Calculation of ionic charging free energies in simulation ...

Calculation of ionic charging free energies in simulation systems withatomic charges, dipoles, and quadrupolesDavid H. Herce, Thomas Darden, and Celeste Sagui Citation: J. Chem. Phys. 119, 7621 (2003); doi: 10.1063/1.1609191 View online: http://dx.doi.org/10.1063/1.1609191 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v119/i15 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 15 15 OCTOBER 2003

ARTICLES

Calculation of ionic charging free energies in simulation systemswith atomic charges, dipoles, and quadrupoles

David H. HerceDepartment of Physics, North Carolina State University, Raleigh, North Carolina 27606

Thomas DardenNational Institute of Environmental Health Sciences Research Triangle Park, North Carolina 27709

Celeste SaguiDepartment of Physics, North Carolina State University, Raleigh, North Carolina 27606

~Received 22 April 2003; accepted 21 July 2003!

The ionic charging free energy is a very sensitive probe for the treatment of electrostatics in anygiven simulation setting. In this work, we present methods to compute the ionic charging free energyin systems characterized by atomic charges and higher-order multipoles, mainly dipoles andquadrupoles. The results of these methods for periodic boundary conditions and for sphericalclusters are then compared. For the treatment of spherical clusters, we introduce a generalization ofGauss’ law that links the microscopic variables to the measurable macroscopic electrostatics via awork function. © 2003 American Institute of Physics.@DOI: 10.1063/1.1609191#

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I. INTRODUCTION

The study of solvated ions is fundamental to a larnumber of chemical and biological systems. For instanmetal ions play a crucial role in the structure and stabilitynucleic acids, enzymatic catalysis, and a variety of chemreactions. In particular, the structure of the solvation shethe dynamics of ions in water, and the ionic charging frenergy have become classical research problems, givingto a large amount of experimental1–7 and theoretical studiesusing both classical and quantum mechanical molecularnamics approaches.8–20 The molecular dynamics simulationcomplement the experimental results and lead to a deinsight into the solvation and diffusion of ionic specieThese days, quantum studies of the solvation sheespecially those involving the relatively inexpensive densfunctional theory methods—have become fairly popularthe description of solvation shells around ions. However, istill questionable whether these simulations can achieve‘‘bulk’’ behavior. Classical models, on the other hand, easyield bulk behavior as well as those quantities that requextensive conformational sampling, such as the free eneYet these models have intrinsic inaccuracies, as is repeatpointed out when they are compared to quantum results.ten, authors try to capture ‘‘the best of both worlds’’ throuthe use of quantum mechanics and molecular mecha~QM-MM ! approaches. However, the fact still remains ththe majority of the biomolecular simulations—due to thsheer size—employ classical water models. Ideally, then,would like to build into these classical models as many qutum features as possible. This opens the door to a diffesort of problems. Apart from all the issues concerningrametrization, the correct treatment of electrostatics is scially relevant21 and generally far from trivial. In addition

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when size becomes a real problem for simulations, muscale methods are used to interface the atomistic solugenerally surrounded by some necessary explicit waterand a continuum representation of the solvent. The mchallenge in these approaches is how to build the interfbetween the cluster and a continuous solvent. Perhaps onthe simplest systems where one can test the correctnesthe interface consists of an ion in the center of a sphereexplicit waters.

The ionic charging free energy is a very sensitive proof the quantitative behavior of ions and solvent withinsimulation frame and for the correct treatment of electrosics. In the past,22,23 ionic charging free energies have beused to test different electrostatic approaches: to test thelidity of ‘‘P summations’’ against that of ‘‘M summations’’24

and to study the role of cluster and periodic boundary cditions. However, current trends in classical force fieldsto include multipolar approaches for a more accurate repsentation of electrostatics. Since accurate descriptions ofvent and ionic behavior requireat leastthe inclusion of po-larization, in this paper we generalize previous treatmentionic charging free energies to include permanent or indupoint multipoles, mainly dipoles and quadrupoles. In adtion, the role of cluster and periodic boundary conditionsinvestigated. Note that the emphasis is on the developmof the correct treatment of electrostatics with multipoles uder the two boundary conditions, and not on the testingany particular water model.

In the calculation of Coulombic lattice sums under peodic boundary conditions where only charges are presentsum ofqi /r i depends on the order of summation—that is,the order in which the individual atomic interactions aaccumulated.25 For example, for 0,r ,`, let S1(r ) ~known

1 © 2003 American Institute of Physics

license or copyright; see http://jcp.aip.org/about/rights_and_permissions

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7622 J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 Herce, Darden, and Sagui

as P summation! denote the sum ofqi /r i over all atoms withr i,r and letS2(r ) ~known as M summation! denote the sumof qi /r i over all water molecules whose oxygen distanfrom the ion center is less thanr. In the past, the use ofS1(r )or S2(r ) has been contentious, as discussed in Ref. 23. Hmer et al.26 and Ashbaugh and Wood27,28 have argued thathe choice ofS1(r ) is physically correct. The use ofS1(r )leads to a physically correct value of 0 for the electrostapotential of an ideal gas of water molecules, while othapproaches likeS2(r ) lead to artifactual nonzero values. Ifinite clusters, on the other hand, the sum is always zerothere are no problems of convergence. The conventiovalue of the electrostatic potential for spherical clust~SBCs! is calculated by taking the ensemble average offinite sum and adding a Born correction2q(121/e)/R,whereR is a measure of the cluster radius. Unfortunately,waters near the vacuum–solvent boundary have artifacbehavior that affects the dynamics of nearby waters.avoid these artifacts that affect the electrostatic potential,important to consider theasymptoticbehavior ofS1(r ) orS2(r ), for r large, but not approaching the cluster radiuBoth S1(r ) andS2(r ) are observed to reach asymptotic pteaus within the cluster interior, but the asymptotic valuachieved are different.26

Darden et al.23 demonstrated that when Gauss’ lawapplied to a spherical cluster, this leads to a definition ofelectrostatic potential of ions in clusters that involves conbutions only from the interior solvent. Thisinterior potentialagrees with the P-summation result. In fact, ionic chargfree energies based on the interior potential in SBCs wshown to agree with the results obtained in systems unperiodic boundary conditions~PBCs! using Ewald summa-tions with finite-size corrections. Furthermore, the differenbetween the electrostatic potential calculated in this manand the conventional value of the SBC electrostatic potenis given ~except for a small continuum correction! by thework required to take a test charge through the vacuusolvent interface. This potential difference is essentially cstant as a function of ionic charge and radius, with a nonzlimit for infinite clusters. For pure solvent clusters, this wois known as the surface potential of water and depends onwater model~its experimental value is not known!. The ex-istence of this surface potential bears important conquences for the interface between an atomistic cluster acontinuum solvent. The surface potential represents an etrostatic potential discontinuity which is independent of tstatistical mechanics method used to build the interfaTherefore, unless improved water models are used cluscontinuum multiscale efforts are doomed to spurious forat the interface.

In this paper we propose a method to compute iocharging free energies in the presence of point dipolesquadrupoles. We apply Gauss’ law to a spherical cluster leing to a definition of aninterior potential inside the clusterThis, in turn, leads to an ionic charging free energy in agrment with that obtained in systems under PBCs using Ewsummations with finite-size corrections. To test our formism, we chose RPOL~Ref. 29! as a convenient model fopolarizable water~this model has been presented in the


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erature with different van der Waals parameters; the oneschose are in Table I!. The issue of van der Waals parameteis crucial in classical measurements of ionic charging fenergies, as well as for other structural and diffusioproperties.30 With the parameters in Table I, the results otained with RPOL in this work agree fairly well with thosobtained using the SPC model.

A brief outline of the paper is as follows: first we discuionic free energies in a solvent with multipoles under SBCUsing Gauss’ law for a system with point charges, dipoland quadrupoles, we derive the radial work function whoasymptotic limit is the bulk electrostatic potential of the ioat the center of the cluster. We next discuss PBCs by definthe electrostatic potential of the ion, both in terms of papotential and self-potential contributions. Next, we discumethods for computing the ionic free energies: thermonamic integration using the trapezoidal rule with many intmediate charge states and free energy perturbation withsiderably less intermediate charge states. Next, we descour molecular dynamics simulations and the parameused. Section V presents our results. The paper ends wshort conclusion.

II. SPHERICAL CLUSTER AND PERIODICBOUNDARY CONDITIONS IN IONIC CHARGING FREEENERGIES IN SOLVENTS WITH ELECTROSTATICMULTIPOLES

The free energy due to the polarization of the mediumchanging the charge state of an ion fromq5q(0) to q(1) isgiven by evaluating the electrostatic potential of the ionCq

at intermediate charge statesq(1)<q<q(2) and integratingagainstdq:

DG5Eq~0!

q~1!

Cqdq. ~2.1!

The simplest case to consider is that of a continumodel of charging an ion. A spherical cavity with interndielectric constant 1 and radiusa is immersed in an infiniteisotropic linear dielectric continuum with dielectric constae. A point charge at the center of the cavity is slowly chargfrom chargeq(0) to q(1). If the charge of the ion isq, thework of bringing a small bit of additional chargedq frominfinity to a point rÞ0 is given byCq(r )dq, wherer 5ur uand whereCq(r ) denotes the electrostatic potential at tpoint r . Using Gauss’ law and the spherical symmetry of tsystem, the potential when the ion has chargeq is Cq(r )5q/(er ) for r .a and Cq(r )5@q/r 2q/a1q/(ea)# for 0,r ,a. HereCq(r ) includes contributions from the polarized dielectric medium as well as the direct Coulomb intaction with the point charge at the center of the cavity. Ththe energetic cost, due to the polarization of the mediumbring the chargedq from infinity to the pointr is given by@Cq(r )2q/r #dq, and the work in incrementing the chargof the ion is given by taking the limit asr→0. Finally, thework due to the polarization of the medium in charging tion is given by the familiar Born expression

DG52~121/e!@q~1!22q~0!2#

2a. ~2.2!


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7623J. Chem. Phys., Vol. 119, No. 15, 15 October 2003 Ionic charging free energies in simulation systems

Similarly, the free energy of charging a simple point ioat the center of a bath of explicit solvent can be obtainedevaluating, for each intermediate ionic charge stateq, thework required to increment the ionic charge by a smamountdq and then summing these work contributions. Twork required to increment the ionic charge is given by mtiplying the electrostatic potentialCq at the ion bydq. If thesystem is described by higher-order multipoles in additioncharges, the electrostatic potential also depends on thesethe charging free energy is still given by Eq.~2.1!. Onemight also be interested in considering the work due topolarization of the medium when incrementing the dipolethe ion bydp. If E0 represents the electric field at the ioposition ~minus the field produced by the ion itself!, thiscontribution is

DGp52Ep~0!

p~1!

E0•dp. ~2.3!

By symmetry, this contribution is zero in a spherical ion.

A. Spherical cluster simulations

For the case of an ionfixedat the center of a sphericacluster of water molecules, Gauss’ law provides a straigforward path to a correct evaluation of the electrostatictential of the solvated ion at different charge states.23 In thecontext of explicit solvent, withe51, Gauss’ law states thathe surface integral of the electric fieldE over a closed sur-face equals 4p times the total charge contained within thsurface. Given a point ion at the center of a spherical cluof solvent, with interactions modeled by a simple atom–ateffective pair potentials, the total charge contained withisphere about the ion is obtained by summing the atopartial charges of the atoms contained within the sphere

In this work, we consider the physically important cawhere both dipoles and quadrupoles are present in thetem. We consider an ion in a spherical cluster ofN watermolecules. We assume for simplicity that each atomiccarries a chargeqi , a dipolepi , and a quadrupoleQi . Pois-son’s equation in this case is

¹2F~r !524p (i 51

3N11

~qi1pi•“ i1Qi :“ i“ i !d~r2r i !,

~2.4!

whereq15q is the fractional charge of the ion, andp1 andQ1 its dipole and quadrupole. Next, we integrate overspherical volumeV of radiusr centered at the origin:

EV¹2F~r !dV52E

SE~r !• r r 2dV, ~2.5!

wherer 5r /ur u is the unit radial vector anddV the differen-tial solid angle. We define thetime and angular average othe electrostatic field as

E~r !51

4p K E E~r !• r dV L , ~2.6!

where the brackets indicate time average. With this defition, the volume integral of Eq.~2.4! becomes


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r 2E~r !5q1Cq~r !2K EV“•S (

i 51

3N11

pid~r2r i !D dVL1K E

V“•S (

i 51

3N11

Qi•“d~r2r i !D dVL , ~2.7!

whereCq(r ) is thetime-averagedsum of the partial chargeof the water molecules contained in the volume 4pr 3/3. Wethen define a vectorPp related to the instant water dipoldensity and a vectorPQ related to the instant water quadrupole density by

Pp~r !5 (i 52

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pid~r2r i !,

~2.8!

PQ~r !52 (i 52

3N11

Qi•“d~r2r i !.

When the individual dipolespi are induced, the time averagof Pp(r ) is the standard polarization of the system. The timaverage of the volume integrals in Eq.~2.7! represents effec-tive chargesCp and CQ . These volume integrals can btransformed in surface integrals~the contributions of thespherically symmetric ion vanish!:

Cp5K 2EV“"Pp~r !dVL

5K 2 (i 52

3N11 ESpi• rd~r2r i !r

2dVL5K 2 (

i 52

3N11

pi• r id~r 2r i !L , ~2.9!

whered(r2r i) is the volumedelta function whiled(r 2r i)only depends on the radial coordinate, and

CQ5K 2EV“"PQ~r !dVL

5K (i 52

3N11 ESr •Qi•“d~r2r i !r

2dVL . ~2.10!

Finally, the time- and angular-averaged electric fieldE(r ) inthe radial direction is given by

E~r !5q1C~r !

r 2, ~2.11!

where C(r ) is the total charge, C(r )5Cq(r )1Cp(r )1CQ(r ), due to the water molecules.

The workWq(r 1 ,r 2)dq done in bringing a small chargdq from a distancer 2 to r 1 , 0,r 1,r 2,`, is obtained byintegratingE(r )dq over r 1<r<r 2 . Carrying out the inte-gration, we obtain

Wq~r 1 ,r 2!5Er 1

r 2E~r !dr5

q1C~r 1!

r 12

q1C~r 2!

r 2

1Er 1

r 2 dC~r !

dr

1

rdr. ~2.12!


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Noting thatC(r )50 for r sufficiently small and that thework done by the solvent when the charge of the ionincremented bydq does not include the bare Coulomb inteaction between the ion and incremental chargedq, we definethe radial work functionWq

0(r ) for 0,r ,` by

Wq0~r !5 lim

r 1→0@Wq~r 1 ,r !2q/r 1#

52q1C~r !

r1E

0

r dC~s!

ds

1

sds. ~2.13!

Using dCq(r )/dr5^( i 523N11qid(r 2r i)& and Eqs.~2.9!

and ~2.10!, Wq0(r ) can also be expressed as

Wq0~r !52

q

r1

^( r i<rqi&

r1

^2( r i5rpi• r i&

r 2

1^( r i5r r i•Qi• r i&

r 31K (

r i,rS qi

r i2

pi• r i

r i2

1@3r i•Qi• r i2Tr~Qi !#

r i3 D L , ~2.14!

where the last term in this expression is the P summatioIf the cluster were infinite, the bulk value of the electr

static potential of the partially charged ion, which is the eergetic cost~in units of dq) due to the bath of solvent, tobring a test chargedq in from infinity would clearly be givenby the limit of Wq

0(r ) as r tends to infinity. For larger thetime-averaged electric field strengthE(r ) is well approxi-mated byq/(er 2), wheree is the dielectric constant of wateThus, for larger 1 and r 2 , r 1<r 2 , Eq. ~2.12! becomes

Wq0~r 2!2Wq

0~r 1!5Wq~r 1 ,r 2!'~q/e!~1/r 121/r 2!'0~2.15!

andWq0(r ) should converge rapidly to a limit.23 In fact, it has

been shown that for sufficiently large clusters thesize-corrected radial work functionWq

0(r )1q/(er ) reaches anasymptotic plateau within the interior of the cluster, and tplateau value closely approximates the bulk value ofelectrostatic potential obtained in infinite clusters. Thobservation23 is also valid in solvents having lower dielectrconstants, where the sumWq

0(r )1q/(er ) converges rapidlyto the bulk limit. The dielectric constante needed for thiscalculation can be obtained from simulations of the pure svent.

For larger, q1C(r ) becomes approximatelyq/e. Thusby Eq. ~2.14!, for larger ~inside the cluster!,

Wq0~r !1q/~er !'K (

0,r i,rS qi

r i2

pi• r i

r i2


r i3 D L . ~2.16!

On the other hand, ifr is outside the cluster, Eq.~2.14!becomes


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i 52

3N11 S qi

r i2

pi• r i

r i2


r i3 D L

2q~121/e!

r. ~2.17!

It is possible to define theinternal electrostatic potentiain SBCs by averaging over the oscillations:

Cq~SBC,int!51

r 22r 1E

r 1

r 2@Wq

0~r !1q/~er !#dr. ~2.18!

To define theexternalelectrostatic potential, we observe thin Eq. ~2.17! the electrostatic potential at the ion is given bthe time average of the sum of the potential producedevery atom of the solvent, plus a Born correction. As longr is outside the cluster,only the Born correction depends or. The most appropriate value ofr outside the cluster is onethat is as small as possible, but still outside the charge dsity of the cluster. We definer ] to be the radial distance awhich the water density drops below some small predefilevel. Thus, noting that the conventional value of the electstatic potential has some small contribution due to denoutsider ] , we define theexternalelectrostatic potential inSBCs by

Cq~SBC,ext!5Wq0~r ]!1q/~er ]!

'K (i 52

3N11 S qi

r i2

pi• r i

r i2


r i3 D L

2q~121/e!

r ]. ~2.19!

In this expression, the term in brackets would represent thsummation evaluated outside the cluster. It is trivial to shthat the differenceCq(SBC,ext)2Cq(SBC,int) is not zero,but it is the work~perdq) of taking a test chargedq throughthe vacuum–water interface—i.e.,the surface potential ofthe water cluster—plus correction terms that are small forhigh dielectric solvent like water. We find thaCq(SBC,ext)2Cq(SBC,int) is essentially invariant to ioniccharge, polarization, and radius.

B. Periodic boundary conditions: Ewald summation

Under PBCs, the simulation cell consisting of a simppoint ion in a solvent bath ofN water molecules is replicateon a lattice to form a very large array of copies, with toriginal simulation cell at its center. The array of replicatcells is immersed in a dielectric continuum. We assume tthe dielectric constant of this continuum is infinite~‘‘con-ducting’’ boundary conditions!. Consider a charged ion inbath ofN water molecules. The ion is identified by the suffi‘‘1’’ and has a chargeq15q, a dipolep1 , and a quadrupole


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Q1 . The system can therefore be described by a set of pcharges, dipoles, and quadrupoles$qi ,pi ,Qi% with i51,2,...,3N11, at positionsr1 ,r2 ,...,r3N11 within the unitcell. The water charges satisfy total neutrality—i.e.,q21q3

1¯1q3N1150—and a uniform plasma is added to neutrize the charge of the ion. The edges of the unit celldenoted by vectorsaa , a51, 2, 3, which need not beorthogonal. The conjugate reciprocal vectorsaa* are definedby the relationsaa* "ab5dab ~the Kronecker delta!, for a,b51, 2, 3. The point chargeqi at positionr i has fractionalcoordinatessa i , a51, 2, 3, defined bysa i5aa* "r i . The po-tential energy function for the system in PBCs is that foratoms in the central simulation cell. The atoms in the cencell interact with each other, with the atoms in the replicacells, with the induced charge distribution in the surrounddielectric continuum, and finally with the uniform neutraliing plasma. In the limit of an infinite array of copies of thcentral cell, the Coulomb interactions, including those wthe neutralizing plasma and with the surrounding dieleccontinuum, are treated by Ewald summation. This sumsplit into a short-range term which is handled in the dirsum, plus a long-range, smoothly varying term, handledthe reciprocal sum by means of Fourier methods. Len5n1a11n2a21n3a3 be a translation of the lattice and definthe reciprocal lattice vectorsm by m5m1a1* 1m2a2*1m3a3* with m1 , m2 , m3 integers not all zero. The direcpart can be expressed as a sum of pair interactions over mmum image pairs. For atomsi and j in the central cell, hav-ing coordinatesr i and r j , respectively, letr i j denote theminimum image distance between them—that is,r i j 5ur i j u5minnur i2r j1nu, where the minimum is over all integetriples (n1 ,n2 ,n3). If the Ewald parameterb, which controlsthe relative rates of convergence of the infinite series giby the direct and reciprocal sums, is chosen so that the dsum is restricted to the minimum image interaction, theninstantaneous electrostatic potential of the ion can be wrias23,31

Cq~x!5 (i 52

3N11

$qiB0~r 1i !1@pi•~r12r i !1Tr~Qi !#B1~r 1i !

1Qi :~r12r i !~r12r i !B2~r 1i !%11

pV

3 (mÞ0

exp~2p2m2/b2!

m2exp~22p im"r1!

3S~m!2q2b

Ap2q

p

b2V2Tr~Q1!

4b3

3Ap, ~2.20!

with

B0~r !5erfc~br !

r, ~2.21!

Bl~r !51

r 2 F ~2l 21!Bl 21~r !1~2b2! l

bApexp~2b2r 2!G ,

l .0. ~2.22!


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In the equations above,x denotes the multidimensional vector of instantaneous atomic positionsr i , 1< i<3N11 andV5a1"a2Ãa3 is the volume of the unit cell. In this expression forCq , the self-potential22 is already accounted for anshould not be added. The structure factorS(m) is given by

S~m!5(j 51

N

@qj12p ipj "m24p2Qj :mm#

3exp~2p im"r j !, ~2.23!

where exp(2pim"r j )5exp@2pi(m1s1j1m2s2j1m3s3j)# andsa j , a51, 2, 3, are the fractional coordinates of sitej, de-fined above.

Thus, under PBCs, the simulation dynamics for the stem consisting of a point ion having chargeq in a periodi-cally replicated cube of water molecules are determinedthe electrostatic potential of the ionCq(x) and the Lennard-Jones interactions, which converge absolutely when sumover the array of periodic images of atoms in the censimulation cell. The electrostatic potential of the ion is ttime average of the instantaneous electrostatic potentialCq

and can be written as

Cq~PBC!5^Cq~x!&. ~2.24!

In summary, we have presented expressions for the etrostatic potentialCq(PBC) of the ion under PBCs, for thinternal electrostatic potential of the ion in a spherical cluter, Eq.~2.18!, and for theexternalelectrostatic potential ofthe ion in a spherical cluster, Eq.~2.19!. The equations forthe SBC potential are expressed in terms of thesize-corrected radial work function given by Eqs.~2.16! and~2.17!. These expressions for the electrostatic potential wbe then used to compute the ionic charging free energydescribed in the next section.

III. CHARGING FREE ENERGIES

A. Thermodynamic integration

Here, we consider the problem of calculating the frenergy of changing the charge state of a simple point iona bath of water molecules using thermodynamic integratiThe system in each of the two ionic charge states~neutral orfully charged! is characterized by a Hamiltonian consistinof a kinetic energy term and a potential energy termV(l,x),wherel50, 1 denotes the charge state andx the multidimen-sional vector of atomic positionsr i in the (3N11)-particlesystem.

To calculate the free energy of charging the ion usthermodynamic integration~TI!, it is necessary to definepotential energy function for the intermediate~nonphysical!values of the ionic charge statel. Since the perturbation duto changing the atomic charges does not generally leasingularities, the intermediate potentialV(l,x), for 0<l<1,can be given by simple linear interpolation:

V~l,x!5~12l!V~0,x!1lV~1,x!. ~3.1!


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In more complex cases where the perturbed system differmore than the atomic charges—for example, if atoms‘‘appearing’’ or ‘‘disappearing’’—it may necessary to invokmore complex interpolation rules.

The free energy difference between the two ionic chastates is given by

DG5E0

1K ]V~l,x!

]l Ll

dl5E0

1

^V~1,x!2V~0,x!&ldl,

~3.2!

where^ &l , 0<l<1, denotes the ensemble average oversystem at charge statel.

For PBCs the above equation simply becomesDG5*q1(0)

q1(1)Cq(PBC)dq. Under SBCs, the free energy calc

lated by TI will agree with that obtained using the convetional or finite cluster value for the electrostatic potent~without the Born correction!. For this reason we largelyrelied on standard molecular dynamics methods to obtime-averaged electrostatic potentials at the ions, which wthen used to obtain charging free energies in the two bouary conditions.

B. Free energy perturbation theory

Another way to compute the free energy is through pturbation theory. For instance, Eq.~3.1! may be rewrittensuch that the Hamiltonian of the system is expressedH(l,x)5H0(x)1lU(x). In this expression, ifH0 is, forinstance, the Hamiltonian for the uncharged state, tU(x)5V(1,x)2V(0,x). In this section we will review somemethods for an efficient computation of free energy diffences as proposed by Hummer and Szabo.32 These methodsare based on free energy perturbation theory~FEPT!.

The free energy difference between two states characized byl andl0 can be expressed as an equilibrium averain a canonical ensemble:

DG~l0→l!52kBT lnêxp@2b~l2l0!U~x!#&l0,~3.3!

whereb5(kBT)21 and the brackets indicate a canonical aerage with the ‘‘equilibrium’’ HamiltonianH0(x)1l0U(x).A Taylor’s expansion inl gives

DG~l0→l!5 (n51

`~l2l0!n

n!G~n!~l0!, ~3.4!

whereG(n)(l)5]nG(l)/]ln is thenth derivative of the freeenergy, which in the first three terms of the expansiongiven by

G~1!~l!5Û&l ,

G~2!~l!52b^~U2Û&l!2&l , ~3.5!

G~3!~l!52b2^~U2Û&l!3&l .

In the TI method described in the previous section,free energy difference is computed between the states dmined byl50 andl51:


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e

e

-l

inred-

r-

as

n

-

r-e

-

s

eer-

DG~0→1!5E0

1

G~1!~l!dl5E0

1

Û&ldl

5E0

1

^V~1,x!2V~0,x!&ldl. ~3.6!

This integral can be calculated by the simplest methodcomputing the integrand in very smalll intervals or by usinga weight scheme:

DG~0→1!'(i 51

N

wiG~1!~l i !, ~3.7!

wherewi and l i , i 51,2,...N, are the weights at the pointwhere the integrand is evaluated. Ifexact values ofG(1)(l i) are available, then the approximation forDG(0→1) is exact for a polynomial expansion ofDG(l) in l ofdegree one less than the number of free parameters. Ifl i andwi are taken as free parameters, theN-point Gauss–Legendre integration formula forDG(0→1) is exact to2Nth order of FEPT. For instance, the expression for tpoints (N52),

DG~0→1!'1

2~Û&1/221/A121Û&1/211/A12!, ~3.8!

is exact to fourth order of FEPT.Often, one wants to compute changes in the free ene

based on simulations of the initial~l50! and final ~l51!states. The literature~see, for instance, Ref. 32 and refeences therein! provides easy-to-use formulas for these casIn particular, the addition of an adjustable, intermediate scan improve the procedure. In this case, the free paramecomprise the weightsw0 and w1 at the end points, the adjustable intermediate statel int , and its corresponding weighwint . An example of this procedure is Simpson’s rule, whiis exact to fourth order of FEPT:

DG~0→1!'1

6~Û&01Û&1!1

2

3Û&1/2. ~3.9!

One may generalize Eq.~3.7! by including not just thefirst derivative, butm derivatives of the free energy in thapproximation

DG~0→1!'(i 51

N

(j 51

m

wi~ j !G~ j !~l i !. ~3.10!

The parameters in this equation are determined by requiit to be exact for a polynomial of the highest possible degrWhen all points are adjustable, it can be shown that the sond derivatives do not contribute toDG(0→1). The sim-plest expression that employs higher-order derivatives isone-point equation

DG~0→1!'Û&1/21b2

24^~U2Û&1/2!

3&1/2, ~3.11!

which is exact to fourth order of FEPT.The results reviewed so far are exact as long as the

rivativesG(n)(l i) of the free energy are exact. The statisticerrors that result from experimental measurements or cputer simulations are not taken into account. This situat


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can be corrected by using a polynomial approximation tofree energy. If in addition to the derivativesG( j )(l i) of thefree energy, the statistical errorss i

( j ) are also provided at thepoints l i , i 51,2,...,N, the polynomial approximationpk(l)5a1l1a2l21¯1akl

k to DG(l) can be obtainedby minimizing ax2 functional:

x2~a1 ,...,ak!5(i 51

N

(j 51

mi S pk~ j !~l i !2G~ j !~l i !

s i~ j ! D 2

, ~3.12!

where pk( j )(l i) is the j derivative of pk(l) evaluated inl i

and s i( j ) is the statistical error ofG( j )(l i). The coefficients

a1 ,...,ak are obtained by minimizing Eq.~3.12!—i.e., bysolving the set of linear equations]x2/]a150,...,]x2/]ak

50.

IV. SIMULATIONS

We calculated the free energy of charging ions in wausing molecular dynamics simulations of systems consisof a single ion in either a cubic cell of water~PBCs! or in acluster of water molecules~SBCs!. The free energies werestimated by calculating the electrostatic potential of theat intermediate charge states and then integrating using eEq. ~2.1! or one of the perturbative approaches. These simlations were all performed using theSANDER module of theAMBER 7 ~Refs. 33 and 34! simulation package. In this version, the Ewald formalism is not supported in the Gibmodule; instead, thermodynamic integration with particmesh Ewald~PME! treatment is provided inSANDER. Thisgave us a check on the results obtained usingSANDER andour computations using a custom subroutine of the timaveraged electrostatic potentials at the ions for a systemder PBCs at different charge statesl, 0<l<1.

To test our formalism we used theRPOL ~Ref. 29! watermodel. This model has charges and induced dipoles. Itsigns point charges of20.730 and 0.365 and polarizabilitieof 0.528 and 0.170 to the oxygen and hydrogen nuclei,spectively. The bond length between oxygen and hydroge1 Å, while the bond angle is 109.47°. This geometry wconstrained throughout all simulations by using the analversion ofSHAKE.35 The water–water and ion–water interations are described by Coulomb and Lennard-Jones~LJ! in-teractions. The LJ interactions betweenRPOL waters act onlybetween the oxygens. Similarly the ion–water LJ intertions act between water oxygens and the ion. We noticethe LJ parameters of the model presented in the literavary considerably. The chosen LJ parameters, charges,polarizabilities for oxygen and hydrogen in water, along wthose for sodium, potassium, fluoride, and chloride,shown in Table I.

The induced-dipole interactions are computed accordto the method introduced in previous work.31 Essentially, weimplemented classical Ewald36 and PME based37,38 treat-ments of fixed and induced point dipoles into the sanmolecular dynamics~MD! module of AMBER 6 and 7.33,34

During MD, the induced dipoles can be propagated alowith the atomic positions either by iteration to seconsistency at each time step or by a Car–Parrinello tenique using an extended Lagrangian formalism. The us


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PME for electrostatics of fixed charges and induced dipotogether with a Car–Parrinello treatment of dipole propation in MD simulations leads to a cost overhead, for a givtime step, of only 33% above that of MD simulations usistandard PME with fixed charges, allowing the study of plarizability in large macromolecular systems.

Features common to both SBC and PBC simulationsthe following. To compute the ion charging free energaccording to Eq.~2.1!, we defined 21 intermediate chargstates for each ion. For example, for sodium we considereneutral sodium ion together with ions having charges 0.0.1,..., 0.95 and fully charged. To carry out the calculatiowith any of the perturbative methods, we computed the fenergy derivatives in the prescribed points for a givenproximation. For example, we considered a neutral, hcharged, and fully charged sodium for Eq.~3.9!. Simulationswere performed at constant temperature and volume.measured the potential at each intermediate state averagin time. A 1-fs time step was used. Temperature was ctrolled by using a Berendsen thermostat.39 For the calcula-tion of the induced dipoles, we used the Car–Parrinemethod31 with a dipolar mass of 0.33.

The cluster simulation systems consisted of a single ifixed at the origin and immersed in a sphere of 372 wamolecules, corresponding to a cluster radius of appromately 14 Å. These simulations were run using theCAP op-tion in theSANDER module ofAMBER, which imposes a weakhalf-harmonic restoring potential on waters outside the clter radius. We used the defaultCAP force constant of 1.5kcal mol21 Å21. For each of the 21 intermediate charge stawe ran 500 ps of molecular dynamics to equilibrate, flowed by 200 ps of data collection. For the perturbative aproaches, we took 300 extra picoseconds of data collectiothe prescribed points. The nonbond cutoff was set suciently large~999 Å! to ensure all pairwise nonbond interations were calculated.

For the PBC simulations, the systems consisted osingle ion in a cubic cell of 256 RPOL waters. The box siwas fixed at 19.755 Å, corresponding to a number density0.0333 Å23. As above, 21 intermediate charge states wdefined, and for each of these 500 ps of equilibration andps of data collection were performed using a 1-fs time stFor the perturbative approaches, we took 300 extra picoonds of data collection at the prescribed points, wherealso computed average higher-order derivativesG( j )(l i) ofthe free energy. The LJ interactions were truncated using

TABLE I. Lennard-Jones interaction coefficients, partial charges, and poizabilities for the water oxygen and hydrogen and the ions used insimulations.

Atom type s ~Å! e ~kcal/mol! q (e) a ~Å3!

O~H2O) 3.205 0.160 20.730 0.528H~H2O) ¯ ¯ 0.365 0.170Na1 2.350 0.130 1.0000 0.240K1 3.154 0.100 1.0000 0.830F2 3.359 0.100 21.0000 1.050Cl2 4.450 0.100 21.0000 3.250Cl1 4.450 0.100 1.0000 3.250


ede

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TABLE II. Charging free energy for ions in water under PBCs~cube of 256 RPOL waters!, obtained fromsimulations with 21 intermediate states. The subindices inDG indicate the number of intermediate states usin the calculation, and the supraindices in curly brackets, the order of the polynomial used in fitting thfirstderivative of the free energy, i.e.,G(1)(l)5^U&l5^V(1,x)2V(0,x)&l5Cl(PBC). The second column(DG21), gives the result using numerical integration over the 21 intermediate states with the trapezoidaThe third, fourth, and fifth columns give the results obtained after integration of a polynomial fitting toG(1)

(l) of first (DG21$1%), second (DG21

$2%), and third (DG21$3%) order, respectively, obtained by minimizing Eq.~3.12!

using the 21 intermediate charge states. For instance, the integration of the quadratic fitting isDG21$2%5a1l

1a2l21a3l3, and the corresponding coefficients (a1 ,a2 ,a3) are given in the last column. The sixth anseventh columns give the results obtained using Eq.~3.9! (DG3) and Eq.~3.8! (DG2), respectively.

Ion type DG21 DG21$1% DG21

$2% DG21$3% DG3 DG2 (a1 ,a2 ,a3)

Na1 291.7 291.9 291.6 291.6 291.1 292.5 ~8.0,285.4,214.2!K1 271.8 272.0 271.7 271.7 271.0 272.0 ~6.7,267.9,210.6!F2 2126.1 2126.6 2126.1 2126.1 2126.6 2125.3 ~21.8,2104.6,219.7!Cl2 294.0 294.3 294.0 294.0 294.1 294.1 ~26.1,273.8,214.1!

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atom-based cutoff of 8.5 Å and a correction term. The eltrostatic interactions were evaluated using the smooth Palgorithm.38 The Ewald convergence coefficient was 0.3Å21, and a 20320320 grid was used, with B-spline interpolation of order 6. The ‘‘ischarged’’ option inSANDER wasused, which implements the version of the Ewald sum apppriate for non-neutral unit cells. The electrostatic potentiathe partially charged ion was evaluated and time averageimplementing a custom subroutine.

V. RESULTS AND DISCUSSION

First, we consider the ionic charging free energy for stems with PBCs. In each of the MD simulations under PBthe instantaneous electrostatic potential at the ion was evated at each time step, using the PME approximation ofpotentialcq(x) given by Eqs.~2.20! and~2.23!. The value ofCl(PBC) is estimated by the average at the end of theps of data collection. From these potentials the chargingenergiesDG were obtained by integration, using the traezoidal rule with 21 equally spaced points. The charging fenergies calculated this way are displayed in Table II, incolumn headed byDG21. Figure 1 shows the cumulativradial average of the electrostatic potential~averaged over altime windows! for partially charged Na and Cl ions inperiodic box of 256 RPOL waters, for each of the 21 intmediate charge states, from neutral~top curve! to fullycharged~bottom curve!.

Next, we consider the ionic charging free energy for stems with SBCs. In each of the cluster simulations, the cter volume was divided into spherical shells of thickne0.04 Å about the cluster center. During the 200-ps datalection phase of the simulations, the time-averaged chadue to both the permanent charges and the induced dipoleach spherical shell were obtained, and from this the timaveraged charge distribution about the center was calculaThis finally led to the calculation of the size-corrected radwork functionWq

0(r )1q/(er ), whereWq0(r ) is given by Eq.

~2.13! or ~2.14!, and the dielectric constante was set to thereported value of 106.29 Results for the charging free energobtained by integration over 21 charge states using theinter-nal electrostatic potential@Eq. ~2.18!# are shown in Table III,in the column headed byDG21. Figure 2 shows the size

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corrected work functionWq0(r )1q/(er ) ~averaged over all

time windows! for partially charged Na and Cl ions, incluster of 372 RPOL water molecules. Each of the 21 intmediate charge states is shown, from neutral~top curve! tofully charged~bottom curve!. SinceWq

0(r )1q/(er )→2` asr→0, the function was cut forr ,1. These plots indicate thathe size-corrected radial work function for each charge sis essentially constant over an intervalr 1<r<r 2 , with r 1

57 Å, and r 2510 Å for the 14-Å cluster simulations. Fothese clusters, the root-mean-square~rms! deviation of thework function about its average over the range 7<r<10 isabout 0.4 kcal mol21 e21. Reference 23 shows that the rgion of constant size-corrected radial work function i

FIG. 1. Cumulative radial average of the electrostatic potential~averagedover all time windows! for partially charged Na and Cl ions in a periodibox of 256 RPOL water molecules. Each of the 21 intermediate chastates is shown, from neutral~top curve! to fully charged~bottom curve!.


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TABLE III. Charging free energy for ions in water under SBC~spherical cluster of 372 RPOL waters!, obtainedfrom simulations with 21 intermediate states. The subindices inDG indicate the number of intermediate stateused in the calculation, and the supraindices in curly brackets, the order of the polynomial used in fittifirst derivativeof the free energy, i.e.,G(1)(l)5Û&l5^V(1,x)2V(0,x)&l . The second column (DG21) givesthe result using numerical integration over the 21 intermediate states with the trapezoidal rule. The third,and fifth columns give the results obtained after integration of a polynomial fitting toG(1)(l) of first (DG21

$1%),second (DG21

$2%), and third (DG21$3%) order, respectively, obtained by minimizing Eq.~3.12! using the 21 inter-

mediate charge states. For instance, the integration of the quadratic fitting isDG21$2%5a1l1a2l21a3l3, and the

corresponding coefficients (a1 ,a2 ,a3) are given in the second to last column. The sixth and seventh colugive the results obtained using Eq.~3.9! (DG3) and Eq.~3.8! (DG2), respectively.

Ion type DG21 DG21$1% DG21

$2% DG21$3% DG3 DG2 (a1 ,a2 ,a3)

Na1 291.9 292.2 291.8 291.8 291.7 294.5 ~8.5,286.7,213.6!Cl2 294.1 294.4 294.1 294.1 293.2 292.9 ~25.8,274.6,213.7!

r

feot

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creases as the radius of the cluster increases, while thedecreases.

The standard deviations of the charging free energiesDG21 were estimated using the values of the 21 independpotential determinations. To first decimal place the romean-square deviations in all cases were 0.1 kcal mol21. Asfound previously for nonpolarizable models, Figs. 1 and 2well as the columnDG21 in Tables II and III indicate that thevalues of the charging free energy computed under PBusingCq(PBC) agree with those obtained in the interiorthe cluster in SBCs, i.e., whenCq(SBC,int) is used. TableIV compares these values against those obtained withSPC water model. The results obtained using theexternalelectrostatic potential@Eq. ~2.19!# are also shown in thistable. For singly charged cations the free energies calcul

FIG. 2. Size-corrected work functionWq0(r )1q/(er ) ~averaged over all

time windows! for partially charged Na and Cl ions, in a cluster of 37RPOL water molecules. Each of the 21 intermediate charge states is shfrom neutral~top curve! to fully charged~bottom curve!. The function hasbeen set to zero forr ,1.


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using theexternalvalue of the electrostatic potential are'12kcal mol21 more negative than those calculated using thein-ternal value. The differenceCq(SBC,ext)2Cq(SBC,int) isessentially thesurface potential of the water cluster, plus asmall continuum correction.23 It depends weakly on the cluster radius and appears to extrapolate to a value of'210.7 kcal mol21 e21 or '2460 mV for an infinite clusterradius. This is comparable to Sokhan and Tildesley’s40 esti-mate of2543 mV for pure SPC water in planar slab geometry. In this sense,Cq(SBC,ext)2Cq(SBC,int) ~when ex-trapolated! can be interpreted as the surface potential ofinfinite water cluster.

Next, we consider the results of applying FEPT. For thwe notice that the first derivative of the free energy isG(1)

(l)5Û&l5^V(1,x)2V(0,x)&l ; for systems under PBCsÛ&l5Cl(PBC), while for systems under SBCs,Û&l

5Wl0(r )1l/(er ). We consider Table II for PBCs and Tab

III for SBCs. The second column (DG21) in these tablesshow the charging free energies obtained by integration,ing 21 equally spaced charge states. The third, fourth,fifth columns give the results obtained after integration o

n,

TABLE IV. Comparison of ionic charging free energies for systems wPBCs and SBCs for two water models, RPOL and SPCs, and three diffeexperiments.

System Na1 K1 Cl2 F2

256RPOLPBC 291.7 271.8 294.0 2126.1

372RPOLSBC,int 291.9 ¯ 294.1 ¯

372RPOLSBC,ext 2104.7 ¯ 281.6 ¯

256SPCPBCa 295.9 270.2 293.4 2140.8

372SPCSBC,inta 296.1 270.0 293.7 2141.6

372SPCSBC,exta 2108.4 282.3 281.5 2129.4

Expt. 1b 287.2 264.7 281.2 2138.5

Expt. 2c 288.6 271.2 266.2 294.1

Expt. 3d 288.9 271.2 277.4 2105.3

aSimulation with the SPC water model, Dardenet al., J. Chem. Phys.109,10089~1998!.

bExperimental value. Y., Marcus, J. Chem. Soc. Faraday Trans.87, 2995~1991!.

cExperimental value. Friedmanet al., Water: A Comprehensive Treatis~Plenum, New York, 1973!, Vol. 3, pp. 1–118.

dExperimental value. B. E. Conway, J. Chem. Phys.7, 721 ~1978!.


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TABLE V. Charging free energy for ions in water under PBCs, obtained from simulations with three intediate states. The subindices inDG indicate the number of intermediate states used in the calculation. In botsecond and third columns, the results are obtained as a cubic fitting to the free energy. These columnsresults from a polynomial fitting of third order obtained by minimizing Eq.~3.12!, where derivatives of the freeenergy calculated numerically have been included. The second column (DG3 @G(1)#) gives the values whereonly the first derivative is considered in the fitting, while the third column (DG3 @G(1),G(2)#) also includes thesecond derivative. This last free energy can be written asDG(l)5a1l1a2l21a3l3, where the coefficients(a1 ,a2 ,a3) are given in the sixth column. The fourth and fifth columns give the values obtained using Eq.~3.9!(DG3) and Eq. ~3.11! (DG1), respectively.DG1 uses first- and third-order derivatives. The last colum(DG3

mod@G(1)#) gives a modified version of the second column, where the weight of the ‘‘transition’’ pointl50has been diminished.

Ion type DG3 @G(1)# DG3 @G(1),G(2)# DG3 DG1 (a1 ,a2 ,a3) DG3mod@G(1)#

Na1 291.1 291.3 291.3 291.0 ~8.4,286.9,212.8! 291.5K1 271.5 271.5 271.5 271.5 ~8.0,269.6,29.9! 271.8F2 2126.8 2125.0 2126.8 2110.8 ~28.7,291.1,225.2! 2126.3Cl2 293.2 291.2 293.2 288.8 ~29.9,263.1,218.2! 294.0

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polynomial fitting to G(1)(l) of first (DG21$1%), second

(DG21$2%), and third (DG21

$3%) order, respectively, obtained bminimizing Eq. ~3.12! using the 21 intermediate chargstates. For instance, the integration of the quadratic fittinDG21

$2%5a1l1a2l21a3l3u015a11a21a3 , and the corre-

sponding coefficients (a1 ,a2 ,a3) are given in the eighth column. The results in these tables show that a quadratic pnomial fitting ofG(1)(l) gives results in agreement with thnumerical integration shown in the second column (DG21),and one can use the given set of coefficients to computecharging free energy for a given ion between any two chastates~as long as the charge does not change sign, as weshow below!. The sixth and seventh columns give the resuobtained using Eq.~3.9! (DG3) and Eq.~3.8! (DG2), re-spectively. These equations are exact to fourth order of FEand show fairly good agreement withDG21, although theyonly use three and two points, respectively.

Tables V and VI show results corresponding to an adtional 300-ps simulation, where only three charge statesmeasured. Since the quadratic fitting to the first derivativethe free energyG(1)(l) in Tables II and III gives good results with respect toDG21, here we only consider a cubifitting to the free energyG(l). The second and third columns give the results from this cubic fitting, obtainedminimizing Eq. ~3.12!, where derivatives of the free energcalculated numerically have been included. The secondumn (DG3 @G(1)#) gives values where only the first derivative is considered in the fitting, while the third colum


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l-

(DG3 @G(1),G(2)#) also includes the second derivative. Thlast free energy can be written asDG(l)5a1l1a2l2

1a3l3, where the coefficients (a1 ,a2 ,a3) are given in thesixth column. The fourth and fifth columns give the valuobtained using Eq.~3.9! (DG3) and Eq.~3.11! (DG1), re-spectively ~this includes first- and third-order derivativeevaluated at one point!. DG3 is the same quantity as thacomputed in Tables II and III; the slight difference in valuis related to the behavior of the neutral particle~at l50!,associated with the transition in the orientation of the wamolecules that happens near the neutral ion. Figure 4 shthat the behavior of this point is not representative oflarge positive or negative charge domains, but pertains tosmall transition region. Taking advantage of this observatione can diminish the weight of this point~say, by one-third!and obtain a better fitting, as shown in the last columnTable V.

Figure 3 compares the size-corrected radial work fution to a ‘‘generalized’’ P summationS1(r ) defined by

^S1~r !&5K (0,r i,r

S qi

r i2

pi• r i

r i2


r i3 D L . ~5.1!

As Eq. ~2.16! shows, for larger ~inside the cluster,r 1<r<r 2) the size-corrected radial work functionWq

0(r )

nter-bothns give

ined

TABLE VI. Charging free energy for ions in water under SBCs, obtained from simulations with three imediate states. The subindices inDG indicate the number of intermediate states used in the calculation. Inthe second and third columns, the results are obtained as a cubic fitting to the free energy. These columthe results from a polynomial fitting of third order obtained by minimizing Eq.~3.12!, where derivatives of thefree energy calculated numerically have been included. The second column (DG3 @G(1)#) gives the valueswhere only the first derivative is considered in the fitting, while the third column (DG3 @G(1),G(2)#) alsoincludes the second derivative. This last free energy can be written asDG(l)5a1l1a2l21a3l3, where thecoefficients (a1 ,a2 ,a3) are given in the last column. The fourth and fifth columns give the values obtausing Eq.~3.9! (DG3) and Eq.~3.11! (DG1), respectively.DG1 uses first and third-order derivatives.

Ion type DG3 @G(1)# DG3 @G(1),G(2)# DG3 DG1 (a1 ,a2 ,a3)

Na1 291.0 291.1 291.0 285.4 ~8.4,285.9,213.6!Cl2 293.2 292.9 294.3 294.0 ~210.1,265.4,217.4!


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1q/(er) approachesS1(r ). Even though both functions reacasymptotically the same plateau~in fact, the agreement iseven better for larger clusters: see Ref. 23!, it is clear fromthe figure thatS1(r ) oscillates considerably more than thradial work function. The reason for this smooth behavioillustrated in the lower panel. In this figure, the solid linrepresents the last term in Eq.~2.14!, while the dotted linerepresents the first terms. In the regionr 1<r the amplitudeof the oscillations is quite small, and the oscillations in tfirst term exactly cancel the oscillations in the second teThe sum of the two terms givesWq

0(r )1q/(er )'*0

r 1@dC(s)/ds# s21 ds ~where the upper limit of the integral is r 1 and not r!. In other words, afterr 1 , the size-corrected work function becomes approximately constaand this happens long beforeS1(r ) reaches its constanvalue, even though both expressions give the saasymptotic results.

Finally, Fig. 4 shows the average electrostatic potenat the position of the ion in a PBC box with 256 waters. Tion in question has the same van der Waals parameters aCl21 ion, but its charge varies froml521 to l51. The dotsrepresent simulation points, while the solid line showsresults of a quadratic fitting. The data indicate a transitbetween the negative-charge region and positive-chargegion at l'20.12. This behavior was previously found bseveral authors22,41,42 for nonpolarizable water potentialsThis transition is explained in terms of the different behavof the structural waters around positive and negative io

FIG. 3. Top: comparison between the size-corrected radial work funcWq

0(r )1q/(er ) and the P summationS1(r ) for a partially charged sodium~neutral, top curve; half-charged, middle curve; fully charged, bottom cu!in a cluster of 372 RPOL water molecules. Bottom: contributions tosize-corrected radial work function for a fully charged sodium ion. The soline represents the last term in Eq.~2.14!, while the dotted line representthe first terms.


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The asymmetry of the structural waters has several diffeeffects on electrostatic potential surrounding the ion:~i! Thefluctuations of the waters around positive ions are considably smaller than the fluctuations of the vecinal watearound negative ions. In fact, around positive ions the strrepulsive potential of the water oxygens restrict the oxygmotions. Around negative ions, the structures with one ofhydrogens pointing toward the ion dominate, but since bhydrogens are equivalent and the hydration shell has a filifetime, transitions are allowed.~ii ! There is a positive po-tential at the site of theneutral particle. This is due to thepotential used; both SPC and RPOL waters do not attacrepulsive potential to the hydrogens and therefore the hydgens penetrate the Lennard-Jones sphere of the neutraticle. Consequently, there is a cost of free energy associwith increasing the charge in this region.~iii ! As a conse-quence of the two previous considerations, negative ionsmore stably solvated than positive ions. For instance,computed the ionic charging free energyDG4@G(1),G(2)#~i.e., using four intermediate states and the first and secderivatives in the FEPT calculation! for the intermediatestatesl50, 0.3, 0.5, 0.75, and 1 for a chloride-type ion C1

and obtained a result of274.9 kcal mol21. This is to becompared with293.2 kcal mol21 in Table V.

VI. CONCLUSIONS

In this work we have introduced methods to computeionic charging free energy in systems characterizedatomic charges, dipoles, and quadrupoles and comparedresults obtained for periodic boundary conditions and sphcal clusters. For the latter, we have introduced a generation of Gauss’ law that links the microscopic variables to tmeasurable macroscopic electrostatics via a work functStrictly speaking, this is different from P summations whethe averaged electrostatic potential is computed; i.e., thetistical average is applied to the solution of Poisson’s eqtion. The approach first introduced in Ref. 23 is to apply tstatistical average to the differential equation itself and thobtain the macroscopic results via the work necessarybring a charge from infinity.

At present, we are working in the implementation ofrelatively sophisticated water model that includes dipo

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e

FIG. 4. Average electrostatic potential for a Cl-like ion as a function ofchargel in a periodic box of 256 RPOL water molecules. Dots repressimulation points and the solid lines, quadratic fittings.


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and quadrupoles. This implementation43 efficiently computeshigher-order multipolar interactions, using PME~Ref. 38!and multigrid~Ref. 44! based methods. In the meantime, whave tested our formalism with the RPOL model, which hpoint charges and polarizabilities. For the RPOL model,results are in relative agreement with those obtained usinnonpolarizable water models23 and with accepted experimental values. Naturally, the particular numbers depend onmodel, but at present we are more interested in the treatmof electrostatics. Ionic charging free energies calculatedEwald summation in PBCs agree with those calculatedSBCs using theinternal electrostatic potential defined by thsize-corrected radial work function in Eq.~2.18!. Chargingenergies calculated with both approaches are appropriations in bulk. Furthermore, these studies including pointpoles also corroborate the conclusion in Ref. 23 that at por outside the first few water layers about ions in clust~e.g.,r .7 Å), the electric field strength is on average givby q/(er 2), where the dielectric constante is calculated byEwald summation in PBCs. Thus dielectric screening in bsolvent is accurately described by this simple continuum pture. We have also assessed the validity of different FEapproaches, finding that in fact they provide an inexpensvery accurate way of computing the charging free energy

Simulations are very valuable in determining the ablute free energies of ionic solvation, since these are difficto characterize experimentally.45 Unfortunately, the calcu-lated values are a function of the empirically chosen vanWaals parameters for ion and water, which are generallyjusted to agree with accepted experimental values. In ation, the calculated free energies are also sensitive totreatment of long-range electrostatics. It is therefore imptant to solve all issues concerning electrostatics beforeparametrization is carried out. At present, the need for maccurate water models has been established. These mgenerally involve polarization and higher order multipoleThey will also become very important in QM and MM caculations of ionic solvation free energies. Another importaissue is the existence of a surface electrostatic potential incluster, which has been found in our work and in previowork using nonpolarizable models. At present, there is csiderable research effort invested in multiscale methodinterface atomistic solvents with a continuous representatAt this point, it is understood that effective two-body modedo not work well at the interface and that polarizable modare needed. The strict testing of the models’ behavior atsurface of the cluster is highly recommended: no amounstatistical mechanical insight into building the interface wbe able to eliminate the electrostatic potential discontinuat the surface of the cluster, since this is a direct result ofwater model employed. Therefore, unless improved wamodels are used, such efforts are doomed to spurious foat the interface.


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ACKNOWLEDGMENTS

Support for this work has been provided by NSF-ITAward No. 0121361 and the North Carolina SupercomputCenter~NCSC!.

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