An atomic capacitance–polarizability model for the calculation of molecular dipole moments and polarizabilities

An Atomic Capacitance–PolarizabilityModel for the Calculation of MolecularDipole Moments and Polarizabilities

LASSE JENSEN,1,∗ PER-OLOF ÅSTRAND,2 KURT V. MIKKELSEN1

1Chemical Laboratory III, Department of Chemistry, H. C. Ørsted Institute, University of Copenhagen,DK-2100 Copenhagen Ø, Denmark2Materials Research Department, Risø National Laboratory, DK-4000 Roskilde, Denmark

Received 3 July 2000; revised 11 December 2000; accepted 12 March 2001

ABSTRACT: A classical interaction model for the calculation of molecularpolarizabilities has been investigated. The model is described by atomic capacitancies,polarizabilities, and a parameter related to the size of the atom, where one set ofparameters has been employed for each element. The model has been parameterized forthe elements H, C, N, O, F, and Cl from quantum chemical calculations of the molecularpolarizability and dipole moment for 161 molecules at the Hartree–Fock level. The atomiccharge has been divided into a nuclear charge and an electronic contribution, which alsoallows for modeling the permanent molecular dipole moment. Results are presented forpolyenes. Excellent agreement with quantum chemical calculations is obtained for thecomponents of the polarizability perpendicular to the chain, but the results are lesssatisfying for the component along the chain. An inherent deficiency of using atomiccapacitancies for large molecules and long chains is discussed. © 2001 John Wiley &Sons, Inc. Int J Quantum Chem 84: 513–522, 2001

Key words: polarizabilities; dipole moments; atomic capacitances; interaction models;polyenes

Correspondence to: P.-O. Åstrand; e-mail: [email protected].

∗Present address: Theoretical Chemistry, Materials ScienceCentre, Rijksuniversiteit Groningen, Nijenborgh 4, 9747 AGGroningen, The Netherlands.

Contract grant sponsor: Statens Naturvidenskabelige Forskn-ingsråd (SNF) (K.V.M.).

International Journal of Quantum Chemistry, Vol. 84, 513–522 (2001)© 2001 John Wiley & Sons, Inc.

JENSEN, ÅSTRAND, AND MIKKELSEN

Introduction

E vidently it is becoming more and more im-portant to be able to understand and calculate

conducting and optical properties of organic mole-cules and materials. Conjugated organic moleculeswith delocalized electron systems are especially in-teresting because of their potentially large opticalproperties. From a technological point of view, theinterest lies in the potential exploitation of thesemolecules in optoelectronics and molecular-basedphotonics devices [1 – 5]. These materials are de-signed on an atomistic scale by considering the elec-tronic structure of the molecular components. Froma theoretical point of view, the molecular responseto an external electromagnetic field is calculatedmost efficiently by applying quantum chemical re-sponse theory [6]. Accurate quantum chemical cal-culations of molecular properties can, however, onlybe carried out for rather small molecules due tothe large requirements of computer resources. Forlarger molecules and assemblies of molecules, mod-eling is restricted to less sophisticated methods.

For a long time, it has been known that theisotropic part of molecular static polarizabilities to alarge extent can be described by an additive model,which means that the molecular polarizability canbe calculated as a sum of transferable atom-type orbond contributions [7 – 10]. The static polarizabilityof organic molecules has recently been investigatedby an additive model [11, 12], and it also has beendemonstrated that the frequency-dependent molec-ular polarizability tensor of halogen derivatives ofbenzene can be modeled by transferable and addi-tive atomic polarizability tensors [13].

A more elaborate model, but yet very simplecompared to quantum chemical calculations, is theso-called interaction model. In its simplest form,a set of atomic polarizabilities interact according toclassical electrostatics in the limit of no external elec-tric field. This approach was introduced 85 yearsago [14 – 19], and has, to a large extent, been ex-ploited by Applequist and coworkers [20 – 23]. Animportant extension of this approach is the inclu-sion of a damping term of the internal electric fields[24, 25]. The model by Thole [25] has turned out tobe more attractive, and it has been investigated indetail recently [26 – 30]. With a minor modificationof the damping term by Thole, the interaction modelhas been extended to also include the frequency de-pendence of the dipole polarizability tensor in anUnsöld-type of model [29]. This model has been

used recently to study the static and frequency-dependent polarizabilities of carbon nanotubes [30].The frequency dependence of the interaction modelhas also been discussed in terms of the calculationof absorption spectra [31, 32].

The Applequist model has been extended toalso include atomic capacitances, and thereby in-tramolecular charge-transfer effects are also mod-eled [33, 34]. This atomic monopole–dipole inter-action (AMDI) model is supposed to give a betterdescription of π-conjugated molecules in particular,and it has been applied to the investigation of themolecular polarizabilities of fullerenes, heterocyclicmolecules, and polyenes, respectively [35 – 37].

In this work, we have modified the capacitance–polarizability model for the calculation of molecularstatic polarizabilities by Olson and Sundberg [33].Here, the atomic charge is divided into the nuclearcharge, which gives a permanent electric field, aswell as a charge due to the electron distribution,which is modeled by an atom-type capacitance. Aswill be described, this also enables that permanentmolecular electric moments can be modeled, andthat an apparant atomic capacitance can be definedwhich is different from the parameterized atom-type capacitancies. Furthermore, the damping ofthe potential and electric field in a modified Tholemodel is included [29].

As discussed in a previous work, it is advan-tageous to parameterize an interaction model formolecular polarizabilities from quantum chemicalcalculations [29]. The reason is that both an inter-action model and quantum chemical calculationsgive the electronic part of the polarization, whereasexperimental polarizabilities also contain both zero-point and pure vibrational contributions as well astemperature effects. Here, we have extended the setof 115 aliphatic and aromatic molecules employedin a previous work [29] with 46 olefines, and haveparameterized a capacitance–polarizability modelfrom ab initio molecular polarizabilities and dipolemoments.

Theory

A theory for a capacitance–polarizability interac-tion model of molecular polarizabilities and dipolemoments is presented. Formally, it is only a mi-nor modification of the approach by Olson andSundberg [33], and we will therefore use the samenotation and follow their derivation as closely aspossible. We will, however, use the sign conventions

514 VOL. 84, NO. 5

ATOMIC CAPACITANCE–POLARIZABILITY MODEL

of Buckingham [38], for example, for electric fields,which are consistent with the definitions in classi-cal electrostatics. It should be noted that, even if ourmethod formally is very similar to the approach byOlson and Sundberg [33], it will be apparant that itis conceptually quite different.

The potential energy of N atomic capacitances ap

in an external electric field can be written as

Vqq = 12

N∑p

q2p

ap+ 1

2

N∑p

N∑q �= p

qpqqT(0)pq

+N∑p

qpϕextp − λ

N∑p

qp (1)

where qp is an induced atomic charge. The first termis the self-energy required for creating an inducedatomic charge, and the second term is the electro-static interaction between all induced charges. Here,T(0)

pq is the so-called interaction tensor of rank 0,and it is defined as 1/Rpq where Rpq is the distancebetween the particles. The third term is the inter-action between the induced atomic charges and anexternal potential ϕext

p . If ϕextp is assumed to be a po-

tential of a uniform external electric field Eextα , it is

given as

ϕextp = rp,αEext

α (2)

where rp,α is a component of the coordinate vector.Here and in the remaining part of this work, the Ein-stein summation convention is employed for Greekindexes. The fourth term in Eq. (1) is a Lagrangemultiplier, which is included to ensure electricalneutrality. Here, the molecules are assumed to beneutral, but it is trivial to include charged species inthe approach. A molecule is thus assumed to consistof atomic capacitances connected by conducting,but noncapacitive wires. Analogously, the potentialenergy of a system of N polarizabilities αp,αβ is writ-ten as

Vµµ = 12

N∑p

α−1p,αβµp,αµp,β − 1

2

N∑p

N∑q �= p

T(2)pq,αβµp,βµq,α

−N∑p

Eextα µp,α (3)

where µp,α is a component of an induced atomicdipole moment, and T(2)

pq,αβ is the interaction tensorof rank 2, ∇α∇β(1/Rpq). The first term is the cor-responding self-energy, the second term is thedipole–dipole interaction, and the last term is the in-teraction with an external electric field. If the atoms

are considered to be spherically symmetric, theatomic polarizability is isotropic, and may be writ-ten as

αp,αβ = αpδαβ . (4)

If we consider a system of both capacitances andpolarizabilities, a charge–dipole interaction term isalso obtained:

Vqµ =N∑p

N∑q �= p

qpT(1)pq,αµq,α = −

N∑p

N∑q �= p

µp,αT(1)pq,αqq

(5)

where T(1)pq,α = −T(1)

qp,α is the interaction tensor ofrank 1, ∇α(1/Rpq).

A damping term of the intramolecular interac-tions according to Thole [25] is employed. In hisapproach, the interaction tensors are modified as

T(2)pq,αβ = 3v4

pqrpq,αrpq,β

R5pq

− (4v3pq − 3v4

pq)δαβ

R3pq

(6)

where vpq = Rpq/spq if Rpq < spq. If Rpq > spq, vpq = 1,and the normal relation is recovered. Accordingly,

T(1)pq,α = − (4v3

pq − 3v4pq)rpq,α

R3pq

(7)

and

T(0)pq = v4

pq − 2v3pq + 2vpq

Rpq. (8)

It is noted that spq is a distance, and is assumed to bedependent on the size of the atoms. In our previouswork [29], we assumed that spq is proportional to theatomic second moment as

spq = (pq)1/4 (9)

where p is assumed to be an atom-type parameter.In comparison, Thole defined spq as [25]

spq = cd(αpαq)1/6 (10)

which only includes one additional fitting parame-ter compared to the Applequist model [20].

The total potential energy V of the molecule isgiven as

V = Vqq + Vqµ + Vµµ. (11)

Normally, the molecular polarizability is deter-mined by minimizing V with respect to inducedelectric moments qp and µp,α as well as the Lan-grange multiplier according to classical responsetheory [33]. In this work, it is instead noticed that amolecule consists of both nuclei and electrons, andthat it is only the electrons that may give a charge

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 515


flow within a molecule. Thus, it is only the electronicpart that should be described by a capacitance. Con-sequently, the atomic charge is partitioned as

qp = Zp + Np (12)

where Zp is the nuclear charge and Np is the numberof electrons, and is assumed to be a negative charge.Instead of minimizing V with respect to qp as in theregular approach, V is instead minimized with re-spect to Np as

∂V∂Np

= 0 = Np + Zp

ap+

N∑q �= p

NqT(0)pq + ϕ

permp

+N∑

q �= p

T(1)pq,αµq,α − λ (13)

where the permanent electrostatic potential ϕpermp is

given as

ϕpermp = ϕext

p + ϕintp = ϕext

p +N∑

q �= p

ZqT(0)pq (14)

and thus consists of both an external potential andan internal part arising from the nuclear charges.Furthermore,

∂V∂µp,α

= 0 = α−1p,αβµp,β −

N∑q �= p

T(2)pq,αβµq,β − Eperm

p,α

−N∑

q �= p

T(1)pq,αNq (15)

where the permanent electric field Epermp,α analo-

gously consists of the external field and an internalfield arising from the nuclear charges:

Epermp,α = Eext

α + Eintp,α = Eext

α +N∑

q �= p

T(1)pq,αZq. (16)

Finally,

∂V∂λ

= 0 =N∑p

Np +N∑p

Zp (17)

which ensures electrical neutrality. A new parame-ter bp is introduced as

Np

bp= Np + Zp

ap(18)

which results in that Eqs. (13), (15), and (17) canbe written as a matrix equation analogously to thework by Olson and Sundberg [33]. Adopting the fol-

lowing notation for the matrix elements,

App,αβ = α−1p,αβ ; Apq,αβ = −T(2)

pq,αβ (p �= q) (19)

Mpp,α = 0; Mpq,α = T(1)pq,α (p �= q) (20)

and

Cpp = b−1p ; Cpq = T(0)

pq (p �= q), (21)

Eqs. (13), (15) and (17) can be rewritten as

Epermp,α =

∑q

Apq,αβµq,β −∑

q

Mpq,αNq (22)

ϕpermp = −

∑q

Mpq,αµq,α −∑

q

CpqNq + λ (23)

−∑

q

Zq =∑

q

Nq. (24)

These equations are rewritten in a matrix form asEperm

ϕperm

−Z

=

A −M 0

−MT −C 10 1 0

µ

Nλ

(25)

where Z = ∑q Zq. Thus,

µ

Nλ

=

A −M 0

−MT −C 10 1 0

−1 Eperm

ϕperm

−Z

=

B g h1

gT D h2

hT1 hT

2 h3

Eperm

ϕperm

−Z

. (26)

In comparison to the work by Olson and Sund-berg [33], the main difference is that the right-handside contains the permanent potentials and elec-tric fields, which also include contributions fromnuclear charges. The molecular dipole moment isgiven as

µmolα =

∑p

µp,α + rp,α(Np + Zp)

=∑p,q

[Bpq,αβEint

q,β + rp,αgqp,βEintq,β + rp,αDpqϕ

intq

]+

∑p

rp,α(Zp − h2,pZ) − h1,p,αZ

+∑p,q

[Bpq,αβ + rp,αDpqrq,β]Eextp,β (27)

where we have used Eqs. (2), (14), and (16) as wellas the conditions∑

p

gpq,α =∑

q

gpq,α = 0 (28)

516 VOL. 84, NO. 5


and ∑q

h2,q = 1. (29)

The permanent molecular dipole moment is identi-fied as

µpermα =

∑p,q

[Bpq,αβEint

q,β + rp,αgqp,βEintq,β + rp,αDpqϕ

intq

]+

∑p

rp,α(Zp − h2,pZ) − h1,p,αZ (30)

and the molecular polarizability as

αmolαβ =

∑p,q

[Bpq,αβ + rp,αDpqrq,β] (31)

where it has been assumed that the external field isuniform. The atomic charges qp are given as

qp = Zp + Np = Zp − h2,pZ

+∑

q

(gqp,βEperm

q,β + Dpqϕpermq

)= Zp − h2,pZ +

∑q

(gqp,βEint

q,β + Dpqϕintq

)+

∑q

(gqp,β + Dpqrq,β)Eextβ (32)

and in the limit of no external field,

qp = Zp − h2,pZ +∑

q

(gqp,βEint

q,β + Dpqϕintq

). (33)

We have thus derived an electrostatic interactionmodel for the permanent molecular dipole and themolecular polarizability. It is also possible to obtainatomic charges, dipole moments, and polarizabili-ties from this approach, which may be used as a rep-resentation of the electrostatics in, for example, thecalculation of intermolecular interactions [39, 40].These properties are described by three types ofatom-type parameters: an atomic capacitance bp andatomic polarizability αp, and a size parameter p,which may be parameterized from quantum chemi-cal calculations of the molecular properties. Finally,it is noted that an apparant atomic capacitance canbe obtained by rearranging Eq. (18) as

ap = bp

(1 + Zp

Np

). (34)

Thus, different atomic capacitances are obtained foreach atom, and they can be used to analyze, for ex-ample, the conductivity and reactivity of molecules.

Calculational Details

Quantum chemical calculations of the molecularpolarizability tensor and dipole moment of 46 ole-fines∗ have been carried out, and the molecules havebeen added to the set of 115 molecules used inthe previous investigation of aliphatic and aromaticmolecules [29]. The Dalton program package hasbeen used for all of the quantum chemical calcula-tions [41], and response theory has been employedto obtain the molecular polarizability [6]. The cal-culations have been performed at the self-consistentfield (SCF) level with the basis set by Sadlej [42].Furthermore, standard bond lengths and bond an-gles have been employed [43, 44].

The model parameters have been optimized byminimizing the root-mean square as

rms =√∑N

i = 1∑3

α,β = 1(αmodelαβ,i − α

QCαβ,i)2

N − 1(35)

where αQCαβ,i is a component of the quantum chem-

ically derived polarizability tensor, αmodelαβ,i is the

corresponding polarizability derived from an inter-action model, and N is the number of molecules.A similar rms is defined for the molecular dipolemoment. When both properties are optimized at thesame time, we have simply added the two rms forthe polarizability and the dipole moment.

Results and Discussion

The atomic parameters have been optimized forseveral different models. Existing models have beenincluded mainly for comparison with our newmodel. The Applequist model [20] consists of onlyatomic polarizabilities without any damping of theinternal electric fields. The results of the optimiza-tion are given in Table I. As expected, and alsodiscussed previously [29, 34], the Applequist model

∗The additional molecules are: ethene, propene, trans-2-butene, 1,3-butadiene, 1,3,5-hexatriene, ethyne, difluoroethyne,trans-1,2-difluoroethene, dichloroethyne, trans-1,2-dichloroet-hene, fluoroethene, chloroethene, fluoroethyne, nitroethene,nitroethyne, trans-1-fluoropropene, 3-fluoropropene, trans-1-chloropropene, 3-chloropropene, trans-1-nitropropene, 3-nitro-propene, trans-1-hydroxypropene, 3-hydroxypropene, cyano-ethene, trans-3-cyano-2-propene, 3-cyano-1-propene, propyne,3-hydroxypropyne, 3-nitropropyne, 3-chloropropyne, 3-fluoro-propyne, 3-cyanopropyne, butadiyne, fluorobutadiyne, amino-ethene, aminoethyne, trans-1-aminopropene, 3-aminopropene,3-butenal, 3-butenoic acid, trans-2-butenoic acid, trans-2-butenal,3-butenon, propenal, propenoic acid, and 1-hydroxyethene.



TABLE IApplequist polarizabilities (in au, 1 au = 0.1482 Å3).

Atom α αa αb

H 3.34 1.61 0.91–1.13C 13.86 4.20 4.16, 5.92N 9.38 8.44 3.58O 6.15 8.78 2.93–3.14F 1.41 2.44 2.16Cl 16.42 12.65 12.89

rms 149.40 139.44

a See Ref. [29].b See Ref. [20].

cannot describe different types of molecules suchas both aliphatic and aromatic molecules by onlyone parameter for each element. Normally, severaltypes of atoms are used for each element [20, 23].Adding olefines to the set of molecules changes theatomic polarizabilities quite drastically. In our pre-vious work, where we only included aliphatic andaromatic molecules [29], it was fair to state that thevalues of the atomic polarizabilties were close to theApplequist polarizabilities [20, 23], but that is notthe case when olefines are included. To add olefinesthus only further emphasizes the need for using sev-eral parameters for each element in the Applequistmodel.

We have also included the Thole model [25],which includes a damping of the internal electricalfields according to Eq. (10) in addition to the atomicpolarizabilities in the Applequist model. The opti-mized parameters in Table II are compared to results

TABLE IIThole polarizabilities (in au, 1 au = 0.1482 Å3).

Atom α αa αb αc

H 1.24 1.83 3.47 3.50C 14.06 12.19 9.46 10.18N 9.51 7.88 7.46 7.60O 4.82 5.78 5.82 6.39F 1.53 2.54 2.94Cl 15.82 16.21 16.11

cd 2.05 1.991 1.662 1.7278rms 10.02 8.26

a See Ref. [29].b See Ref. [25].c See Ref. [27]. Fitted to experimental polarizabilities.

from our previous work where olefines where notincluded [29], to the original work by Thole [25],as well as to the recent work by van Duijnen andSwart [27]. The general observation is that the Tholemodel gives an order of magnitude lower rms thanthe Applequist model, even though only one extrafitting parameter cd has been added. The rms be-comes about 20% larger when the set of moleculesis extended with olefines and the parameters arealso changed. In particular, the hydrogen polariz-ability has become smaller. It is 1.24 au comparedto 1.83 au in our previous work [29], and about3.5 au elsewhere [25, 27]. Furthermore, the carbonand nitrogen polarizabilities have increased whenthe olefines are added, whereas the magnitude ofthe oxygene, fluorine, and chlorine polarizabilitieshas decreased. In our previous work, we carriedout a preliminary calculation by adding 13 olefinesto the set of molecules, but then we did not findany reasonable results [29]. There are two reasons.A dominant part of the error originated from theall-trans polyenes C8H10, C10H12, and C12H14, andthese molecules are not included here. Secondly,since only a few olefines were included, they hada small weight in the optimization. Here, where thenumber of aliphates, aromates, and olefines is ap-proximately the same, the additional error to therms by including the olefines is distributed over allof the molecules.

In our previous work, we modified the Tholemodel according to Eq. (9) (here, it is termed themodified Thole model) [29]. There are thus two pa-rameters for each element. As also noted in ourprevious work, the improvement of the optimiza-tion is significant compared to the Thole model, butnot especially large considering the number of pa-rameters that has been added (see Table III). If the

TABLE IIIModified Thole model (units in au).

Previous Work [29] This Work

Atom α α

H 1.84 2.75 1.86 3.18C 11.52 20.99 12.05 20.99N 10.55 26.55 10.94 22.70O 5.64 12.16 5.46 9.75F 2.25 4.78 2.05 4.98Cl 16.08 17.64 15.20 14.64

rms 6.67 8.85

518 VOL. 84, NO. 5


values of the parameters are compared, it is notedthat the polarizabilities are quite similar, whereassome of the size parameters change quite substan-tially when olefines are added. In particular, Oand Cl are affected, and O has been altered by asmuch as 20%. Accidentally, the value of C is iden-tical to the value found in our previous work.

We have carried out two optimizations of amodel, also including atomic capacitances, wherewe, in one case, only optimize the molecular polar-izabilities, and in the other case, both the molecularpolarizabilities and dipole moments. It should benoted that, in the case where only the molecularpolarizabilities are optimized, the approach pre-sented here is equivalent to the Olson–Sundbergmodel [33] apart from the damping of the inter-nal electric fields. The partitioning of the atomiccharge according to Eq. (12) does not affect the cal-culation of molecular polarizabilities; it only allowsus to define permanent electric moments. There-fore, the approach including atomic capacitances,but where we only optimize the molecular polar-izability, is termed the Olson–Sundberg/modifiedThole (OS/MT) model. The results of the optimiza-tion are given in Table IV. This approach gives animprovement of about 15% of the rms, which shouldbe considered as minor since one additional fit-ting parameter has been added for each element.This is in line with the findings of Shanker andApplequist [35, 37], who also found small contribu-tions to the molecular polarizabilities from addingatomic capacitances. It is noted that the capacitanceof carbon bC is negative, and therefore unphysical.Furthermore, the expected relative magnitudes ofthe parameters are not obtained. For example, bH islarger than both bF and bCl. Furthermore, F is al-most a factor 2 larger than in the modified Tholemodel, and is now larger than both N and O.

Finally, we have optimized both the molecularpolarizabilities and dipole moments in a capac-itance model. This approach is termed µα/bα,which denotes, respectively, the molecular prop-erties being optimized and the atomic parametersemployed. The results of the optimization are givenin Table IV. Since we also have included the molecu-lar dipole moments in the optimization, the rms forthe molecular polarizability is about 10% larger thanfor the OS/MT model. Nonetheless, the accuracy ofthe molecular polarizabilities is comparable to theresults of the other models. The rms of the moleculardipole moments is 0.85 au. Obviously, the error inthe molecular dipole moments of the present modelis too large for an accurate model of the electrosta-tics of the molecule. For example, the accuracy hasto be approximately an order of magnitude betterfor an accurate estimate of the electrostatic and in-duction contributions to intermolecular interactionenergies [39, 40]. Nonetheless, the inclusion of themolecular dipole moments in the optimization actu-ally improves the parameterization of the molecularpolarizabilities because the expected relative mag-nitudes of the parameters are obtained in almost allcases. For the capacitances, bH is an order of mag-nitude smaller than the other capacitancies, and wefind bC < bN < bO < bF. Furthermore, bCl is con-siderably larger than the other atomic capacitancies.For the polarizabilties, similar values are obtainedas for the modified Thole model. The largest dif-ference is obtained for αN, which has been reducedby about 30%. For the size parameters , we findlarger differences than for the atomic polarizabili-ties as compared to the modified Thole model. Inparticular, N has been reduced by about a fac-tor of 3. The expected and the same relative orderof the magnitudes of αp and p are obtained, i.e.,αC > αN > αO > αF. It should, however, have been

TABLE IVCapacitance models (units in au).

OS/MT Model µα/bα Model

Atom α b α b

H 2.55 4.79 0.1186 1.95 4.31 0.0073C 14.21 20.32 −0.0491 12.96 20.99 0.0526N 8.93 8.34 0.1712 7.33 7.79 0.0615O 6.01 7.76 0.1696 4.73 6.59 0.0703F 2.13 9.40 0.0827 2.07 3.95 0.0815Cl 14.87 15.99 0.0746 15.68 13.17 0.1581

rms 7.72 8.49 (in αmol)/0.85 (in µmol)



FIGURE 1. Static polarizability tensor for the all-trans polyenes as a function of double bonds. The parameters weretaken from Table IV. � denotes the QM component along (‖) the chain. ◦, � denotes the QM components perpendicular(⊥) to the chain. For the modified Thole model, (—), (- -) denote ‖ and ⊥ components, respectively. Results from theOS/MT model are denoted by (•) and (�, �) for ‖ and ⊥ components, respectively. The µα/bα model is denoted by(+) and (×, ∗) for ‖ and ⊥ components, respectively.

expected that Cl is larger than C, but that is notthe case.

Calculations for all-trans polyenes (—CH=CH—)n are presented for n ≤ 10, and are comparedto quantum chemical calculations for n ≤ 6 (Fig. 1).It is noted that only some of the polyenes, n ≤ 3,have been included in the optimization of theparameters. For the quantum chemical calculations,it is noted that the component of the molecularpolarizability along the chain α‖ is superlinearwith respect to n. For the modified Thole model,α‖ is almost linear with respect to n, whereas bothcapacitance models present problems for large n.The large errors for the capacitance models willbe discussed in the next paragraph. Furthermore,the limitations for the Applequist and Tholemodels to describe π-conjugated systems havebeen discussed elsewhere [29]. For the componentsof the polarizability perpendicular to the chains,the results for the various interaction models are,however, in excellent agreement with quantumchemical calculations.

For the two capacitance models, OS/MT andµα/bα, it is noted that the discrepancies comparedto the quantum chemical calculations become larger

with increasing n. This is an inherent error in adopt-ing atomic capacitancies, and it can be realized fromthe following model system. If we regard a systemof only two equal capacitancies b situated infinitelyfar away from each other, the relay matrix in Eq. (26)becomes

−1/b 0 1

0 −1/b 11 1 0

−1

= 12

−b b 1

b −b 11 1 1/b

.

(36)

A component of the molecular polarizability isgiven according to Eq. (31) as

αmolαβ =

2∑p,q = 1

rp,αDpqrq,β = r2,α−b2

r2,β (37)

where we have chosen �r1 as the origin. The diago-nal components of the molecular polarizability thusapproach minus infinity for large distances if it isassumed that b > 0. Obviously, this is not phys-ically correct, and would give problems for largemolecules as well as molecular complexes over awide range of intermolecular distances. One solu-tion would be to include hypercapacitances in line

520 VOL. 84, NO. 5


with hyperpolarizabilities according to

q = a1ϕ + 12 a2ϕ

2 + · · · (38)

where higher order terms also include, for exam-ple, electric fields and field gradients. The inclusionof a2 instead of a1 in Eq. (1) gives a nonlinear set ofequations, and this will be investigated in detail in afuture work. It remains, however, to be investigatedif any interaction model can include the contribu-tions from the σ → σ ∗ and π → π∗ excitations to α‖of polyenes, which clearly cannot be described byan additive model (as, for example, seen in Fig. 1).Here, it is only noted that the capacitance may bewritten as

a = 12 a2ϕ + · · · (39)

and thus it becomes zero for the model systemwhere the particles are placed infinitely far awayfrom each other since the potential then is zero.It thus provides a physically correct model of mole-cular polarizabilities. It should, however, be men-tioned that a different choice of parameters canhide this inherent error in the sense that is pushedtoward longer intermolecular distances. To some ex-tent, the problem has been considered by Shankerand Applequist since they have adopted a limiteddelocalization length [37]. A related problem hasbeen discussed by Applequist for the origin depen-dence in optical activity of chiral molecules [45].

The physical significance of atomic capacitanceshas also been discussed by Applequist in his par-tial neglect of ring interactions (PNRI) approxima-tion [34]. In the PNRI approach for aromatic mole-cules, one approximation is that one in-plane andone out-of-plane polarizability are adopted for thecarbon atom, and another approximation is thatsome elements in the T tensors are neglected. Analternative approach would be to include atomicparameters dependent on the electric field, as, e.g.,hyperpolarizabilities, since the out-of-plane compo-nent to the internal electric fields is zero for a planarmolecule due to symmetry reasons. However, suchan approach would also lead to a set of nonlinearequations.

Conclusions

In this work, we have demonstrated that, by par-titioning the atomic charge into a nuclear and anelectronic contribution also, the permanent molec-ular dipole moment can, in principle, be modeledin an interaction model. A model has been parame-terized for the elements H, C, N, O, F, and Cl from

quantum chemical calculations of 161 molecules.In a comparison of a capacitance model with andwithout the optimization of the molecular dipolemoments, it is found that the atomic capacitancesget the expected magnitudes relative to each other.The errors in the molecular dipole moments are,however, too large for the present model to be usedfor calculating electrostatic and induction energiesin intermolecular interactions. Nonetheless, this ap-proach would, in principle, be a suitable approachfor obtaining atom-type parameters to describe theelectrostatic part of intermolecular interactions, butits use in practice has to be investigated in moredetail. Results are presented for polyenes, and theyare in reasonable agreement with quantum chemicalcalculations. Problems of applying atomic capaci-tances for large molecules are discussed.

ACKNOWLEDGMENTS

K.V.M. thanks Statens NaturvidenskabeligeForskningsråd (SNF) for support. P.-O.Å. thanksProf. Jon Applequist for fruitful comments on themanuscript.

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An atomic capacitance–polarizability model for the calculation of molecular dipole moments and polarizabilities