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Atomic Radii: Incorporation ofSolvation Effects

BRIAN J. SMITH,1 NATHAN E. HALL1, 2

1 Biomolecular Research Institute, 343 Royal Parade, Parkville, VIC 3052, Australia2 School of Chemistry, University of Melbourne, Parkville, VIC, Australia

Received 22 December 1997; accepted 22 April 1998

ABSTRACT: Atomic radii used to define the solute cavity in continuum-basedmethods are determined by reproducing the solvent-accessible surface defined

Ž .as the loci of minima in a potential solvent interaction potential between theŽsolute and a probe. This potential includes electrostatic interaction ion]dipole,

.ion]quadrupole, and ion-induced dipole terms as well as a Lennard]Jonesenergy term. The method alleviates the need to distinguish solute atoms indifferent chemical environments. These radii, when used in the calculation ofsolvation free energies, are shown to be superior to fixed atom-specific radii orto radii obtained from the electron isodensity surface from quantum-mechanicalcalculations. Q 1998 John Wiley & Sons, Inc. J Comput Chem 19: 1482]1493,1998

Keywords: atomic radii; solute cavity; solvation; electrostatics; solventinteraction potential

Introduction

Ž .ontinuum macroscopic models have be-C come a popular method for describing theeffect of solvents on molecular systems, and havebeen successfully incorporated into quantum-mechanical calculations and simple electrostaticmodels. A common feature of all these methods isthe requirement to determine the solute cavity.The cavity is, however, simply a conceptual sim-plification; there is no physical boundary betweenthe solute and solvent. Consequently, all sets of

Correspondence to: B. J. Smith

radii will depend on the model for which theyhave been determined, and there cannot be anoptimal set of atomic radii for any solute. Nonethe-less, these models do provide an efficient andreliable method for determining solvation effects,provided the cavity is of a realistic size and shape.

The Onsager model1 provides the simplest ap-proach, employing a spherical cavity. This methodcan be easily incorporated into quantum-mechani-cal calculations, which makes it particularly attrac-tive.1 ] 5 Ellipsoidal cavities can provide a moregeneral boundary between the dielectric regions.6 ] 9

Simple geometrical shapes like the sphere or el-lipse are, however, unrealistic for many molecularspecies. Several methods, such as the polarizable

( )Journal of Computational Chemistry, Vol. 19, No. 13, 1482]1493 1998Q 1998 John Wiley & Sons, Inc. CCC 0192-8651 / 98 / 131482-12

ATOMIC RADII

Ž . 10, 11continuum model PCM , the conductor-likeŽ . 12 ] 18screening model COSMO , and the finite dif-

Ž .19ference Poisson]Boltzmann FDPB approach,use atom-centered radii to generate the solute cav-ity. Many different sources of atomic radii have

Ž .20been considered. Ionic Bondi radii determinedfrom the analysis of crystal structures, and radiiderived by Pauling,21 are two such sets that havebeen used extensively to define the solute bound-ary. For ionic solutes, it was found that the radii

˚needed to be increased by 0.1 A for anions and by˚0.85 A for cations to reproduce experimental solva-

tion energies.22 Subsequently, an empirical factorof 7.0% was chosen for scaling crystallographicallyderived radii to best fit experimental solvationenergies of ionic species.23

Standard Pauling radii are often enlarged by20% in PCM24, 25 calculations. The choice of a scal-ing factor of 1.2 has been rationalized throughmolecular dynamics simulations.26 Implementa-tion of a simple scaling factor of 1.2, however, wasfound to produce erroneous results for the glycinezwitterion.27 To overcome this discrepancy, theradii of the NH and O spheres needed to be3

˚reduced by 0.4 A. Further investigations of manysystems has revealed that different scaling factorsare necessary, depending on the charge of thesolute: a scaling factor of 1.10]1.15 was found tobe appropriate for ions28 and a factor of 1.20]1.25appropriate for neutral species.29 A further distinc-tion can be applied between polar and nonpolarhydrogens.29, 30

Gavezzotti has reported an alternative set ofatomic radii,31 based on x-ray crystallographic re-sults, which have been used successfully in FDPBcalculations.32 A set of radii developed by Rashin33

has also been implemented in a number of sol-vation studies.15, 34, 35 To satisfactorily reproduceexperimental solvation energies it was foundnecessary to distinguish between atoms that arehydrogen bonding and nonhydrogen bonding.Other parameterized sets of atomic radii have beendeveloped specifically for other methods, includ-ing the Langevin dipole method of Warshel,36 anda molecular mechanics method by Bliznyuk andGready.37 More complicated and extensive sets ofradii, based on the many atom types used in somemolecular mechanics force fields, have also beendeveloped, although these radii are largely depen-dent upon the charge sets employed.38

Individual collections of atomic radii, havingonly a small number of atom types, are generallynot sufficiently flexible to account for different

atomic environments or charges. With many dif-ferent atom types, however, difficulties arise whenthe assignment of atom type becomes ambiguousor inappropriate, particularly in transition statesand exotic minima. An approach that eliminatesthese problems is one in which the atomic radii aredetermined as a function of the partial atomiccharge, although additional factors of basis setdependence and choice of method for determiningthe charges is introduced where charges are ob-tained quantum mechanically. Several relation-ships between partial atomic charge and radii havebeen determined previously.39 ] 45 This approachappears general and can be easily applied, al-though there can still be large errors in the calcula-tion of solvation energies. In the very successfulSMx models of Cramer and Truhlar,46 the atomicradii are a function of the partial atomic charge,which is determined self-consistently.

An alternative to using atomic-centered radii togenerate the solute cavity is to use the electronisodensity surface from quantum-mechanical cal-culations. One of the benefits of such a scheme isthat the extensive parameterization that atomicradii methods often require is replaced by a singleparameter, the electron density cutoff, typically in

Ž y3 . 25the range 0.0004]0.001 a.u. e ? bohr . Usingthis approach, however, a new variability is intro-duced as the isodensity surface is dependent uponthe basis set and level of theory used to derive theelectron density.

Investigations of the solvation energies of sim-ple neutral, anionic, and cationic species have beenperformed where the electron density has beenused to generate the cavity surface for FDPB calcu-lations.47 A single cutoff value is found not to beappropriate for all molecular species. Anions, withmore diffuse electron distributions, require largercutoff values than do neutral molecules. Similarly,cations require a different cutoff, again to bestreproduce experimental solvation energies. Thestudy of the effect of basis set variation also foundthat, depending upon the basis set being used,different cutoff values are required. Similar con-clusions to these were drawn by Stefanovich16 whoused an isodensity cutoff value of 0.001 a.u. tocalculate the solvation energies of neutral and ionicspecies. The solvation energies of anions were

Žfound to be underestimated resulting from the.diffuse electron distribution and solvation ener-

Žgies of cations overestimated resulting from the.more contracted electron distribution . The assign-

ment of a cutoff value to be used for zwitterionic

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systems becomes ambiguous as neither the cutoffapplicable for neutral, positive, nor negativespecies would appear to be appropriate.

It is clear that there is no automatic method fordetermining the solute cavity that is general

Ženough to deal successfully with all systems in-cluding transition states, neutral, charged, and

.zwitterionic species equally well. Problems stemfrom the need to either define atom types forassignment of parameterized atomic radii or deter-mine appropriate values for the electron densitycutoff.

Solvent Interaction Potential

A general method that has been designed toovercome the current deficiencies in determiningthe solute cavity is described here. The method

Ždetermines atomic radii to be applied with FDPB.calculations from a potential established between

the solute and a solvent probe, the solvent interac-Ž .tion potential SIP . The radii, r , are obtained bySIP

reproducing the solvent-accessible surface, definedas the loci of minima in the potential.

The solvent interaction potential incorporatesboth electrostatic and nonelectrostatic interactionsbetween the solute and the probe. The nonelectro-

Ž .static van der Waals energy is calculated using aŽ .Lennard]Jones LJ potential, E , while the elec-LJ

trostatic energy is determined from the interactionof the solute partial atomic charges with the dipole,E , quadrupole, E , and ion-induced dipole,iyd iyqE , of the probe. The total solvent interactioniy i dpotential energy E of the probe with the soluteSIPis given by:

Ž .E s E q E q E q E 1SIP LJ iyd iyq iyi d

Here, many factors which also contribute to thetotal solvation free energy are assumed to cancel.49

The electrostatic potential of the solute is repre-sented using atomic charges that are determinedquantum mechanically, while the probe is as-signed the experimental electrostatic propertiesŽ .dipole, quadrupole, and polarizability of the sol-vent. Although the method should be generalenough to deal with any solvent, it is demon-strated here using water.

A Lennard]Jones potential, established betweenthe probe and each atom of the solute, is used torepresent the van der Waals repulsive and attrac-tive dispersion forces between the solute and theprobe. The total LJ energy at any point on the

solvent interaction potential is given by the inter-action of the probe with each of the solute atoms:

N 12 6s si i Ž .E s 4« y 2ÝLJ i ½ 5ž / ž /r ri

This LJ potential is defined by two parameters: thewell depth of the interaction between the probeand solute atom, « ,U and the separation at whichi

Žthe potential is zero, s , where s is related to thei iy1r6 .equilibrium bond separation by 2 r . The equi-e

librium separation, r , is equal to the sum of theeLennard]Jones radii, r , and the solvent probeLJradii, r . For all interactions between the probePand hydrogen atoms, « is assigned a value of 0.17ikJ moly1, whereas, for the first-row atoms, C, N,O, and F, a well depth of 0.34 kJ moly1 is used.These values are taken from parameters for He]Neand Ne]Ne interactions respectively.49 The r ,LJvalues need be generated for each individual atomtype. The atomic radii determined here dependonly on the element type so that no assignment ofatoms in different chemical environments needs tobe made. This restriction allows for the equal im-plementation of the solvent interaction potentialfor all systems over any potential energy surface,including exotic minima and transition states.Consequently, different charged states, includingneutral, cationic, anionic and zwitterionic systemsand charge transfer processes are all dealt with inan identical manner.

The experimental gas-phase dipole moment, m,and quadrupole moment tensors, Q , of the sol-aavent are scaled by a Langevin function, L, to

Ž Xprovide the average effective moments dipole m ,X .and quadrupole Q in the direction of the fieldaa

Ž .generated by the solute partial atomic charges Fwhen it is subject to electrical orientation andthermal randomizing forces:

X Ž .m s Lm 3X Ž . Ž .Q s LQ a s x , y , z 4aa aa

where:

e x q eyx 1L s y and x s mFrkTx yxe y e x

Under the influence of low electric fields, theLangevin function tends toward zero, correspond-ing to totally random orientations of the solventdipole.

* The well depth, « , should not be confused with theidielectric constant of either the solute or solvent.

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ATOMIC RADII

The Langevin dipole of the probe is assumed tolie in the opposite direction to the electrostaticpotential produced by the solute partial atomiccharges, q . At any point about the solute, eachiatom of the solute makes an angle u with theiprobe dipole, at a distance of r . Based on theiassumption that the quadrupole is able to freelyrotate about its principle axis, the total dipole andquadrupole interaction energies are given by thefollowing expressions50 :

N Xym qi Ž .E s cos u 5Ýiyd i2« riiXN1 Q qz z i 2Ž . Ž .E s 3 cos u y 1 6Ýiyq i32 « rii

The experimental dipole value of water, 1.86 De-bye,51 and the experimental quadrupole tensor,Q s y0.13 Buckinghams,51 have been used here.z z

A polarizable body situated in the electric fieldarising from a charge will have induced in it asmall dipole. The interaction of the charge withthis induced dipole leads to an additional interac-tion energy. Again, assuming free rotation about

Ž .the molecular axis z , the equation for the ion-in-duced component of the solvent interaction poten-tial is given by:

2N1 qi 2E s a cos uÝiy i d z z i4 2 ž2 r «ii

Ž .a q ax x y y 2 Ž .q sin u 7i /2

The Langevin function is not applied to the polar-izability components a , a , and a , as ther-x x y y z zmal motions do not effect the ion-induced dipoleinteraction energy between the solute and theprobe. The experimental polarizabilities of waterused here are a s 1.528 = 10y24, a s 1.415 =x x y y

10y24, and a s 1.468 = 10y24 cm3.51z z

Solvent-Accessible Surface

Ž .The solvent-accessible surface SAS is deter-mined from the loci of minima of the solvent

Ž .interaction potential SIP , evaluated over a gridsurrounding the solute; a regular grid spacing of

˚0.1 A has been applied here. Because the center ofmass and the probe center do not necessarily coin-

Ž .cide, an offset G is applied in the direction of theelectric field. Thermal motions are incorporated

using the Langevin of the offset value:X Ž .G s LG 8

When the solute atomic charges create only a smallfield, the offset will tend to zero. Conversely, theoffset will be greatest under the influence of alarge electric field. The effect of the offset underthe influence of a positive charge is to expand theSAS, while the SAS is contracted near a negativecharge. To determine whether a grid point is part

Ž .of the SAS, the solvent interaction energy ESIPmust project a minimum along the direction to thenearest atom. This analysis is performed at a dis-tance from the grid point given by the Langevinoffset, but applied in the opposite direction to the

Ž .electrostatic field Fig. 1 .

Atomic Radii fromSolvent-Accessible Surface

Atomic radii, r , are obtained from the sol-SIPvent-accessible surface generated from the solvent

Ž .interaction potential SAS by determining theSIPradii that best reproduce this surface. Grid points,

Žwhich lie within the calculated SAS each repre-.senting a cubic volume element , yield the solvent-

excluded volume, SEV . Initial estimates ofSIP

FIGURE 1. The grid point X is part of the SAS if theenergy at the point X X projects a minimum along thedirection to the nearest atom; the interaction energy atX X must be less than the energy at points B and C.

˚ X( )Points B and C are 0.1 A single grid spacing from X .

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the radii are used to generate a representativesolvent-accessible surface, SAS , and the corre-repsponding solvent-excluded volume, SEV . Therepmismatch in the two volumes is given by thoseelements that do not belong to both the SEV andSIPSEV . Each atomic radius is adjusted in turn untilrepthe sum of the mismatch elements is minimized,leading to the optimized r values.52

SIPAn estimate of the surface area, A , definedSIP

by the SAS , is obtained from the sum of the areaSIPof the exposed faces of the volume elements. Anexposed face is defined as a face shared betweentwo volume elements of which one belongs to theSEV and the other the solvent.SIP

Calculation of Solvation Free Energies

The electrostatic contribution to the solvationfree energy is calculated here using the FDPBapproach. The radii generated from the SIP, r ,SIPare applied in conjunction with the atomic chargesused to define the SIP of the solute. Calculations ofthe nonelectrostatic component of solvation, incor-porating the cavitation and dispersion terms, areperformed using the relationship based on thesolvent-accessible surface area:

Ž .DG s g A q b 8non-elec SIP

y1 ˚y2Values of the constants g and b, 20.0 J mol Aand 3.5 kJ moly1, respectively, are taken fromSitkoff et al.53 The total solvation free energy isgiven by the sum of the electrostatic and nonelec-trostatic components:

Ž .DG s DG q DG 9solv elec non-elec

It is worth noting that, although the electrostaticcomponent of the solvation energy is size consis-tent, the nonelectrostatic component is not.

Computational Details

Standard molecular orbital calculations54 havebeen performed using the Gaussian-94 program.55

Geometries of the solutes have been optimized atŽ . 56the MP2r6-31G d level. CHELPG charges calcu-

Ž .lated at the HFr6-31qG d level are used to de-fine the atomic charges of the solute. CHELPGcharges are preferred as they are generated fromthe electrostatic potential rather than Mullikencharges, which are derived from the arbitrary par-

titioning of electron density of the electronic wavefunction.

FDPB calculations are performed using the Del-Phi program57 with an internal solute dielectricof 1 and an external dielectric of 78.54. A grid ex-

˚ ˚tent of 25 A with 0.25 A resolution is used inŽ .these calculations. The CHELPG HFr6-31qG d

charges used in the FDPB calculations are thoseused to define the SIP. Calculation of atomic radiifrom the SIP takes a comparable amount of time asthe FDPB calculations.†

Parameterization

The Lennard]Jones component of the SIP re-quires the van der Waals radii of each atom. Inaddition, an appropriate offset needs to be deter-mined. For the atoms H, C, N, O, and F, r thatLJlead to r are determined so as to best reproduceSIPthe solvation free energies of a set of solutemolecules comprising the first-row hydrides andmethylhydrides, and their anions and cations.These systems are chosen as their experimentalsolvation free energies are relatively well deter-mined. The solvation energies of CHy and NHy

3 2are unreliable, and therefore not included in thisset. The aim of this procedure is to calculate solva-tion free energies within an accuracy of 10 kJmoly1 of experimental solvation free energies.

The difficulties in determining accurate solva-tion energies of ions are well recognized, withuncertainties in pK values and gas-phase acidi-aties and basicities being the main sources of error.Solvation energies for ions have been reevaluatedhere using recent experimental gas-phase aciditiesand basicities.58 These are presented in Table I.The solvation energies obtained are very similar tothose recently obtained by Florian and Warshel.63´Experimental DG values of the neutral speciessolvare based on values from Hine et al.59 and Wag-man et al.60 with standard state corrections.61

Ž .The optimized Lennard]Jones radii r andLJŽ .offset G for the parameterization set of molecules

˚Ž .are as follows in A ; 0.17 for the probe offset andr of C s 1.90, N s 2.10, O s 1.80, F s 1.72, andLJH s 0.80. The average error in the solvation freeenergies calculated for the parameterization setusing the SIP method is y0.3 kJ moly1, with anaverage unsigned error of 4.1 kJ moly1. Theseresults are presented in Table II. All calculated

† These calculations require only a few minutes on a SGIR10000 for the systems presented here.

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TABLE I.Evaluation of Experimental Free Energies of Solvation of Ions.a

b d0 c 0( )DG DG HX pK DGacid/base solv a solv

yOH 1607.0 y26.4 15.74 y458.0yCH O 1567.0 y21.3 15.5 y414.33

yF 1530.5 y31.4 3.18 y458.0+NH y817.1 y18.0 9.27 y339.44

+CH NH y867.1 y19.2 10.62 y298.33 3+H O y660.9 y26.4 y1.75 y441.33

+CH OH y729.6 y21.3 y3.0 y360.43 2

a Experimental free energies of solvation of ions are determined using the following expressions:

0 ( y) ( ) 0 ( +) 0 ( ) ( y)DG X = yDG HX y DG H + DG HX + 5.69 pK anions, Xsolv acid solv solv a

0 ( +) ( +) 0 ( +) 0 ( ) ( +)DG BH = yDG BH + DG H + DG B y 5.69 pK cations, BHsolv base solv solv a

where DG is the gas-phase acidity of HX and DG the gas-phase basicity of B, and DG0 the solvation free energies of theacid base solvy 1 ( y 1)proton or parent base, HX or B. The solvation energy of the proton is taken as y1085.8 kJ mol units are in kJ mol .

b From ref. 58.c From refs. 59 and 60, with standard state corrections.61

d From ref. 62.

solvation energies lie within 10 kJ moly1 of theexperimental values, including the methoxide an-ion, which has generally proven difficult to repro-duce via computational methods.32, 64, 65 Becausethere is no accurate experimental solvation ener-

Ž y y.gies for nitrogen anions NH or CH NH pa-2 3rameterization of the nitrogen atom has been ap-plied to neutral and cationic species only. Thus,while the solvation energies for the nitrogen speciesare well reproduced, the parameterization may be

TABLE II.( y1) aComparison of Calculated Solvation Free Energies kJ mol .

bFDPBc d eDG r r r AM1-SM2 ILD PCMexpt SIP A r

CH 9.2 y2.8 y2.9 y1.7 y5.9 y1.7 y9.24CH CH 7.5 0.6 0.6 2.0 2.5 0.9 y7.53 3NH y18.0 5.5 5.6 0.8 0.0 1.3 y6.03CH NH y19.2 8.4 10.4 7.2 y6.5 7.1 y3.53 2H O y26.4 y9.2 y13.3 y5.6 0.0 y10.4 4.22CH OH y21.3 y4.4 y5.3 1.8 y3.0 y3.8 y6.23HF y31.4 2.0 y0.8 y0.6 31.9 10.9 11.1CH F y0.8 y6.8 y9.6 y6.1 2.5 y10.9 y18.03

yOH y458.0 y3.9 2.4 85.5 5.7 y23.2 y15.6yCH O y416.7 8.8 43.4 94.4 67.3 10.9 y13.43

yF y458.0 1.1 1.1 62.4 10.3 22.9 y12.6+NH y337.1 y4.2 y0.5 y48.6 6.7 y1.8 y39.74

+CH NH y300.9 y1.5 y2.9 y24.0 0.1 y8.7 y34.73 3+H O y440.9 1.7 1.2 y15.6 6.6 18.3 30.83

+CH OH y365.5 5.8 6.8 3.7 13.2 18.2 17.13 2

a ( )Results for each of the computational methods are presented as differences theory minus experiment .b ( ) ( )Finite difference poisson ]Boltzmann method using SIP derived radii r , fixed atom-specific radii r , and radii from theSIP A

( )isodensity surface r .rc From ref. 64.d From ref. 63.e ( ) ( ) (Calculated at the HF / 6-31+G d level upon geometries optimized at the MP2 / 6-31G d level. van der Waals radii H:1.2, C:1.5,

)N:1.5, O:1.4, F:1.35 were unscaled.

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slightly biased toward the positive species. Thismay result in nitrogen having an artificially largeradii, but the lack of reliable experimental dataleaves no avenue for testing this. Similarly, thecarbon radius is parameterized only in a methylŽ .CH environment, which may have biased the3determination of the carbon r value. The param-LJeterization is also generally biased toward the ionicsystems, because the solvation energy of ions ismore sensitive to changes in the radii than neutralsystems.

The largest components of the solvent interac-tion potential are the dipole interaction and theLennard]Jones energies, with the magnitudes ofthe quadrupole and ion-induced dipole energiesmuch smaller by comparison. Presented in TableIII are the average energies for each of the individ-ual components of the SIP, which define the SASŽ . yminima in the potential for H O, OH , and2H Oq. Omitting the quadrupole and ion-induced3dipole terms in the calculation of the solvent inter-action potential gives rise to almost identical rSIP

˚values. The largest variation in radii is 0.02 A, withmost radii remaining unchanged.

Radii determined from the SIP using MP2r6-Ž .311qG d, p CHELPG charges are almost identical

Ž .to those obtained using the HFr6-31qG d CHEL-ŽPG charges, despite quite large differences in some

.cases 0.1 e . There appears, therefore, no benefitassociated with using more elaborate methods to

Ž .calculate charges than HFr6-31 q G d forthese calculations. Only minor changes were ob-

Ž .served when MP2r6-31 q G d -optimized ge-Ž .ometries were used in place of MP2r6-31G d

geometries, showing no requirement for the com-putationally more expensive geometry optimiza-tions. In general, the solvent-excluded volumegenerated by the atom-centered spheres repro-duces the volume generated by the SIP to within1%. Of the systems studied here, the largest devia-

˚3Ž . Žtion in volumes SEV y SEV is 2.2 A of aSIP rep˚3.total solvent-excluded volume of 130 A for the

water molecule, a difference of 1.7%. The atomicradii, however, are derived by minimizing the

TABLE III.Components of Total Solvent Interaction

y1( )Potential kJ mol .

E E E ELJ iy d iy q iy id

H O y0.410 y0.138 0.000 y0.0022yOH y0.181 y0.785 0.018 y0.004

+H O y0.480 y0.646 y0.014 y0.0023

total difference between the SEV and the SEV .SIP repThus, some regions of the SEV extend beyondrep

the SEV , and some regions of the SEV are notSIP SIPcovered by the SEV . The total mismatch in vol-rep

˚3 ˚3ume for water, for example, is 6.2 A ; 2.0 A existsin the SEV , which the SEV does not cover, andSIP rep

˚3the SEV extends beyond the SEV by 4.2 A .rep SIPThe total mismatch in volumes does not exceed5% of the SEV for any of the systems here.SIP

To assess the effect of the SIP method a set offixed atom-specific radii, r , which best repro-Aduced the solvation free energies of the parameter-ization set of solutes, were determined. The samesolute geometries, CHELPG charges, and FDPBconditions were used for these calculations to en-able a direct comparison of solvation energies. The

Ž . Žoptimized atomic radii r were found to be inA

˚.A H s 1.04, C s 1.69, N s 2.12, O s 1.57, andF s 1.48. The errors in the calculated solvation freeenergies using these radii are presented in Table II.These radii are generally quite similar to those ofBondi,20 with the exception of hydrogen and nitro-

˚Ž .gen atoms, which are considerably smaller 0.16 A˚Ž .and larger 0.57 A , respectively. For neutral

species, the solvation free energies using r valuesAcompare quite favorably with the energies ob-tained through r . Errors in solvation free ener-SIPgies are found to be only slightly larger using rAradii, although the largest value for water, 13 kJmoly1, is larger than the target value of 10 kJmoly1. For the anions, the calculated solvation freeenergies of OHy and Fy are in good agreementwith experiment. The error for methoxide, how-ever, is almost 45 kJ moly1. The cationic species allperform well, with no errors in excess of 10 kJmoly1. The average and unsigned-average errorsusing these r radii are 2.4 and 7.1 kJ moly1,Arespectively, significantly larger than those fromthe SIP method.

Atomic radii can also be obtained from theelectron isodensity surface of quantum-mechanicalcalculations, r . In this case the only variable is ther

cutoff. A cutoff of 0.0017 a.u. was found to mini-Ž .mize the total unsigned error in the calculated

solvation free energies of the set of molecules inthe parameterization set. Electron densities were

˚Ž .calculated over a regular 0.1 A cubic grid sur-rounding the solute molecule at the HFr6-31qŽ .G d level. The molecular volume includes all grid

points having an electron density that is greaterthan the cutoff. The solvent-excluded volume in-cludes the molecular volume plus all other gridpoints within the probe radius of the molecular

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ATOMIC RADII

volume. Radii were obtained from the solvent-ex-cluded volume by the same method used to gener-

Ž .ate radii from the SIP above . Solvation energiesŽ .were calculated FDPB using these radii and

Ž .HFr6-31qG d CHELPG charges. The errors inthe calculated solvation free energies using theseradii are presented in Table II.

Energies calculated for the neutral species usingthe r radii all have errors of less than 10 kJ moly1.r

The anions and cations, however, are all extremelypoorly represented. Apart from CH OHq , the er-3 2rors for all the charged species are greater than 10kJ moly1, with errors as large as 86 and 94 kJmoly1 for hydroxide and methoxide, respectively.Clearly, a uniform value for the electron densitycutoff is not appropriate to generate atomic radiifor species of different charge. The unsigned aver-age error using radii from the electron density is24.0 kJ moly1, making this a particularly unsatis-factory method for determining radii.

Also presented in Table II are the errors in thecalculated solvation free energies from three alter-native methods, including AM1-SM2,64 ILD,63 andPCM.24, 25 AM1-SM2 does quite well with the ex-ception of HF and CH Oy, although it has been3parameterized using solvation energies of ions thatdiffer significantly from those used here. ILD doesreasonably well for CH Oy, but rather poorly for3the other anions. PCM performs least well of thealternative methods, although it does much betterwith anions than FDPB with r radii. The discrep-r

ancy can, in part, be attributed to the outlyingŽcharge portion of the electron density that lies

.beyond the cavity boundary that is present in allquantum-mechanical calculations.13 For this lim-ited set of solutes the SIP approach might beexpected to perform better than the other methodssince it has been parameterized using these sol-utes, although AM1-SM2 has been parameterizedwith CH Oy and yet it still performs rather poorly3for this system.

SIP Atomic Radii

Radii and CHELPG charges of the set of mole-cules used in the parameterization of the SIP Len-nard]Jones radii are presented in Figure 2. It isapparent from this figure that there is a widevariety of radii predicted for atoms in differentenvironments. Thus, for example, the radius of

˚ ˚carbon ranges from 1.62 A on methoxide to 1.90 Aon methylfluoride and, for hydrogen, values range

˚ ˚between 0.63 A on methoxide and 1.16 A on am-monia.

Alignment of the probe dipole with the electro-static field ensures a favorable interaction with thesolute charges that will cause a reduction in the

Žradii, especially of charged species both negative.and positive , because the Langevin function tends

to reduce severely the magnitude of the effectiveprobe dipole surrounding neutral systems. Theeffect of the offset, however, depends on the localelectrostatic field. Generally, the electrostatic fieldincreases in the direction toward a negative charge,and decreases in the direction toward a positivecharge. In regions where the electrostatic potentialincreases in the direction toward the solute, theSAS is moved inward, reducing the radii of nearbyatoms. Conversely, the SAS is moved outward,increasing the radii of nearby atoms, where theelectrostatic potential decreases in the directiontoward the solute. The Lennard]Jones interactionenergy between the probe and hydrogen atoms ismuch weaker than the interaction with oxygen. Asa result, the radii of hydrogen atoms are largelydependent on the atoms to which they are bonded.This produces the large radii for hydrogen atomsbonded to nitrogen.

The effects of the offset and dipole interaction isbest illustrated through inspection of the trend inradii of the series H Oq, H O, and OHy. Without3 2an offset being applied, the oxygen atom radius in

˚ ˚H O is found to be 1.76 A, just 0.03 A shorter than2the Lennard]Jones radius for the oxygen atom.The oxygen atom radii in H Oq and OHy are3

˚substantially smaller, 1.67 and 1.71 A, respectively,as a result of stronger ion]dipole interactions. Theoffset decreases the oxygen atom radius in H O by2

˚0.07 A and increases the hydrogen atom radius by˚ q0.20 A. The oxygen atom radius in H O actually3

˚increases by 0.16 A despite the oxygen atom carry-ing a negative charge; the electrostatic potentialdecreases in the direction of the solute at all pointssurrounding the solute. Thus, the direction inwhich the offset is applied does not necessarilycorrespond to the charge of the nearest atom. InOHy the oxygen atom radius is quite small, result-ing from the cooperative effects of both favorableion]dipole interactions and the probe offset.

Radii from Electron Density

The hydrogen atom radii obtained from theelectron isodensity surface are generally larger thanthe radii from the SIP method. They range from

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SMITH AND HALL

( ) ( )FIGURE 2. Calculated HF / 6-31+G d CHELPG charges and SIP atomic radii in square brackets .

˚ ˚0.96 A in the hydronium ion to 1.39 A in methox-ide. This is in accordance with radii determinedpreviously from electron densities66 and the expec-tation that anions possess a more diffuse electrondistribution than neutral and cationic species.Atomic radii decrease along the CH , NH , H O,4 3 2HF series for the non-hydrogen atom. The carbon

˚atom radius in methane is 1.91 A, nitrogen in˚ ˚ammonia is 1.85 A, oxygen in water is 1.72 A, and

˚fluorine in HF is 1.57 A. The hydrogen atom radiialso decrease along this series.

Applications of Radii from SIP

The differential free energy of solvation of thetautomer equilibrium between 2-hydroxypyridineŽ . Ž .A and 2-pyridone B has been measured experi-mentally in water.67 The observed difference, 18.0kJ moly1, is well reproduced by the semiempirical

Ž y1 .methods, AM1-SM1 and PM3-SM3 18 kJ mol ,Ž y1 . 46but underestimated by AM1-SM2 11 kJ mol .

Calculations using radii obtained from the SIP

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ATOMIC RADII

yield a differential solvation energy of 22.6 kJmoly1, also in good agreement with experiment.

˚Ž .The nitrogen radius in A 1.82 A is significantly˚Ž .smaller than in B 1.99 A . The oxygen radius in A

is identical to the radius found in methanol,˚whereas the radius in B is 0.06 A smaller. The

N-bonded hydrogen in B has a much smaller ra-˚ ˚Ž . Ž .dius 1.01 A than in methanamine 1.13 A :

N OH N O

H

A B

The experimental solvation energies of the E andZ isomers of methylacetamide are roughly equal,41.8 kJ moly1, very similar to that found for ac-etamide, 40.6 moly1.68 FDPB calculations usingcharges and radii from the OPLS force field69 un-derestimate the solvation energy of the E isomerŽ y1 .25.9 kJ mol , and overestimate the solvation

Ž y1 .energy of the unsubstituted amide 54.4 kJ mol ,but reproduce quite well the solvation energy of

Ž y1 . 70the Z isomer 41.0 kJ mol . Using radii andcharges from the SIP, the solvation energies of theE and Z isomers are predicted to be 42.8 and 39.0kJ moly1, respectively, in quite good agreementwith experiment. For acetamide, however, the cal-culated solvation energy is 47.4 kJ moly1, still

Žsomewhat larger than experiment yet within they1 .10 kJ mol target . The solvation energies ob-

tained using the radii obtained from the SIPmethod are very similar to the results from AM1-SM2.46

The solvation free energies for the methylamines and their cations are generally rather poorlyreproduced using continuum-based methods.63, 71

Solvation energies obtained using radii and chargesfrom the SIP method are no exception to this.Ammonia, methanamine, and their protonated ca-tions were used in determining the Lennard]Jonesradii of the SIP, and good agreement with experi-ment is guaranteed. The solvation free energies forŽ . Ž .CH NH and CH N, however, are calculated3 2 3 3

Ž y1to be too small by 13.4 and 12.1 kJ mol , respec-. Ž . q Ž . qtively , whereas for CH NH and CH NH3 2 2 3 3

the calculated solvation energies, y283.3 andy264.4 kJ moly1, are considerably greater thanexperimental estimates, y268.6 and y236.9 kJ

moly1, respectively.71 Increasing the radii of thecations to improve the agreement with experimentcan only make the comparison for the neutralspecies worse.

Conclusions

A new method, based on a solvation interactionpotential, has been successfully developed to de-termine atomic radii for use in FDPB calculationsof solvation free energies. The method is designedto be general and applicable at any point on apotential energy surface, including exotic minimaand transition states, irrespective of the totalcharge. The method has been shown to yield radiithat produce solvation free energies of a parame-terization set of molecules more accurately thanradii obtained from more conventional ap-proaches, such as fixed atom-specific radii or fromthe electron isodensity surface.

References

1. M. W. Wong, M. J. Frisch, and K. B. Wiberg, J. Am. Chem.Ž .Soc., 113, 4776 1991 .

2. M. W. Wong, K. B. Wiberg, and M. J. Frisch, J. Am. Chem.Ž .Soc., 114, 523 1992 .

3. M. W. Wong, K. B. Wiberg, and M. J. Frisch, J. Am. Chem.Ž .Soc., 114, 1645 1992 .

4. M. W. Wong, K. B. Wiberg, and M. J. Frisch, J. Chem. Phys.,Ž .95, 8991 1991 .

5. K. B. Wiberg and M. W. Wong, J. Am. Chem. Soc., 115, 1078Ž .1993 .

6. D. Rinaldi, M. F. Ruitz-Lopez, and J.-L. Rivail, J. Chem.Ž .Phys., 78, 834 1983 .

7. D. Rinaldi, J.-L. Rivail, and N. Rguini, J. Comput. Chem.,Ž .13, 675 1992 .

8. C. Chipot, D. Rinaldi, and J.-L. Rivail, Chem. Phys. Lett.,Ž .191, 287 1992 .

Ž .9. G. P. Ford and B. Wang, J. Comput. Chem., 13, 229 1992 .

Ž .10. J. Tomasi and M. Perisco, Chem. Rev., 94, 2027 1994 .

11. S. Miertus, E. Scrocco, and J. Tomasi, Chem. Phys., 55, 117Ž .1981 .

12. A. Klamt and G. Schuurmann, J. Chem. Soc. Perkin Trans.¨¨Ž .II, 799 1993 .

Ž .13. A. Klamt and V. Jonas, J. Chem. Phys., 105, 9972 1996 .

14. J. Andzelm, C. Kolmel, and A. Klamt, J. Chem. Phys., 103,¨Ž .9312 1995 .

15. T. N. Truong and E. V. Stefanovich, Chem. Phys. Lett., 240,Ž .253 1995 .

JOURNAL OF COMPUTATIONAL CHEMISTRY 1491

SMITH AND HALL

16. E. V. Stefanovich and T. N. Truong, Chem. Phys. Lett., 244,Ž .65 1995 .

17. E. V. Stefanovich and T. N. Truong, J. Chem. Phys., 105,Ž .2961 1996 .

Ž .18. K. Baldridge and A. Klamt, J. Chem. Phys., 106, 6622 1997 .

19. See A. R. Leach, Molecular Modelling. Principles and Ap-plications, Longman, New York, 1996.

Ž .20. A Bondi, J. Phys. Chem., 68, 441 1964 . Atomic radii are˚H s 1.2, C s 1.70, N s 1.55, O s 1.52, and F s 1.47 A.

21. L. Pauling, The Nature of the Chemical Bond, CornellUniversity Press, Ithaca, NY, 1960.

22. W. M. Latimer, K. S. Pitzer, and C. M. Slansky, J. Chem.Ž .Phys., 7, 108 1939 .

23. A. A. Rashin and B. J. Honig, J. Phys. Chem., 89, 5588Ž .1985 .

24. R. Cammi, M. Cossi, and J. Tomasi, J. Chem. Phys., 104,Ž .4611 1996 .

25. J. B. Foresman, T. A. Keith, K. B. Wiberg, J. Snoonian, andŽ .M. J. Frisch, J. Phys. Chem., 100, 16098 1996 .

26. F. J. Luque, M. J. Negre, and M. Orozco, J. Phys. Chem., 97,Ž .4386 1993 .

27. R. Bonaccorsi, P. Palla, and J. Tomasi, J. Am. Chem. Soc.,Ž .106, 1945 1984 .

Ž .28. M. Orozco and F. J. Luque, Chem. Phys., 182, 237 1994 .

29. M. Bachs, F. J. Luque, and M. Orozco, J. Comput. Chem., 15,Ž .446 1994 .

30. M. Cossi, V. Barone, R. Cammi, and J. Tomasi, Chem. Phys.Ž .Lett., 225, 327 1996 .

Ž .31. A. Gavezzotti, J. Am. Chem. Soc., 105, 5220 1983 .

32. I. Alkorta, H. O. Villar, and J. Perez, J. Comput. Chem., 14,Ž .620 1993 .

33. A. A. Rashin and K. Namboodiri, J. Phys. Chem., 91, 6003Ž .1987 .

34. A. A. Rashin, M. A. Bukatin, J. Andzelm, and A. T. Hagler,Ž .Biophys. Chem., 51, 375 1994 .

35. T. Furuki, A. Umeda, M. Sakurai, Y. Inoque, and R. Chujo,ˆ ˆŽ .J. Comput. Chem., 15, 90 1994 .

36. V. Luzhkov and A. Warshel, J. Comput. Chem., 13, 199Ž .1991 .

37. A. A. Bliznyuk and J. E. Gready, J. Phys. Chem., 99, 14506Ž .1995 .

38. D. Horvath, D. van Belle, G. Lippens, and S. J. Wodak, J.Ž .Chem. Phys., 104, 6679 1996 .

39. M. A. Aguilar and F. J. Olivares del Valle, Chem. Phys., 129,Ž .439 1989 .

40. F. J. Olivares del Valle and M. A. Anguilar, J. Comput.Ž .Chem., 13, 115 1992 .

Ž .41. F. J. Olivares del Valle, Chem. Phys., 170, 161 1993 .

42. B. Roux, H.-A. Yu, and M. Karplus, J. Phys. Chem., 94, 4683Ž .1990 .

43. S. W. Rick and B. J. Berne, J. Am. Chem. Soc., 116, 3949Ž .1994 .

44. S. Miertus, J. Bratos, and M. Trebaticka, J. Mol. Liquids, 33,´Ž .139 1987 .

45. O. Takahashi, H. Sawahata, Y. Ogawa, and O. Kikuchi, J.Ž . Ž .Mol. Struct. Theochem , 393, 141 1997 .

46. C. J. Cramer and D. G. Truhlar, In Reviews in Computa-tional Chemistry, Vol. VI, K. B. Lipkowitz and D. B. Boyd,Eds., VCH, New York, 1995.

47. C. Lim, S. L. Chan, and P. Tole, In Structure and Reactivityin Aqueous Solution, C. J. Cramer and D. G. Truhlar, Eds.,ACS, Washington, DC, 1994.

48. R. Bonnaccorsi, C. Ghio, and J. Tomasi, Studies Phys. Theor.Ž .Chem., 21, 407 1982 .

49. G. C. Maitland, M. Rigby, E. B. Smith, and W. A. Wakeham,Intermolecular Forces, Clarendon Press, Oxford, 1981.

Ž .50. A. D. Buckingham, Disc. Faraday Soc., 24, 151 1957 . Seealso J. O’M. Bockris and A. K. M. Reddy, Modern Electro-chemistry 1, Plenum, New York, 1970.

51. C. G. Gray and K. E. Gubbins, Theory of Molecular Fluids,Clarendon Press, Oxford, 1984.

52. In the process of minimization, the radii of each atom isincremented in turn until the total mismatch increases fromthe previous step. Following this, the radii are decrementeduntil the mismatch again increases. This process is repeatedwith decreasing increments until the desired accuracy isobtained.

53. D. Sitkoff, K. A. Sharp, and B. Honig, J. Phys. Chem., 98,Ž .1978 1994 .

54. W. J. Hehre, L. Radom, P. v. R. Schleyer, and J. A. Pople, AbInitio Molecular Orbital Theory, John Wiley & Sons, 1986.

55. M. J. Frisch, G. W. Trucks, H. B. Schlegel, P. M. W. Gill,B. G. Johnson, M. A. Robb, J. R. Cheeseman, T. Keith, G. A.Petersson, J. A. Montgomery, K. Raghavachari, M. A. Al-Laham, V. G. Zakrzewski, J. V. Ortiz, J. B. Foresman, J.Cioslowski, B. B. Stefanov, A. Nanayakkara, M. Challa-combe, C. Y. Peng, P. Y. Ayala, W. Chen, M. W. Wong, J. L.Andres, E. S. Replogle, R. Gomperts, R. L. Martin, D. J. Fox,J. S. Binkley, J. D. Defrees, J. Baker, J. P. Stewart, M.Head-Gordon, C. Gonzalez, and J. A. Pople, Gaussian-94,Revision C.3, Gaussian, Inc., Pittsburgh, PA, 1995.

56. C. M. Breneman and K. B. Wiberg, J. Comput. Chem., 11,Ž .361 1990 .

57. DelPhi, Version 2.50, Biosym Technologies, San Diego, CA,1993.Ž .58. a J. E. Bartmess, In NIST Standard Reference DatabaseNumber 69, W. G. Mallard and P. J. Linstrom, Eds., Na-tional Institute of Standards and Technology, Gaithersburg,

Ž . Ž .MD, 20899 August 1997 . b E. P. Hunter and S. G. Lias, InNIST Standard Reference Database Number 69, W. G. Mal-lard and P. Linstrom, Eds., National Institute of Standards

Ž .and Technology, Gaithersburg, MD, 20899 August 1997Ž .http:rrwebbook.nist.gov .

Ž .59. J. Hine and P. K. Mookerjee, J. Org. Chem., 40, 292 1975 .

60. D. D. Wagman, V. B. P. Parker, R. H. Schumm, I. Halow,S. M. Bailey, K. L. Churney, and R. L. Nuttall, J. Phys.

Ž . Ž .Chem. Ref. Data, 11 Suppl. 2 1982 .

61. C. Lim, D. Bashford, and M. Karplus, J. Phys. Chem., 95,Ž .5610 1991 .

62. A. Albert and E. P. Sergeant, Ionization Constants of Acidsand Bases, Metheun, London, 1962.

63. J. Florian and A. Warshel, J. Phys. Chem. B, 101, 5583´Ž .1997 .

64. C. J. Cramer and D. G. Truhlar, J. Comput.-Aid Mol. Des., 6,Ž .629 1992 .

VOL. 19, NO. 131492

ATOMIC RADII

65. J. Chen, L. Noodleman, D. A. Case, and D. Bashford, J.Ž .Phys. Chem., 98, 11059 1994 .

66. R. F. Hout Jr. and W. J. Hehre, J. Am. Chem. Soc., 105, 3628Ž .1983 .

Ž .67. P. Beck, Acc. Chem. Res., 10, 186 1977 .Ž .68. R. Wolfenden, Biochemistry, 17, 210 1978 .

69. W. L. Jorgensen and J. Tirado-Rives, J. Am. Chem. Soc., 110,Ž .167 1988 .

70. A. Jean-Charles, A. Nicholls, K. Sharp, B. Honig, A. Tem-

pczyk, T. F. Hendrickson, and W. C. Still, J. Am. Chem.Ž .Soc., 113, 1454 1991 .

71. I. Tunon, E. Silla, and J. Tomasi, J. Chem. Phys., 96, 9043˜´Ž .1992 .

Ž . q72. Experimental solvation energies for CH NH and3 2 2Ž . qCH NH were determined from the gas-phase basicities3 3

Ž y1 .taken from ref. 59 896.5 and 918.1 kJ mol , neutralŽsolvation free energies from refs. 60]62 18.0 and 13.4 kJ

y1 . Ž .mol , and pK s from ref. 63 10.77 and 9.8, respectively .a

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