibu ion ha is app oxima ed e y well by a Gaussian p j ~D!51 A 2 ps j 2e2~D2D j !2/2 s j 2.~14! Va ious s a es j o he solu e a e simula ed, each simula ion p o iding a p obabili y dis ibu ion p j (D) a ound D j . These dis ibu ions can be pieced oge he 34 o yield he dis ibu ion p0(D) o e a wide ange o alues D. Zhou and Szabo35 ha e p oposed a simpli ied me hod ha pe mi s ob aining p0(D) by simula ing only wo s a es o he solu e: j 50 and j 51. In hei wo k, he eo ganiza ion ene gy is gi en by l5D02DG10 ,~15! wi h he ee-ene gy di e ence be ween s a e j and he eac- an s a e o he solu e DG j 05 E 0 j d j 8D j 8.~16! The mean alue o he ene gy gap D j o a ce ain alue o he cha ging pa ame e j can be ob ained by a cubic in e - pola ion polynomial in j , whe e all coe icien s a e de e - mined by he mean alues D0and D1and he dispe sions b s 0 2and b s 1 2in he eac an and p oduc s a e, espec i ely. Inse ing he polynomial exp ession o Zhou and Szabo in Eqs. ~15!and ~16!leads o lCR51 2~D02D1!11 12 ~ b s 0 22 b s 1 2!,~17! lCS51 2~D02D1!21 12 ~ b s 0 22 b s 1 2!.~18! As poin ed ou by Zhou and Szabo, he accu acy o he in- e pola ion polynomial app oxima ion o D j can be checked by ca ying ou an addi ional simula ion o j 51/2. The alue o he ene gy gap D1/2 ob ained is hen compa ed o he alue p o ided by he in e pola ion o mula. We ound a de ia ion o less han 1% o all cases conside ed he e. F om he in e pola ion polynomial, he p obabili y densi y p0(D) can also be cons uc ed.35 B. Simula ion de ails The inclusion o he cha ged solu e pa icles in he simu- la ion in oduces a di icul y when dealing wi h sys ems o ini e size: The o ien a ion o he dipole momen s o he sol- en molecules is aniso opic and he pe iodic eplica ion o he cen al simula ion cell does no ep esen a physical pic- u e o he sol en . This issue has been widely discussed in he li e a u e37 and a ious me hods ha e been de eloped o a oid pe iodic bounda y condi ions ~pbc! o such sys ems. Sphe ical simula ion cells in conjunc ion wi h a sui able me hod o a oid sel -pola iza ion on he su ace o he cell a e usually employed.38 We ha e conside ed simula ions wi h pe iodic bounda y condi ions and wi hin a sphe ical geome y. In he pbc simu- la ions, he cha ge–dipole in e ac ions can be desc ibed ia an e ec i e in e ac ion po en ial ha includes he eac ion ield in a simila manne o Eq. ~5!: 477J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. Vi,j qd5qi S 2 ij 2 a D m j– ij,~19! whe e a is he sc eening cons an as de ined in Eq. ~3!. The solu e cha ges in e ac wi h all dipoles inside hei co e- sponding cu o sphe es. The cu o adius o he cha ge– dipole in e ac ions was se o be he same as o he dipole– dipole in e ac ions. In he sphe ical model, he simula ion cell is a sphe ical essel wi h he solu e si ua ed a he cen e o he essel. Each molecule in e ac s wi h all molecules inside he essel and elec os a ic in e ac ions a e aken ully in o accoun @se a 50 in Eqs. ~5!and ~19!#. The simula ion sphe e is di ided in o wo egions: an inne sphe e, whe e he molecules a e allowed o mo e and o a e and an ou e shell, whe e he molecules a e kep ixed du ing he simula ion. The mol- ecules cons i u ing he ou e shell a e si ua ed ini ially in a cc s uc u e wi h andomly dis ibu ed dipole momen s. The ixed dipoles in he ou e shell p e en an unphysical pola - iza ion o he dipoles close o he su ace o he essel. A co ec ion o he ene gy gap D0has o be applied in he case o he sphe ical model. Con ibu ions om he sol en ou - side he sphe e can be pa ially aken in o accoun by con- side ing he sphe e o be imme sed in a con inuous dielec ic medium wi h dielec ic cons an e ou . The in luence o his dielec ic medium is no aken in o accoun du ing he simu- la ion ~as i would equi e he ull solu ion o he Laplace equa ion inside he sphe e o each con igu a ion!, bu i s e ec on D0may be es ima ed by elec os a ic conside - a ions. Fo a CR p ocess, he con ibu ions o D0 om ou - side he simula ion cell wi h adius Ra e D0 ou 52~ e ou 21! 2 e ou 11 ~ed!2 R3,~20! whe e dis he dis ance be ween he solu e cha ges. We s udied he dependency o D0on he size o he sys em o bo h models. A highly pola sol en (y52.90) was used o sol a e wo solu e ions wi h equal adii d5 a 5a51.44Å, cha ges qd52e,qa51eand sepa a ed by a dis ance o d56 Å. The sol en adius was se o be he same as he solu e adius and a packing ac ion o h 50.417 was used. In he case o he pbc geome y we simula ed sys ems wi h N5246 490 854 sol en molecules. The cu o adius o he dipole–dipole and cha ge–dipole in e ac ions was se o c5min(L/2,8 s) o each sys em, whe e Lis he side- leng h o he cubic simula ion cell. Fo he sys em wi h N 5854 sol en molecules we conduc ed a simula ion wi h he ull cu o c5L/2 in o de o check he in luence o he educed cu o on D0. In he sphe ical geome y we simula ed he same sys em wi h a o al o N o 5673 sol en molecules. In his case, he adius o he inne sphe e ~whe e he molecules a e allowed o mo e!was a ied. The esul s a e combined in Table II. Fo small sys em sizes, he wo geome ies p oduce qui e di e en alues o he ene gy gap D0. The pbc simula ions o e es ima e D0, whe eas he sphe ical model esul s in oo small alues o he gap. As he sys em size inc eases, bo h models yield simila esul s. As he size o he ou e shell in he sphe ical model dec eases ~las alue in Table II!, he pola iza ion o he cell su ace esul s in lowe alues o D0. The educed cu o c58 s511.52Å o he pbc sys em wi h N5854 sol en molecules gi es p ac ically he same alue o D0as he sys em wi h c5L/2514.75Å. Al hough he sphe ical geome y pe mi s he usage o smalle sys em sizes, we ha e op ed o he pbc geome y wi h N5854 sol en molecules and a educed cu o c 58 s o ou simula ions. Ou choice is based on he ac ha he sphe ical model was no able o ep oduce he dielec- ic cons an o he sol en . Thus he usage o a single ge- ome y o he de e mina ion o bo h he dielec ic cons an and he eo ganiza ion ene gy is clea ly p e e able. The pbc geome y also allows us o calcula e he sol en adial dis i- bu ion unc ions. Wi h ou choice o he sys em size and he cu o adius, a molecule close o he simula ion cell bound- a y only in e ac s wi h a small ac ion o he pe iodic im- ages o he sol en molecules in he i s sol a ion shell. When applying he con en ional Mon e Ca lo me hod o c ea ing sys em con igu a ions ~ ansla ion and o a ion o a molecule!in he case o mix u es, we no iced a e y slow elaxa ion o he sys em. I a cha ged solu e is p esen , he ini ial cc s uc u e o he sol en elaxes apidly o a si ua- ion whe e he solu e is sol a ed by he molecules ha a e ini ially in he icini y o he solu e. Thus, he mola compo- si ion o he i s sol a ion shells depends on he ini ial po- si ions o he wo species wi hin he cc g id. The high den- si y o sol en molecules close o he solu e now ende s a es uc u e o he sol a ion shell ex emely di icul . I he wo componen s o he mix u e a e equally sized, one can adop a much mo e e icien me hod o c ea ing con igu a- ions: The con en ional ‘‘mo e’’ is al e na ed wi h a ‘‘swap’’ o molecules: wo molecules o di e en species a e selec ed andomly. Now, he iden i ies o he molecules a e in e changed, i.e., he dipole momen o he wo molecules is changed. We only change he absolu e alues o he dipole momen s; he dipole o ien a ions a e no al e ed. Each new con igu a ion is gene a ed ei he by a mo e o by a swap and he p obabili y o a swap was se o 10%. V. RESULTS AND DISCUSSION The eo ganiza ion ene gies lCR and lCS o a cha ge ecombina ion and a cha ge sepa a ion p ocess ha e been TABLE II. Values o D0 o pbc and he sphe ical model. The sol en and solu e pa ame e s a e speci ied in he main ex . Fo he sphe ical model N e e s o he numbe o molecules ha a e allowed o mo e. In his geome y he o al numbe o molecules was held ixed a N o 5673. ND0~kcal/mol! cGeome y 246 226.9 c5L/2 pbc 490 223.1 c58 spbc 854 221.9 c58 spbc 854 222.1 c5L/2 pbc 239 217.9 ¯sphe ical 371 222.4 ¯sphe ical 521 223.5 ¯sphe ical 593 218.6 ¯sphe ical 478 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. calcula ed o a ious sol en s using he me hods desc ibed abo e. The solu e consis s o wo ha d sphe es wi h adius d5 a5a51.44Å and dis ance d56Å. A. Pu e sol en s We i s s udy he case o ET eac ions in pu e sol en s. We ha e conside ed sol en s wi h pola i ies yin he ange o 0.75<y<3.0. The sol en adius was chosen o be he same as he solu e adius. A packing ac ion o h 50.417 and a empe a u e o T5300K was used in all simula ions. We will compa e ou simula ion da a wi h he eo gani- za ion ene gy exp essions p o ided by he Ma cus heo y and he MSA. In he Ma cus con inuum desc ip ion, he e- o ganiza ion ene gy o a cha ge ans e o an elemen a y cha ge be ween wo equally sized solu e molecules o adius aa a dis ance din a nonpola izable sol en wi h s a ic di- elec ic cons an e 0is gi en by l5e2 S 121 e 0 DS 1 a21 d D .~21! The MSA heo y akes in o accoun he molecula aspec o he sol en . In he case o in ini ely sepa a ed cha ge cen e s (d→`), he eo ganiza ion ene gy is jus he sum o he sol a ion ee ene gies o he wo ions8,9 l5e2 S 121 e 0 D 1 a~11 d !.~22! He e, d is a co ec ion o he sol a ed ion adius, which o a e y good app oxima ion is gi en by39 d 53 s/a~1081/3 e 0 1/622!21.~23! I he dis ance be ween he ions is no in ini e, he inclusion o he ion–ion in e ac ion in Eq. ~22!wi hin he amewo k o he MSA would equi e he knowledge o he mean ion– ion po en ial. We will app oxima e his e m by he sc eened in e ac ion in a con inuum dielec ic medium, as i has been done by o he s.36 The inclusion o his e m in Eq. ~22! e- sul s in lMSA5e2 S 121 e 0 DS 1 a~11 d !21 d D .~24! We should keep in mind ha his app oxima e lMSA does no eme ge om a pu ely mic oscopic pic u e o he sol en , as i s in luence on he ion–ion in e ac ion is aken in o accoun by a con inuum desc ip ion. This is expec ed o be a good app oxima ion, as long as he dis ance o he ions is su i- cien ly la ge. When e alua ing Eqs. ~24!and ~23!, we ha e o p o ide he solu e adius a, he ion dis ance d, he sol en adius s and he s a ic dielec ic cons an o he sol en e 0. The la e may be ob ained di ec ly om he MSA, using he ollowing ela ions:23 3y5~114 j !2 ~122 j !42~122 j !2 ~11 j !4, ~25! e 05~114 j !2~11 j !4 ~122 j !6. Fo compa ison wi h expe imen al da a i is o en mo e con- enien o use he expe imen al alue o e 0in Eqs. ~24!and ~23!. We will hus compa e h ee heo e ical exp essions wi h ou simula ion da a: ~a! he Ma cus exp ession Eq. ~21!,~b! he ‘‘consis en ’’ MSA esul as gi en by Eq. ~24!using he sol en pola i y yin o de o de e mine e 0 om he MSA and ~c! he ‘‘expe imen al’’ MSA esul lMSA ex , using he dielec- ic cons an s as ob ained om he simula ions in Eq. ~24!. The esul s a e shown in Fig. 6, whe e we ep esen he e- o ganiza ion ene gy e sus he Peka ac o 121/ e 0. In his ep esen a ion, he Ma cus esul @Eq. ~21!# is a s aigh line. We ind ha lCS.lCR o all simula ed pu e sol en s. Simila esul s ha e been ob ained by o he au ho s o com- pa able sys ems.36 This is due o he combina ion o wo e ec s:34,40 Fi s , he ee ene gy su aces a e no s ic ly pa abolic, as i would be equi ed by linea esponse heo y. Second, e en i he de ia ions om pa abolic su aces a e small, he cu a u e o he pa abolas may be di e en in he eac an and p oduc s a es. Bo h ea u es gi e ise o a di - e en eo ganiza ion ene gy o he cha ge sepa a ion and cha ge ecombina ion p ocesses. The Ma cus exp ession o l esul s in an o e es ima ion o he eo ganiza ion ene gy. Equa ion ~21!gi es l(y 53.0)5172.6 kcal/mole o he sol en wi h he highes po- la i y. In he plo we ha e scaled he co esponding cu e o coincide wi h lMSA ex a his pola i y. This is equi alen o using an e ec i e ion adius o a52.0Å in Eq. ~21!. When he geome ic ac o is adjus ed in his way, he con inuum desc ip ion s ill does no desc ibe well he o e all beha io FIG. 6. Reo ganiza ion ene gies o a pu e sol en . The squa es ep esen lCS, he ci cles co espond o lCR as ob ained om he simula ions. The dashed line is he esul as ob ained by he con inuum desc ip ion ~Ma cus! scaled o coincide wi h lMSA ex o he highes pola i y. The do ed line co - esponds o he heo e ical MSA esul o he simula ed sys em. The MSA esul lMSA ex , using e 0as ob ained om he simula ions as an inpu pa am- e e , is ep esen ed by he solid line. 479J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al. o he eo ganiza ion ene gy. This ea u e is expec ed. The exp ession o lin he con inuum desc ip ion, Eq. ~21!, in- dica es ha he eo ganiza ion ene gy depends only on e 0 o ixed adii and sepa a ion dis ance and his dependence is e y weak. Fo he MSA esul s, we ind a much be e ag eemen be ween he heo e ical exp essions and he simula ion da a. The heo e ical MSA esul ~b!gi es good ag eemen o small pola i ies, bu unde es ima es he eo ganiza ion en- e gy o high pola i ies. This is no su p ising, as i is well known ha he MSA unde es ima es he dielec ic cons an o high sol en pola i ies. This esul s in an unde es ima ion o he Peka ac o 121/ e 0on one hand and o he geome ic ac o 1/a(11 d ) on he o he . The MSA exp ession lMSA ex wi h e 0 aken om he simu- la ion da a yields eo ganiza ion ene gies which a e close o he simula ion esul s o he cha ge ecombina ion p ocess. By cons uc ion, he MSA does no dis inguish be ween he cha ge ecombina ion and cha ge sepa a ion p ocesses, which as shown by he simula ion esul s, ha e di e en e- o ganiza ion ene gies. Thus, he good ag eemen o lMSA ex wi h he esul s o he cha ge ecombina ion p ocess in he case p esen ed is p obably o ui ous. Howe e , his exp es- sion desc ibes he o e all shape o bo h cu es much be e han he Ma cus app oxima ion. The heo e ical exp ession o lwi hin he amewo k o MSA includes he geome ic ac o 1/a(11 d ) which depends on e 0 ia Eq. ~23!. The abili y o lMSA ex o ep oduce he end o he simula ion da a o bo h p ocesses sugges s ha he inclusion o his geome - ic ac o is an impo an co ec ion o he Ma cus exp ession o he eo ganiza ion ene gy, Eq. ~21!. The geome ic ac o , as gi en by he MSA, can be in e p e ed as an e ec i e solu e adius a(11 d ) and is a consequence o he sol en molecula i y in he MSA pic u e. This emphasizes he need o a molecula desc ip ion o he sol en , e en in he case o pu e sol en s. F om hese esul s one may be emp ed o conclude ha he MSA is able o p edic he sol en s uc u e a ound he solu es. We a e cu en ly examining his ques ion o single ions in solu ion. P elimina y esul s indica e ha he pola - iza ion densi y o he sol en a ound an ion is no e y well ep esen ed by he MSA exp ession. Howe e , he ee en- e gy o sol a ion, gi en as an in eg al o e he pola iza ion densi y, is p edic ed qui e well by he MSA exp ession. B. Mix u es We will now p oceed o he case o bina y mix u es o pola sol en s. We ha e calcula ed he eo ganiza ion ene gy o he wo mix u es s udied in he p e ious sec ion @mix u e ~A!wi h simila pola i ies o he wo componen s: yH53.0 and yL52.18 and mix u e ~B!wi h a he di e en pola i ies o he wo componen s: yH53.0 and yL50.75#. The esul s o mix u e ~A!a e shown in Fig. 7 and o mix u e ~B!in Fig. 8. When he solu e is no p e e en ially sol a ed by one o he wo componen s o he mix u e, one would expec a lin- ea beha io o he sol a ion ene gy, as he mola ac ion o he componen s is a ied.10 Fo nonpola izable sol en s, he heo e ical exp essions o he eo ganiza ion ene gy educe o a sol a ion ene gy in he limi d→`. Thus, a linea ~o close o linea !dependence o lon he mola ac ion H would indica e ha no p e e en ial sol a ion is p esen in he p ocess. Fo bo h mix u es we obse e de ia ions om a linea beha io in he eo ganiza ion ene gy. In mix u e ~A! we obse e an almos linea beha io o lCS, while all al- ues o lCR o he mix u es lie abo e a s aigh line connec - ing he esul s o he pu e sol en s. Fo mix u e ~B! his beha io is much mo e p onounced. When adding a small mola ac ion o he componen wi h he highe pola i y, he eo ganiza ion ene gy shows a d as ic inc ease, and e y quickly he eo ganiza ion ene gy l eaches a alue close o ha o he pu e sol en ( H51). As i was al eady obse ed o mix u e ~A!, he eo ganiza ion ene gy o he cha ge ecombina ion p ocess lCR de ia es mo e om he linea FIG. 7. Reo ganiza ion ene gies o mix u e ~A!(yH53.0 and yL52.18). The squa es ep esen lCS, he ci cles co espond o lCR. The linea depen- dence, as expec ed in he limi d→` o ideal sol a ion, is ep esen ed as a dashed line o each case. The xsymbols ep esen he MSA esul s lMSA ex . FIG. 8. Reo ganiza ion ene gies o mix u e ~B!(yH53.0 and yL50.75). The squa es ep esen lCR, he ci cles co espond o lCS. The solid lines connec ing he da a poin s ha e been ob ained by spline in e pola ion. The linea dependence, as expec ed in he limi d→` o ideal sol a ion, is ep esen ed as a dashed line o each case. The xsymbols ep esen he MSA esul s lMSA ex . 480 J. Chem. Phys., Vol. 110, No. 1, 1 Janua y 1999 Denk e al.