Free Energy of Ionic Hydration

Transcription

1 1206 J. Phys. Chem. 1996, 100, Free Energy of Ionic Hydrtion Gerhrd Hummer,*,, Lwrence R. Prtt, nd Angel E. Grcí Theoreticl Biology nd Biophysics Group T-10, MS K710, Center for Nonliner Studies, MS B258, nd Theoreticl Chemistry nd Moleculr Physics Group T-12, MS B268, Los Almos Ntionl Lbortory, Los Almos, New Mexico ReceiVed: April 11, 1995 X The hydrtion free energies of ions exhibit n pproximtely qudrtic dependence on the ionic chrge, s predicted by the Born model. We nlyze this behvior using second-order perturbtion theory. The verge nd the fluctution of the electrosttic potentil t chrge sites pper s the first coefficients in Tylor expnsion of the free energy of chrging. Combining the dt from different chrge sttes (e.g., chrged nd unchrged) llows clcultion of free-energy profiles s function of the ionic chrge. The first two Tylor coefficients of the free-energy profiles cn be computed ccurtely from equilibrium simultions, but they re ffected by strong system-size dependence. We pply corrections for these finite-size effects by using Ewld lttice summtion nd dding the self-interctions consistently. An nlogous procedure is used for the rection-field electrosttics. Results re presented for model ion with methne-like Lennrd-Jones prmeters in simple point chrge wter. We find two very closely qudrtic regimes with different prmeters for positive nd negtive ions. We lso studied the hydrtion free energy of potssium, clcium, fluoride, chloride, nd bromide ions. We find negtive ions to be solvted more strongly (s mesured by hydrtion free energies) compred to positive ions of equl size, in greement with experimentl dt. We scribe this preference of negtive ions to their strong interctions with wter hydrogens, which cn penetrte the ionic vn der Wls shell without direct energetic penlty in the models used. In ddition, we consistently find positive electrosttic potentil t the center of unchrged Lennrd-Jones prticles in wter, which lso fvors negtive ions. Regrding the effects of finite system size, we show tht even using only 16 wter molecules it is possible to clculte ccurtely the hydrtion free energy of sodium, if self-interctions re considered. 1. Introduction A qudrtic dependence on the ionic chrge of the electrosttic free energy of solvtion of simple ion in queous solution is bout the simplest resonble possibility for tht behvior. The Born model predicts tht qudrtic dependence. 1 Severl computer simultion clcultions hve shown tht it is pproximtely correct for the simplest monovlent ions in wter. 2-4 Theoreticl simplifictions hve been dvnced to tke dvntge of such behvior. 3,5-7 If tht qudrtic behvior were correct with sufficient ccurcy, it would indeed permit importnt simplifictions of the difficult tsk of moleculr clcultions of solvtion free energies owing to electrosttic interctions in complex solutions. The theoreticl simplifictions identified on tht bsis cn be viewed either s perturbtion theory through second order in the electrosttic interctions or s Gussin modeling of certin therml fluctutions of those interctions. With the doption of either view, these methods would hve wide pplicbility nd gret simplicity. The question of the ccurcy of the qudrtic dependence on the chrge of the free energy owing to electrosttic interctions deserves to be rised for its own ske nd given precise nswer s generl s possible. This qudrtic behvior is not universl truth, nd previous simultion clcultions hve given helpful informtion on the circumstnces where this qudrtic dependence cn be expected * To whom correspondence should be ddressed t the Theoreticl Biology nd Biophysics Group. Theoreticl Biology nd Biophysics Group. Center for Nonliner Studies. Theoreticl Chemistry nd Moleculr Physics Group. X Abstrct published in AdVnce ACS Abstrcts, December 15, to fil. 2 However, previous simultion clcultions re sufficiently disprte tht high precision nswer to the question of the ccurcy of second-order perturbtion theory for the free energy owing to electrosttic interctions is not vilble. The disprte chrcter of the vilble simultion results is lrgely cused by lck of uniformity with respect to the tretment of finite-system-size effects on electrosttic interctions in queous solutions. It is not typicl for finite-system-size correction nd the electrosttic solvtion free energy to be of similr size. In contrst to the role of computer experiments in nswering this question, lbortory experiments hve been useful mostly for frming the question The difficulty of using lbortory experiments for the present purpose resides in our inbility to extrct generlly n electrosttic contribution from contributions of the other interctions present. Becuse of these points, this work clcultes the free energy owing to electrosttic interctions of simple, sphericl ions in wter by Monte Crlo methods nd gives prticulr ttention to the methodologicl issue of correction for finite system size. The moleculr models used re simple, but they hve been widely tested. Becuse the gol of this work is to ddress the question of qudrtic dependence on chrge of the electrosttic solvtion free energy, these models re sufficiently relistic for the present purposes. However, we will compre our computed free energies with experimentl results nd thus provide informtion on how these models might be simply improved for prediction of electrosttic free energies. Before proceeding with the technicl developments, it is worthwhile to give some discussion of the ide for the present tretment of system-size effects on solvtion free energies of ions. There is no generlly vlid recipe tht llows determi /96/ $12.00/ Americn Chemicl Society

2 Free Energy of Ionic Hydrtion J. Phys. Chem., Vol. 100, No. 4, ntion of the effects of finite system size on the clculted physicl quntity in computer simultions. Wht must generlly be done is to nlyze the observed size dependence empiriclly. If, s is the cse for Coulomb interctions of long rnge, different procedures re vilble, then we should expect consistent thermodynmic limiting (N f ) results for different methods of treting the finite-size system. It is well understood tht certin quntities involving integrls over the whole smple, such s the dipole-moment fluctutions, depend intrinsiclly on exterior conditions or constrints. 11 Those conditions must then be properly understood theoreticlly. For the present problem involving the interctions nd ssocited thermodynmics of n ion immersed in dielectric liquid, resonble view is the following: Tretment of electrosttic interctions in truly periodic formt, e.g., by Ewld procedures, is consistent with the periodic boundry conditions tht re nerly inevitble for other resons. In periodic boundry conditions the interctions t the longest rnge tht must be tken seriously occur t n pprecible frction of the distnce to the surfce of the simultion cell. For typicl simulted system sizes, ionic interctions t tht longest rnge re lrge. Tretment of electrosttic interctions in truly periodic formt thoroughly tempers those lrge interctions. But mthemticl price for true periodicity of electrosttic interctions is self-interction ssocited with interctions with imges nd uniform neutrlizing chrge bckground. For neutrl systems this self-interction cn be sometimes ignored. For nonneutrl systems, such s those studied here, there my be prcticl dvntges of consistency obtined for explicit considertion of the self-interction. We will ccount for these self-interctions explicitly in the clcultions below. This rgument permits tretments of the ionic interctions other thn Ewld summtion. In fct, the work below tests generlized rection-field (GRF) method nd lso finds tht consistent results cn be obtined if self-interctions re treted on similr bsis. 2. Theoreticl Methods 2.1. Clcultion of the Free Energy of Chrging. The vrious methods for computing free energies using computer simultions hve been reviewed extensively We strt here from the potentil distribution theorem for the excess chemicl potentil µ ex, 15 µ ex (q 1 ) - µ ex (q 0 ) ) -k B T ln exp{-β[u(q 1 ) - u(q 0 )]} q0 (1) where q 0 nd q 1 re the two chrge sttes nd β ) 1/k B T;... q denotes therml configurtion-spce verge in the chrge stte q; nd u(q) is the configurtion-dependent interction energy of the ion in chrge stte q with the solution. Aprt from finitesize corrections to be discussed lter, u(q) is given by qφ(r), where φ(r) is the electrosttic potentil t the chrge position r. We next nlyze eq 1 utilizing cumulnt expnsion 16 with respect to β, exp(-β u) q0 ) exp[ (-β) nc n (2) n)0 n!] where u ) u(q 1 ) - u(q 0 ). This defines the cumulnts C n of order n ) 0, 1, 2 s C 0 ) 0 C 1 ) u q0 C 2 ) ( u - u q0 ) 2 q0 (3) (3b) (3c) We cn interpret eq 2 s Tylor expnsion in q ) q 1 - q 0 if we set u ) qφ + (q q 0 2 )ξ/2, where ξ ccounts for finite-size effects s self-interction to be discussed further below, µ ex ) q( φ q0 + q 0 ξ) - β 2 q2 [ (φ - φ q 0 ) 2 q0 - ξ β] +... where µ ex ) µ ex (q 1 ) - µ ex (q 0 ). The men nd the fluctution of the electrosttic potentil t the chrge site q (corrected for finite-size effects) yield the derivtives of the free energy with respect to q. The informtion bout the derivtives cn therefore be extrcted from equilibrium simultions. In principle, higher-order cumulnts could be used to obtin informtion bout the other Tylor coefficients. However, s ws observed by Smith nd vn Gunsteren, 4 higher-order cumulnts re incresingly difficult to extrct from computer simultions of limited durtion. Therefore, we will evlute C 1 nd C 2 t few discrete chrge sttes nd combine this informtion bout the derivtives, either by constructing n interpolting polynomil or by using χ 2 fit to polynomil expression (or ny other functionl form) for the free energy s function of q. The χ 2 fit minimizes the men squre devition of the observed dt with respect to the coefficients { k } of the fitting function µ ex (q; { k }), n χ 2 ) i)1 {[ µ ex (q i ; { k }) - µ ex obs (q i ) σ i ]2 + [ µ ex (q i ; { k }) - µ ex obs (q i ) where σ i nd F i re the estimted errors (stndrd devitions) ex ex of the observed first nd second derivtives µ obs nd µ obs t chrge stte q i Long-Rnge Coulomb Interctions nd Finite-Size Effects. To minimize finite-size effects on energetic properties of Coulombic systems, we dopt the following strtegy: 17 We use lttice summtion for clculting the electrosttic interctions to ccount for the periodic boundry conditions employed in the computer simultions, nd we consistently include the selfinterctions rising from lttice summtion. We point out tht side from forml consistency the numericl results cn motivte this pproch by demonstrting in finite-size nlysis tht the devitions from the thermodynmic limit (N f ) re smll. The Coulomb energy of periodiclly replicted system of chrges q i t positions r i (i ) 1,..., N) cn be expressed s U ) where r ij ) r j - r i + n, with the lttice vector n chosen such tht r ij is vector in the unit cell. The effective, positiondependent potentil φ EW (r) is obtined by lttice summtion using Ewld s method, 13,18,19 φ EW (r) ) n 1ei<jeN q i q j φ EW (r ij ) + 1 / 2 erfc(η r + n ) + r + n k*0 4π ( exp - k Vk 2 where V is the volume of the box, erfc is the complementry error function, nd k ) k. The two lttice sums extend over rel- nd Fourier-spce lttice vectors n nd k, respectively. F i 1eieN 2 (4) ]2} (5) q i 2 ξ EW (6) 4η 2 + ik r ) - π Vη 2 (7)

3 1208 J. Phys. Chem., Vol. 100, No. 4, 1996 Hummer et l. The self-term ξ EW ) lim rf0 [φ EW (r) - 1/r] is the Wigner potentil: Using Green s theorem nd (1/r) ) -4πδ(r), we find ξ EW ) lim rf0[ φ EW (r) - 1 r] ) - 1 4π lim r >ɛ dr 1 ɛf0 r φ EW (r) (8) The integrtion region is infinite nd includes ll bckground chrge nd lttice imge chrges, φ EW (r) ) -4π [ δ(r - n) - 1 (9) n V] Equtions 8 nd 9 estblish tht ξ EW is the electrosttic potentil in Wigner lttice t chrge site owing to the lttice imges nd the neutrlizing bckground. For Ewld summtion in cubic lttice the self-term is ξ EW ) /L, where L is the length of the cube. It will be interesting to remember tht ξ EW cn lso be expressed in terms of quntities ssocited with the primitive simultion cell ξ EW ) - 1 4π lim ɛf0 V: r >ɛ dr 1 r φ EW (r) - The first term on the right is explicitly the interction with the bckground density in the primitive simultion cell. The second term on the right is n integrl over the surfce of the primitive simultion cell. It describes electrosttic interctions of the centrl unit chrge with dipolr surfce distribution φ EW (r)nˆ, where nˆ is the surfce norml pointing outwrd. Eqution 6 cn lso be used for nonneutrl system since chrges re implicitly compensted by homogeneous bckground in the Ewld formultion. This results in n expression for the energy difference u between two configurtions with different chrge sttes q 0 nd q 1 of n ion t position r, In the following, we will use this expression contining selfinterction which is qudrtic in the chrge to clculte the free energy of chrging; i.e., we ssume tht the self-interction ccounts for the finite-size corrections. 24 In our clcultions, we will lso use generlized rection field (GRF). 25,26 The GRF Coulomb interction depends only on the distnce r of the chrges nd hs cutoff distnce r c, Θ is the Heviside unit-step function; p(x) is screening polynomil: 1 4π V d2 r φ EW (r)nˆ ( 1 r) (10) u ) qφ EW (r) + 1 / 2 ξ EW (q q 0 2 ) (11) φ GRF (r) ) 1 r p(r/r c ) Θ(r c - r) + C (12) p(x) ) (1 - x) 4 (1 + 8x/5 + 2x 2 /5) (13) By nlogy with the Ewld summtion, we define the self-term for the GRF s the potentil t the chrge site, ξ GRF ) lim rf0 [φ GRF (r) - 1/r]. The totl energy of neutrl systems, if defined s in eq 6, is independent of the constnt C. However, in nonneutrl systems C ffects the totl energy. We define C in nlogy with the Ewld potentil, which stisfies 22 V dr φ EW (r) ) 0 (14) TABLE 1: Lennrd-Jones Prmeters of the Ion-Wter Interctions ion ɛ/(kj mol -1 ) σ/nm N K C F Cl Br Me such tht the verge potentil in the cell vnishes. If we require the normliztion condition eq 14 lso for the GRF interction, we obtin C ) -πr c2 /5V. The GRF self-term is ξ GRF ) -12/ 5r c + C. For r c ) L/2, the normliztion condition eq 14 ccounts for only smll dditionl correction, yielding ξ GRF ) -24/5L - π/20l. It is interesting to mke connection with the correction method proposed by Sloth nd Sørensen. 27 These uthors use the minimum-imge Coulomb interction. To eliminte the system-size dependence in their clcultion of chemicl potentils of restricted-primitive-model ions, they introduce bckground neutrlizing the test-prticle chrge. This is done by dding constnt ξ 1/r to the bre Coulomb potentil, 28 This corresponds to enforcing eq 14 nd dding self-term ξ 1/r ) lim rf0 [φ(r) - 1/r] for the minimum-imge interction. ξ 1/r is lso precisely the first term on the right side of eq 10. It ccounts for lrge correction since ξ 1/r -2.38/L. 3. Computer Simultions ξ 1/r ) - 1 V V dr 1 r (15) We clculted the free energy of chrging ions in wter using Metropolis Monte Crlo simultions. 13,30 The systems comprise single ion nd N wter molecules. For wter we used the simple point chrge (SPC) model. 31 The ion-wter interctions were described by Coulomb nd Lennrd-Jones (LJ) interctions. The Coulomb terms involve the prtil chrges of oxygens nd hydrogens on SPC wter. The LJ interctions ct only between wter oxygen nd the ion. We studied the ions N +, K +, C 2+, F -, Cl -, nd Br -. The LJ prmeters for these ions were those of Strtsm nd Berendsen. 32 We lso studied the chrging of model ion Me with methne LJ prmeters s given by Jorgensen et l. 33 Lorentz-Berthelot mixing rules 13 were pplied to obtin LJ prmeters between wter nd Me. The LJ prmeters re compiled in Tble 1. The chrge interctions in the simultions were clculted using the Ewld lttice summtion (eqs 6 nd 7) nd the generlized rection-field potentil (eqs 12 nd 13). In both cses, the rel-spce interctions were truncted on n tom bsis using L/2 s the cutoff nd pplying the periodic boundry conditions on n tom bsis. For the Ewld Fourier-spce clcultion, cutoff of k 2 e 38(2π/L) 2 ws used, resulting in k vectors. To correct the bckground dielectric constnt from infinity to ɛ RF ) 65, term 2πM 2 /(2ɛ RF + 1)V ws dded to the potentil energy (in both Ewld nd GRF clcultions), where M is the net dipole moment of the wter molecules. The rel-spce dmping fctor ws set to η ) 5.6/L. Electrosttic potentils t the ion sites were clculted using φ EW nd φ GRF. The potentils were clculted fter ech pss (one ttempted move per prticle) nd stored for nlysis. For ech system psses were used for verging. Rndom configurtions or configurtions of previous runs were used s initil structures nd lwys extensively equilibrted. The temperture ws 298 K. The totl number density ws F ) nm -3 in ll simultions. Cubic boxes were used s simultion cells with

4 Free Energy of Ionic Hydrtion J. Phys. Chem., Vol. 100, No. 4, edges L ) [(N + 1)/F] 1/3. The Monte Crlo move widths were chosen so tht n pproximte cceptnce rtio of 0.5 ws obtined. In ddition, thermodynmic integrtion (TI) ws used to clculte directly the free energy of chrging. Within Monte Crlo psses, the chrge of the ion ws linerly chnged from 0 to its full mgnitude ((e, 2e, where e is the elementry chrge). The free-energy chnges were then clculted s n µ ex (q 1 ) - µ ex (q 0 ) ) (q 1 - q 0 )n -1 φ i + ξ(q q 2 0 )/2 (16) i)1 where the sum extends over n ) Monte Crlo psses nd the lst term is finite-size correction. Eqution 16 pproximtes the exct expression µ ex (q 1 ) - µ ex (q 0 ) ) q0 q 1 dq u(q) q q (17) TI ws lso performed using the reverse pth, i.e., decresing the chrge to 0. We lso performed Monte Crlo simultions of ion-wter clusters comprising one ion nd N SPC wter molecules (4 e N e 256). The strting structure ws rndom configurtion with the bulk density of wter in cubic box round the ion. The cluster ws equilibrted for t lest psses (with n cceptnce rte of bout 0.5) nd then verged over psses t 298 K. We used the bre Coulomb interction 1/r nd did not pply distnce cutoff. No periodic boundry conditions were employed in the cluster simultions. TABLE 2: Results for the Men nd the Fluctution of the Potentil O (with nd without Finite-Size Corrections) t the Position of Methne-like Lennrd-Jones Prticle Me Crrying Chrge q N Coulomb q/e m f m c f c 256 EW EW GRF EW EW EW GRF EW EW EW GRF EW EW EW GRF EW EW EW GRF Coulomb refers to the tretment of the electrosttic intections (Ewld or GRF). N is the number of wter molecules. The men nd the fluctution re listed s m ) e φ nd f ) βe 2 (φ - φ ) 2, both in units of kilojoules per mole. The corrected vlues re m c ) m + qeξ nd f c ) f - e 2 ξ. The sttisticl errors of m nd f re estimted from block verges s pproximtely 4.0 nd 30 kj mol Results nd Discussion 4.1. Chrging of Methne-like Lennrd-Jones Prticle. The free energy of chrging methne-like LJ prticle in SPC wter ws determined from series of simultions with N ) 128 nd 256 wter molecules nd with Ewld nd GRF chrge tretment. A rnge of chrges from -e to +e ws covered in steps of 0.25e (N ) 128) nd 0.5e (N ) 256). The results for the men m nd the fluctution f of the potentil t the ion site (with nd without finite-size correction) re compiled in Tble 2. In the clcultions, the potentil φ t the ion site (r ) 0) is defined s N 3 φ ) i)1r)1 q ir φ(r ir ) (18) where the double sum extends over ll wter oxygen nd hydrogen sites; φ is either φ EW or φ GRF. The men m nd the fluctution f re clculted from Monte Crlo psses s m ) e φ f ) βe 2 (φ - φ ) 2 (19) (19b) The corrected vlues for men nd fluctution re defined s m c ) m + qeξ nd f c ) f - e 2 ξ. The Tylor expnsion of the free energy of chrging round chrge stte q ssumes the following form: µ ex ) ( q e ) m c - 1 2( q e ) 2 f c +... (20) From the results of Tble 2 we see tht the finite-size corrections re of mgnitude similr to the uncorrected results m nd f. The uncorrected results of the different methods nd system sizes re widely spred. If however the finite-size corrections re pplied, we obtin consistent results for ll methods nd Figure 1. Probbility distributions P(eφ) of the electrosttic energy eφ t the site of methne-like ion Me with chrge q ) +e from Ewld summtion with N ) 256 (), s), N ) 128 (0, - -), nd GRF with N ) 256 wter molecules (+, - -), respectively. The lines re Gussin distributions. Also shown re Gussin distributions corrected for finitesize effects, which re peked ner eφ ) 550 kj mol -1 ; they gree closely in position nd vrince. system sizes over the rnge of ion chrges considered. With estimted errors (1 stndrd devition, s clculted from block verges) of 4.0 nd 30 kj mol -1 for m c nd f c, we find dt of different methods within 2 stndrd devitions from ech other. The only exception is the fluctution f c for q ) -e, where the two extreme vlues (Ewld, N ) 128 nd 256) differ by bout 3 stndrd devitions. In the following, we will restrict the discussion to the corrected vlues m c nd f c. Figure 1 shows the probbility distribution P(eφ) of the electrosttic energy eφ for n ionic chrge of q ) e, s clculted from histogrms. The P(eφ) curves follow closely Gussin distributions with the men nd vrince clculted from the φ dt. This reflects the pproximte vlidity of second-order perturbtion theory in the ionic chrge. However, the P(eφ) curves for the Ewld summtion with N ) 128 nd 256, s well s GRF with N ) 256 wter molecules, differ widely, both in the pek position nd in the width. To illustrte the importnce of the finite-size correction, we included in Figure

5 1210 J. Phys. Chem., Vol. 100, No. 4, 1996 Hummer et l. Figure 2. Averge electrosttic potentil φ t the site of the negtively chrged ion Me (q ) -e) clculted from the pir correltions of Monte Crlo simultion using Ewld summtion nd N ) 256 wter molecules. The results of the integrtion using the GRF interction with cutoff r c ) L/2 nd the bre Coulomb interction 1/r re shown with long- nd short-dshed lines, respectively. Finite-size corrections re dded s constnts. The Ewld-summtion result is shown s reference with solid line. 1 the Gussin distributions corresponding to the corrected vlues m c nd f c for men nd vrince. The ppliction of the finite-size corrections brings the three curves to very close greement, yielding results tht re pproximtely independent of system size nd tretment of electrosttic interctions. To further illustrte the importnce of the finite-size correction, we clculted φ from the pir correltions of the Ewldsummtion simultion with N ) 256 wter molecules s Figure 3. Averge electrosttic potentil m c t the position of the methne-like Lennrd-Jones prticle Me s function of its chrge q. m c contins corrections for the finite system size. Results re shown from Monte Crlo simultions using Ewld summtion with N ) 256 (+) nd N ) 128 ( ) s well s GRF clcultions with N ) 256 wter molecules (0). Sttisticl errors re smller thn the size of the symbols. Also included re liner fits to the dt with q < 0 nd q g 0 (solid lines). The fit to the tnh-weighted model of two Gussin distributions (eq 22) is shown with dshed line. φ (R) ) 4πF H2 O 0 R dr r 2 φ(r)[q O g IO (r) + 2q H g IH (r)] (21) F H2O is the wter density, q O nd q H re the oxygen nd hydrogen chrge, nd g IO nd g IH re the ion-oxygen nd ion-hydrogen pir correltion functions. Figure 2 shows the results for the chrge stte q ) -e of the ion Me s function of the integrtion cutoff R for the bre Coulomb potentil φ(r) ) 1/r nd φ GRF with r c ) L/2. In both cses we included the finitesize correction s constnt. The integrtion of the 1/r interction extended into the corners of the cube, using the correct weights. As reference, the Ewld result is shown s stright line. All three methods converge to within bout 1 kj mol -1, which hs to be compred with the estimted sttisticl error of 4 kj mol -1 of the dt. The integrted 1/r interction shows strong oscilltions, nd only in the corners of the cube does it pproch its finl vlue. The GRF interction on the other hnd contins lrge self-term nd within two oscilltions reches its limiting vlue. This illustrtes n importnt point regrding the correction of finite-size effects in the clcultion of chrge-relted quntities. We chieve greement between different methods of treting Coulomb interctions (Ewld summtion, rection field, bre Coulomb interction) if we (i) normlize φ ccording to eq 14 nd (ii) dd self-term ξ ) lim rf0 [φ(r) - 1/r] to the energy. Further demonstrtions of the vlidity of these finitesize corrections will be given in the discussion of the results for sodium nd fluoride ions in SPC wter. Figure 3 shows m c s function of the chrge. We observe two liner regimes with different chrcteristics for q < 0 nd q g 0. Liner behvior of m c on the whole rnge of q would reflect vlidity of the second-order perturbtion theory. It would imply Gussin sttistics of φ nd, correspondingly, tht the coefficients in the Tylor expnsion of order 3 nd higher vnish. However, since we observe trnsition in the liner behvior between chrges of -0.25e nd 0, the sttistics re only Figure 4. Fluctution of the electrosttic potentil f c t the position of methne-like Lennrd-Jones prticle s function of its chrge q. f c contins corrections for the finite system size. Error brs indicte 1 estimted stndrd devition of the dt. For further detils see Figure 3. pproximtely Gussin. We note tht from the φ dt of psses it proved impossible to extrct relible informtion bout the Tylor coefficients (cumulnts) of order 3 nd higher. The second Tylor coefficient f c cn however be extrcted ccurtely. Figure 4 shows f c s function of q/e. Included in Figure 4 s lines re the vlues of f c estimted from the liner fits of m c for q < 0 nd q g 0. We hve fitted the m c nd f c dt by model with two Gussin regimes. Included in Figures 3 nd 4 is χ 2 fit of the whole set of derivtive dt (38 dt points) to µ ex (q) - µ ex (0) ) ( + q + b + q 2 )[1 + tnh(c + dq)]/2 + ( - q + b - q 2 )[1 - tnh(c + dq)]/2 (22) where χ 2 is defined s in eq 5 with prmeters +, b +, -, b -, c, nd d. This model cn nicely reproduce the dt. We find trnsition t q ) c/d -0.2e between the two regimes of pproximtely Gussin behvior with qudrtic q dependence. We scribe this trnsition to differences in the structurl orgniztion of wter molecules ner negtively nd positively chrged ions. A possible explntion for the observed behvior is tht for positive ions, the oxygen tom of wter is pointing towrd the LJ prticle. The strongly repulsive forces of the

6 Free Energy of Ionic Hydrtion J. Phys. Chem., Vol. 100, No. 4, TABLE 3: Free Energy (in kj mol -1 ) of Chrging the Methne-like Lennrd-Jones Prticle Me from 0 to (e function µ ex (0f+e) µ ex (0f-e) p p p p p tnh The free energy ws clculted from fitting to polynomils p n of degree n nd tnh-weighted model of two Gussin regimes (eq 22). r -12 interction prevent lrge fluctutions of φ becuse of the restricted oxygen motions. The hydrogens re pointing wy so tht rerrnging them hs only comprbly smll effect on φ. For negtive ions, the structures with one of the hydrogens pointing towrd the ion will dominte. Becuse of the symmetry between the wter hydrogens nd the finite life time of the hydrtion shell, trnsitions will occur which could explin the lrger fluctutions in the negtive chrge rnge. Similrly, trnsition to different Gussin behvior for highly-chrged positive ions ws observed by Jyrm et l. 2 These uthors studied the free energy of chrging of sodium ion in the chrge rnge 0 to 3e. When incresing the ion chrge, trnsition occurs to more wekly decresing qudrtic freeenergy regime t chrge of bout 1.1e. This trnsition hs lso been discussed by Figueirido et l. 34 We lso find nonvnishing potentil t the methne site even t zero chrge. 5 In dipolr solvent, φ q)0 is zero becuse of chrge-reversl symmetry. However, the symmetry of the chrge distribution on the wter molecule gives rise to positive potentil for q ) 0; this is primrily cused by the hydrogens penetrting the LJ sphere of the methne prticle, since they do not hve protecting repulsive shell in the model used. As consequence, there is smll chrge region in which incresing the chrge costs free energy. A positive potentil t the center of n unchrged prticle ws lso observed by Rick nd Berne. 35 As consequence of both the positive potentil t zero chrge nd the lrger potentil fluctutions for negtive ions, negtive ions re more stbly solvted compred to positive ions. Tble 3 compiles the free energies of chrging s clculted from fitted polynomils p n of degree n to the derivtive dt m c nd f c. Except for the simple Gussin model p 2, different fitting functions give consistent results for the free energies of chrging. For ions with chrge +e nd -e we find µ ex ) -250 nd -431 kj mol -1. Interpreted within Born model for the free energy, 1 i.e., ex µ Born ) -(1-1/ɛ)q 2 /2R (23) we obtin Born rdii R + ) 0.27 nm nd R - ) 0.16 nm. (A vlue of ɛ ) 80 is used for the dielectric constnt, but this hrdly ffects the results.) The difference between R + nd R - is somewht smller if we use the ctul coefficients of the q 2 term in the free-energy expnsion, s obtined from eq 22 giving 0.23 nd 0.16 nm for the Born rdii of positive nd negtive ions. We emphsize the model chrcter of the interction potentils used in this study. A repulsive shell of the hydrogen tom might reduce the free energy difference between positive nd negtive ions. The fvoring of negtive ions however should persist. The lower free energy of negtive ions compred to positive ions of equl size grees with the experimentl observtions. The hydrtion free-energy dt compiled by Mrcus 36 for lkli metl nd hlide ions show power-lw dependence with respect to the ion rdius. Using these fitted curves, we find Figure 5. Pir correltion functions g IO (top pnel) nd g IH (bottom pnel) of the Me ion with wter oxygen nd hydrogen. The g(r) curves re shifted verticlly ccording to the ionic chrge by q/e, i.e., by 1 for q ) +e, 0.5 for q ) 0.5e, etc. The g(r) curves of Ewld summtion nd GRF simultions with N ) 256 wter molecules re shown with solid nd dshed lines, respectively. differences of 150 nd 240 kj mol -1 for the solvtion free energy between negtive nd positive ions of the size of potssium nd sodium, respectively. The LJ prticle Me studied here hs vn der Wls rdius between those of K + nd N +. The clculted free energy required to go from -e to +e is 180 kj mol -1, which is indeed brcketed by the experimentl dt. The revised Born model by Ltimer et l. 37 lso yields lower free energies for negtive ions. For lkli metl nd hlide ions, it uses effective Born rdii R ) r p +, where r p is the Puling rdius nd is nd nm for ctions nd nions. This smller effective-rdius correction for nions in eq 23 results in considerbly lower free energies of negtive ions compred to positive ions of equl size, in greement with our clcultions. The difference of the effective Born-rdius correction s defined by Ltimer et l. 37 is nm, which grees with wht we find for the Me ion. The energetic differences in the hydrtion of positive nd negtive ions go long with differences in the structurl orgniztion of wter molecules in the hydrtion shell. Figure 5 shows the ion-wter pir correltion functions for different ionic chrges. Going from q ) 0 to positive chrges does not chnge the qulittive properties of the ion-oxygen nd ionhydrogen correltion functions g IO nd g IH. An increse of the ionic chrge results in higher first pek. However, going from chrge q ) 0 to negtive chrges ffects strongly the structurl orgniztion of the first hydrtion shell. Alredy t q ) -0.5e g IH shows the buildup of second pek t bout r ) 0.2 nm distnce. This pek reches vlue of lmost 5 t q ) -e, compred to g IH essentilly being zero in this distnce region for chrge q ) 0. This strong interction of the negtively chrged ion with the hydrogens of wter in turn ffects the ionoxygen correltion functions. Despite the negtive chrge of both the ion nd oxygen site, g IO hs first pek with height of bout 5 for q ) -e compred to only 3 for q ) +e. The strong chrge repulsion between wter oxygen nd the ion with q ) -e is overcome by lrge ttrction cused by wter hydrogen pointing towrd the ion nd penetrting the ionic vn der Wls shell without energetic penlty Free Energy of Chrging of the Ions N +, K +, C 2+, F -, Cl -, nd Br -. Using the LJ prmeters of Strtsm nd Berendsen 32 (see Tble 1), we computed solvtion free energies

7 1212 J. Phys. Chem., Vol. 100, No. 4, 1996 Hummer et l. TABLE 4: Results for the Men m c nd Fluctution f c of the Potentil (with Finite-Size Corrections Included) t the Position of Sodium, Potssium, Clcium, Fluoride, Chloride, nd Bromide Ions t Different Chrge Sttes q ion q/e m c f c N N N K K K C C C F F F Cl Cl Cl Br Br Br The dt were clculted from Monte Crlo simultions using N ) 128 wter molecules nd Ewld summtion over psses. The men nd the fluctution re listed s m c ) e( φ + qξ) nd f c ) βe 2 (φ - φ ) 2 - e 2 ξ, both in units of kilojoules per mole. The sttisticl errors of m c nd f c re estimted from block verges to be pproximtely 4.0 nd 30 kj mol -1. of ions representing N +, K +, C 2+, F -, Cl -, nd Br -. Agin, we emphsize the model chrcter of this study. Its purpose is not to provide ccurte theoreticl vlues for the free energies but rther to chrcterize the theory. We cn expect to obtin ccurte vlues only fter considerble improvement of the currently rther crude descriptions of the interction potentils used here nd similrly in most other studies. Some of tht work hs indeed been guided by using free energies of hydrtion. 38,39 However, controversies bout certin technicl spects, primrily regrding the correct tretment of long-rnge interctions, need to be resolved to obtin conclusive results. 40,41 We extensively studied the solvtion free energy of the sodium ction using the model described in section 3. Monte Crlo simultions using N ) 128 wter molecules were crried out for chrges 0, 0.5e, nd 1.0e to clculte the men m c nd the fluctution f c of the electrosttic potentil φ t the ion site. As in the previous clcultions, psses were used for verging. The results re listed in Tble 4. As for the unchrged methne, the potentil t the unchrged sodium site is slightly positive. The decrese of m c with incresing chrge is stronger thn liner, nd, correspondingly, the fluctution f c increses slightly with the chrge. This indictes tht simple Gussin model using n expnsion round the unchrged prticle is of limited utility. We use the informtion bout the derivtives to clculte the free energy of chrging using polynomil fits. The results for the sodium ion using polynomils of degrees 2, 4, nd 6 re compiled in Tble 5. Also included in Tble 5 re results obtined from TI, s described in section 3. TI ws performed using Ewld summtion nd N ) 8, 16, 32, 64, 128, nd 256 wter molecules s well s using the GRF Coulomb interction nd N ) 32, 64, nd 128 wter molecules. We observe excellent greement of the free-energy dt from polynomil fits nd TI, except for the p 2 fit which cnnot fully ccount for the incresing potentil fluctutions with incresing chrge. The TI dt of chrging from 0 to +e nd unchrging from +e to 0 show vritions of bout 5 kj mol -1. Regrding the system-size dependence, Ewld summtion gives ccurte results even for s few s N ) 16 wter molecules. The GRF shows more pronounced system-size dependence with the N ) 64 dt TABLE 5: Results for the Free Energy µ ex (kj mol -1 ) of Chrging the Sodium Ction from q ) 0 to +e in SPC Wter method Coulomb N ex µ self µ ex p 2 EW p 4 EW p 6 EW TIv EW TIV EW TIv EW TIV EW TIv EW TIV EW TIv EW TIV EW TIv EW TIV EW TIv EW TIV EW TIv GRF TIV GRF TIv GRF TIV GRF TIv GRF TIV GRF µ ex includes the finite-size corrections which re listed s µ ex self. The free energies were clculted from polynomil fits to the derivtive dt of Tble 4 (polynomils p n of degree n). Also included re results of thermodynmic integrtion (TI). Liner chrging pths from 0 to +e nd from +e to 0 re denoted by upwrd (v) nd downwrd (V) rrows, respectively. Ewld (EW) nd generlized rection-field (GRF) interctions were used for the chrges. TABLE 6: Results for the Free Energy µ ex (kj mol -1 ) of Chrging the Potssium, Clcium, Fluoride, Chloride, nd Bromide Ions from q ) 0 to (e, 2e in SPC Wter ion p 2 p 4 p 6 TIv TIV K C F Cl Br µ ex includes finite size corrections. Detils s in Tble 5. (cutoff r c ) 0.62 nm) being slightly too low. These results indicte tht the free energy of chrging is unexpectedly insensitive to the system size if the electrosttic interctions re treted ppropritely. In prticulr, it is importnt to pply the correct finite-size corrections. For Ewld summtion with N ) 16, for instnce, the finite-size correction ccounts for bout 60% of the free energy. Without the self-terms the Ewld results for N ) 256 nd N ) 16 differ by bout 63 kj mol -1 ; with the self-terms included the difference is only 5 kj mol -1. Tble 6 lists the results of polynomil fits of the free energy to the derivtive dt for the other ions studied (K +, C 2+, F -, Cl -, nd Br - ). Also included re results of TI clcultions using Ewld summtion nd N ) 128 wter molecules. Except for the polynomil fit of degree 2, we obtin consistent results from the derivtive dt nd TI. The p 2 results re lwys somewht too negtive, but this is more pprent for the negtive ions. The two TI dt per ion typiclly brcket the p 4 nd p 6 results for the free energy. Interestingly, there is no simple trend for the free energy of chrging of monovlent ctions with the ion size (s mesured by σ of the LJ interction). The positive ions N + nd K + s well s the negtive ions F -, Cl -, nd Br - show the expected increse of µ ex with incresing σ. However, only the negtively chrged methne-like LJ prticle Me - fits into this ordering. The positively chrged Me + hs less negtive µ ex thn K +, even though the vn der Wls dimeter σ of K + is considerbly lrger. However, the LJ interction of the K + ion

8 Free Energy of Ionic Hydrtion J. Phys. Chem., Vol. 100, No. 4, TABLE 7: Results for the Excess Chemicl Potentil µ ex (kj mol -1 ) of Trnsferring n Unchrged LJ Prticle from Idel Gs into SPC Wter LJ prticle µ ex LJ prticle µ ex N 9.2(1) Cl 21(3) K 23.7(5) Br 24(3) C 10.2(3) Me 10.2(9) F 9.7(2) The LJ prmeters re those of Tble 1. Errors re estimted from block verges. is more shllow thn tht of Me + with the LJ ɛ vlues differing by fctor of bout 150. We lso clculted the excess chemicl potentil of inserting unchrged LJ prticles in SPC wter of density F ) nm -3 t temperture T ) 298 K. This ws done using test-prticle insertion. 15,42-50 A set of 5000 SPC wter configurtions ws used of simultion run extending over Monte Crlo psses. The simultion ws performed using N ) 256 wter molecules nd GRF Coulomb interction with cutoff of r c ) 0.9 nm. We clculted exp(-βu) using 100 test prticles per configurtion, where u is the interction energy of LJ test prticle with the wter molecules. For the LJ interction, sphericl cutoff distnce of L/2 ) nm ws used. A cutoff correction for the r -6 term ws pplied, ssuming homogeneous wter density beyond the cutoff. The excess chemicl potentil is clculted s µ ex ) -k B T ln exp(-βu) (24) Results re listed in Tble 7. We find positive vlues for µ ex between 9 nd 25 kj mol -1, fvoring the gseous stte. Adding µ ex to the free energy of chrging, we obtin single-ion free energies of hydrtion. Experimentl dt for single-ion free energies of hydrtion hve been compiled by, for instnce, Friedmn nd Krishnn, 51 Conwy, 52 nd most recently Mrcus. 36 The first two references report vlues for the stndrd molr Gibbs free energy G 0, i.e., for hypotheticl trnsfer from 1-tm gs stte to 1 mol/l solution. Mrcus lists vlues for G* which is the Gibbs free energy of bringing n ion from n empty box into solution. The theoreticl clcultions determine the excess free energy of hydrtion, i.e., the trnsfer from n idel gs of given density to solution of equivlent solute density. This process corresponds to tht used by Mrcus, so tht G* is the experimentl equivlent of the theoreticl free energy tht we hve referred to s µ ex, disregrding volume contributions. Becuse Mrcus used G* for the experimentl free energies of hydrtion, we will retin tht nottion here for those quntities. Conversion from G to G* requires djustment for the differences in stndrd sttes: we dd to G the free energy of n idel gs going from pressure p 0 corresponding to density of 1 mol/l to pressure p 1 ) 1 tm, which is k B T ln(p 1 /p 0 ), i.e., G* ) G kj mol Another correction ccounts for differing vlues for the reference ion H +. We tke the most recent vlue by Mrcus 36 G*[H + ] ) ( 6 kj mol -1 nd djust the other vlues [ nd ( 17 kj mol ] ccordingly. Results for the clculted free energy of ionic hydrtion µ ex ) µ ex (q)0) + µ ex (0fq) nd the experimentl vlues G* re compiled in Tble 8. For the clculted vlues we use those obtined from fit of sixth-order polynomil p 6 to the derivtive dt, s listed in Tble 6. The experimentl dt were djusted s described bove. The experimentl dt for ctions show little vrition between the three sources. However, the nion dt vry by s much s 70 kj mol -1, with the Conwy TABLE 8: Results for the Clculted Free Energy of Ionic Hydrtion (kj mol -1 ) Compred with Experimentl Dt G* ion µ ex b c d e f g N K C F Cl Br The experimentl dt were djusted to give G* ) kj mol -1 for H +, s used by Mrcus. 36 Also included re computer simultion results by Strtsm nd Berendsen 32 nd Migliore et l. 55 b Experimentl dt of Mrcus. 36 c Experimentl dt of Friedmn nd Krishnn. 51 d Experimentl dt of Conwy. 52 e Computer simultion dt of Strtsm nd Berendsen clculted using moleculr dynmics of N ) 216 wter molecules. 32 The results contin Born-type correction pplied by the uthors to their rw dt. f Computer simultion dt of Strtsm nd Berendsen without the Born correction. 32 g Computer simultion dt of Migliore et l. clculted using moleculr dynmics of N ) 342 wter molecules. 55 dt 52 brcketed by the those of refs 36 nd 51, but generlly closer to the dt of Mrcus. 36 The clculted free energy dt for ctions do not show cler trend. The results for N + nd K + re too low nd too high by bout 10%, respectively. The hydrtion free energy of C 2+ is too high by bout 15%. The nions on the other hnd show cler tendency with the mgnitudes of the clculted free energies generlly being too lrge. The reltive errors re 26, 10, nd 15% for F -, Cl -, nd Br -, respectively, compred to the dt of Mrcus. These significntly too negtive vlues of the hydrtion free energy of nions might be consequence of the unprotected hydrogen toms in the wter-ion interction model used. The positively chrged hydrogen tom cn penetrte the LJ shell of the ions without direct energetic penlty. The interction with the negtive point chrge t the center of the ion strongly binds the wter molecule, resulting in lrge enthlpic contribution to the free energy of hydrtion. But lso effects of nondditive interctions might ply considerble role. 54 Also included in Tble 8 re computer simultion results by Strtsm nd Berendsen. 32 These uthors used thermodynmic integrtion in conjunction with isotherml-isobric moleculr dynmics simultions to compute hydrtion free energies of ions. The interction potentils used here re identicl with those of Strtsm nd Berendsen, except for the tretment of the electrosttic interctions. We used Ewld summtion, wheres Strtsm nd Berendsen used sphericl cutoff nd Borntype correction for finite-size effects. These uthors (nd others 39 ) rgue tht the ppliction of Born-type correction is rther crude, pproximting the solvent molecules beyond the cutoff by dielectric continuum. Nevertheless, in the bsence of better lterntive it hs been widely dopted. Migliore et l. 55 clculted the free energy of ionic hydrtion bsed on perturbtion formul from Monte Crlo simultions using MCY wter nd b initio ion-wter potentils. These uthors lso used sphericl cutoff. Tble 8 includes the results of Migliore et l., who did not pply finite-size correction. Qulittively, our free-energy dt gree with those of Strtsm nd Berendsen 32 nd Migliore et l. 55 We observe the sme ordering of the free energies with respect to ion size. The quntittive greement is however poor. Our vlues for the ctions N + nd K + re closer to the experimentl dt of Mrcus. The ction free energies of Strtsm nd Berendsen (with Born correction) re consistently more negtive thn those of our clcultions. On the other hnd, our nion free energies re significntly more negtive thn those of Strtsm nd µ ex

9 1214 J. Phys. Chem., Vol. 100, No. 4, 1996 Hummer et l. Figure 6. Energetics of clusters of fluoride ion nd SPC wter. Results re shown for the interction energy u s of the fluoride ion with the wter (]), s well s the men φ (+) nd vrince φ 2 (0) of the electrosttic potentil t the ion position. The figure shows differences of these quntities with respect to the bulk vlues clculted from Monte Crlo simultion of n N ) 128 wter-molecule system using Ewld summtion: u s ) u s - u s,ew, φ ) φ - φ EW, nd φ 2 ) φ 2 - φ 2 EW. The lines re fitted curves s explined in the text. Error brs indicte 1 stndrd devition estimted from block verges. The stndrd devitions of the bulk nd cluster dt were dded. Berendsen s well s of Migliore et l. The results of Strtsm nd Berendsen for Cl - nd Br - re somewht closer to the experimentl dt of Mrcus when the Born correction is included. Without the correction they re significntly too high. The most pronounced discrepncies between the nion dt of Strtsm nd Berendsen 32 nd ours re those of the fluoride ion, with our µ ex vlues being lower by 83 kj mol -1. This is somewht surprising since Strtsm nd Berendsen used the sme prmeters for the wter-wter nd wter-ion interctions. The difference cn be consequence of using different ensembles (NVT versus qusi-npt), or, more likely, it is cused by the different tretment of the electrosttic interctions (Ewld versus sphericl cutoff). The fluoride ion lso shows the lrgest reltive devitions from the experimentl results. For further investigtion of these discrepncies, we hve studied the energetics of clusters of different size formed by single fluoride ion nd wter. We hve performed Monte Crlo simultions using one F - ion tht nucletes N ) 4, 8, 12, 16, 32, 64, 128, nd 256 wter molecules t 298 K, s described in section 3. We clculted the interction energy u s of the fluoride ion with the SPC wter molecules over psses. Figure 6 shows the differences u s ) u s - u s,ew with respect to the bulk simultion using N ) 128 wter molecules nd Ewld summtion s function of N -1/3. u s cn be fitted to third-order polynomil in N -1/3 over the whole rnge of system sizes. Extrpoltion to N f yields limit for u s close to zero. (However, the nontrivil dependence on N -1/3 limits the ccurcy of the extrpoltion.) The result obtined from Ewld summtion, u s,ew ) ( 4 kj mol -1, lso grees with the vlue u s ) kj mol -1 obtined from integrting the pir correltion functions of the bulk simultion using φ(r) ) 1/r in eq 21, dding the LJ contributions, nd pplying the finite-size correction -e 2 ξ 1/r. The integrtion shows tht the LJ contributions re strongly repulsive ( 90 kj mol -1 ) but compensted by lrge electrosttic interctions. The vlue for the solvtion energy reported by Strtsm nd Berendsen, u s ) -823 kj mol -1, is however considerbly smller. The observed differences in u s of bout 150 kj mol -1 gree in mgnitude nd sign with those of the free energies (83 kj mol -1 ). If we truncte the integrtion of 1/r weighted with the pir correltion functions obtined from Ewld summtion t R ) 0.9 nm (which is the cutoff Strtsm nd Berendsen used) nd do not pply finite-size correction, we obtin vlue of -867 kj mol -1 in much closer greement with Strtsm nd Berendsen s. This indeed indictes tht the tretment of the electrosttic interctions (Ewld summtion versus sphericl cutoff) is the mjor source of the discrepncy. Also included in Figure 6 re the results for the men nd the vrince of the electrosttic potentil t the ion site. Figure 6 shows differences with respect to the bulk vlue. The differences of the men vlues φ closely follow the solvtionenergy differences u s nd cn lso be fitted to third-order polynomil in N -1/3. The differences of the fluctution φ 2 depend linerly on N -1/3 for N between 8 nd 256. Both fitted curves extrpolte to pproximtely 0, indicting tht the clculted bulk vlues re the correct limits for N f. From the cluster-size dependence of the solvtion energy nd the men nd vrince of the electrosttic potentil, s well s the results for Me nd N +, we conclude tht the use of periodic boundry conditions in conjunction with Ewld-summtion (or rection-field) electrosttics closely pproximtes the correct bulk behvior of the system; however, to get correct energetics, it is importnt to include the self-interctions in the Coulomb energy. 5. Conclusions We hve shown tht free energies cn be ccurtely clculted from equilibrium simultions by extrcting derivtive informtion with respect to coupling prmeter. We hve studied the free energy of electrosttic chrging in wter, which ccounts for most of the free energy of ionic solvtion for typicl ion sizes. The choice of the ionic chrge s coupling prmeter results in free-energy expressions involving cumulnts of the electrosttic potentil φ t the chrge sites. We find tht the sttistics of φ re pproximtely Gussin. This mens tht only the first nd second moment of the distribution cn be clculted ccurtely, with higher moments dominted by the poorly smpled tils. Correspondingly, only informtion bout the first nd second derivtive of the free energy cn be clculted ccurtely for ny given chrge stte. The informtion for different chrge sttes (e.g., unchrged nd fully chrged) cn then be combined using interpoltion or polynomil fitting. We hve studied methne-like Lennrd-Jones prticle in SPC wter. We observe two lmost Gussin regimes seprted by q ) 0 with different chrcteristics. Negtive ions re more stbly solvted compred to positive ions of equl size, in greement with the experimentl dt. 37 The system shows further symmetry, since the verge electrosttic potentil t the position of the unchrged prticle is positive. This mens tht incresing the ion chrge first costs energy. We relte these symmetries of the energetics (lower free energy of negtive ions, positive potentil) to the structurl symmetry of the wter molecule. The hydrogen toms cn penetrte the ionic vn der Wls shell, wheres the oxygen tom is better protected. For the unchrged prticle, this results in net positive potentil, nd the point chrge t the center of negtive ions exerts strong electrosttic interctions with the tightly bound hydrogen of wter. However, prticulrly for smll nions this effect might be exggerted by the interction potentils used. This potentil model does not give protective vn der Wls sphere to the chrge on the hydrogen tom. In principle, this is fundmentl difficulty, but in computer simultions, the heights of energetic brriers usully exclude the singulrity. The development of interction potentils for nion-wter interctions nevertheless hs to ccount for these problems. The strong interctions with the hydrogens pull the wter closer, nd the first mxim of