atomic radii in molecules for use in a polarizable force field

Atomic Radii in Molecules for Use in aPolarizable Force Field

MARCEL SWART,1,2 PIET TH. VAN DUIJNEN3

1Institut de Quımica Computacional and Departament de Quımica, Universitat de Girona,Campus Montilivi, 17071 Girona, Spain2Institucio Catalana de Recerca i Estudis Avancats (ICREA), Pg. Lluıs Companys 23,08010 Barcelona, Spain3Theoretical Chemistry, Zernike Institute for Advanced Materials, University of Groningen,Nijenborgh 4, 9747 AG Groningen, The Netherlands

Received 8 March 2010; accepted 13 May 2010Published online 8 July 2010 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/qua.22855

ABSTRACT: We report here the results for an ab initio approach to obtain theparameters needed for molecular simulations using a polarizable force field. Theseparameters consist of the atomic charges, polarizabilities, and radii. The former two arereadily obtained using methods reported previously (van Duijnen and Swart, J Phys ChemA 1998, 102, 2399; Swart et al. J Comput Chem 2001, 22, 79), whereas here we report a newapproach for obtaining atomic second-order radii (SOR), which is based on second-orderatomic moments in scaled Voronoi cells. These parameters are obtained from quantum-chemistry calculations on the monomers, and used without further adaptation directlyfor intermolecular interactions. The approach works very well as shown here for fourdimers, where high-level coupled cluster with singles and doubles, and perturbativetriples (CCSD(T)) and density functional theory (DFT) Swart-Sola-Bickelhaupt functionalincluding Grimme’s dispersion correction (SSB-D) reference data are available forcomparison. The energy surfaces for the three methods are very similar, which is also thecase for the interaction between a water molecule with either a chloride anion or asodium cation. These latter systems had previously been used to criticize Thole’s dampedpoint-dipole method, but here we show that with the correct use of the method, it isperfectly able to describe the intermolecular interactions. This is most obvious for theinduced dipole moment as function of the chloride–oxygen distance, where the direct(discrete) reaction field results are virtually indistinguishable from those obtained atCCSD(T)/aug-cc-pVTZ. VC 2010 Wiley Periodicals, Inc. Int J Quantum Chem 111: 1763–1772, 2011

Key words: polarizable force field; parameterization; density functional theory;coupled cluster methods; atomic radii

Correspondence to: M. Swart; e-mail: [email protected] grant sponsor: Ministerio de Ciencia e Innovacion

(MICINN).Contract grant number: CTQ2008-06532/BQU.Contract grant sponsor: DIUE, Generalitat de Catalunya.Contract grant number: 2009SGR528.

International Journal of Quantum Chemistry, Vol 111, 1763–1772 (2011)VC 2010 Wiley Periodicals, Inc.

Introduction

P olarization occurs in all matter and resultsfrom the fact that the electronic charge dis-

tribution within a molecule depends on the pres-ence of other nearby ions and molecules [1].Within quantum-chemistry, this many-body effectis taken into account automatically, i.e., the inter-actions are in the Hamiltonian H, which makesthat the wavefunction W responds to them (polar-izes). However, current state-of-the-art (bio)molec-ular simulation packages use effective pairwisenonbonding interactions that do not take polariza-tion into account properly. Instead, they use frac-tional atomic charges that are either scaled-upfrom their gas-phase values or obtained in calcu-lations in which the solvent environment is pres-ent implicitly (dielectric continuum models) or ex-plicitly. Both approaches are flawed, because theyassume that polarization merely leads to anincrease of the molecular dipole moment (implicitpolarization). Indeed, the dipole moment of waterincreases from 1.8 Debye in the gas phase to anaverage of 2.6 Debye in the liquid phase at roomtemperature. However, the real instantaneousmoment oscillates between 2.2 and 2.8 Debye [2],if polarization is taken into account correctly. Thisoscillation of the molecular dipole moment ofwater results from the strong dependence ofpolarization on the orientation with respect toother molecules. Changes in these mutual orienta-tions lead to changes in the electrostatic field,which affects the induced dipole moments consid-erably. These latter contribute to the electric fieldsexperienced by other polarizable particles, thusleading to changes in the dipole moments thatfluctuate with time. Therefore, the second-genera-tion nonpolarizable (and nonfluctuating) forcefields are poor approximations to the true fluctu-ating polarization.

A number of corrections to these nonpolariz-able force fields have been proposed amongwhich models based on charges on a spring (e.g.,Drude oscillators [3–7]) and those based onatomic or molecular polarizabilities [8–29]. Theformer have several disadvantages, one of themost important of which is that they are fitted toelectrostatic potentials (ESPs) outside the mole-cules [3]. This makes them very dependent on thechoice of grid, where the ESPs are evaluated. Thesame is true for ESP charge analyses, where e.g.,atomic charges of buried atoms are obtained with

great uncertainty, because the potential is mainlydetermined by atoms near the surface [30, 31].Furthermore, these ESP charge analyses are oftenorientation dependent [32, 33] and usually do notpreserve molecular (let alone atomic) multipolemoments. Some time ago, we presented a moreconsistent charge analysis [34] without the needfor fitting to ESPs. It uses atomic multipolemoments, which are used within the AmsterdamDensity Functional (ADF) program [35] also (witha short-range function [36]) for obtaining thepotential inside the molecule and thus give reli-able atomic charges. Moreover, these multipole-derived charges preserve the atomic and molecu-lar multipole moments [34]. A further complica-tion of the models with charges on a spring isthat atomic charges (on a spring) are not observ-ables, in contrast to the molecular polarizability.Therefore, force fields using this latter molecularproperty (polarizability) are much more straight-forward, easily obtained, and physically sound.

In third-generation polarizable force fieldsbased on atomic or molecular polarizable par-ticles, such as the direct (discrete) reaction field(DRF) approach as developed in our group in thepast decades [1, 2, 37–47], the electrostatic(source) field induces dipole moments at the par-ticles. These in turn generate additional (dipole)fields at the other polarizabilities, which leads toadditional induced dipoles. These can enhance orreduce the initial induced dipoles, depending onthe mutual orientation of the polarizable particles.Because the particles interact through theirinduced dipole moments, which in turn dependon each other, one needs to iteratively solve a setof equations in similar fashion as the self-consist-ent field equations in quantum chemistry. Forinstance, for a system of N polarizable particles,the dipole moment lp of particle p depends on itspolarizability ap, the electrostatic field Eelst andthe dipole field Tpq resulting from the otherinduced dipoles lq [1, 47]:

lp ¼ apEel ¼ ap Eelst þXN

q 6¼p

Tpqlq

24

35 (1)

An efficient and accurate way of representingthe molecular polarizabilities is given by Thole’smodel [48, 49] of interacting atomic polarizabil-ities; more recently, Jensen et al. [50] proposed amodel with Gaussian charge distributions. In bothmodels, the total molecular polarizability can be

SWART AND VAN DUIJNEN

1764 INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY DOI 10.1002/qua VOL. 111, NO. 7/8

constructed from atomic polarizabilities, i.e., byrewriting Eq. (1), one obtains l ¼ [a�1�T]�1E (inmatrix form). Hence, the linear response to agiven electrostatic field, i.e., the polarizability ofthe total system, is obtained as the inverse of therelay matrix R ¼ [a�1�T]. This inverse matrix R�1

of size 3N � 3N can be reduced to a normal 3 � 3polarizability tensor amn:

amn ¼XN

t;j¼1

ðAijÞmn ; A � R�1 ¼ ½a�1 � Tpq�1;

m; n 2 fx; y; zg(2)

Therefore, the molecular polarizability amn isdirectly obtained from (the positions of) the(atomic) polarizable particles.

These polarizabilities can then directly be usedwithin a polarizable force field [47], using appro-priate and physically correct expressions for elec-trostatics, dispersion and polarization/inductionenergy, which can all be derived from second-order perturbation theory [1]. The only missingterm is the short-range repulsion for which wehave in the past borrowed the energy expressionpresent in CHARMM [51, 52], or as a modifiedSlater–Kirkwood term [47]. A vital part for thisshort-range repulsion is formed by the values forthe atomic radii that are needed for these repul-sion energy expressions. We had used either theCHARMM values [51], Bondi values [53], orFrecer’s charge-dependent radii [personal commu-nication, 1995; 47]. The latter have the advantagethat they depend on the fractional charge of theatom, and should thus be better suited fordescribing a wide variety of cationic, neutral, andanionic species. However, these radii are availableonly for H-Ar and much depends on the value ofthe fractional charge chosen. Therefore, it wouldbe more advantageous if these values for theatomic radii could be obtained directly from thesame quantum chemical calculation. Here, wereport such an approach, in which we computeatomic second-order (R2) moments and use theseas measure of the atomic size. Subsequently, weapply these radii within our fully classical DRFapproach to study the intermolecular interactionsof four dimers (ammonia, water, formic acid, andformamide) for which reference data at coupledcluster with singles and doubles, and perturbativetriples (CCSD(T)) are available [54], and for theinteraction between a water molecule and either asodium cation or chloride anion. For these latter

systems, we have computed reference interactionenergies at CCSD(T)/aTZ to be able to comparewith reliable reference data.

Computational Details

All density functional calculations have beenperformed using a locally modified version of theADF [55] program. The CCSD(T) calculationsreported here for the potential energy surface ofthe Naþ��H2O and Cl��H2O systems have beenobtained with the aug-cc-pVTZ basis (aTZ) usingthe CFOUR program (version 1.2) [56, 57].

Atomic Radii from Second-OrderMoments

There are a number of different ways in whichatomic second-order moments can be obtained,e.g., either based on a Mulliken-like approach [58],or within density functional programs making useof the numerical integration grid. In the latter case,one could use the same approach as done withinADF to generate the atomic multipole moments[34, 36], or using the separation of the total gridsinto atomic contributions using Voronoi cells [35,59]. The latter have the advantage that they areeasily obtained and can be used straightforwardlywithin any density functional program that usesnumerical integration grids. A disadvantage ofstandard Voronoi cells is, however, that the spacebetween two atoms is divided equally between thetwo, irrespective of the character of these atoms(Fig. 1). For atoms of roughly the same size, thisposes no problem, but whenever smaller atoms(e.g., hydrogen) or larger atoms (e.g., third row orhigher) are involved this may lead to odd behav-ior. A fast and equally simple change of the Voro-noi setup would be to take a reference atom-sizeinto account while making the decision to whichatom the different integration grid-points belong.i.e., in the standard Voronoi setup, for any givenintegration point i, the distance is computedbetween the point and all atoms a, and the point isassigned to the atom with the smallest distance d:

dstdi;a ¼ ri;a (3)

Now, by taking into account a reference atom-size sa, which is specific for each element, oneobtains the scaled Voronoi setup:

ATOMIC RADII IN MOLECULES FOR USE IN A POLARIZABLE FORCE FIELD

VOL. 111, NO. 7/8 DOI 10.1002/qua INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 1765

dscaledi;a ¼ ri;asa

(4)

As a result, the space between two atoms isdistributed more evenly in that elements of smallsize (e.g., hydrogen) are assigned a much smallerregion of the integration grid than elements oflarge size (e.g., gold). For the reference atom-sizessa, we take the values by O’Keeffe and Brese [60]who compiled a table for most elements of theperiodic table (for the missing elements, we sim-ply take standard covalent radii).

This setup of using scaled Voronoi cells, indeed,seems to work well, as for instance, for water atthe Swart-Sola-Bickelhaupt functional includingGrimme’s dispersion correction (SSB-D)/TZ2P level

[61], we find Voronoi charges of þ0.7350 on oxygenand �0.3675 on hydrogen, if we sum up all thecharge density within the Voronoi cells and use thesesums as atomic charges. These signs are completelyopposite to what should be expected based on thedipole moment of 1.80 Debye, which is by construc-tion reproduced exactly by the Multipole DerivedCharge analysis with representation up to dipoles(MDC-d) charges of �0.6386 on oxygen and þ0.3193on hydrogen. For the scaled-Voronoi cells, the behav-ior is now consistent, i.e., the scaled charges showvalues of þ0.361 on hydrogen and�0.722 on oxygen.

We can now obtain atomic radii from thesescaled Voronoi cells as well by looking at the sec-ond-order moments (R2) moments as obtained fromnumerical integration over the integration grid:

R2� �

a¼

Xi2a

wiqi � ðri � raÞ2 (5)

where wi is the weight of the grid-point within theintegration grid, qi the charge density in the grid-point, and ri and ra, the grid-point and atomic posi-tion, respectively. Subsequently [62], the second-order moments are transformed into atomic sec-ond-order radii (SOR) with the following equation:

Ra ¼ a0 þ a1 � qþ arcsin h R2� �

aþb0 þ b1 � q

� �(6)

where a0, a1, b0, and b1 are parameters, and q is thetotal molecular charge. For the systems studied inthis article, the corresponding atomic SOR radiiand fractional charges are reported in Table I.

APPLICATION TO FOUR DIMERS

In 2006, Hobza and coworkers [54] presented abenchmark set of 22 dimers (S22 set), where the

FIGURE 1. Standard Voronoi cells setup. [Color figurecan be viewed in the online issue, which is available atwileyonlinelibrary.com.]

TABLE IAtomic radii and charges for systems studied here (at SSB-D[60]/TZ2P).

q a (au) R (au) q a (au) R (au)

Naþ þ1 1.743 3.1883 Cl� �1 27.274 3.6499Ammonia Water

N �0.8838 9.0615 3.2683 O �0.6386 6.3464 3.0974H 0.2946 1.3766 0.9578 H 0.3193 1.3766 1.0924

Formic Acid FormamideC1 0.3262 9.4469 3.1724 C1 0.1273 9.4469 3.1094H4 0.2046 1.3766 1.0016 H6 0.1507 1.3766 1.0202O2 �0.6263 6.3464 2.8054 O2 �0.4739 6.3464 2.8068H5 0.5382 1.3766 1.0769 N3 �0.1879 9.0615 3.0697O3 �0.4427 6.3464 2.7933 H4 0.1522 1.3766 0.9910

H5 0.2316 1.3766 0.9938



interaction energies have been obtained withCCSD(T) and, in some cases, also the geometry.For four of these dimers (Scheme 1) for which thedimer geometry had been obtained at CCSD(T),we have performed a potential energy surface(PES) scan using the SSB-D/TZ2P density func-tional, a recent development consisting of a smallmodification of the nonempirical Perdew-Burke-Ernzerhof functional (PBE) [63] functional, whichalso contains Grimme’s empirical dispersion cor-rection [64, 65] for correctly treating p–p stacking.This new density functional theory (DFT) methodwas recently [61] shown to perform well for theseweak interactions, and thus can serve as a refer-ence for the PES curve. In general, our DRF resultsdo well for the energy surfaces (Fig. 2), followingthe SSB-D surfaces very well apart from the short-range region, where the DRF surface increases toofast. This is not at all that surprising because thispart is described by a (screened) Lennard–Jones(R�12) potential, i.e., the only part of our force fieldthat does not result directly from second-orderperturbation theory [1]. In future studies, we willinvestigate if other forms such as an exponentialform A�exp(�B�r) would be needed for improvingthis part of the DRF energy expression. However,it is comforting that by using an ab initio approachfor model parameter development, as done here,in which all monomer properties are obtainedfrom quantum chemistry calculations, the DRFapproach correctly predicts the differences in stabi-lization for these four dimers.

DEFENSE OF THOLE’S MODEL OFINTERACTING POLARIZABILITIES

In a recent letter [66], Masia, Probst, and Rey(MPR)-I suggested that the damped point dipolemethod as proposed by Thole in 1981 [48], needsa new implementation to describe the dipolemoments of halide–water complexes correctly.This letter followed earlier articles (MPR-II [67]and MPR-III [68]), in which they expect Thole’sparameters, originally proposed for intramolecu-lar damping, to be inappropriate for intermolecu-lar interactions within completely classical molec-ular mechanics (MM) descriptions. In general,Thole’s scheme works excellent for constructingmolecular polarizabilities [49, 69], i.e., at distancesat or within normal chemical bond distances.Then why should it not work at the much longerintermolecular distances where the overlap ismuch smaller? MPR did not give a clue to whythey expect it to fail there. Instead, they refer onlyto Thole’s original paper [48] and its first quan-tum mechanics (QM)/MM application [38, 70]with the message (MPR-III): ‘‘We will not analyzethe extremely useful work on polarizable atomsdesigned to incorporate reaction fields into quan-tum chemical calculations.’’ They, therefore,ignore many studies with either the DRFapproach [37, 42, 45, 48] or related approaches byother groups [8–29]. In these studies, Thole’smodel has been applied successfully to a widerange of systems and problems, applications that

SCHEME 1. Four dimers studied here. [Color figure can be viewed in the online issue, which is available atwileyonlinelibrary.com.]



were fully or in part based on classical simula-tions. With the DRF approach, we showed thatThole’s model can be used to mimic quantumchemical calculations [71, 72] including many-body interactions [43], and we addressed solvent

effects on spectra [73–76], chemical reactions [77–79], and (hyper)polarizabilities [2, 74, 80–85]. Allthese DRF studies were already based on ‘‘good’’ab initio atomic charges [34, 58] and polarizabil-ities [49, 69] but not yet atomic radii (vide supra).

FIGURE 2. Energy surfaces (kcal mol�1) obtained for dimerization of ammonia (A), water (B), formic acid (C), andformamide (D) with SSB-D/TZ2P and DRF; also indicated with red/yellow dot is the interaction energy as obtained atCCSD(T)/aTZ level. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

FIGURE 3. Energy surfaces (kcal mol�1) obtained for Cl��H2O (A) and Naþ��H2O (B) with SSB-D/TZ2P, CCSD(T)/aTZ and DRF. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]



By definition, (hyper)polarizabilities are (higherorder) responses of an object to an external electricfield. By putting sources at distances comparablewith the dimensions of the object, the resultingfields are no longer external, and the sources haveto be considered as part of the object, whichrequires more sophisticated methods than the sim-ple ‘‘product of field and polarizability.’’ Therefore,the (induced) moments of such complexes if calcu-lated with undamped interactions between polariz-able, charged and/or dipolar species at short dis-tances are no longer representative, and indeedtend to be overestimated if Thole’s model is notbeing used. To show that Thole’s model is able tocorrectly describe these interactions, we haveinvestigated the systems used by MPR. Given thatMPR used CCSD and not the more adequateCCSD(T) method, we repeated the calculations ofthe sodium cation and chloride anion with a watermolecule (Scheme 2) using an adequate basis set,i.e., CCSD(T)/aTZ.

The results are given in Figure 3 together withthe SSB-D/TZ2P [61] and our classical DRF ener-gies using the same halide–water and sodium–water geometries (parameters are given in Table I,which were obtained from SSB-D/TZ2P calcula-tions on the NaCl molecule). It is immediatelyobvious that there is complete agreement betweenthe three methods for the interaction energy, apartfrom the short-range region, where the Lennard–Jones potential is too repulsive (vide supra). Asmentioned above, this short-range potential is theonly energy term that does not result directly fromsecond-order perturbation theory (unlike the otherterms in the DRF energy expression), and mayneed to be improved in future studies.

The dipole moments of the chloride–water sys-tem are given in Figure 4, as function of the

ClAO distance, as obtained from the CCSD(T)/aTZ data, and those obtained with our DRFapproach. These dipole moments were obtainedwith the chloride in the origin, the oxygen ofwater on the negative z-axis, and the hydrogenson the positive y-axis (Scheme 2). The dipolemoments of the coupled cluster and DRF showsimilar trends, both for the y- and z-directions.Both increase upon shortening of the ClAO

FIGURE 4. Dipole moment (Debye) obtained forCl��H2O with CCSD(T)/aTZ and DRF as function of theClAO distance (A) where Cl is at the origin, O on thez-axis and the water molecule is in the yz-plane. [Colorfigure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]

SCHEME 2. Cl��H2O and Naþ��H2O orientations.

FIGURE 5. Dipole moment (Debye) obtained forX��H2O (X ¼ Cl�, Naþ) with CCSD(T)/aTZ as function ofthe X–O distance (A) where X is at the origin, O on thez-axis and the water molecule is in the yz-plane. [Colorfigure can be viewed in the online issue, which isavailable at wileyonlinelibrary.com.]



distance, which results from the increase in polar-ization as the chloride–water distance decreases.This phenomenon is easily understood and iswell covered by the polarizable DRF approach,where this polarization is an explicit componentof the interaction. Therefore, the divergence assuggested by MPR does not occur when the Tholemodel is used properly within the DRF approach.The main argumentation for the criticism by MPRof the damped point-dipole method is shown tobe based on wrong assumptions about thedamped point-dipole method. If used correctly,Thole’s model works perfectly.

Interestingly, the dipole moment of theNaþ��H2O system goes through a maximum ataround 2.2 A (see Fig. 5), which MPR attributedto the hyperpolarizability. However, it is not nec-essary to go to the hyperpolarizability to explainthe maximum, as it is related to the maximum inthe polarizability itself. For instance, for thehydrogen molecule as a function of the HAH dis-tance by using a minimal basis set that cannotgenerate the hyperpolarizability, we also observethe same trend (Fig. 6, left) with a maximum inthe polarizability. Starting from the hydrogens atinfinite separation, where the total polarizabilityis simply the sum of the atomic parts, the polariz-ability increases along the axis, because the elec-

tron of one atom ‘‘feels’’ the nucleus of the otheratom. In other words, the electrons have morespace to move. At shorter interatomic distances,this space is reduced by the concentrating nuclearcharge leading to a decreasing polarizability,which should end up at that of the (here badlyrepresented) helium atom. Note that, in this basisset, there is no off-axis component of the polariz-ability. Included are the ‘‘classical’’ DRF polariz-ability components, by using the parameters forthe STO basis [21]. [This model polarizability islarger than (half of) the asymptotic value of thequantum mechanics (QM) case, because it isrelated to bound H-atoms in various molecules.]The parallel (xx) component behaves qualitativelycorrectly, whereas both DRF components are inagreement with the (screened) Silbersteins equa-tions ([86] and [48]), showing that Thole’sapproach catches the essential physics.

In the right-hand side part of Figure 6, thedipole induced by a point charge in a hydrogenatom is presented, as obtained from variousapproaches. The QM calculation with a DZP basisresults in a maximum, similar to that in Figure 5,and again, the DRF result is qualitatively correct.The unscreened interaction between ‘‘undressed’’charges and point polarizabilities obviously leadsto a ‘‘polarization catastrophe.’’

FIGURE 6. Left: Polarizability of the H2 molecule as function of the interatomic distance, calculated with a minimal(STO) basis, and as obtained from DRF90 using the parameters for this basis [21]. Right: Induced moment in a hydro-gen atom by a point charge obtained from classical and QM/MM (RHF/DZP) calculations. [Color figure can be viewedin the online issue, which is available at wileyonlinelibrary.com.]



Conclusions

We report here the results for an ab initioapproach to obtaining the parameters needed forcarrying out molecular simulations using a polar-izable force field. These parameters consist of theatomic charges, polarizabilities, and radii. The for-mer two are readily obtained using methodsreported previously [34, 49], whereas here wereport a new approach to obtain atomic radiidirectly from quantum chemistry calculations.The approach for obtaining the radii is based onscaled Voronoi cells and the second-order atomicmoments, which result in the SOR. These parame-ters are obtained from quantum-chemistry calcu-lations on the monomers, and used without fur-ther adaptation directly for intermolecularinteractions. The approach works very well asshown here for four dimers, where high-levelCCSD(T) and DFT (SSB-D) reference data areavailable for comparison. The energy surfaces forthe three methods are highly similar, which isalso the case for the interaction between a watermolecule with either a chloride anion or a sodiumcation. These latter systems had previously beenused to criticize Thole’s damped point-dipolemethod, but here we show that with the correctuse of the method, it is perfectly able to describethe intermolecular interactions. This is mostobvious for the induced dipole moment as func-tion of the chloride–oxygen distance, where theDRF results are virtually indistinguishable fromthose obtained at CCSD(T)/aug-cc-pVTZ.

References

1. van Duijnen, P. Th.; Swart, M.; Jensen, L. The DiscreteReaction Field approach for calculating solvent effects. InSolvation Effects on Molecules and Biomolecules: Compu-tational Methods and Applications; Challenges and Advan-ces in Computational Chemistry and Physics, Vol. 6;Canuto, S., Ed.; Springer: Dordrecht, 2008; p 39.

2. Jensen, L.; Swart, M.; van Duijnen, P. Th. J Chem Phys2005, 122, 034103.

3. Harder, E.; Anisimov, V. M.; Vorobyov, I. V.; Lopes, P. E.M.; Noskov, S. Y.; Mackerell, A. D., Jr.; Roux, B. J ChemTheory Comp 2006, 2, 1587.

4. Yu, H.; Whitfield, T. W.; Harder, E.; Lamoureux, G.; Voro-byov, I. V.; Anisimov, V. M.; Mackerell, A. D., Jr.; Roux, B.J Chem Theory Comp 2010, 6, 774.

5. Baker, C. M.; Lopes, P. E. M.; Zhu, X.; Roux, B.; Mackerell,A. D., Jr. J Chem Theory Comp 2010, 6, 1181.

6. Zhao, D.-X.; Liu, C.; Wang, F.-F.; Yu, C.-Y.; Gong, L.-D.;Liu, S.-B.; Yang, Z.-Z. J Chem Theory Comp 2010, 6, 795.

7. Baker, C. M.; Mackerell, A. D., Jr. J Mol Model 2010, 16, 567.

8. Mayer, A.; Lambin, P.; Astrand, P.-O. Nanotechnology2008, 19, 025203.

9. Aidas, K.; Mikkelsen, K. V.; Kongsted, J. Phys Chem ChemPhys 2010, 12, 761.

10. Sun, X.; Yoo, S.; Xantheas, S. S.; Dang, L. X. Chem PhysLett 2009, 481, 9.

11. Jiang, J.; Wu, Y.; Wang, Z.-X.; Wu, C. J Chem TheoryComp 2010, 6, 1199.

12. Isegawa, M.; Kato, S. J Chem Theory Comp 2009, 5, 2809.

13. Magnus Olsen, J.; Aidas, K.; Mikkelsen, K. V.; Kongsted, J.J Chem Theory Comp 2010, 6, 249.

14. Archambault, F.; Chipot, C.; Soteras, I.; Luque, F. J.; Schul-ten, K.; Dehez, F. J Chem Theory Comp 2009, 5, 3022.

15. Babin, V.; Baucom, J.; Darden, T. A.; Sagui, C. Int J QuantChem 2006, 106, 3260.

16. Paterson, M. J.; Kongsted, J.; Christiansen, O.; Mikkelsen,K. V.; Nielsen, C. B. J Chem Phys 2006, 125, 184501.

17. Nielsen, C. B.; Christiansen, O.; Mikkelsen, K. V.;Kongsted, J. J Chem Phys 2007, 126, 154112.

18. Aidas, K.; Møgelhøj, A.; Nilsson, E. J. K.; Johnson, M. S.;Mikkelsen, K. V.; Christiansen, O.; Soderhjelm, P.;Kongsted, J. J Chem Phys 2008, 128, 194503.

19. Smalø, H. S.; Astrand, P.-O.; Jensen, L. J Chem Phys 2009,131, 044101.

20. Li, H. J Chem Phys 2009, 131, 184103.

21. Kumar, R.; Wang, F.-F.; Jenness, G. R.; Jordan, K. D. JChem Phys 2010, 132, 014309.

22. Nakano, H.; Yamamoto, T.; Kato, S. J Chem Phys 2010,132, 044106.

23. Aidas, K.; Møgelhøj, A.; Kjaer, H.; Nielsen, C. B.; Mikkel-sen, K. V.; Ruud, K.; Christiansen, O.; Kongsted, J. J PhysChem A 2007, 111, 4199.

24. Mayer, A.; Astrand, P.-O. J Phys Chem A 2008, 112, 1277.

25. Borodin, O. J Phys Chem B 2009, 113, 11463.

26. Ponder, J. W.; Wu, C.; Ren, P.; Pande, V. S.; Chodera, J. D.;Schnieders, M. J.; Haque, I.; Mobley, D. L.; Lambrecht, D. S.;DiStasio, R. A., Jr.; Head-Gordon, M.; Clark, G. N. I.; John-son, M. E.; Head-Gordon, T. J Phys Chem B 2010, 114, 2549.

27. Bedrov, D.; Borodin, O.; Li, Z.; Smith, G. D. J Phys Chem B2010, 114, 4984.

28. Aidas, K.; Kongsted, J.; Sabin, J. R.; Oddershede, J.; Mikkel-sen, K. V.; Sauer, S. P. A. J Phys Chem Lett 2010, 1, 242.

29. Lopes, P. E. M.; Roux, B.; Mackerell, A. D., Jr. Theor ChemAcc 2009, 124, 11.

30. Jensen, F. Introduction to Computational Chemistry; Wiley:New York, 1998.

31. Nistor, R. A.; Polihronov, J. G.; Muser, M. H.; Mosey, N. J.J Chem Phys 2006, 125, 094108.

32. Chirlian, L. E.; Francl, M. M. J Comput Chem 1987, 8, 894.

33. Breneman, C. M.; Wiberg, K. B. J Comput Chem 1990, 11, 361.

34. Swart, M.; van Duijnen, P. Th.; Snijders, J. G. J ComputChem 2001, 22, 79.

35. te Velde, G.; Bickelhaupt, F. M.; Baerends, E. J.; FonsecaGuerra, C.; van Gisbergen, S. J. A.; Snijders, J. G.; Ziegler,T. J Comput Chem 2001, 22, 931.



36. Fonseca Guerra, C.; Snijders, J. G.; Velde, G. te.; Baerends,E. J Theor Chem Acc 1998, 99, 391.

37. Thole, B. T.; van Duijnen, P. Th. Theor Chim Acta 1980, 55,307.

38. Thole, B. T.; van Duijnen, P. Th. Chem Phys 1982, 71, 211.

39. van Duijnen, P. Th.; Thole, B. T. Chem Phys Lett 1981, 83,129.

40. van Duijnen, P. Th.; Juffer, A. H.; Dijkman, J. P. J MolStruct (THEOCHEM) 1992, 260, 195.

41. van Duijnen, P. Th. In New Challenges in ComputationalQuantum Chemistry; Broer, R.; Aerts, P. J. C.; Bagus, P. S.,Eds.; Department of Chemical Physics and Material Sci-ence: Groningen, 1994; p 71.

42. van Duijnen, P. Th.; de Vries, A. H. Int J Quant Chem1996, 60, 1111.

43. Grozema, F. C.; Zijlstra, R. W. J.; van Duijnen, P. Th. ChemPhys 1999, 246, 217.

44. Grozema, F. C.; Swart, M.; Zijlstra, R. W. J.; Piet, J. J.; Sieb-beles, L. D. A.; van Duijnen, P. Th. J Am Chem Soc 2005,127, 11019.

45. de Vries, A. H.; van Duijnen, P. Th.; Juffer, A. H.; Rull-mann, J. A. C.; Dijkman, J. P.; Merenga, H.; Thole, B. T. JComput Chem 1995, 16, 37.

46. de Vries, A. H.; van Duijnen, P. Th.; Zijlstra, R. W. J.;Swart, M. J. El. Spec Rel Phen 1997, 86, 49.

47. Swart, M.; van Duijnen, P. Th. Mol Simul 2006, 32, 471.

48. Thole, B. T. Chem Phys 1981, 59, 341.

49. van Duijnen, P. Th.; Swart, M. J Phys ChemA 1998, 102, 2399.

50. Jensen, L.; Astrand, P. O.; Osted, A.; Kongsted, J.;Mikkelsen, K. V. J Chem Phys 2002, 116, 4001.

51. Brooks, B. R.; Bruccoleri, R. E.; Olafson, B. D.; States, D. J.;Swaminathan, S. J.; Karplus,M. J Comput Chem. 1983, 4, 187.

52. de Vries, A. H. Modelling condensed-phase systems. Fromquantum chemistry to molecular models, PhD thesis; Rijk-suniversiteit Groningen: Groningen, The Netherlands, 1995.

53. Bondi, A. J Phys Chem 1964, 68, 441.

54. Jurecka, P.; Sponer, J.; Cerny, J.; Hobza, P. Phys ChemChem Phys 2006, 8, 1985.

55. Baerends, E. J.; Autschbach, J.; Berces, A.; Berger, J. A.;

Bickelhaupt, F. M.; Bo, C.; de Boeij, P. L.; Boerrigter, P. M.;

Cavallo, L.; Chong, D. P.; Deng, L.; Dickson, R. M.; Ellis,

D. E.; van Faassen, M.; Fan, L.; Fischer, T. H.; Fonseca

Guerra, C.; van Gisbergen, S. J. A.; Groeneveld, J. A.; Grit-

senko, O. V.; Gruning, M.; Harris, F. E.; van den Hoek, P.;

Jacob, C. R.; Jacobsen, H.; Jensen, L.; Kadantsev, E. S.; van

Kessel, G.; Klooster, R.; Kootstra, F.; van Lenthe, E.;

McCormack, D. A.; Michalak, A.; Neugebauer, J.; Nicu, V.

P.; Osinga, V. P.; Patchkovskii, S.; Philipsen, P. H. T.; Post,

D.; Pye, C. C.; Ravenek, W.; Romaniello, P.; Ros, P.; Schip-

per, P. R. T.; Schreckenbach, G.; Snijders, J. G.; Sola, M.;

Swart, M.; Swerhone, D.; te Velde, G.; Vernooijs, P.; Ver-

sluis, L.; Visscher, L.; Visser, O.; Wang, F.; Wesolowski, T.

A.; van Wezenbeek, E. M.; Wiesenekker, G.; Wolff, S. K.;

Woo, T. K.; Yakovlev, A. L.; Ziegler, T. ADF2007.01; SCM:

Amsterdam, The Netherlands, 2007.

56. Stanton, J. F.; Gauss, J.; Harding, M. E.; Szalay, P. G.CFOUR 1.2, Austin, TX; Mainz, Germany, 2010.

57. Harding, M. E.; Metzroth, T.; Gauss, J. J Chem TheoryComp 2008, 4, 64.

58. Thole, B. T.; van Duijnen, P. Th. Theor Chim Acta 1983, 63,209.

59. Fonseca Guerra, C.; Handgraaf, J.-W.; Baerends, E. J.; Bick-elhaupt, F. M. J Comput Chem 2004, 25, 189.

60. O’Keeffe, M.; Brese, N. E. J Am Chem Soc 1991, 113, 3226.

61. Swart, M.; Sola, M.; Bickelhaupt, F. M. J Chem Phys 2009,131, 094103.

62. Swart, M. J Comput Chem 2010 (in preparation).

63. Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys Rev Lett 1996,77, 3865; Erratum Phys Rev Lett 1997, 78, 1396.

64. Grimme, S. J Comput Chem 2004, 25, 1463.

65. Grimme, S. J Comput Chem 2006, 27, 1787.

66. Masia, M.; Probst, M.; Rey, R. Chem Phys Lett 2006, 420,267.

67. Masia, M.; Probst, M.; Rey, R. J Chem Phys 2005, 123.

68. Masia, M.; Probst, M.; Rey, R. J Chem Phys 2004, 121, 7362.

69. Swart, M.; van Duijnen, P. Th.; Snijders, J. G. J Mol Struct(THEOCHEM) 1999, 458, 11.

70. Thole, B. T. All but tedious: Quantum Chemistry andBiomolecules, PhD thesis; Rijksuniversiteit Groningen:Groningen, The Netherlands, 1982.

71. van Duijnen, P. Th.; de Vries, A. H. Int J Quant Chem,QCS 1995, 29, 523.

72. van Duijnen, P. Th.; Rullmann, J. A. C. Int J Quant Chem1990, 38, 181.

73. Grozema, F. C.; van Duijnen, P. Th. J Phys Chem A 1998,102, 7984.

74. Jensen, L.; van Duijnen, P. Th.; Snijders, J. G. J Chem Phys2003, 119, 3800.

75. van Duijnen, P. Th.; Netzel, T. L. J Phys Chem A 2006, 110,2204.

76. Jensen, L.; Swart, M.; van Duijnen, P. Th.; Autschbach, J.Int J Quant Chem 2006, 106, 2479.

77. de Vries, A. H.; van Duijnen, P. Th. Bioph Chem 1992, 43,139.

78. van Duijnen, P. Th.; Swart, M.; Grozema, F. C. In HybridQuantum Mechanical and Molecular Mechanics Methods;Gao, J.; Thompson, M. A., Eds.; ACS Symposium Series712: Washington, DC, 1999; p 220.

79. Remko, M.; van Duijnen, P. Th.; Swart, M. Struct Chem2003, 14, 271.

80. Jensen, L.; van Duijnen, P. Th. Int J Quant Chem 2005, 102,612.

81. Jensen, L.; van Duijnen, P. Th. J Chem Phys 2005, 123,074307.

82. van Duijnen, P. Th.; de Vries, A. H.; Swart, M.; Grozema,F. C. J Chem Phys 2002, 117, 8442.

83. Jensen, L.; van Duijnen, P. Th.; Snijders, J. G.; Chong, D. P.Chem Phys Lett 2002, 359, 524.

84. Jensen, L.; Swart, M.; van Duijnen, P. Th.; Snijders, J. G. JChem Phys 2002, 117, 3316.

85. Jensen, L.; van Duijnen, P. Th.; Snijders, J. G. J Chem Phys2003, 118, 514.

86. Silberstein, L. Philos Mag 1917, 33, 521.



atomic radii in molecules for use in a polarizable force field

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