HFF: a force field for liquid crystal molecules

HFF: a force field for liquid crystal molecules

Edgardo Garciaa,* , Matthew A. Glaserb, Noel A. Clarkb, David M. Walbac

aDepartamento de Quimica, Universidade de Brasılia, Brasılia DF 70910-900, BrazilbCondensed Matter Laboratory, Department of Physics, Univeristy of Colorado, Boulder, CO, USA

cDepartment of Chemistry and Biochemistry, University of Colorado, Boulder, CO, USA

Abstract

Soft material simulations require an accurate representation of the intermolecular and intramolecular potential energy surfacein order to achieve realistic predictions for their properties. The conformational potential of the molecules has a profound effecton many properties of molecular aggregates. This is usually a serious weakness of most commercial empirical force fields. Inthis paper we describe a force field (HFF), specifically designed for liquid crystal molecules. Intramolecular parameters areobtained from ab initio quantum mechanics calculations on model molecules which are substructures of liquid crystal mole-cules. HFF reproduces ab initio conformational energy profiles for model molecules with errors much smaller than commercialforce fields. Our focus is not on creating general purpose force fields but on strategies to develop reliable force fields on-demandfor specific needs.q 1999 Elsevier Science B.V. All rights reserved.

Keywords:Molecular mechanics; Force field design; Liquid crystals

1. Introduction

For the last three years we were working on compu-ter simulations of ferroelectric liquid crystals (FLCs).These are promising materials for building fastswitching flat displays, optoelectronic and non-linear-optic devices [1,2]. Our goal is to create toolsfor the computer-aided design of FLC molecules toguide the laboratory synthesis of new materials.Liquid crystals are organic molecules usually contain-ing more than ten rotational single bonds. Propertiesof such materials depend strongly on their shape andelectrostatic potential. The shape is determined by themolecule’s bond lengths, angles, and, in particular, itsconformation, which has a significant effect on theshape of the molecule and therefore on the materialproperties. The electrostatic potential also seems toplay an important role in many FLC properties.

Computer simulations based on classical mechanicsrely on the quality of the force field. We need a forcefield capable of giving a good representation of theground state molecular energy hypersurface and alsoof the molecular charge distribution. Most general-purpose commercial force fields available on themarket are not appropriate for these needs becausethey are not specifically parameterized to reproduceconformational energies. Average errors of at least1.0 kcal/mol are common [3]. In some cases commer-cial force fields even fail for qualitative purposes, ascan be seen by the examples presented in this paper.Moreover, many empirical methods employed incommercial force fields to obtain atomic partialcharges, such as charge equilibration [4], are notcapable of predicting molecular dipoles with goodaccuracy, so they cannot be expected to correctlypredict intermolecular electrostatic interactions. Oursolution to this problem was to create a force field(Hybrid Force Field, or HFF) specifically parameterized

Journal of Molecular Structure (Theochem) 464 (1999) 39–48

0166-1280/99/$ - see front matterq 1999 Elsevier Science B.V. All rights reserved.PII: S0166-1280(98)00533-8

* Corresponding author.

to reproduce ab initio geometries and rotationalenergy profiles of model organic molecules whichare substructures of FLC molecules. The parametersobtained for these substructures are then used in thesimulations. Recently FLC polarization was success-fully explained for 28 molecules by the use of HFF inone-molecule mean-field simulations [5]. Large-scalesimulations were also performed [6] and more appli-cations of HFF, as well as variants of it, will bepresented in future papers.

2. General description of HFF

The phase behavior of liquid crystal materials isgoverned by the intramolecular potential and by asubtle interplay of intermolecular interactions. Theintramolecular potential, in particular the conforma-tional contribution, modulates a variety of entropicand energetic effects. In developing force fields ourefforts are directed toward description of the intramo-lecular potential, giving special attention to theconformational potential. Electrostatic intermolecularinteractions are treated as coulomb potentials employ-ing atomic partial charges derived from quantummechanics on each FLC molecule. Weak intermolec-ular interactions such as dispersion and induction aretreated empirically rather than quantum mechanically

for practical reasons. In our research we have oftendealt with mean field simulations in which the inter-molecular interactions do not need to be considered.However, we are aware of the fact that the mean fieldapproach limits the number of properties that can bepredicted and that many-molecule large-scale simula-tions would require a better treatment of such weakinteractions. Proving intermolecular interactions iscertainly the main obstacle in deriving force fieldsfrom quantum mechanics and was less well-studiedthan the intramolecular potentials [7]. For thepreviously outlined reasons dispersion interactionsare represented, as is done in most current force fields,by a Lennard-Jones potential. Induction effects are nottreated at all. A key part of our future efforts will bethe study of quantum derived potentials for treatmentof dispersion and induction interactions. The namehybrid force field(HFF) comes from the fact that weemploy a hybrid molecular model with both implicitand explicit hydrogen atoms. In this model hydrogensattached to sp3 carbons are treated as implicit atoms,while hydrogens attached to sp2, spl and aromaticcarbons are treated explicitly (Fig. 1). This strategyallows faster simulations, since it implies a reductionin the number of interaction sites in an FLC moleculeof around a factor of two, and is still able to reproducethe electrostatics of aromatic rings and unsaturatedregions.

E. Garcia et al / Journal of Molecular Structure (Theochem) 464 (1999) 39–4840

Fig. 1. Hybrid model for the ferroelectric liquid crystal molecule W314. Explicit hydrogen atoms are used only in aromatic rings.

3. Functional form

Since our force field is designed for specific mole-cules we need a strategy for rapid, on-demand deriva-tion of parameters from ab initio data. To attain thisgoal the number of parameters to optimize has to bekept as low as possible. The so calledclass I, or diag-onal, force fields have the simplest functional formand for this reason was our choice. Cross-terms canbe added in the future if necessary. We do not seereasons for their immediate inclusion since ourresearch involves study of the equilibrium statisticalmechanics of soft condensed matter, which does notneed spectroscopic accuracy. HFF’s functional formis based on the Dreiding II force field [8]. This choicewas made because of its simple functional form, butmodifications like new atom types and a more sophis-ticated torsion term extend its capabilities. The totalpotential energy for HFF is written as:

U � Ustr 1 Ubend1 Uinv 1 Utor 1 Uvdw 1 Ucoul:

The functional form for each term is as follows:BondedBonds

Ustr �Xij

12:Kr rij 2 req

� �2;

Angles

Ubend�Xij

12:Ku uij 2 ueq

� �2;

Umbrellas

Uinv �Xijkl

Kf cosfeq 2 cosfijkl

� �m;

m� 1; feq� 0; m� 2;feq ± 0� �

;

Dihedrals

Utor �Xijkl

Xn

Cnccos2cijkl ; n� 0;1;…;6:

Nonbondedvan der Waals

UvdW�X*i,j

41ijsij

rij

!12

2sij

rij

!6" #

1ij � 1i1j

� �1=2; sij � sisj

� �1=2� �

;

Coulomb

UCoul �X*i,j

qi ·qj

rij:

For the two nonbonded sums, interactions of types1-2, 1-3 and 1-4 are excluded, which is indicated byan asterisk. These interactions are implicitly includedin the bond, angle, and dihedral intramolecular termsrespectively. A cosine–sine series with up to 13 termswill be employed in the future for fitting complextorsional energy profiles.

4. Parametrization

Parameters have to be optimized against some kindof data that representsreality. Experimental molecu-lar geometry from microwave, NMR, X-ray, and othertechniques could be used as a source of data.However, experimental data depends on the molecu-lar environment within the material. The equilibriumgeometries of a molecule in the crystal, solution andgas phases could be different from each other. Toobtain molecular parameters that are independent ofthe medium, gas phase experiments or ab inito calcu-lations must be employed. Since the former are avail-able only for a limited number of molecules and ourwork requires large amount of data, ab initio methodsseems to be a practical choice ofreality. Good qualityab initio calculations can now be performed for mole-cules of up to , 15 heavy atoms on cheap work-stations and even personal computers. Liquid crystalmolecules are too large for such calculations to becarried out, so our strategy is to break the moleculesin smaller fragments, perform ab inito calculations onthem, and then transfer the parameters derived forthese fragments to the parent molecules (Fig. 2).The results of these calculations are stored in adatabase from which parameters are obtained. Our

E. Garcia et al / Journal of Molecular Structure (Theochem) 464 (1999) 39–48 41

original calculations were performed at the HF//6-31G*//HF/6-31G* level, using Gaussian94 [9] andSpartan [10]. Currently we are improving the levelof calculation to MP2/6-31G*//HF/6-31G*. We alsointend to study the performance of density functionalmethods in the near future. Parameters need to bedetermined for each energy term. In our approachwe optimize specific parameters for each type of dihe-dral in a molecule. The dihedral angle contributionneeds seven parameters for each possible combinationof four atoms. Considering that by atoms we reallymean atom typesand there could be many atomtypes for an element, we come out with an incredible

number of parameters to optimize. For our researchpurposes we adopt a parametrization strategy in whichwe focus on the most critical parameters namely,equilibrium bond lengths and bond angles, torsionalparameters and atomic partial charges. The parame-trization is performed in several steps allowing fasterdevelopment of new parameters. A description of thisstrategy follows.

4.1. Generic fixed parameters

In order to simplify the parametrization we employgeneric force constants for bonds, angles, and


Fig. 2. Each liquid crystal molecule is decomposed into child molecules that contain the necessary torsions. A relaxed dihedral drive is carriedout for each child molecule using ab initio methods for the dihedrals marked with an arrow.

improper torsion (umbrella). Their values are thesame as used in the Dreiding II force field [8], namely:Kr � b·700 kcal/mol/A2 (b � bond order),b � 1(single bond), 2 (double bond), 3 (triple bond) and1.5 (aromatic, resonant bonds). For all angle forceconstants, Ku � 100 kcal/mol/rad2. Generic Dreidingvalues are also used forkf andfeq, as well as for non-critical torsion parametersCnc , like the internal dihe-drals in benzene rings. Strictly speaking, forceconstants are dependent on the atomic neighborhood,however, the use of generic values seems to be areasonable approximation for the goal of reproducingthe ab initio equilibrium geometry.

4.2. Specific fixed parameters

The equilibrium geometrical parametersreq andueq

are obtained from the corresponding bond lenghts andangles in the minimum-energy ab initio structurescalculated at the HF/6-31G* level. The van derWaals parameterse i ands i are taken from the litera-ture, mostly from OPLS [11–14], where parametersare optimized to reproduce liquid state thermody-namic data. Atomic partial charges are obtainedfrom fits to the electrostatic potentials of wave func-tions by the CHELPG procedure [15], at the semiem-pirical AM1 level [16]. Charges have somedependence on molecular conformation. HFF partialcharges on topologically equivalent atoms are aver-aged, so that the conformation dependence isremoved. This is necessary for deriving torsion para-meters without distortions produced by differentcharges on equivalent atoms. Charge symmetrizationis done by an automatic algorithm for finding auto-morphism classes in a molecule based on theMATCHEM system [17,18]. Effective charges areobtained for alkyl carbons after removing hydrogenatoms by adding their charges to the carbon’s charge.

4.3. Specific optimized parameters

The torsion parametersCnc are obtained from fitsto the ab initio torsion potential. A relaxed dihedraldrive is performed to get the ab initio energy profilesfor each fragment molecule in the data base. Therelaxed dihedral drive consists in constraining a dihe-dral at a given value and energy minimizing the rest ofthe molecular structure, repeating the procedure fordifferent values. In general we use 158 intervals in

the angle values. The same procedure is performedusing the force field, and the torsion parameters areoptimized to fit the corresponding ab initio energyprofile. We use molecular substructures that containthe same torsion as that required in the parent liquidcrystal molecule. In cases where there is more thanone different torsional potential term for a given rota-table bond we use several substructures from whichthe necessary torsion parameters can be determinedindependently. If this is not possible, the totaltorsional potential is divided equally among the indi-vidual torsional potential terms of the bond.

5. Performance

5.1. Bond lengths and angles

The ab initio equilibrium bond lengths and anglesare well reproduced, maximum discrepancies beingwithin the accuracy of HF/6-31G* calculations,0.03 A and 1.58 for bond lengths and angles respec-tively [19]. The worst observed case is a 58 deviationfor the carbonyl ester angle in benzoates [6]. Bondsand angles vary with molecular conformation. Duringa torsion dihedral drive HFF bond length variation canbe as large as 0.03 A˚ . In the case of a torsion with-out non-bonded interactions involved, butane forexample, no deviations from equilibrium values areobserved. This is owing to the fact that 1-4 interac-tions are excluded in our functional. HFF angle varia-tions are in most cases not greater than 58 during atorsion drive. Examples for geometrical variationswith dihedral angle were shown previously for sometest cases [6]. We certainly do not expect bonds andangles to match the corresponding ab initio variations.If this happens it is likely to be simply fortuitous.Specific force constants and cross-terms betweenbonds, angles and dihedrals would be necessary toaccomplish this task. However, the ability of HFF toreproduce equilibrium geometry within the ab initioaccuracy is what matters to us since we are interestedin the equilibrium properties of liquid crystals.

5.2. Dihedrals

This is certainly the most critical geometricalcharacteristic. The shape of a liquid crystal is mostlyaffected by equilibrium dihedral values. HFF is


specifically designed to fit torsional profiles so itsaccuracy and reliability is far beyond the commercialgeneral purpose force fields that we have tried. Over-all, HFF performance is very good, with the position,depths and curvature of the minima correctly repro-duced. The energy profile average deviations are lessthan 0.1 kcal/mol, and at worst 0.3 kcal/mol for somecomplex asymmetric profiles. It has to be pointed outthat the quality of the fits can be greatly improved if

we use a full Fourier series for the torsion term, withboth cosine and sine terms and more parameters. Ourstrategy for the future is to use as many parameters asnecessary to get a perfect fit. In the followingexamples we compare HFF, the ab initio data usedin the parametrization, and results from two commer-cial force fields, one with cross terms CVFF [20], andthe other a simple diagonal force field SYBYL [21]. Ascan be seen by the example in Figs. 3–8, the use of


Fig. 3. Energy versus dihedral angle for the C_33–C_32–C_32–C_33 torsion.

Fig. 4. Energy versus dihedral angle for the O_3–C_32–C_32–C_33 torsion.

cross-terms does not guarantee a better performancewith regard to torsional profiles. For all force fieldcalculations the relaxed dihedral drives wereperformed using the same dihedral constraint forceconstant and the same convergence criterion. TheCVFF v2.3 force field implemented in the programDiscover [22] was used. Charges and cross-termoptions were included in all calculations. Spartan’simplementation of the Sybyl force field was employed[10]. For the ab initio calculations we used optimizedHF/6-31G* geometries in all cases. This basis set hasa good quality–cost relation and was also chosen byother authors to obtain torsional profiles [23]. Whenavailable, MP2/6-31G* energies calculated with HF/6-31G* geometry were employed. In Figs. 3–8 only aportion of the torsional potential is shown, the rest isnot included for clarity, and because it is not necessaryowing to the symmetry of the torsions.

In Fig. 3 we see that even for a simple alkane,butane, there are quite significant discrepancies. TheSybyl force field works surprisingly well, reproducingeven the barriers. CVFF overestimates thetrans–gauche energy difference by 1 kcal/mol and thebarrier by , 2 kcal/mol. The position of thegaucheminimum is also shifted by about 98. HFF has aperfect fit in this case.

For the ether torsion (C_33–C_32–C_32–O_3), in

Fig. 4, both force fields perform qualitatively well.Sybyl underestimates the dihedral angle of thegaucheconformer by 58 while CVFF overestimates it by thesame amount. The most important fact here is thatCVFF incorrectly points to a difference in stabilitybetween thegauche–transconformers as large as, 1.8 kcal/mol. On the contrary, the ab initio HF/6-31G* results predict thegaucheconformation to be alittle more stable than thetrans (by 0.1 kcal/mol).MP2 level calculations support this conclusion, witha difference of 0.5 kcal/mol.

The results shown in Fig. 5 confirm that generalpurpose force fields can not be trusted for torsionalpotential. Both CVFF and Sybyl give completelywrong predictions for the stable conformations ofthe chloroester torsion (Cl–C_32–C_RCO–O_2), asshown in Fig. 5. A free rotation is predicted by Sybyl,probably owing to the lack of specific parameters forthis case. CVFF predicts a global minimum at, 1428and a barrier at 08. These results are in disagreementwith the HF/6-31G* data which gives a global mini-mum at 08 and a secondary minimum at, 1188.These results are confirmed by MP2 calculations,where the energy of the secondary minimum at,1188 is about 0.2 kcal/mol larger than the HF result.Also, the barrier at 608 is 0.4 kcal/mol lower. In thiscase both general purpose force fields fail even at a


Fig. 5. Energy versus dihedral angle for the Cl–C_32–C_RCO–O2 torsion.

qualitative level. HFF has a maximum deviation of0.15 kcal/mol.

The torsions shown in Figs. 6 and 7 are present inmany liquid crystal molecules. In Fig. 6, for the alkylbenzene torsion, both commercial force fields do a

fairly good job. Sybyl correctly represents the curva-ture of the potential around the minimum and also thebarrier height, but gives a small secondary minimumat 908 that is not predicted in ab initio calculations upto the MP2 level. CVFF does not present a secondary


Fig. 7. Energy versus dihedral angle for the C_R–C_R–C_33–F torsion.

Fig. 6. Energy versus dihedral angle for the C_R–C_R–C_32–C_33 torsion.

minimum, but does less well in representing thepotential profile in general, giving a steeper curveand a barrier more than twice the observed ab initiovalue. HFF shows and almost perfect fit in this case,the maximum deviation being 0.03 kcal/mol.

Results for the anisole torsion are presented in Fig.7. MP2 energies were employed in this case because asmall secondary minima is predicted at 908 by HFlevel calculations. Here we can see that Sybyltorsional potential is completely wrong, owing tothe use of general parameters. The Sybyl profile isthe same as that predicted for the alkybenzene torsion.CVFF does very well, the only problem being a smal-ler curvature near the minimum. However, it is impor-tant to compare the CVFF results in Figs. 6 and 7.Even though they are qualitatively correct in bothcases, the aklylbenzene torsion is represented asbeing less flexible than the anisole torsion, whereasthe ab initio results show the opposite behavior. So inanalyzing force field results it is also important tocompare relative energy profile of the differenttorsions in the molecule. HFF presents some problemsto fit this profile, with the 7-term consine series, whichis an indication of the need for more parameters. Abetter general fit than the one shown in Fig. 7 ispossible [6], but this fit incorrectly displays asecondary minimum at 908. In the present case we

lose a little in the fitting but gain in the overallshape of the potential. The HFF maximum deviationis , 0.1 kcal/mol.

A fluorinated alkylbenzene torsion is shown in Fig.8. In this case both Sybyl and CVFF fail completely.Sybyl apparently doesn’t have any torsion parametersfor this case, and CVFF seems to be overestimatingthe van der Waals repulsion producing a minimum at908 instead of the correct one at 08. HFF presents aperfect fit.

5.3. Molecular dipoles

Our hybrid molecular representation includes twoapproximations, one in the use of implicit hydrogensin sp3 carbon atoms, the other is charge symmetriza-tion at structurally equivalent atoms. Even using thebest partial charges we can afford, the approximationsdescribed earlier limit the quality of the molecule’sdipole and also the electrostatic potential. For a set of38 liquid crystal molecules we calculated AM1CHELPG charges, at the optimized AM1 geometry,and compared the dipoles obtained from our hybridmodel against the AM1 dipoles. AM1 dipoles for themolecules studied range from 0.1 to 9.4 D. An aver-age relative error of 14% and absolute average error of^ 0.24 D were observed, which is within the accuracy


Fig. 8. Energy versus dihedral angle for the C_R–C_R–C_32–F torsion.

of the AM1 method (,0.35 D [24]). For 34 of themolecules in the set, with dipoles bigger than 1.0 D,the average relative error was only 7%. These resultsshow that even within the hybrid model approxima-tions the molecular dipoles are well reproduced,although some caution is always recommended foreach particular case. The use of better quality quan-tum mechanical methods to obtain partial charges andthe implementation of a charge generator based onthis data are in our future plans.

6. Conclusions

A good representation of conformational profiles isnecessary for good quality atomic level soft-materialssimulations. Our experience to date shows thatcommercial force fields cannot be trusted for thispurpose. Even force fields that include cross-termsin their functional form do not perform well if notspecifically parametrized to reproduce torsionalprofiles. Reliable predictions in molecular simulationsare only possible if reliable force fields are used. Thisis especially important in the case of soft-materialssimulations where materials properties reflect a subtleinterplay of intramolecular and intermolecular forces.Our strategy for deriving reliable force fields is basedon the use of ab initio data and on-demand derivationof parameters for target molecules. A simple diagonalform force field, such as HFF, can give a correctrepresentation of molecular conformational relativeenergy, geometry and electrostatics. With HFF weachieved a very good accuracy relative to the ab initiodata. Average bond length and angle deviations arewithin the accuracy of the ab initio methods used inthe derivation of the parameters. We believe that abinitio data will play a very important and ever incres-ing role in parametrizing force fields. Quantummechanics offers large amounts of data at an ever-decreasing computational cost. The main challengein force field development will be the derivation ofintermolecular parameters from quantum mechanicalcalculations. The difficulties here lie not only in thedevelopment of intermolecular potentials using abinitio methods, but also in the mapping of molecularobservables into classical functionals with transfer-able atomic parameters.

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HFF: a force field for liquid crystal molecules

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