a wannier-function-based ab initio hartree–fock study of polyethylene

11 September 1998

Ž .Chemical Physics Letters 294 1998 126–134

A Wannier-function-based ab initio Hartree–Fock study ofpolyethylene

Alok Shukla a,1, Michael Dolg a, Hermann Stoll b

a Max-Planck-Institut fur Physik komplexer Systeme, Nothnitzer Straße 38 D-01187 Dresden, Germany¨ ¨b Institut fur Theoretische Chemie, UniÕersitat Stuttgart, D-70550 Stuttgart, Germany¨ ¨

Received 18 May 1998; in final form 17 July 1998

Abstract

We report the extension of our Wannier-function-based ab initio Hartree–Fock approach – meant originally forthree-dimensional crystalline insulators – to deal with quasi-one-dimensional periodic systems such as polymers. The systemstudied is all-transoid polyethylene, and optimized lattice parameters, cohesive energy and the band structure utilizing6-31G)) basis sets are presented. Our results are shown to be in agreement with those obtained with traditionalBloch-orbital-based approaches. q 1998 Elsevier Science B.V. All rights reserved.

1. Introduction

Polyethylene, with its rather simple geometric andelectronic structure, can be characterized as a proto-type of all insulating polymers. Perhaps for thatreason, and certainly because of its technologicalimportance, it is one of the most extensively studiedsystems, both experimentally and theoretically. Itsgeometrical structure has been investigated in a num-

w xber of experiments both by X-ray diffraction 1–4w xand by the neutron-scattering method 5 . The elec-

tronic structure of polyethylene has been investigatedin photoconduction and photoemission experiments

w xresulting in the determination of its band gap 6 , andin angle-resolved photoemission experiments leadingto the measurement of its valence-band structurew x7 .Theoretically it has been investigated by a num-

1 E-mail address: [email protected]

w xber of semiempirical methods 8–12 , by the abŽ . winitio crystal-orbital Hartree–Fock HF method 13–

x17 and by subsequent many-body perturbation the-Ž . w xory MBPT based correlation methods 18–20 .

Correlation effects have also been accounted for inw x Žstudies using density-functional theory 21–23 for a

w xreview see, e.g., Ref. 24 .Recently we have proposed an approach to study

the electronic structure of periodic insulators in termsw xof Wannier functions 25 , with the long-term goal of

a wavefunction-based treatment of electron correla-Ž .tion effects in three-dimensional 3D crystalline

solids. The approach has been made much moreefficient as compared to its original implementation,and has since been applied, at the HF level, to studythe electronic structure and various related properties

w xof a variety of 3D ionic insulators 26–28 . As afurther improvement, we have extended it to deal

Ž .with quasi-one-dimensional 1D periodic systemssuch as polymers, as demonstrated in the present

0009-2614r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 98 00850-1

( )A. Shukla et al.rChemical Physics Letters 294 1998 126–134 127

Letter by means of an all-electron study of all-trans-oid polyethylene. The details of the approach per-taining to its implementation, along with its applica-tions to an infinite LiH chain and all-trans poly-

w xacetylene, are reported elsewhere 29 .w xAs is evident from the results of Refs. 18–20 ,

impressive successes can be achieved in the ab initiotreatment of electron-correlation effects for the caseof 1D systems such as polymers, even within Bloch-orbital-based approaches. However, generalization toa 3D solid is far from trivial because, for such asystem, owing to the higher dimensionality of itsBrillouin zone, the sum over the virtual states ink-space will involve a huge number of terms. More-over, the larger degree of degeneracy generally asso-

Žciated with Bloch orbitals of a 3D crystal band.crossing , will make such an approach even more

difficult to implement. This is the reason behind ourbelief that Wannier functions offer the most naturallanguage in which to formulate the ab initio treat-ment of electron-correlation effects, as their utility isnot restricted to systems of any particular dimension-ality. And which electrons will contribute to impor-tant correlation effects can be decided almost intu-itively, as electrons localized on Wannier functionswhich are far-apart will interact only weakly. How-ever, being a relatively new approach, it is importantfor us to check its applicability on a variety ofsytems even at the HF level. Since a large body ofhigh-quality theoretical and experimental data al-ready exists on polyethylene, any new theoreticalstudy of the compound can be judged against it. Thisis the reason that we present a HF study of all-trans-oid polyethylene in the present Letter, utilizing therelatively large 6-31G)) basis set employing polar-ization functions. In order to be absolutely certainabout the correctness of our method, we also used

w xthe Bloch-orbital-based CRYSTAL program 30 tostudy the same compound employing the identical6-31G)) basis set, and those results are also pre-sented for comparison. We note that Wannier func-tions have been employed in earlier ab initio studies

w xof electron correlation effects in polymers 31 . How-ever, unlike the past studies where the Wannierfunctions were obtained by localizing the Bloch or-bitals, the present work leads to direct determinationof Wannier functions of the polymers, without invok-ing the Bloch orbitals.

The remainder of the Letter is organized as fol-lows. In Section 2, we briefly describe the theorywith particular emphasis on the treatment of theCoulomb lattice sums which differs from theEwald-summation based approach adopted for ourearlier studies on 3D crystals. In Section 3, wepresent the results of our calculations which includeoptimized geometry, cohesive energy, band structureand contour plots of some of the Wannier functions.Finally, Section 4 contains our conclusions.

2. Theory

w xIn our previous papers 25,26 we showed that,for a crystalline insulator with 2n electrons per unitc

cell, one can obtain its n restricted-Hartree–cŽ .Fock RHF Wannier functions localized in the refer-

� < : 4ence unit cell, a ;as1,n , by solving the equa-c

tion

TqUq 2 J yKŽ .Ý b bžb

k < : ² < < : < :q l g R g R a se a ,Ž . Ž .Ý Ý g k k a/gkgNN

1Ž .

where T represents the kinetic-energy operator, Urepresents the interaction of the electrons of refer-ence cell with the nuclei of the whole of the crystalwhile J , K are the Coulomb and the exchangeb b

operators defined as

¶1< : ² : < :J a s b R b R aŽ . Ž .Ýb j jr12j •,

1< : ² : < :K a s b R a b RŽ . Ž .Ýb j j ßr12j

2Ž .

Ž .The first three terms of Eq. 1 constitute the canoni-cal Hartree–Fock operator, while the last term is aprojection operator which makes the orbitals local-ized in the reference cell orthogonal to those local-ized in the unit cells in the immediate neighborhoodof the reference cell, in the limit of infinitely highshift parameters lk ’s. These neighborhood unit cells,g

( )A. Shukla et al.rChemical Physics Letters 294 1998 126–134128

whose origins are labeled by lattice vectors R ,k

collectively define an ‘‘orthogonality region’’ andare denoted by NN. The projection operators alongwith the shift parameters play the role of a localizingpotential in the Fock matrix, and once self-con-sistency has been achieved, the occupied eigenvec-

Ž .tors of Eq. 1 are localized in the reference cell, andare orthogonal to the orbitals of NN – thus making

w xthem Wannier functions 25,26 . The size of NN canbe denoted by specifying the number N of nearestneighbors that are included in NN. For example,Ns3 shall imply that NN contains up to third-nearestneighbors of the reference cell, and so on. From the

ŽWannier functions localized in the reference cell cf.Ž ..Eq. 1 one can obtain the corresponding orbitals

localized in any other unit cell by the expression

< : < :a R sTT R a 0 , 3Ž . Ž . Ž . Ž .i i

< Ž .:where a 0 represents a Wannier orbital localizedin the reference unit cell assumed to be located at the

< Ž .:origin while a R is the corresponding orbital ofi

the i-th unit cell located at lattice vector R , and thei

corresponding translation is induced by the operatorŽ . Ž . Ž .TT R . Thus Eqs. 1 and 3 provide us with thei

complete HF solution of the infinite crystal.We have computer implemented the approach

outlined above within a linear-combination of atomicŽ .orbital LCAO approach, utilizing Gaussian lobe-

w xtype basis functions 32 . The nontrivial aspect of theprogram mainly involves the evaluation of the infi-nite lattice sums contained on the right hand side of

Ž .Eq. 2 . The evaluation of exchange lattice sums canbe performed entirely in the real-space for insulatorsquite inexpensively, utilizing the rapidly decayingnature of both the lattice sums, as well as that of thesingle-particle density matrix. However, the Coulomblattice sums are divergent on their own. The contri-bution of the Coulomb interaction to the Fock matrixconverges only when the electron-repulsion latticesums are appropriately combined with the latticesums involving electron-nucleus attraction. For the3D systems studied studied earlier, we used anEwald-summation based approach to perform the

w xlattice sums involved in the Coulomb series 25–28 .However, for the treatment of quasi 1D systems suchas polymers, we have adopted an entirely real-spacebased approach towards the evaluation of the

Coulomb series, the details of which have beenw xpresented elsewhere 29 . For the sake of complete-

ness we will briefly outline the salient features of theapproach here as well. In this approach, all thematrix elements needed to represent the canonical

Ž .HF operator included in Eq. 1 in the LCAO formcan be generated from the translationally invariant

w xskeleton of the corresponding Fock matrix 29

² < < :Ž .F t s p t T q 0Ž . Ž .p q p q p q

M atoms ZA² :Ž .y p t q 0Ž .Ý Ý p q < <ry R y rj Ajsy M A

M 1² .q2 p t r t q RŽ . Ž .Ý ÝÝ p q r s j r12r , sjsy M t r s

: ²Ž . .Ž . Ž .= q 0 s R D t y p t s RŽ . Ž .Ý ÝÝj r s r s p q kr , sk t r s

1:Ž .Ž . Ž .= r t q R q 0 D t . 4Ž .r s k r s r sr12

The notation used in the equation above is consistentw xwith the one introduced in our earlier work 26

< Ž .:where, e.g., p t denotes the p-th basis functionp q

located in the unit cell labeled by lattice vector tp q< Ž .:and q 0 denotes the q-th basis function located in

the reference cell. Additionally, R , R , t and tj k p q r sŽ .represent arbitrary translation vectors of the lattice

Ž .and D t represents the elements of the one-par-r s r s

ticle density matrix of the infinite system assumingthat r-th function is located in the cell t while ther s

s-th function is in the reference cell. The second andthe third elements of the equation represent theelectron-nucleus and electron-electron interactionparts of the Coulomb series, respectively. The latticesums representing both types of contributions to theCoulomb series are denoted by the sums over indexj. The Coulomb contribution to the Fock matrixelement displayed above is brought to convergenceby choosing to terminate both the lattice sums at thesame and a ‘‘sufficiently-large’’ value of the indexj. We choose to denote this value by variable M,which clearly implies the inclusion of the Coulombinteraction of the reference cell electrons with thosein up to its M-th nearest neighbors. As a conse-quence of the fact that the unit cell is electricallyneutral, the divergences inherent in the two Coulomb


contributions are equal and opposite in nature andshould cancel themselves for large values of M. Inorder to obtain the correct value of the total energyper unit cell, the nucleus-nucleus repulsion energy isalso computed by restricting the corresponding sumto M nearest neighbors of the reference cell. The last

Ž .term of Eq. 4 represents the contribution of theexchange interaction. The corresponding lattice sumsrepresented by index k converge rapidly in the realspace with increasing value of k. The lattice sumscorresponding to the cell index t also converge fastr s

for insulating systems because of the rapidly decay-ing nature of the corresponding density matrix ele-

Ž . < < w xments D t , with increasing value of t 30 .r s r s r s

The present, entirely real-space-based approach tothe infinite lattice sums is quite common in polymerstudies and can be found in the literature dating backto the seventies. An excellent review of the generalreal-spaced-based approaches can be found, e.g., in

w xthe work of Delhalle et al. 33 .In the theory described above there are two free

parameters viz., N which represents the size of theorthogonality region of the Wannier functions of thereference cell, and M which represents the range ofthe Coulomb interaction included in the Fock matrix.

w xIn another paper 29 we have presented a detailedstudy of the influence of these parameters on thetotal energy per unit cell for the infinite LiH chainand all-trans polyacetylene. In all the cases wefound that Ns3 and Ms75 yielded total energies

converged up to f 0.1 mHartree. In the presentstudy we have used the values Ns5 and Ms85.

The band structure presented in this work wasobtained using the approach outlined in our previous

w xwork 27 . This is achieved by performing a Fouriertransform on the final converged real-space Fock

Ž Ž ..matrix cf. Eq. 4 to obtain its reciprocal-spaceŽ .k-space representation. Then the correspondingFock matrix is diagonalized at different k points toobtain the band structure.

3. Calculations and results

In this section we present the results of calcula-tions performed on polyethylene in its all-transoidgeometry. The polymer was modeled as a singlechain oriented along the x-axis, as shown in Fig. 1.Since our program is not yet able to utilize thepoint-group symmetry, the reference unit cell wasassumed to consist of a single C H unit, as against2 4

a CH unit if the use of the point-group symmetry2

were possible. The bonds included in the referencecell during the calculations are also indicated in Fig.1. There are four geometrical parameters involved inthe structure of all-transoid polyethylene namely,

Ž .carbon–carbon bond length R , carbon–carbon–CCŽ .carbon bond angle a , carbon–hydrogen bond

Ž .length R and hydrogen–carbon–hydrogen bondCHŽ .angle b . In the present calculations all four param-

eters were optimized.

Fig. 1. Structure of all-transoid polyethylene as considered in the present work. The polymer is assumed to be oriented along the xdirection. CC bonds are in the xy plane while the CH bonds are in the yz plane. Bonds included in the reference cell in the calculations areenclosed in the dashed box; the two CC bonds crossing the borderline are translationally equivalent, and only one of them is included in thereference cell.


We adopted a 6-31G)) basis set, the same basisset which we also used to study all-trans polyacety-

w xlene in our previous work 29 . The polarizationfunctions used in the basis set consisted of onep-type exponent of 0.75 Bohry2 on hydrogen and asingle d-type exponent of 0.55 Bohry2 on carbon.

w xThus the basis set on carbon was 3s,2p,1d typew xwhile the one on hydrogen was of 2s,1p type. In

our calculations, of course, we used the lobe repre-sentation of the corresponding p- and d-type Carte-sian basis functions, while the true Cartesian basisfunctions were used in the calculations performedwith the CRYSTAL program. For the six atomsincluded in the reference cell the present choice ofthe basis set amounted to 48 basis functions per unitcell. Since we chose to orthogonalize the Wannierfunctions of the reference cell to those residing in itsup to fifth-nearest neighbor cells, we had to includethe basis functions of those cells so as to be able tosatisfy the orthogonality condition. This increases thetotal number of basis functions associated with thereference cell Wannier functions to 528, which, ofcourse, amounts to an enormous increase in the

Ž Ž ..dimension of the Fock matrix cf. Eq. 1 to bediagonalized. However, the integral evaluation time,which dominates the calculations, is unaffected bythis proliferation in the basis set size associated withthe Wannier functions. The reason for that is that theone- and two-electron integrals are evaluated withfull regard to the translational symmetry of the prob-

w xlem 26,29 , and therefore depend only on the num-ber of basis functions per unit cell.

Our results on the optimized geometry and thetotal energy per unit cell along with the correspond-ing cohesive energy obtained with the 6-31G))

basis set are presented in Table 1 which also com-pares them to the calculations performed by us –employing the same basis set – with the CRYSTALprogram. As far as the comparison with the works ofother authors is concerned, we will restrict it only tocalculations in which the ab initio crystal HF ap-proach was adopted. To the best of our knowledge,

w x w xonly Karpfen 13 and Teramae et al. 14 haveperformed full geometry optimization for all-trans-oid polyethylene within an ab initio crystalHartree–Fock approach. These authors, of course,employed a Bloch-orbital-based formalism and their

w xresults are also summarized in Table 1. Karpfen 13

Table 1A summary of our HF results on all-transoid polyethylene per-formed with the 6-31G)) basis set and its comparison with thecorresponding calculations performed by us with the CRYSTALprogram and the HF results of other authors

R R a b E ECC CH total coh

aThis work 1.532 1.092 113.0 106.6 y78.0728 9.85CRYSTAL 1.533 1.092 113.2 106.5 y78.0723 9.85

bKarpfen 1.562 1.102 112.2 107.4 – –cKarpfen 1.547 1.089 112.6 107.0 – –

dTeramae et al. 1.565 1.102 115.3 104.7 – –eExp. 1.53 1.069 112.0 107.0 – –fExp. 1.578 1.06 107.7 109.0 – –

The Wannier-function-based calculations were performed withup to fifth-nearest neighboring cells included in the orthogonality

Ž .region Ns5 and 85 neighbors included in the Coulomb seriesŽ .Ms85 . Experimental values are also listed for comparison. The

˚lengths are expressed in A, the bond angles are in degrees, theŽ .total energy per C H unit E is in Hartrees while the2 4 totalŽ . acohesive energy per CH unit E is in eV. Performed with2 coh

the lobe representation of the 6-31G)) basis set. b Performedw x cwith an extended basis set without polarization functions 13 .

w x dPerformed with the STO-3G basis set 13 . Performed with thew x e w x fSTO-3G basis set 14 . X-ray scattering results at 4 K 1–4 .

w xNeutron scattering results at 4 K 5 .

used both the STO-3G basis set as well as an ex-w xtended basis set employing a 4s,2p set for carbon

w xand a 2s set for hydrogen. Teramae et al. used onlythe STO-3G set in their calculations. Thus, polariza-tion functions were used in neither of the papers. It isclear from the table that the agreement between theresults obtained with our approach, and the ones withthe CRYSTAL program, is very close. The agree-ment between our results and those of other authorsis generally satisfactory, i.e., the deviations are withinthe limits of uncertainties associated with quantum-chemical calculations, except for the carbon–carbon

˚Ž .bond length R which is about 0.03 A shorterCCw xthan the best values reported by Karpfen 13 and

w xTeramae et al. 14 . A possible reason is the use ofpolarization functions in our calculations. Experi-mental results on polyethylene are available from

w xX-ray scattering experiments 1–4 as well as fromw xneutron-scattering experiments 5 , and are summa-

rized in Table 1. As far as the comparison with theexperiments is concerned our results for all parame-ters, except for the carbon–hydrogen bond lengthŽ .R , are in good agreement with the experimentalCH

results obtained with the X-ray scattering techniques


w x1–4 . R in the present calculations is overesti-CH˚mated by about 0.03 A compared to the X-ray

scattering results. However, for this parameter, re-sults of other authors also differ by at least thatamount. Thus, in our opinion, this discrepency maybe due to electron-correlation effects. We also pre-sent results for the cohesive energy per CH unit2

obtained from ours as well as CRYSTAL calcula-tions. For that purpose we used Hartree–Fock refer-ence energies for carbon and hydrogen ofy37.677838 au and y0.498233 au, respectively.These energies were obtained by performing atomicHF calculations employing the same 6-31G basis setas used in the polymer calculations. Our results forthe cohesive energy are in excellent agreement withthe one obtained with the CRYSTAL program, how-ever, we could not locate any other theoretical orexperimental results on the cohesive energy of poly-ethylene.

In Fig. 2 we present the band structure of all-

transoid polyethylene computed with the 6-31G))

basis set determined by our approach as well as theone computed using the CRYSTAL program. Al-though, in what follows, we have not compared ourHF band structure with those presented in the works

w xof other authors 13,15 , even a cursory comparisonreveals that the qualitative features of all theHartree–Fock energy bands are essentially the same.Our band structure was computed with Ns5 andMs100 while in the CRYSTAL calculations 100 kpoints were used to perform the integration over the

w xBrillouin zone 30 . Since our results for the geome-try were in better agreement with the experimentalone determined with the X-ray scattering techniques,we chose to evaluate the band structure for the

w xcorresponding geometrical parameters 1–4 . Sixhighest occupied bands along with the seven lowestunoccupied bands are presented in Fig. 2. The abso-lute values of the band energies naturally differedsomewhat owing to the different treatment of the

Ž .Fig. 2. Band structure of polyethylene obtained using our approach solid lines compared to that obtained using the CRYSTAL programŽ . Ž . w xdashed lines . The experimental geometry X-ray scattering 1–4 and a 6-31G)) basis set were used in both cases. The close agreementbetween the two sets of bands is obvious.


Coulomb series in the two approaches. Therefore weshifted all the CRYSTAL band energies so that thetops of the valence bands obtained from the two setsof calculations coincided. Clearly the agreement be-tween our band structure and the one obtained withthe CRYSTAL program is very good. The value of

Ž .the direct band gap at ks0 point is 0.6088 auŽ .16.57 eV obtained with our approach is in excellentagreement with the corresponding CRYSTAL value

Ž .of 0.6077 au 16.54 eV . We believe that the smalldifferences between our band structure and the onecomputed using the CRYSTAL program are mainlydue to the use of lobe functions in our approach toapproximate the Cartesian basis functions used bythe CRYSTAL program. The value of the direct gapfor all-transoid polyethylene was measured to be 8.8

Fig. 3. Contour plots of the charge density of the p-type valenceWannier function of the reference cell. Contours are plotted in the

Ž .xy plane with zs0.25 au x is the axis of the polymer . Themagnitude of the contours is on a natural logarithmic scale. Thetwo carbon atoms of the unit cell are located at the positionsŽ ..1.2,"0.81,0.0 au, while the four hydrogen atoms are at

Ž .locations .1.2,"2.01,"1.62 au, approximately. The orbitalclearly has a bonding character with respect to both the carbon–carbon as well as the carbon–hydrogen bond. The rapidly decay-ing strength of the contours testifies to the localized nature of theWannier function.

Fig. 4. Contour plots of the charge density of the second p-typevalence Wannier function of the reference cell. The geometricalinformation is same as in the caption of Fig. 3. This contour plotclearly indicates an antibonding character of the Wannier functionwith respect to the two carbon atoms of the unit cell, and abonding character with respect to the carbon–hydrogen bonds.Some weaker contours can be seen to reach the carbon atomsbelonging to the nearest-neighbor unit cells.

w xeV by Less et al. 6 . Therefore, as is generally thecase with HF bands, the band gap of polyethylene isoverestimated by a factor of two, pointing to theimportance of the electron correlation effects. Withour approach the highest point of the valence bandoccurs at f10.46 eV which, according to the Koop-mans theorem, can be directly compared to the ion-ization potential of the compound. The experimentalvalue of the ionization potential of polyethylene is

w xreported to be in the range 9.6–9.8 eV 7 . Althoughthe agreement between the HF value and the experi-mental one is better for the ionization potential ascompared to the band gap, the differences still pointto the importance of electron-correlation effects. Sun

w xet al. 20 recently studied the influence of electroncorrelation effects on the valence band of poly-ethylene using a Bloch-orbital-based second-ordermany-body perturbation theory. The improvementover the HF results which they observed testifies to


the general belief that electron correlation effects areessential to describe the quasi-particle properties insuch systems.

One can obtain an intuitive understanding of thechemical bonding in the system by examining theWannier functions. Such views are presented in Figs.3 and 4 which display the charge densities of theWannier functions corresponding to the bonds of the

Ž .unit cell, which are antisymmetric p-like under thereflection about the xy plane. The contour plotscorrespond to the charge density associated with thecorresponding Wannier function in the xy plane withzs0.25 atomic units. These orbitals were evaluatedat the experimental geometry used also to computethe band structure. Fig. 3 clearly implies a Wannierfunction which has a bonding character with respectto the two carbon atoms of the unit cells, while thesecond p-like shown in Fig. 4 displays an antibond-ing character. Both the Wannier functions possessbonding characters with respect to the C-H bonds ofthe unit cell. From the contour plots the localizednature of the orbitals is quite obvious. The s-type

Ž .Wannier functions not presented here , although stilllocalized within the unit cell, are mixtures of boththe traditional C-C and the C-H bonds.

4. Conclusions and future directions

In conclusion, an ab initio Wannier-function-basedHartree–Fock approach developed originally to treatinfinite 3D crystalline systems has been applied to a1D system, namely, polyethylene. The main differ-ence as compared to the case of 3D systems has beenan entirely real-space based treatment of the Coulombseries. The approach yields results which are inexcellent agreement with those obtained using theCRYSTAL program, where Ewald summation is usedto perform the Coulomb lattice sums. In future, weintend to make our treatment of the Coulomb seriesmore efficient by incorporating real-space multipoleexpansion techniques.

The discrepancy between our Hartree–Fock re-sults for all-transoid polyethylene and the experi-mental ones was found to be most significant for theband structure. For example, the Hartree–Fock valueof the direct band gap is wrong by a factor of two as

compared to the experimental one. These differencespoint to the importance of electron-correlation ef-fects. In a future publication, we will include thesewithin a local-correlation approach to study theirinfluence on the quasi-particle properties of poly-

w xethylene and other insulating systems 34,35 .

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a wannier-function-based ab initio hartree–fock study of polyethylene

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