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Article
Accurate Molecular Polarizabilities Based on Continuum Electrostatics
JeanFranc#ois Truchon, Anthony Nicholls, Radu I. Iftimie, Benoît Roux, and Christopher I. Bayly
J. Chem. Theory Comput.
,
2008
, 4 (9), 14801493• DOI: 10.1021/ct800123c • Publication Date (Web): 13 August 2008
Downloaded from http://pubs.acs.org on May 13, 2009
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Accurate Molecular Polarizabilities Based on ContinuumElectrostatics
JeanFranc¸ois Truchon,
†,‡
Anthony Nicholls,
§
Radu I. Iftimie,
†
Benoît Roux,

andChristopher I. Bayly*
,‡
De´partement de chimie, Uni
V
ersite´ de Montre´al, C.P. 6128 Succursale centre
V
ille, Montre´al, Que´bec, Canada H3C 3J7, Merck Frosst Canada Ltd., 16711 TransCanada Highway, Kirkland, Que´bec, Canada H9H 3L1, OpenEye Scientiﬁc Software, Inc.,Santa Fe, New Mexico 87508, and Institute of Molecular Pediatric Sciences, GordonCenter for Integrati
V
e Science, Uni
V
ersity of Chicago, Illinois 929 East 57th Street,Chicago, Illinois 60637
Received April 7, 2008
Abstract:
A novel approach for representing the intramolecular polarizability as a continuumdielectric is introduced to account for molecular electronic polarization. It is shown, using a ﬁnitedifference solution to the Poisson equation, that the electronic polarization from internal continuum(EPIC) model yields accurate gasphase molecular polarizability tensors for a test set of 98challenging molecules composed of heteroaromatics, alkanes, and diatomics. The electronicpolarizationoriginatesfromahighintramoleculardielectricthatproducespolarizabilitiesconsistentwith B3LYP/augccpVTZ and experimental values when surrounded by vacuum dielectric. Incontrast to other approaches to model electronic polarization, this simple model avoids thepolarizability catastrophe and accurately calculates molecular anisotropy with the use of veryfew ﬁtted parameters and without resorting to auxiliary sites or anisotropic atomic centers. Onaverage, the unsigned error in the average polarizability and anisotropy compared to B3LYPare 2% and 5%, respectively. The correlation between the polarizability components from B3LYPand this approach lead to a
R
2
of 0.990 and a slope of 0.999. Even the F
2
anisotropy, shownto be a difﬁcult case for existing polarizability models, can be reproduced within 2% error. Inaddition to providing new parameters for a rapid method directly applicable to the calculation ofpolarizabilities, this work extends the widely used Poisson equation to areas where accuratemolecular polarizabilities matter.
1. Introduction
The linear response of the electronic charge distribution of a molecule to an external electric ﬁeld, the polarizability, isat the srcin of many chemical phenomena such as electronscattering,
1
circular dichroism,
2
optics,
3
Raman scattering,
4
softness and hardness,
5
electronegativity,
6
and so forth. Inatomistic simulations, polarizability is believed to play animportant and unique role in intermolecular interactions of heterogeneous media such as ions passing through ionchannels in cell membranes,
7
in the study of interfaces,
8
andin protein

ligand binding.
9
Polarizability is considered to be a difﬁcult and importantproblem from a theoretical point of view. Much effort hasbeen invested in the calculation of molecular polarizabilityat different levels of approximation. At the most fundamentallevel, electronic polarization is described by quantummechanics (QM) electronic structure theory such as extendedbasis set density functional theory (DFT) and ab initiomolecular orbital theory. However, the extent of the com
* Corresponding author phone: (514) 4283403; fax: (514) 4284930; email: christopher_bayly@merck.com.
†
Universite´ de Montre´al.
‡
Merck Frosst Canada Ltd.
§
OpenEye Scientiﬁc Software, Inc.

University of Chicago.
J. Chem. Theory Comput.
2008,
4,
1480–1493
1480
10.1021/ct800123c CCC: $40.75
©
2008 American Chemical Society
Published on Web 08/13/2008
putational resources required is an impediment to the wideapplication of these methods on large molecular sets or onlarge molecular systems such as druglike molecules.
10
Tocircumvent these limitations, empirical physical models basedon classical mechanics have been parametrized to ﬁtexperimental or quantum mechanical polarizabilities.In this article, we explore a new empirical physical modelto account for electronic polarizability in molecules. Theelectronic polarization from internal continuum (EPIC) modeluses a dielectric constant and atomic radii to deﬁne theelectronic volume of a molecule. The molecular polarizabilitytensor is calculated by solving the Poisson equation (PE)with a ﬁnite difference algorithm. The concept that adielectric continuum can account for solute polarizability hasbeen examined previously. For example, Sharp et al.
11
showed that condensed phase induced molecular dipolemoments are accounted for with the continuum solventapproach and that it leads to accurate electrostatic free energyof solvation. More recently Tan and Luo
12
have attemptedto ﬁnd an optimal inner dielectric value that reproducescondensed phase dipole moments in different continuumsolvents. In spite of these efforts, we found that none of thesemodels can account correctly for molecular polarizability.Here, the concept is explored with the objective of producinga high accuracy polarizable electrostatic model. Therefore,we focus on the optimization of atomic radii and innerdielectrics to reproduce the B3LYP/augccpVTZ polarizability tensor.In this preliminary work, we seek to establish the soundness and accuracy of the EPIC model in the calculation of the molecular polarizability tensor on three classes of molecules: homonuclear diatomics, heteroaromatics, andalkanes. These molecular classes required special attentionwith previous polarizable models as a result of their highanisotropy.
13

15
Overall, 53 different molecules are used toﬁt our model and 45 molecules to validate the results. Thesespeciﬁc questions are addressed: Can the EPIC modelaccurately calculate the average polarizability? If so, can itfurther account for the anisotropy and the orientation of thepolarizability components? How few parameters are neededto account for highly anisotropic molecules, and how doesthis compare to other polarizable models? How transferableare the parameters obtained with this model? Is the modelable to account for conformational dependency? In answeringthese questions, we obtained a fast and validated methodwith optimized parameters to accurately calculate the molecular polarizability tensor for a large variety of heteroaromatics not previously considered.The remainder of the article is organized as follows. Inthe next section, we brieﬂy review the most successfulexisting polarizable approaches, focusing on aspects relevantto this study. Then we introduce the dielectric polarizablemethod with a polarizable sphere analytical model. Amethodology section in which we outline the computationaldetails follows. The molecular polarizability results are thenreported. This is followed by a discussion and conclusion.
2. Existing Empirical Polarizable Models
2.1. Point Inducible Dipole.
The point inducible dipole(PID) model was ﬁrst outlined by Silberstein in 1902.
16
Thismodel has been extensively used to calculate molecularpolarizability
14,15,17

22
and to account for manybody effectsin condensed phase simulations.
23

25
Typically, in the PIDmodel, an atom is a polarizable site where the electric ﬁelddirection and strength together with the atomic polarizabilitydeﬁne the induced atomic dipole moment. Since the electricﬁeld at an atomic position is in part due to other atoms’induced dipoles, the set of equations must be solvediteratively (or through a matrix inversion). In 1972, Applequist
19
showed that the PID can accurately reproduce averagemolecular polarizability of a diverse set of molecules butalso that the mathematical formulation of the PID can leadto a polarizability catastrophe. Brieﬂy, when two polarizableatoms are close to each other, the solution to the mathematical equations involved is either undetermined (with the matrixinversion technique) or the neighboring dipole momentscooperatively increase to inﬁnity. To circumvent this problem, Thole
14,22
modiﬁed the dipole ﬁeld tensor with adamping function, which depends on a lengthscale parametermeant to represent the spatial extent of the polarizedelectronic clouds; his proposed exponential modiﬁcation isstill important and remains in use.
13,14,26
2.2. Drude Oscillators.
The Drude oscillator (DO) represents electronic polarization by introducing a masslesscharged particle attached to each polarizable atom by aharmonic spring.
27
When the Drude charge is large andtightly bound to its atom, the induced dipole essentiallybehaves like a PID. The DO model is attractive because itpreserves the simple charge

charge radial Coulomb electrostatic term already present and it can be used in moleculardynamics simulation packages without extensive modiﬁcations. The DO model has not yet been extensively parametrized to reproduce molecular polarizability tensors, butrecent results suggest that it could perform as well as PIDmethods. Finally, the DO model also requires a dampingfunction to avoid the polarizability catastrophe.
26
2.3. Fluctuating Charges.
A third class of empiricalmodel, called ﬂuctuating charge (FQ), was ﬁrst publishedin a study by Gasteiger and Marsili
28
in 1978 to rapidlyestimate atomic charges. Subsequently, FQ was adapted toreproduce molecular polarizability and applied in moleculardynamic simulations.
29,30
It is based on the concept thatpartial atomic charges can ﬂow through chemical bonds fromone atomic center to another based on the local electrostaticenvironment surrounding each atom. The equilibrium pointis reached when the deﬁned atomic electronegativities areequal. The FQ model, like the DO, has mainly been used incondensed phase simulations and not speciﬁcally parametrized to reproduce molecular polarizabilities. A majorproblem with FQ is the calculation of directional polarizabilities (eigenvalues of the polarizability tensor). For planaror linear chemical moieties (ketones, aromatics, alkanechains, etc.) the induced dipole can only have a componentin the plane of the ring or in line with the chain. For instance,the outofplane polarizability of benzene can only be
Polarizabilities from Continuum Electrostatics
J. Chem. Theory Comput., Vol. 4, No. 9, 2008
1481
correctly calculated if outofplane auxiliary sites are built.For alkane chains, though, there is no simple solution.
31
Forthis reason, the ability of the FQ model to accuratelyrepresent complex molecular polarizabilities is clearly limited.
2.4. Limitations with the PID Related Methods.
ThePID and the related models have been parametrized and showan average error on the average polarizability around 5%.However, errors in the anisotropy are often around 20% orhigher.
15,20
Diatomic molecules are not handled correctly byany of these methods, leading to errors of 82% in theanisotropy for F
2
, for example.
13,14
Heteroaromatics, whichare abundant moieties in drugs, are often poorly describedby PID methods. This limitation is due to the source of anisotropy in the PID model, that is, the interatomic dipoleinteraction located at static atom positions. It is neverthelesspossible to improve these models. For example, using fullatomic polarizability tensors instead of isotropic polarizabilities has reduced the errors in polarizability componentsfrom 20% to 7%.
20,21
In the case of the DO model, acetamidepolarizabilities have been corrected by the addition of atomtypedependent damping parameters and anisotropic harmonic springs.
32
In these cases, the improvement required asigniﬁcant amount of additional parameters which brings anadditional level of difﬁculty in their generalization. Asillustrated below, our model seems to address most of thesecomplications without additional parameters and complexity.
3. Dielectric Polarizability Model
The mathematical model that we explore in this article isbased on simple concepts that have proved extremely usefulin chemistry.
33

38
We propose a speciﬁc usage that weclarify and describe in this section.
3.1. Model.
Traditionally in Poisson

Boltzmann (PB)continuum solvent calculations, the solute is described as aregion of low dielectric containing a set of distributed pointcharges; the polar continuum solvent (usually water) isdescribed by a region of high dielectric. This theoreticalapproach gives the choice to either include average solutionsalt effects (PB) or to use the pure solvent (PE). Solving PEfor such a system is equivalent to calculating a charge densityaround the solute surface at the boundary where the dielectricchanges.
39
This, among other things, allows the calculationof the free energy of charging of a cavity in a continuumsolvent where, at least in the case of water, polarizationcomes mostly from solvent nuclear motion averaging. Whilethe dielectric boundary is de facto representing the molecularpolarization, the dielectric constants and radii employedtraditionally are parametrized by ﬁtting to energies (such assolvation or binding free energies) without regard for themolecular polarizabilities themselves. These energies are alsodependent on details of the molecular electronic chargedistribution, the solvent/solute boundary, and sometimes thenonpolar energy terms, all of which obfuscate the parametrization with respect to the key property of molecularpolarizability.Our approach is to use an intramolecular effectivedielectric constant, together with associated atomic radii, toaccurately represent the detailed molecular polarizability. Forthis to be a widely applicable model of polarizability, thegenerality between related chemical species of a given setof intramolecular effective dielectric constants and associatedatomic radii would have to be demonstrated. Such apolarizability model, independent per se of the molecule’scharge distribution, could then subsequently be combinedwith a suitable static charge model to produce a polarizableelectrostatic term applicable to force ﬁelds.To evaluate the model, the simplest starting point is gasphase polarizabilities, using a higher dielectric value insidethe molecule and vacuum dielectric outside.
40
This way, thecharge density formed at the exterior/interior boundary comesfrom the polarization of the molecule alone. Comparison of the polarizability tensors from such calculations directly tothose from B3LYP/augccpVTZ calculations allows proofofconcept of the model. The resulting parameters can beused to rapidly calculate molecular polarizabilities on largemolecules.To calculate the molecular polarizability, we ﬁrst solveEPIC for a system in which the interior/exterior boundaryis described by a van der Waals (vdW) surface, an innerdielectric, and a uniform electric ﬁeld. The electric ﬁeld issimply produced from the boundary conditions when solvingon a grid (electric clamp). From the obtained solution, it ispossible to calculate the charge density from Gauss’ law (i.e.,from the numerical divergence of the electric ﬁeld), and theinduced dipole moment is simply the sum of the grid chargetimes its position as shown by eq 1 below.
µ
F
ind
)
∑
i
)
1grid
r
F
i
·
q
i
(1)Knowing the applied electric ﬁeld, it is then possible, asshown in eq 2, to compute the polarizability tensor giventhat three calculations are done with the electric ﬁeld appliedin orthogonal directions; in eq 2,
i
and
j
can be
x
,
y
, or
z
.
R
ij
)
µ
i
ind
/
E
j
(2)
3.2. Spherical Dielectric.
For the sake of clarifyingthe internal structure of the model, let us ﬁrst consider theinduced polarization of a single atom in vacuum under theinﬂuence of a uniform external electric ﬁeld: the EPIC modelfor an atom. Given a sphere of radius
R
, a unitless innerdielectric
ε
in
and the uniform electric ﬁeld
E
, we can exactlycalculate the induced dipole moment with eq 3.
µ
F
ind
)
4
πε
0
(
ε
in

1
ε
in
+
2
)
R
3
·
E
F
(3)Here, the atomic polarizability is given by the electric ﬁeld
E
prefactor, which is a scalar given the symmetry of theproblem. The induced dipole moment originates fromthe accumulation of a charge density at the boundary of thesphere opposing the uniform electric ﬁeld.
39
From eq 3, wesee that the polarizability has a cubic dependency onthe sphere radius and that the inner dielectric can reduce thepolarizability to zero (
ε
in
)
1), while the upper limit of itscontribution is a factor of 1 (
ε
in
.
1). The contribution of
ε
in
to the atomic polarizability asymptotically reaches aplateau as shown in Figure 1. Thus, at high values of
ε
in
,the atomic radius becomes the dominant dependency in the
1482
J. Chem. Theory Comput., Vol. 4, No. 9, 2008
Truchon et al.
electric ﬁeld prefactor; we ﬁnd similar characteristics fornonspherical shapes.It is interesting to make a parallel between eq 3 and thePID model, where the polarizable point would be locatedexactly at the nucleus. In this particular case, it is possibleto equate the polarizability from PE, induced by the radiusand the dielectric, to any point polarizability.
11
However,when the electric ﬁeld is not uniform, the PID induced atomicdipole srcinating from the evaluation of the electric ﬁeld ata single point may not be representative, leading to inaccuracies.
41
This is in contrast with the EPIC model that buildsthe response based on the electric ﬁeld lines passing locallythrough each part of the atom’s surface, allowing a responsemore complex than that of a point dipole. In molecules, theatomic polarizabilities of the PID model do not ﬁnd theircounterparts in the EPIC model since it is difﬁcult to assignnonoverlapping dielectric spheres to atoms and obtain thecorrect molecular behavior. The Cl
2
molecule studied in thiswork is an example.
4. Methods
4.1. Calculations.
PriortotheDFTcalculation,SMILES
42

44
strings of the desired structures were transformed intohydrogencapped threedimensional structures with the program OMEGA.
45
The
n
octane conformer set was alsoobtained from OMEGA. The resulting geometries wereoptimized with the Gaussian ’03
46
program using B3LYP
47

49
with a 631
++
G(d,p) basis set
50,51
without symmetry. Theatomic radii and molecular inner dielectrics were ﬁt basedon molecular polarizability tensors calculated at the B3LYPlevel of theory
52
with the Gaussian ’03 program. Theextended Dunning’s augccpVTZ basis set,
53,54
known tolead to accurate gas phase polarizabilities, was used.
55
Anextended basis set is required to obtain accurate gas phasepolarizabilities that would otherwise be underestimated.The solutions to the PE were obtained with the ﬁnitedifference PB solver Zap
56
from OpenEye Inc. modiﬁed toallow voltage clamping of box boundaries to create a uniformelectric ﬁeld. The electric ﬁeld is applied perpendicularly totwo facing box sides (along the
z
axis). The differencebetween the ﬁxed potential values on the boundaries is setto meet:
∆
φ
)
E
z
×
∆
Z
, where
∆
φ
is the difference inpotential,
E
z
is the magnitude of the uniform electric ﬁeld,and
∆
Z
is the grid length in the
z
direction. The saltconcentration was set to zero, and the dielectric boundarywas deﬁned by the vdW surfaces. The grid spacing was setto 0.3 Å, and the extent of the grid was set such that at least5 Å separated the box wall from any point on the vdWsurface. As detailed in the Supporting Information, gridspacing below 0.6 Å did not show signiﬁcant deteriorationof the results. Small charges of
(
0.001e were randomlyassigned to the atoms to ensure Zap would run, typicallyconverging to 0.000001 kT.In tables where optimized parameters are reported, asensitivity value associated with each ﬁtted parameter is alsoreported. The sensitivity of a parameter corresponds to itssmallest variation, producing an additional 1% error in theﬁtness function considering only molecules using thisparameter. The sensitivity is calculated with a threepointparabolic ﬁt around the optimal parameter value, and thechange required obtaining the 1% extra error is extrapolated.Therefore, the reported sensitivity indicates the level of precision for a given parameter and whether or not someparameters could be eventually merged.
4.2. Fitting Procedure.
Equation 4 shows the ﬁtnessfunction
F
utilized in the ﬁtting of the atomic radii and theinner dielectrics.
F
({
R
},{
ε
})
)
13
N
∑
i
)
1
N
∑
j
)
xx
,
yy
,
zz

R
ij
QM
R
ij

R
ij
QM
+
1
N
θ
∑
i
)
1
N
θ
1


V
F
ij
QM
·V
F
ij
1

cos 45
°
(4)In eq 4,
N
corresponds to the number of molecules usedin the ﬁt,
R
ij
to the polarizability component
j
of the molecule
i
, and
ν
ij
to the eigenvector of the polarizability component
j
of molecule
i
.
N
θ
is the number of nondegenerate eigenvectors found in all the molecules. This ﬁtness function isminimal when the three calculated polarizability componentsare identical to the QM values and when the correspondingcomponent directions are aligned with the QM eigenvectorsof the polarizability tensor.As shown in the Cl
2
example of Figure 2, the hypersurfaceof eq 4 has a number of local minima; it is important thatour ﬁtting procedure allows these to be examined. Becausethe calculations were fast, we decided to proceed in twosteps: First, a systematic search was carried out varying eachﬁtted parameter over a range and testing all combinations.The 30 best sets of parameters were then relaxed using aPowell minimization algorithm, and the set of optimizedparameters leading to the smallest error was kept.
4.3. Deﬁnitions.
The polarizability tensor is a symmetric3
×
3 matrix derived from six unique values. It can be usedto calculate the induced dipole moment
µ
i
(
i
takes the value
x
,
y
, and
z
) given a ﬁeld vector
E
:
µ
i
ind
)R
ix
E
x
+R
iy
E
y
+R
iz
E
z
(5)In this work, we use the eigenvalues and eigenvectors of the polarizability tensor. The eigenvalues are rotationallyinvariant, and their corresponding eigenvectors indicatethe direction of the principal polarizability components.The three molecular eigenvalues are named
R
xx
,
R
yy
, and
R
zz
, and by convention
R
xx
e
R
yy
e
R
zz
. The averagepolarizability (or isotropic polarizability) is calculated witheq 6 below. We also deﬁne the polarizability anisotropy
Figure 1.
Dielectric contribution to the sphere dielectriccontinuum polarizability goes asymptotically to one and mostof the contributions are below
ε
in
)
10.
Polarizabilities from Continuum Electrostatics
J. Chem. Theory Comput., Vol. 4, No. 9, 2008
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