Akira Takahashi and Shaul Mukamel- Anharmonic oscillator modeling of nonlinear susceptibilities and...

8/3/2019 Akira Takahashi and Shaul Mukamel- Anharmonic oscillator modeling of nonlinear susceptibilities and its application to conjugated polymers

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Anharmonic oscillator modeling of nonlinear susceptibilitiesand its application to conjugated polymersAkira Takahashi and Shaul MukamelDepartment of Chemistty, University of Rochester,Rochester,New York 14627(Received 8 September1993;accepted 14 October 1993) IMolecular optical susceptibilities are calculated by deriving equations of motion for the singleelectron reduced density matrix, and solving them using the time dependent Hartree-Fock(TDHF) approximation. The presentapproach ocusesdirectly on the dynamics of the chargesin real space and completely avoids the tedious summations over molecular eigenstates. tfurther maps the system onto a set of coup led harmonic oscillators. The density matrix clearlyshows the electronic structures induced by the external field, and how they contribute to theoptical response. The method is applied to calculating the frequency-dispersed opticalsusceptibility x (3) of conjugated inear polyenes, starting with the Pariser-Parr-Pople (PPP)model. Charge density wave (CDW) like fluctuations and soliton pair like local bond-orderfluctuations are shown to play important roles in the optical responseof these systems.

I. INTRODUCTlONRecently, there has been ncreasing nterest in the non-linear optical properties of r-conjugated polymers, whichare good candidates for optical devices becauseof theirlarge nonlinear optical susceptibilities.- The nonlinearoptical response of conjugated polymers is closely con-nected o some fundamental theoretical problems of one-dimensional systems such as strong electron correlations,and the roles of exotic elementary excitations (solitons orpolarons) 7 Furthermore, they are ideal model systems orstudying exciton confinement effects n nanostructures.The frequency dispersion of nonlinear optical polariz-abilities provides an important spectroscopic ool. A vari-

ety of third, order techniquessuch as third harmonic gen-eration (THG), two photon absorption (TPA), and fourwave mixing- esult in a detailed microscopic probe of elec-tronic and nuclear dynamics. These spectra are tradition-ally calculated using multiple summations over the molec-ular excited states. However, this method has someserious imitations since t requires the computation of allthe excited states n the frequency range of interest, as wellas heir dipole matrix elements.Thesecomputations poseavery difficult many-body problem, particularly since elec-tron correlations are very important in low-dimensionalsystems such as r-conjugated polymers. Large scale nu-merical full configuration interaction calculations showthat nonlinear optical polarizabilities are very sensitive oelectron correlations6 This rigorous approach can be ap-plied in practice only to very small systems (so far poly-eneswith up to 12 carbon atoms have been studied) be-cause of computational limitations. Conjugated polyenesare characterizedby an optical coherence ength, related tothe separationof an electron-holepair of an exciton, whichis typically -40 carbon atoms for polydiacetylene.* t isessential to consider systems larger than the coherencelength in order to account for the scaling and the satura-tion of nonlinear susceptibilitieswith size. Thus, severalauthors calculated the excited states in the independentelectron approximation, *-I4 or by using contiguration in-

teraction including only single electron-hole pair excita-tions.15This method can be carried out for larger systems.However, it is valid only when correlation effectsare weak,which is not the casehere.t6Additional difficulty with thesum over states method is the need to perform tedioussummations over excited states. This forces us to workwith small systems,or to truncate the summations, whichagain limits the accuracy for large systems.The sum overstates method describesoptical processes n terms of theexcitation energiesand transition dipole moments. Thesequantities provide very little physical insight regarding heoptical characteristics of r-conjugated polymers, and donot directly addressquestionssuch as what kind of corre-lation is important, or how characteristic elementary exci-tations such as solitons affect the optical response.Thesynthesis of new optical materials calls for simple guide-lines (structure-property relations) 7 which should allowus to use chemical intuition to predict effects of geometryand various substitutions on the optical susceptibilities.The sum over states method does not offer such simpleguidelines, even when it does correctly predict the opticalsusceptibilities.An alternative view of optical responsemay be ob-tained by abandoning the eigenstate epresentation alto-gether, and considering he material system as a collectionof oscillators. It is well establ ished hat as far as thelinearresponse s concerned,any material system can be consid-ered as a collection of harmonic oscillators.* In fact, theterm oscillator strength of a transition is based on thispicture. It has beensuggested y Bloembergen that opti-cal nonlinearities may be interpreted by adopting an an-harmonic oscillator model for the material degreesof free-dom. This was proposed as a qualitative back of theenvelopemodel. It has been shownZoV2ihat molecular as-semblieswith localized electronic states can indeed be rig-orously representedas a collection of anharmonic oscilla-tors representingnonlocal coherences f Frenkel excitons,although the anharmonicity is more complex than a simplecubic nonlinearity.t9

When applied to molecular assemblies, he sum over2366 J. Chem. Phys. 100 (3), 1 February 1994 0021-9606/94/100(3 )/2366/19/$6.00 @ 1994 American Institute of PhysicsDownloaded 07 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html


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A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers 2367states method showsdramatic cancellations esulting frominterferences etweensingle exciton and two exciton tran-sitions.20-22hesecancellationsmake t extremely difficultto predict trends since he results are very sensitive o ap-proximations such as truncations. In the oscillator repre-sentation, on the other hand , these nterferencesare natu-rally bui lt in from the beginni ng, which greatly facilitatesphysical ntuition. We subsequently xtended he oscillatorpicture to conj ugatedpolyeneswith delocalizedelectronicstates. The calculation was based. n the Pariser-Pople-Parr (PPP) model for ?r electrons, which includes bothshort and long range Coul omb interactions. Many impor-tant properties of. polyenes can b e explained by themode1.23*2425y drawing upon the analogy with semicon-ductors, the Wander representation which requiresperi-odic boundary conditions) was used o developa coupledoscillator picture.* The method was shown to reproducethe size scaling and saturation of conjugatedpolyenes. nthis paper we put the oscillator picture on a-firmer groundand connectit with more traditional quantum chemistrymethods.We calculate he linear and the nonlinear opticalresponse y solving the equationsof motion of the singleelectron reduced density matrix using the time dependen tHartree-Fock (TD HF) approximation.26 he method canbe easily applied o moleculesmuch arger than the excitoncoherence ength, and can therefore reproduce the sizescaling from the small molecules o the bulk (the ther-modynamic imit). .As for the electron correlation prob-lem, since the TDH F approximation describes mall am-plitude collective quantum fluctuations around theHartree-Fock ground state, as well as the coupling be-tween these tluctuations, so me important correlation ef-fects are taken into account by our method. The TDHFapproximation has beenused o calculate nonlinear polar-izabilities of small molecules.27 owever, t should be par-ticularly applicable or large moleculeswhere the energysurface structure is simpler, and collective motions domi-nate their optical response.The density matrix can be expressed sing various rep-resentationswhich provide a complementaryphysical in-sight. These nclude the real space, he molecul ar orbital,and the harmonic oscillator representation.The real spacerepresentations llows us to follow directly the chargeden-sity and bond order fluctuations induced by the externalfield. Using thesequantities, we can explore he electronicstructure of the excitations underlying the optical process.We found that collective CDW like fluctuations andsoliton-pair like bond-order fluctuations dominate the lin-ear and the nonlinear optical responseof polyacetylene.The molecular orbital representationdescribes he nonlin-ear optical process n terms of motions of electrons andholes n the mean ield ground state. Finally, the equationsof motion of the density matrix can be mappedonto a setof coupledharmonic oscillators.Using this transformation,we can describe he nonlinear optical process n terms ofinterferenceamongoscillators.This providesan unconven-tional physical picture which enables s to investigate hemechanismof optical response f various systems includ-ing semiconductors nd nonconjugatedmolecules) rom a

unifled point of view, and clarifies the connectionswithother types of materia ls.In Sec. I we introduce the PPP Hamiltonian, and aclosed equation of motion for the reduced single particledensity matrix is derived n Sec. II using the TDH F ap-proximation. In Sec. V we discuss he real spaceand themolecular orbital representationsof the density matrix,and show how the TDHP equations can be transformedinto a set of coupl edharmonic oscillators.The first and thethird order nonlinear susceptibilitiesare calculated n Sec.V. Numerical calculationspresented n Sec.VI allow us todiscuss he nonl inear responseunctions in terms of chargedensity and bond order fluctuations. Finally, our resultsare summarized n Sec.VII.II. THE PPP HAMILTONIAN

We adopt the PPP Hamiltonian for the 7r electrons.Many properties of polyenes can be reproduced by thisHamiltonian with the appropriate parameters.23We firstintroduce the following set of binary electron operators:lfxm==~~,~n,crt (2.1)

whereZI,,( &,) creates annihil ates) a rr electron of spin uat nth carbon atom. Theseoperatorssatisfy the Fermi an-ticommutation relationccn,oG,rJJ h?&&7~* (2.2)

Using this notation, the PPP Hamiltonian is given byff=&.H+&+&. (2.3)

Hssu is the Su-Schrieffer-Heeger (SSH) Hamiltonian,which consists of the Hiickel Hamiltonian with electron-phonon coupling,&sH= c t,,i%+ ; ;mx,-32. (2.4)n,m,aHere t,,, is the Coulomb integral at the nth atom, tmn(m#n) is the transfer integral between he nth and mthatoms, K is the harmonic force constant representing her-bonds, x, is the deviation of the nth bond ength from themean bond length a long the chain axis z, and X is thedeviation of the equilibrium u-bond ength (in the absenceof T electrons) from that mean.We further assume hat anelectron can hop only between nearest-neighboratoms.Thus,Ll=CYm?2~ (2Sa)nn+lmb+ln=ii_pXn, (2.5b)

and tmn=O otherwise,whereynYnms a repulsionbetweennthand mth sites.a is the mean ransfer integral and p is theelectron-phonon coupling constant.Ho represents he electron-electron Coul omb nterac-tions and is gi ven by n#mHc= c O& ,&,+; c rnml;~,z:,8:m~ (2.6)n n,m,u,,oAn on-site (Hubbard) repulsion U is given.by(2.7)

J. Chem. Phys., Vol. 100, No. 3, 1 February 1994Downloaded 07 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html


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2368 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymersand a repulsion between he nth and the mth sites ynYnmsgiven by the Ohno formula

(2.8)

where Us= 11.13eV is the unscreened n-site repulsion, Eis the dielectric constant which describes he screeningbyu-electrons, ,, is the distancebetweennth and mth sites,and a,= 1.2935A. The parameters re determinedso as toreproduce he correct energy gap for polyacetylene (2.0eV), p=-2.4 eV, p=-3.5 eVA-, K=30 eV Am2, X=0.14 A and E= 1.5.The third term &.. represents he interaction Hamil-tonian between he r-electrons and the external electricfield E(t). The electric field is assumed o be polarizedalong he chain axis z. Within the dipole approximationwethen have

H,,= - E(t)fi, (2.9)where is the molecular polarization operator

I;= --e c Z(n>;;n, (2.10)where -e is the electron charge and z(n) isz-coordinateof nth atom. the

III. EQUATIONS OF MOTlON.FOR THE REDUCE DDENSITY MATRIXStarting with the Schrodinger quation, he equationofmotion of the expectationvalue of our binary electron op-eratorsPL?m=ww Il;;m:,IY(t)), (3.1)

is given byifi&Jt)=WW 1 /jL,Hl IW>), (3.2)

where 1Y(t) ) is the total many-electronwave unction ofthe system.The expectationvaluespzmcan be interpretedas elementsof the single electron reduceddensity matrix.Usually the density matrix is defined o have a unit trace.However, his matrix is normalized as

Trp= 4 5 pZm=ne (3.3)with N being the total number of sites and 12, s the totalnumber of electrons.Utilizing the commutation relations (2.2), we can cal-culate the right-hand side of Eq. (3.2), resulting in

w&Jt> = c [Q.$&) -&&(t>l+ wigy;m:,)I-(P,;Pzm)l,+; $ ?d(/$ljnqn)+G%da, -; -Z %A(~;p;;:m)+ z(m) lE(~)p&W,(3.4)

where(O)=WW lOlW~>), (3.5)

and 0 is an arbitrary operator. Theseequationsof motionare exact, but they are not closed since they contain newhigher order variables (p^;&) etc. in the right-handside.To close he equations,we assume hat I Y(t) ) can berepresented y a single Slater determinant at all times (theTDH F approximation) 26Then. he two-electron densitiescan be factorized into products of single electron densities

+ hT,,4,jP~ ( t), (3.6)and the equationsare closed. Substituting Eq. (3.6) intoEq. (3.4), we obtain the TDH F equation

q(t) =Lfm) +f(t),pU(t) I, (3.7)whereJ? is the Fock operator matrix corresponding oH,,+f+ with spin a,

gmw =Ln+4z,m c %Ynr&) -yn,p&w, (3.8)r,oand fnm(t) is the Fock operator matrix corresponding oH ext 2

f,&> =Sn,,edn)EW. (3.9)Note that so mecorrelation effects,which are very im-portant in low dimensional ystems, re taken into accountby the TDH F approximation. In the f -0 limit, theTDH F coincides with the random phase approximation(RPA) method which describes mall amplitude~quantumfluctuations around the static mean field solution veryweli.26The solution of the TDHF equation further takesthe coupling of the IU?A modes nto account, as will beshown below.

IV, REAL SPACE, MOLECUL AR ORBITAL, ANDHARMONIC OSCILLATOR REPRESENTATIONSWe have solved he equationsof motion by expandingthe singleelectron density matrix in powers of the externalfield.28The zeroth order solution was taken to be the sta-tionary Hartree-Fock (HF) density matrix, which satisfies[ P,p] =o. (4.1)

The HF equation was solved numerically by an iterativediagonalization,as shown n Appendix A.J. Chem. Phys., Vol. 100, No. 3, 1 February 1994

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A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers 2369Since both the P PP Hamiltonian and the stationaryHF solution are symmetric with respect o spin exchange,the TDH F solution must also have that symmetry. Weshall therefore consider he sp in symmetric caseonly, andomit the spin index in the following, denotingp=pf=p~, (4.2)h=h=h. (4.3)We next decomposehe density matrix aspw =p+spw, (4.4)

where 7, represents he H F solution. Then the Fock oper-ator matrix is also decomposedn the formh(t)&+Sh(t),

where(4.5)

Ln=L+2&?l T ml~zl-Ynmp,m~ (4.6)

Sh,mW =2&?2 7 YnYnrSPdQYnmSPnmW- (4.7)Substituting the expansions (4.4) and (4.5) into theTDH F Pq. (3.7), we obtainitip- [&$I - II&p1 = C ,pl+ I: Jpl+ [&Gpl.,(4.8)

All terms in the left-hand side are linear in Sp. The firsttwo terms n the right-hand side, which are zeroth and firstorder in Sp, respectively, describe he coupling with theradiation field and the last term is quadratic in Sp, andcomes rom Coul omb nteraction, as seen rom Eq. (4.7).Hereafter we introduce Liouville space (tetradic) no-tation for the density matrix. To that end we consider6p tobe an M dimensionalvector, rather than an NxN matrixwith N being he numbe r of the atoms and IV=N~.~*~~Wethus introduce a new inear vector space, enoted he Liou-vile space in which ordinary operatorsbecomeM dimen-sional vectors. The TDH F Eq. (4.8) then assumes heform

itip--6p= [ f,Pl + I: Jp1 + [%Spl, (4.9)2Y ,,,(w)=sj,,~i~--6,,~j,+2s,,~(Yi,-Yj,)pi/

-Si,,yi,lS1,+Sj,nYjmPim, (4.10)where 2 is an MxM matrix which is an operator inLiouville space also denotedsuperoperator).We shall usescript letters to denote Liouville spaceoperators.6p in theleft-hand side s a vector. All terms in the right-hand side[Sh,Sp] etc. are consideredM-dimensional vectors.So far, all our equations were written using the realspace (site) representation. To facilitate the numericalcomputations and to gain additional physical insight weshall recast the TDHF equation using two additional rep-resentations.We first introduce he Hartree-Fock molecular orbital(HFMO ) representation. The transformation, ofM-dimensional vectors such as Sp from real space o theHFM O representation s defhmdby

+kk= c, .7;rkk,mnbnn,inn (4.11)where the tetradic transformation matrix Y is

Y kkmn =CmkCnk t (4.12)and c,k is the normalized HF MO coefficient of the H Forbital k at atom m. As shown n Appendix B, the HFM Orepresentationof 2 is given by

P=YYF-T, (4.13)and the TDHF equation n the HFM O representation anbe written as

itip--+== 1 fJ1 + [ f9Ql-t [w$l. (4.14)Here all the M-dimensiona l vectors are in the HFMO rep-resentation,and we regard 6pkk, as NXN matrices whenwe calculatecommutators such as [6h,6p]. An explicit ex-pression or p is given n Appendix B. Note that becauseof the C2, symmetry of the present Hamiltonian, 9 isblock diagonal nto A, and B, symmetry parts, which si m-plifies the numerical calculations.Our equationscan also be mappedonto the equationsof motion of coupled harmonic oscillators. This definesanew.harmonic oscillator (HO) representationwhich pro-vides a tremendousphysical insight. We analyze he HOrepresentation n the following.The density matrix spkk defined by Pq. (4.11) is anM-dimensional vector in Liouville space.The number ofSpeh and 6ph, componen ts (MI) is 2n(N--n), and thenumber of 6p,t and 6p,t components. M2) is (N-n)2+n2, where h,h,... denote occupied HF orbitals, e, e,...denoteunoccupiedHF orbitals, and n is the number of theoccupiedHF orbitals. Sincewe consider he half-filled andspin symmetric caseonly, n is half of the number of sitesIZ N/2. We next introduce the Liouviile spaceprojectionoperator P that projects onto the eh an d he space.Thecomplementaryprojection I-P projects onto the ee andhh space.We thus have

+,=p~p=&%k+~ph,, (4.15)6pz~(I--P)Sp=Sp,,+~phh, (4.16)

wheresp=spl+spz. (4.17)

As shown in Appendix B, the TDHF Eq. (4.8) can bewritten asi~p1--=%6p1= [ f$l+ I: &l + [hid

+ [WQI, (4.18)i~Pz_-fi&~p2= [ f,6p] + [6h,+],

where(4.19)

oh&) =2&t,, 7 YIP&) -~mn~~inrnWt (4120)where = 1,2. The M I X.&f1 matrix PI is the HF stabilitymatrix, and the it4, XM2 matrix a2 is diagonal n Liouvillespaceand its diagonal matrix elementsare given by the



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2370 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymersdifferenceof the HF eigenvalues. hus, if a,,, is a diagonalelement hen -fi2, is an eigenvalue s well. Their explicitexpressions re given in Appendix B. The matrix ??I canbe diagonalized y the M1 XMI matrix w as

wLPIw-l=+it-k~, (4.21)where 0, is an M, XM, diagonal matrix. The Ml/2 diag-onal elements of fi, are RPA energiesa,,,>0 and theother Ml/2 diagonalelementsare a; = - &,.26 We obtainw numerically as described in Appendix B. Then, thetransformation of Liouville spacevectors such as Sp fromreal space o the H O representation s definedby

sp,= C%~,:mn~Pmn*mn (4.22)The transformation matrix % is given by

22 = Y--.rr, (4.23 >and the MXikf matrix %@-s given by

(4.24)where we arrange the M-components of the vectors inLiouville space n the following manner: we put the inter-band 6p,h and 6p,h components,Spl) in the first Mr rows,and he iutraband Sp,r and 6phh,components Sp2) in theremainingM2 rows, namely,

Sp= (4.25)

It is shown n Appendix B, that the TDHF equationcan berecast n the HO representationasitip,-- fi~,Pp,=F,+ ; 6,dp,~ + s R,,Ap,l

+ d;,, sv,vvf~Pv~~Pv~~ (4.26)where a,, is the diagonal elementof a, or a,, the summa-tion in the third term of the right-hand side is done overthe M2 components,and explicit expressions or F, G, R,and S are given in Appendix B.The physical significanceof the right-hand side of Eq.(4.26) is as follows. The first term corresponds o the firstterm of the right-hand side of Eq. (4.18) and representsthe driving force due to the external ield. The second ermcorresponds o the secondand the first terms of the right-hand side of Eqs. (4.18) and (4.19), respectively,and de-scribes he interaction between he external field and Sp.Thus the F and the G terms are induced by the externalfield. The third term corresponds o the third term of theright-hand side of Eq. (4.18) and describes he coupling ofSpl an dSp2. The nonlinear ourth term corresponds o thefourth and the second erms of the right-hand side of Eqs.(4.18) and (4.19), respectively, and represents anhar-

manic coupli ng among the oscillators. As seen rom Eqs.(4.7) and (4.20), theseR and S terms containing Sh areinduced by the C oulomb interaction.To demonstrate he physical significance f this trans-formation, let us temporarily neglect he right-hand side ofthis equation.Then the TDHF equationassumeshe formi8py--f&Sp,=O. (4.27)

As shownbefore,both in fiL, and a,, the diagonalelementsalwayscome n pairs; if a,, is an eigenvaluehen -a, is aneigenvalueas well. We shall denote the correspondi ngeigenvectors p, an d Sp,, respectively.By introducing newvariables,a coordinate

Qv=Sp,+Sp,-,and a momentum

(4.28)

P,= -zn,(Sp,-sp,-), (4.29)we can rewrite these inearized equationsof motion as

&=pv, (4.30)P,= - az,e,. (4.31)This pair of equations epresenta harmonic oscillator withfrequency a,,. We have thus mappe dEq. (4.9) onto theequationsof motions of M/2 coupledharmonic oscillators(4.26).A HO representation ould be most naturally definedby using the normal modesof the entire linear term [left-hand side of Eq. (4.9)], the transformation matrix %could then be definedby the following relation:~2Y2-=fiCi, (4.32)

where fin is a diagonal matrix whose elements are theeigenvalues f 2. The reasonswhy w e do not define henormal modes-ofY as oscillators are as ollows. First, thepresentoscillators definedby our method consist of Ml/2oscillators which are RPA normal modesand M2/2 oscil-lators which are singleelectron-electronor hole-hole pairsas shown n Appendix B. Thus, the oscillators have a clearphysical meaning. Second, we need to diagonalize anM,XM, matrix to obtain oscillators in our method [seeEq. (4.21)], whereas we need to diagonalize the fullMXM matrix to obtain oscillators definedby Eq. (4.32).Thus the presentoscillators are more convenient or prac-tical numerical calculations.Furthermore, the Ml/2 oscil-lators, which come from S&, have a collective nature,22whereas he M,/2 oscillators, which come from Sp,, sim-ply represent single electron-electron or hole-hole pair.Thus, the coupling betweenMl/2 and M,/2 oscillators,namely, the R term is weak. This sugges ts hat the differ-encebetween he presentoscillatorsand the more rigorousset definedby Eq. (4.32) is small.Since p, is block diagonal nto A, and BJ symmetryparts, all the oscillators, which diagonalize Y1, may beclassified nto eitherA, or B, symmetries.As seen rom Eq.(B25), Sv,~y~~#O hen, for example,v is an A, ( B,) os-cillator and v and vN are B, and B,( B, and Ag) oscilla-tors. This indi cates hat A, and B, oscillators do couple nthe equation of motion. This is in contrast to the descrip-

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A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers 2371tion in terms of the eigenstates f the Hamiltonian, whe reii, and B, states do not couple at all. This fundamentaldifferencebetween he oscillator and the eigenstate xpan-sions s related to the nonlinear form of the present equa-tion, as opposed o eigenstate xpansionswhich are linear(a product of an A, and a B, variables can have a B,character). Potentially this allows for a relatively inexpen-sive way of describing complex physical situations, com-pared with eigenstate xpansions.V. NONLINEAR OPTICAL POLARIZA5lLlTlES

To compute the nonl inear optical polarizabilities, weexpand 6p in powers of the external fieldSp(t)=p()(t)+p(2)(t)+p(3)(t)+... , (5.1)

where pc4) t) is the qth order density matrix of the TDHFsolution. The Fock ope rator matrix is further expande d npowers of the external field asSh(t)=h(t)+hc2(t)+h3(t)+... ,

where(5.2)

h~(t)=26,,~y,lpj~)(t)--y,,f~~(t). (5.3)Substituting Fqs. (5.1) and (5.2) into Fqs. (4.9), w e ob-tain the first, the second,and the third order equationsofmotions,

ifip(t> -Liy(t) = [ f(t),p],ikp2(t)-~p2(t)=[h(l)(t),P(1)(t)]

-I- f(thp(t) I,ilip3(t)-~p(3)(t)=[h(*)(t),p(2)(t)]

+ W2wpW 1+[f(f),p(2w.

Taking the Fourier transform of E?qs.5.4)1 +mg(w) = 7-JTr --m g(t)exp(iwt)dt,

(5.4a)

(5.4b)

(5.4c)definedby

(5.5)where g(t) is an arbitrary function of t, we obtain theequationsof motions in the frequency-domain,?%op(*)(o)-Yp(l)(w)= [ f(o),p] , (5.6a)

1= 2?r7-J 1 C[hl(o),p(W--W)]m

where @d(t) is the total polarization to 4th order andP()(t) =O. Fr om Eqs. (5.1) an d (5.9), we see hat Pcq)(t)is given by

+ [ f(o),p()(w-w)l}dw, (5.6b)J. Chem. Phys., Vol. 100, No. 3, t Fkbruary 1994

P(q)(t)=-2eCz(n)p$(t),n

1= 2~T-J ; {[h()(W),f(2)(W--W)]90+ [h2(w) p()(w-co)]+[ f(w>,p2(W--W)]}dw. (5.6~)I iNext, we define a n ew tetradic (MxM) Green func-tion 3 (w) by the following equation:

~,~,(o)=~Si,mSj,n-~ij,mn(W). (5.7)From Fqs . .(5.6) and inverting the matrix, we obtain

p()(o) =s cm> .(whpl , (5.8a)1p2)(~)=~w 2n7-J +m ([,(l)(,>,,(l)(,-,)Im

+ [ f(o),pb-W) 13dw, (5.8b)1p3iw)=%d 2rrSJ +m {[h()(W),f2)(W--W)]co+ [h2(w),p(~)(w--o)]+ [ f(W),p2(W--O)])dw. (5.8~)

In this way, we can obtain interatively the TDH F solution.to arbitrary order in the external field. To reduce compu-fational time, w e have adopted a so mewhatdifferent routefor solving theseequations.The method s outlined in Ap-pendix C. However, the difference s purely technical andthe method is equivalent to the real space epresentationdescribed here. We have added a dampi ng term to theTDHF equation as described n Appendix C. This damp-ing providesa finite linewidth to the optical resonances ndcan represent a simple line broadening mechanism (e.g.,due to coupl ing with phonons ) or a finite spectral resolu-tion.The expectation value of the total polarization opera-tor of a single molecule

9(t)=-(W) ppw, -is

P(t)=---2eCz(n)p,,(t).n (5.9)We shall expandP(t) in powers of the external field

P(t)=P(t)+p2(t)+P(3)(t)+... , (5.10)

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2372 A. Takahashi a nd S. Mukamel: Nonlinear susceptibilities of polymerswhere p$) (t) is obtai ned by taking the inverse Fouriertransform of p$(w). Using P(q)(t) and p(*)(w), we ob-tain our final express ions or the optical polarizabilities(seeAppendix D ),a(--o;m)=--g-+ n = n pnn4e c ( >-y -lBw;w), (5.12)

Y(-3w;.~,o,o)=-~-& n = n pnn4e c ( >-(3'( -3w;W,w,w).(5.13)Equation (5.13) gives he third order polarizability that isresponsible or THG. Other four wave mixing processescan simply be describedby changing he frequencyargu-ments. Extension o higher nonlinearities s also straight-forward.VI. NUMERICAL RESULTS

In this section, we apply our method to the half-ftlledPPP model for polyacetylenewith N=60. In all calcula-tions we used the PPP pa rametersgiven in Sec. II. Thedamping ate [seeEqs. (ClO)] was aken to be I=O.l eV.We shall follow the dynamics of two important physicalquantitieswhich are affectedby the coupling to the radia-tion field. First, the chargedensity at the nth atom, whichdetermines he total polarization is definedbyd,,rl-22p,,. (6.1)

The second uantity is the bond order of the nth bond (p,)which is closely related to the stabilization mechan ismofthe HF ground state, and is definedbyPn= pm+ 1+ Pn+ In * (6.2)We further introduce he bond order parameter,whichmeasureshe strength of bond order alternation

P;=+-lY-l(Pc--fi, (6.3)whereF is the average ond order. It is obtained rom Eq.(A7) as

p=-E .3D,The geometry optimized HF -ground state of thepresentHamiltonian is a bond order wave (BOW), wherep,, alternates etweenevery wo bondsan d d,,=O. The HFground state has an almost uniform bond order parameter@i = 0.24), as show n in Fig. 1. Note that becau se fboundaryeffects, he b ond order parameter ncreases earthe chain edge. As seen rom Eq. (A7), the bond orderparamete r s proportional to the strength of bond lengthalternation, which gives he alternation of tranfer integralas seen rom Eq. (2.5). Thus, the transfer ntegral p,, canbe approxi mated y 8, =B[ 1 ( - 1) S] whereS=0.082 inthis case xcept or the chain edge egion.The BOW struc-ture is stabilized by the ex change , he C oulomb, and theelectron -phonon nteractions.21As indicated .earlier, the TD HF equation is mapped

onto the equations of motion of M/2 coupledharmonic

0.21-----1-----L-.-"'--- -I0 20 n 40 60

FIG. 1. The bond order parameter distribution of the Hartree- Pockground state.

oscillators. These nclude MI/2 oscillators which corre-spond o the eigenvalues f a, and M2/2 oscillators whichcorrespond o the eigenvalues f 0,. Theseoscillatorshavevery different physical properties.Since he MI/2 oscilla-tors are the normal modesof the RPA equation, hey havea collective nature, that is, they are for med by coheren tsuperpositions f many electron-hole airs. This collectiveproperty strongly affects he optical response. s show n nthe following.As far as the linear responses concerned, he systembehaves s a collection of harmonic oscillators (the anha r-monicities only affect the nonlinear response ). Conse-quently, the linear optical susceptibility a( --w;w) can berecast n the Drude form (seeAppendix C),

a( --w;o) =$ CY (6.5)where he summation s performedover the MI/2 oscilla-tors, the oscillator strength ,, of the vth oscillator is givenby

fv=7 [ ; =ehWeh,v+eh,v~]29an d m has the unit of massand determined o give &,f,,=N, m is 1.66m, and 1.59m, in the PPP and SSH models,respectively,wherem, is the massof an electron.From Eq. (6.6), we see hat the collective harmonicoscillatorshavea large oscillator strength coming from thesum of contributions of the various electron and holestates.The extremely arge oscillator strength of the lowestfrequencyMI/2 oscillatorsas show n n Fig. 2 reflects heircollectivenature. Thes eoscillators, herefore,dominate helinear optical respon seunction. On the other hand, eachofthe remain ing n/i,/2 oscillators can be regard edas repre-senting a single electron+lectron or hole-hole pair. Co n-sequently, heir oscillator strengths anish seeEq. (4.19>],and they couple very weakl y with the optically active col-lective oscillators.Thus they play only a secondary ole inthe optical response. ereafter,we consideronly the MI/2oscillators and refer to them simply as the oscillators.The oscillatorscan be further classified nto A, and B,type. The oscillator strengthof the A, oscillators vanishes.

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2374 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymersTABLE I. The frequency (a,), oscillator strength (fv), charge density(d,), and bond order (p,) of the 12 lowest frequency oscill ators.

Oscillator fif2, (eV) fr d, PI3B,(l) 1.99 48.3 0 xA,(l) 2.21 0 0 X4s2) 2.47 4.84 0 X42) 2.75 0 0 XK(3) 3.03 1.78 0 XA,(3) 3.31 0 0 XA,(4) 3.38 0 x 0B,(4) 3.46 0 x 0B,(5) 3.57 0.93 0 XA,(5) 3.58 0 X 04X6) 3.73 0 X 0A,(6) 3.83 0 0 x

cillator s trength. The B,(l), A,(l), B,(2), A,(2), B,(3),A,(3), B,(5), andA,(6) oscillators (in order of,increasingfrequency) have a CD W like electronic structures andhave 0, 1, 2 ..., 7 nodes , respectively. The CDW is stabi-lized by the Madelung energy and is very stable particu-larly in one-dimensional systems. The A,(4), B,(4),A,( 5), and B,(6) oscillators (in order of increasing re-quency) have an oscillating bond order parameter, andhave0, 1, 2,..., 3 nodes, espectively.Since he bond orderparameter,which shows the strength of bond order alter-nation, is locally increasedor decreased,hey have a soli-ton pair like electronic structure.23 he propertiesof the 12lowest frequency oscillators are summarized n Table I.In Fig. 4, we display the linear absorption{Im[o( -w;w)]) and the absolutevalue of the third orderpolarizability connected o THG ( 1 ( -33w;w,o,o) I). Welabel the resonancesn thesespectra by A, B,..., E and a,b,..., g, respectively as indicated in the figure. In order tocompare he three-photon resonanceswith the linear ab-sorption, we have plotted 1 1 vs 31iw.The &J dependence

Bw WI0 3 6

0 3 6

of I y/ for pol yacetylenewas measured n the frequencyrange of 0.4 eV

+Im[~l)(-ol;wl)]si(~l~)}, (6.14a)

/P(t)=& ,Re[j?(2)( -2wl.wl,ol)]cos(2wlt)+Im[~(2)(-201;wl,wl)]sin(2wlr)+...),

(6114b)

ho lev)0 3 6IA

PIG. 4. The linear absorption spectrum Im[a( -o;o)] is plotted vs the frequency o, and compared with the absolute value of the hyperpolarizability1 ( -3qo,o,o) 1 connected to THG which is plotted vs 3~. Left column, P PP model; right column, Hubbel model.



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A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers 2375

0 20 40 60 0.. 10 20n VFIG. 5. Left cohnnn, the.tirst order amplitude of the charge density oscillation induced by the external field at the frequencies of the absorption peaksfiw= (a) 1.99; (b) 2.46; and (c) 3.03eV. We show only the amplitudes Im[i$/(w)] oscillating out of phase with the external field. Right column, thenormalized absolute value of the corresponding first order density matrix in the harmonic oscillator representation, where $(w) is the component ofthe first order density matrix corresponding to the B(y) oscillator and the normaliiation constant is T (l) = E 1$vl) 1. The applied external electric fieldis 10s V/m. y

Note that theseequationshold regardless f the repre-sentation (whether the real space,HFM O or HO). As seenfrom Eqs. (D8a) and (Dl la), thesequantities can be cal-culated successively;Y(-~~o~;o~ wl ,ol) is obtained onlyfrom j5(3)(-30i;ol,wl,wl), which in turn is obtai nedfrom $2( -2wi;wi,wi) and $I( -wi;wi), and $I( --2wl;01,01) is obtained from $I)(---wi;wi). We havekept only terms which contribute to y(-3wl;wl,w1,wl),and all other terms were omitted in theseexpressions. hesingle electron density matrices have a term oscillating inphasewith the external electric field and a term oscillatingout of the phase.The amplitudes of the former terms aregiven by Re[prq)] and they contribute to the real parts ofthe linear and nonli near polarizabilities, and those of thelatter terms are given by Im[jj(q)] and they contribute tothe imaginary parts. Since he charge density is related tothe diagonal elements of the density matrix in the realspace epresentation, t also has terms oscillating in phaseand out of the phasewith the external electric field,.

(p(t) = 1n F CRe[~~q(gwl)lcos(2wlt)an d

+Im[~q(qol)]sin(20tr>+...} , (6.15)

P(& =-2,-(q), --qtii* )nn )... . (6.16)The bond order induced by the external field also hasboth types of terms, and the amplitude is given by~q)(q~l)=~~p,:](-qwl;...)+~~~l,(-qo*;...). (6.17)To analyze the charge dynamics underlying the ab-sorption spectra,we investigate he first order chargeden-

sity induced by the external field. In Fig. 5 we showIm[zi,)] at the frequenciesof the absorption peaksA, B,and C. We show only the imagi nary parts becausehey arestrongly enhancedat the resonance requencies.However,the charge density distributions of the real and the imagi-nary parts are quite similar at every frequency. At thefrequenciesof the peaksA, B, and C, the induced chargedistributions have CD W like structures which are quitesimilar to those of oscillators B,(1), B,(2), and B,( 3),respectively.To see his more directly, we display n Fig. 5the absolutevalue of the components f the density matrixin the HO representation,where [ Sp$*1 shows the com-ponent corresponding o the B,(v) oscillator. At the fre-quency of peakA, the componentcorresponding o B, ( 1)is much larger than the other components.Thus, peak Acan be assignedo the B,( 1) oscillator. At the frequencyof -peaks B and C, the componentscorresponding o B,(2)an d B,( 3) are the largest, respectively,but the B,( 1) os-cillator also has a large,contribution at both frequencies.Thus, peaks B and C can be assigned o the B,(2) an dB,( 3) oscillators, respectively,although the contributionfrom the off-resonantB, ( 1) osci llator is still large becauseof its hug e oscillator strength. At these most prominentpeaks, he componentsof the three lowest energy B, oscil-lators are much larger than the other components.There-fore, we conclude that the absorption spectrum is domi-nated by the characteristic CD W like charge densityfluctuations of these collective B,( 1 , B,( 2), and B,( 3)oscillators. Since hese peaks n absorption are below theHF energy gap, these charge density fluctuations can beregardedas excitons. However, theseexcitons are not sim-ple electron-hole pairs but have the characteristic collec-tive nature of electronic structure of one-dimensional ys-tems.

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2370 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers

FIG. 6. The THG hyperpolarizability 1 ( - 3o;o,w,o) 1 is plotted vs o.%a) One-third of the frequencies of the B, harmonic oscillators and (b)half the frequencies of the A8 harmonic oscillators are also shown in orderto highlight three photon and two photon resonances respectively.

We next consider the frequency dispersion of1 ( -33w;6ww) 1. To compare the frequencies of thepeaksand he oscillators, we display n Fig. 6 the frequencydependence f 171, one-third the frequenciesdf B, oscil-lators, and half the frequencies f A, oscillators. There a resome near resonant oscillators at each peak as seen romFig. 6. However, this cornparis& is not sufficient to iden-tify the origins of the peaks..As will be shown later, weneed o examine he density matrix in each order for suchidentification. This further provides mportant physical in-sight. In Figs. 7, 8, and 9 w e show the density matrices tofirst, second,and third order in the external ield, using hereal spaceand the HO representations. n the real spacerepresentation,we show only zLq qwl) in tlie first and

I , , I I I I0 7.0 40 60n

third order and FL(*) qwl) = ( - l)gnq qwl) in the sec-ond order because ond order is zero in the first and thirdorders and chargedensity is zero n the secondorder. Sim-ilarly, in the HO representation,we show only the B, os-cillator components n the first a nd third orders and onlythe A, oscillator components n the secondorder, since allother componentsvanish. Theseproperties ollow directlyfrom the synimetry of our Hamiltonian.We focusedon the following frequencies i= 1.97eVcorresponding he peakA in absorptionand the peak g inTHG (Fig. 7), tie= 1.63 eV corresponding lie peak e inTHG (Fig. S), and ti=O.67 eV corresponding he peak ain THG (Fig. 9). We first consider he density matrices at+i~= 1.97eV. The frequency of the B,( 1) oscillator is res-onant with this frequency, so that the component corre-spond ing to this oscillator is much larger than the othercomponents n the first o rder. Moreover, the amplitude ofcharge density oscillation is much larger than the othertwo frequencies. n the secondorder, half the frequencyofA,(7). is the closest to 1.97 eV. However, the componentcorresponding o the oscillator is not la rge but that corre-sponding to A,(4) is the largest, and we can observe hecharacteristic so&on pair like bond order oscillation pat-tern of this oscillator in the real space epresentation.OnlyB, oscillators with charge density contribute to the firstorder density matrix. Moreover, only AJY) oscillatorswith bond order (~=4,5,7,...) contribute to the secondorder density matrix becauseA, oscillators with chargedensity do not couple with B, oscillators with chargeden-sity. For the same eason,only B, oscillators with chargedensity contribute to the third order density matrix. SinceA,(4) strongly couples with B,( 1 , which dominates hefirst order density matrix, the A,(4) oscillator is stronglyexcited although t is off resonantat that frequency.There

FIG. 7. Left column (a) the Crst order amplitude of charge density oscillation; (b) second order amplitude of bond order parameter oscillation; and (c)the thud order amplitude of charge density oscillation induced by the external field. Right column, the normalized absolute values of the same orderdensity matrices in the harmonic oscillator representation. $)(qw) is the component of the qth order d&sity matrix corresponding to the B(Y)oscillator when q= 1,3 and corresponding to the AI(v) oscillator when q=2 and the normalization factor is T.,9)= L 1$Yq . Calculations were made forYthe frequency o f the peak f(fio= 1.97 eV).



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A. Takahashi a nd S. Mukamel: Nonlinear susceptibilities of polymers 2377

0.6

0.6--- -:a

lir:"lT@'

n VFIG. 8. Sameas Fig. 7 except that the frequency s fiw= 1.63 eV ( p@ d).

is no unique dominant component, and man y oscillatorscontribute to the density matrix in third order. In spite ofthe strong interferencesamon g oscillators, we clearly seecollective CD W like (charge density) fluctuations in thethird order. The large first order charge density fluctua-tions at this fr&&ency makes second order bond orderparameter oscillations as well as the third order chargedensity fluctuations large, which results in the peak g inTHG. We thus conclude hat this peak s the single-photonresonance orresponding o the absorption peakA.We next consider he density matrix at the frequencyS&=1.63 eV. In first order, although the frequency s offresonancewith respect to B,( 1 , this oscillator is domi-nant and its characteristic charge density distribution isclearly seen. This is because he oscillator strength ofB,( 1) is much larger than all other oscillators, so that it is

-0.06

20 40n

mainly excited even at off-resonance requencies.Becauseof the off-resonance xcitation, the charge density ampli-tude induced by the external field is much smaller co&pared with the single-photon esonant requency%= 1.97eV. In second order, the A,(4) oscillator is dominant.Moreover, the amplitude of bond order oscillations s com-parable o that at the single-photon esonant requencyandmuch larger than for ko=O.67 eV. Thi$ itidicates that thepeak e is a two-photon resonance f AJ?). However, thispeak is not at exactly half the frequency of A,(4) as seenfrom Fig. 6. The shift comes ro& the third order cofitri-butions as will be shown below. Half the frequency of44.3) is closer to that of the peak e than the A&4) oscil-lator. However, this oscillator with no bond order fluctu-ations is not excited, because f the Hamiltonian symme-te. One third the frequencies of the J&(16)-B,(21)

0.4 In ' '. I , _ I

FIG. 9. Sameas Pig. 7 except that the frequency s tiw=O.67 e V (peak a).J. Chem. Phys., Vol. 100, No. 3, 1 February 1994



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2378 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymersoscillators are very close o the peak as seen rom Fig. 6.However, rom 1 i3 1we see hat many B, oscillatorscon-tribute to the density matrix and the B,( 16)-B,(21) -OS-cillators are not the dominant excitations. Therefore, thepeak e cannot be identified as a three-photon resonance.However, becauseof the relatively large contribution ofB,( 18), the peak e is shifted from half the frequency ofA,(4) towards one-third the frequency of B,( 18). Thisillustrates the importance of interferencesamong oscilla-tors.Finally, we consider the density matrix at ti=O.67eV. In the first (second) order, the amplitude of cha rgedensity (bond order) oscillation is much weaker han thatat the single( wo) -photon resonant frequency discussedabove.Therefore, his is not a purely single-or two-photonresonance.Although B,( 1) in first order and A,(4) insecond rder have elatively large contributions, his is alsobecause f the huge oscillator strength of B,( 1) and strongcoupling between B,( 1) and A,(4). In third order, theB,( 1) co mponent s dominant, and the amplitude of thecharge density oscillation is comparab le o those at theother two frequencies.Moreover, peak a is precisely atone-third the frequency of B,( 1) We, therefore;concludethat the peak a in T HG is the three-photon resonancecorresponding o the absorption peak A. In this way, wecan identify all the resonancesn the THG spectrum.In summary, we have made the following identifica:tions: (i) peak b comes rom three-photon resonance oB,( 3) (corresponding o the absorptionpeak C) ; (ii) peakc is a three-photon resonance o B,(7) (corresponding othe absorptionpeak E); (iii) peakd is a three-photon es-onance o B,( 10) and B,( 12). At this frequency,however,also B,(7) contributes to the density matrix significantlyand they strongly interfere; (iv) peak f is a two-photonresonanceo A,(5). Howeve r he contribution from A,(4)is the largest and theseoscillators stronglyinterfere at, hisfrequency.Using these esults, we have dentified the mostimportant oscillators, namely, B,( 1) with CDW like elec-tronic structure and AJ 4) with soliton pair like electronicstructure. However, our analysisclearly shows hat inter-ferencewith the other oscillatorscannot be neglected n theinterpretation of the dispersedTHG spectra.Next, we compare he PPP and the Hiickel results nFig. 4 to illustrate the effect of Coulo mb nterac tion. Thefollowing parameterswhich reproduce he experimentallyobserved nergygap of polyacetylene 2.0 eV) are used nthe Hiickel calculations: U=O, p= -4.4 eV/A, K=20eV A-. Other parametersare taken to b e the sameas forthe PPP model. The TDH F Eq. (4.22) in the oscillatorpicture, shows he following two effectsof Coulomb nter-action. First, since the matrix 9 in the TDH F equationdepends n 3/m,,, he oscillators which diagonalize2, arevery different for the two models; few lowest frequencyoscillators represent collective excitations in the PPPmodel. n contrast, the Hiickel oscillatorssimply representsingleelectron-holepairs. As can be seen rom Figs. 2 a nd4, thesedifferencesprofoundly affect the absorption spec-tra; few lowest requencycollectiveoscillatorscarry almostthe entire transition strength n the PPP model, whereas n

the Hiickel model, the oscillator strength is much moreuniformly distributed. Second, the Coul omb interactionstrongly affects he coupl ing betweenoscillators. In partic-ular, the anharmoni ccoupling [the last term in the right-hand side of Eq. (4.26), which couples he various EPAmodes, takes into account correlation effects beyond theRPA approximation, or beyond configuration interactionwith single electron hole pair states.The anharmoniccou-pling comes rom Coul omb nteractions,and t vanishes orthe Hiickel model where he only sourceof nonlinearity isthe harmonic coupling among modes, nduced by the ex-ternal field (the second erm in the right-hand side of Eq .(4.26)].The Coul omb interaction strongly affects the disper-sion of THG: In the Htickel model, all the major peaksa,b,...,f in the THG spectraare simply three-photon eso-nances orresponding he A, B,...,F peaks n the absorptionspectra, as seen rom Fig. 4. This is quite different for thePPP model.Abe et al. have calculated THG spectra by summingover the excited states obtained by configuration interac-tion including only single electron-holepair states. 5 Theircalculation differs from ours mainly in the following twopoints. Fist, they used he Hiickel ground state as opposedto the H F ground state in the present calculation. There-fore, their method is valid only when the Co ulomb inter-action is very weak. However, since exchangeCoulombinteraction between adjacent sites, which stabilizes theBO W (HF ground state), can be incorporated via therenormalized Hiickel parameters, his probably does notmake a significant difference. Second, heir method candescribecollective excitations but does not take the non-linear coupling between hesecollective excited states ntoaccount. Because f thesedifferences, hey obtaineda verydifferent dispersed HG spectrum.That calculation showsstrongestpeaksat the three-photon esonant requency ofthe lowest frequency B, exciton state, three-photon reso-nanttpeak of the conduction band edge, and two-pho tonresonant peak of the lowest frequency A, exciton state.There is a direct correspondence etween he first peak nboth calculationsbut we find no anal og to the other tworesonances. his shows that the anharmoni c couplings,which represent correlation effects beyond the RPA ap-proximation, strongly affect the THG spectra.The calculatedTPA spec trum m[y( -w;o,-ti,o)] isdisplayed n Fig. 10. t showsa hugenegativepeak-nearhestrongest absorption resonance and two weak positivepeaks at the lowe r and higher energy sides of the peak.Since hesepeaksare close o the absorptionpeak, t i s verydifficult to resolve them experimentally. However, whenwe use parametersappropriate or polydi acetylene stron-ger bond ength alternation), the positive peak at the lowerenergyside shifts towards a lower energy, and the presenttheory can account for the experimental wo-photon ab-sorption spectrum of polydiacetylene.We also show thereal and imaginary parts of the nonlinear optical polariz-abilities connected o TPA and THG, and their phasesdefinedby sin +=Im[r]/]rl .32The phaseprovides a sen-sitive signature or the resonance tructure.



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A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymers

-38ggF4zf$gTsrE

FIG. 10. The real part, the imaginary part and the phase sin q%=Im[y]/l y[ of the third order nonlinear polarizabilities corresponding to third ha rmonicgeneration and two photon absorption are plotted vs COor the PPP model.

VII. DISCUSSIONStarting with exact calculations of small size chainswith up to 12 atoms, severalauthors argued hat there arefour essential tates which almost dominate the nonlinearoptics of r-conjugated polymers.6 They are lA,, m A,,1B,, and n B, states,where, n A, indicates he nth lowestenergyA, state, etc. The lA, state s the ground state andm and n depend on the system size. As indicated in the

previous section, B,( 1) makesa large contribution to theoptical nonlinearity because it has a large oscillatorstrength, and A,(4) does contribute as well because tstrongly coupleswith B,( 1). Thus, the B,( 1) and A,(4 joscillators correspond o the 1B, and m A, excitedstates nthe essential tates picture. We found no oscillator clearlycorresponding to the n B, state. However, since bothB,(2) and B,( 3) have a relatively large contribution, oneof them may correspond o the n B, state. As indicatedearlier, we canno t neglect the contributions from a l argenumber of oscillators to the THG dispersion n our calcu-lation. This is at variancewith the essential tatespicture.There are several possible reasons or these differences.First, the oscillators in our picture are not ju@ diff&entways of specifyi ng excited states; the oscillators interfereand we can have resonances t the differencesof their fre-quencies.When the interference s very strong (which isthe casehere), we cannot establisha clear one to one cor-respondence etweenoscillatorsand excited states.Second,the essentialstates picture is based on the calculation ofshort chains with at mos t 12 atoms. As seen from theelectronic structure of the oscillators shown n Fig. 3, theyhave characteristic length scalesmuch larger than 12 at-oms. Therefore, in such short chains, the chain lengthstrongly affects he electronicstructure of the oscilla tors, aswell as the co rresponding nonlinear optical response.

Third, although some electron correlation effects beyondthe RPA are taken nto account,some of theseeffectscan-not be described n our method. However, since he TDH Fapproximation usedhere can describe mall amplitude col-lective fluctuations and their couplings very well, the ap-proximation is particularly applicable to large systems,where collective motions are expected o be dominan t.We have taken the electron-phonon coupling into ac-count in calculating the geometry optimized HF solution,but dynamical attice motions were neglected n the presentcalculation. Since the mass of a carbon atom is muchheavier than that of an electron, the effect,of lattice mo-tions is usually neglected.However, n the caseof polyacet-ylene, the soliton mass is comparable o that of an elec-tron,33 and soliton like motions strongly affect the linearoptics. Furthermore, Hagler and Heegerhavearguedusinga simplified model that quantu m attice fluctuations signif-icantly increase he o&resonant nonlinear optical suscep-tibilities.7 This is an importan t subject for a future study.Note that it is straightforward to take the dynamics oflattice motions into account in our oscillator picture be-cause his simply involves adding more oscillators to themodel.It is generally accepted hat photoexcitation results inthe formation of chargedsolitons.33 chargedsoliton hasCD W like charge distribution around the soliton center.34Thus the characteristicchargedistributions inducedby theexternal field are very similar to those of a cha rgedsoliton.This suggestshat theseexcitonsmay play some ole in thedecayprocess o chargedsolitons. This could be seenmoredirectly using ultrafast four wave mixing spectroscopy,which will be studied in the future. Both in the presentwork and in Ref. 8, the nonlinear polarizabilities are cal-culated by solving equations of motion for the reduced

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When the expansion of Sp in powers of the external fieldEq. (5.1) is substituted in E?qs. C5) and (C69, and per-forming a Fourier transform to the frequency domain, weobtain

/J(o):;! o, (C7b9

pb>;;?= 1SJ-2rr -mmT p(w9~~)p(w--o9~~!dw,(C8b91.. Q)p(w>;!=- 273

7-Jc [Pb;9~;~pb--o9~!-09 k

1p(w9$)= 2;7-J -m_ c ~pb9:~p(w-o9g

Similarly, we can calculate piq from p(q-),...,p() with-out solving the TDHF equation directly. However, theTDHF equations are required in order to calculate p, .Next, we consider p1q). Substituting Eq. (5.1) into Eq.(4.18) and using EQ. (4.219, we obtain for the TDHFequation for the first order in the HO representation

~fio+~r9pi?bJ9 -4 vp11$9 = 1 f (@9,jd,,(ClOa)where we added the damping term and l? is the dampingconstant. We obtain pl l) from this closed equation. Thesecond and third order density matrices obtained in thesameway as(sm+ir)p~2)(m) -43~~ vp;2(w) .,__

=[h:2(w9,&+ C[h()(o),p()(w-w91~co ---I- 1 W9,p%-a91 Ido 9 (ClOb)

(h+ir9p13;(09 --pin1 yp13b9=V;3(m9 PIY+ C[h()(W9,p(2)(W--O91,m

+ [h2(w9,p%o--o9 I, 5+ [ f (o9,p(2)(w--~l,ldw.

(ClOc)

2382 A. Takahashi and S. Mukamel: Nonlinear susceptibilities of polymersWe can calculate pl q) from piq and lower order densitymatrices using Eqs. (ClO). fn this way, we can calculatedensity matrices to the arbitral order.Finally, we derive the Dr-ude ormula for linear absorp-tion. The qth order polarization Pcq) can be expressed singthe density matrix in the HFMO representation

Pq(~9=-2eCzeh[p~~(W9+p~~(W9],eh (Cl19where

zkk = c Vkkt,nri+)* (Cl29nUsing the matrices X and Y, the first order solution of theTDHF EQ. (ClOa) can also be represented n the HFMObasis. Substituting the solution into Eq. (Cl 19, we obtainEqs. (6.5) and (6.6).

APPENDIX D: OPTICAL SUSCEPTIBILITIES ANDCHARGE DENSITY FLUCTUATIONS

In this appendix we review the basic definitions of non-linear optical polarizabilities and relate them to our equa-tions of motion.We first consider the following single mode opticalelectric field:E(t) =El cos qt.

performing a Fourier transformation, we obtain(Dl9

E(w)= ;&s(o--w*9+ ;Els(w+wl).$ l-Substituting ELq. D2) into Eq. (3.99, we get

f(w) =.7bw(w-019 +7(--wl9S(w+ol),where

0329

CD39

f,A fw9 = f G,,nez(n9El.$Substituting Eq. (D3 9 nto Eq. (5.8a), results in

CD49

p(w9 =~l(w~~--w,)s(w+wl)+p(--opl*)s(w--wl) ,

where(D59

iwro*;*w19=Y(*lo19[ f(&wl),p].Equation (D59 together with Eq. (5.8b) yieldp(2)((39=~(2)(2wl;--w*,--wl)6(w+2wl)

+~~20Khq,w~9S(w9 F( -201;wl,w19x&w-2q). CD71

p is given byJ. Chem. Phys., Vol. 100, No. 3, 1 February 1994



18/19

A. Takahashi and S. Mukamel: Nonlinear susceptibilities of pdymers 2383

jP(=F20~;fwr,fot)=& S(*2wl){[~(rwl;~~~) ,,~)(rwl.~,wl)l+[~(~twl) ,$( =Fq**tw*>) , Wa)

~2)(o-w*,w*)=j& Y(o)c~~~l((--w,;w,),~(w,;--w,)l

-I-~~~~0,;--w,~,~~~-w~;w~~1+f(q),~l%W--l)l+ [f(-wl),~(-ol;ol)l},

(D8b)where we defineg)( --wl;ol) by the following equation:

I

--y $ ( -O*W*)In nm , CD91

[other @ with different frequenciesand with differentorders, are defined n the sameway).Substituting Eqs. (D5), (D7), and (D9) into IQ.(5.8c), we obtainp~3~(o)=p~3~(3w*;--o~,-w~,--o*)~(w+3w~)

+p3)(w*;- q,--wl,qMb+o,)+p( -W*W* w, -Wl)}6(W--WI)7 3+~3)(-3~1;o,,o,,w1)6(o-3~1), (DlO)

where

(Dl la)

I

and so forth. Performing the inverse Fourier transforma-tion of Eq. (DlO) with Eqs. (Dll), we obtain/d3)(t) = [~3)(-301;wl,wl,wl)exp(3iwlt) +j?3

x (--l;wl,ol,--*)exp(iwlt) +h.c.l, 0312)where the relation

ip]Q.o) =pq -w), (D13)h&sbeenused.The optical polar&abilities are defined using the totalpolarization P(t) of a single molecule

P()(t)=~~a(--w*;~l)exp(io,t)+c.c.]E*, (D14a)P()(t)=: [y( -3wl;wl,wl,wl)exp(3i~l t)

=ty( --ol;wl,--l,wl))exp(iwlt) +c.c.l&(D14b)

where a ( --w;w ) is the linear polarizability andy( -33o;w,w,o) and y( -o;o,-~0) are third-order opti-cal polarizabilities connected o TH G and TPA, respec-tively. Note that this definition is the sameas the co mmondefinition in the off-resonant requency region, where theimaginary parts of the polarizabilities vanish. ComparingEqs. (D14) with Eqs. (5.11) and (D12), we obtain FL+.(5.12) and (5.13).Nonlinear Optical Properties of Organic and Polymeric Materials, editedby D. J. Willi ams (American Chemical Society, Washington, 1983),Series 233.Nonlinear Optical E&&s in Organic Polymers, NATO Advanced StudyInstitute, Series E, edited by J. Mrssier, F. Kajzar, P. N. Prasad, and D.R. Uhich (Kluwer Academic, Dordrecht, 1989), Vol. 162.Nonlinear Optical Properties of Organic Molecules and CrystaIs, editedby, D. S. Chemla and J. Zyss (Aca demic, New York, 1987).4Nonlinear Optical and Electroactive Polymers, edited by P. N. Prasa dand D. Uhich (Plenum, New York, 1988).Nonlinear Optical Properties of Polymers, edited by A. J. Heeger, J.Orenstein, and D. R. Ulrich (MRS, Pittsburg, 1988), Vol. 109.6S. Etemad and Z. G. Soos, n SpectroscopyofAdvanced Materials, editedby R. J. H. Clark and R. E. Hester (Wiley, New York, 1991), pp. 87,and references therein.

J. Chem. Phys., Vol. 100, No. 3, 1 February 1 994Downloaded 07 Mar 2001 to 128.151.176.185. Redistribution subject to AIP copyright, see http://ojps.aip.org/jcpo/jcpcpyrts.html


19/19

2384 A. Takahashi and S: Mukamel: Nonlinear susceptibilities of polymersT. W. Hagler and A. J. Heeger, Chem. Phys. Lett. 189, 333 (1992).sS. Mukamel and H. X. Wang, Ph ys. Rev. Lett. 6 9, 65 (1992); H. X.Wang an d S. Mukamel, J. Chem. P hys. 97, 8019 (1992).B. J. Orr a nd J. F. Ward, Mol. Phys. 20, 513 (1971).B. Kirtman, Int. J. Quantu m C hem. 43, 147 (1992).G. P. Agrawal, C. Cojan, and C. Flyzanis, Phys. Rev. B 16,776 (1978).W. K. Wu, Phys. Rev. Lett. 61, 1119 (1988).13J. Yu et aL, Phys. Rev. B 39, 12 814 (1989 ).14X. Sun, K. Nasu, and C. Q. Wu, Appl . Phys. B 54, 170 (1992). -lS;Abe. M. Schreiber, W. P. Su, and J. Yu, Ch em Phvs: Lett. 192, 425(1992); S. Abe, M. Schreiber, W. P. Su, and 3. Yu, Whys. Rev. B 45,

9432 (1992).t61. Ohmine a nd M. Karplus, J. Chem. Phys. 68, 2298 (1978); K.Schulten, I. Ohmine, and M. Karplus, ibid. 64, 4l22 (1976); G. J. B.Hurst, M. Dupuis, and E. Clementi, ibid. 89, 385 (1989).S. R. Marder, J. W. Perry, G. Bourhill, C. B. Gorman, and B. G.Tieman, Scie nce 261, 186 (1993); S. M. Risser, D. N. Beratan, an d S.R. Marder, J. Am. Chem. Sot. 115, 7719 (1993).U. Fano, Rev. Mod. Phys. 29, 74 (1957).19N. Bloembergen, Nonlinear Optics (Benjamin, New York, 1965).20F. C. Spano and S. Mukamel, Phys. Rev. Lett. 66, 11 97 (1991); Phys.Rev. A 40, 5783 (1989).2S. Mukamel, in Molecular Nonlinear Optics, edited by J. Zyss (Aca-demic, New York, 1994), p. 1.=A. F. Garitos, J. R. Heflin, F. Y. Wong, and Q. Zamani-Khamiri, inOrganic Materials for Non linear Optics, Special Publication No. 69,edited by R. A. Hann and D. Bloor (Royal Society of Chemistry,London, 1988); JLR. Heflin, K. Y. Wong, Q. Zamani-Khamiri, and A.

F. Garito, Phys. Rev. B 38, 157 (1988)< Mol. Cryst. Liq. Cryst. 160, 37(1988).23H. Fukutome, J. Mol. Struct. (Theochem) 188, 337 (1989); and refer-ences therein.24Z. G. Soos, in Electrorespon sive and Polymeric Systems, edited by T. A.Marcel (Dekker, Ne w York, 1988), Vol. 1.25D. Yaron and R. J. Silbey, Phys. Rev. B 45, 11655 (1992).26P. Ring and P. Schuck, Th e Nuclear Many-Bod y Problem (Springer,New York, 1980).27See, or example, H. Se kino and R. J. Bartlett, J. Chem. Phys. 94,3665(1991); 98, 3022 (1993).A. Stahl and I. Balslev, Electrodynamics of the Semiconductor BandEdge (Springer, Berlin, Heidelberg, 1987).29S. Mukamel, J. Chem. Phvs. 93, 1 (1982).3oW. S. Fann,.S. Benson, J.-M. Madly, S. Etemad, G. L. Baker, and F.Kajzar, Phys. Rev. Lett. 62, 1492 (1989).31C Halvorson, T. W. Hagler, D. Moses, Y. .Cao, and A. J. Heeger,Chem. Phys. Lett. 200, 364 (1992).W. E. Torruellas, D. Neher, R. Zanoni, G. I. Stegeman, and F. Kajzar,Chem. Phys. Lett. 175, 11 (1990); W. E Torruellas, K. B. Rochford,R. Zanoni, S. Aramaki, and G. I. Stegeman, Opt. Commun . 82, 94(1991); S. Aramaki, W. Torruellas, R. Zanoni, and G. I. Stegeman,ibid. 85, 527 (1991).A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su, Rev. Mod.Phys. 60, 781 (1988), and references therein.34H. Fukutome and M. Sasai, Prog; Theor. Phys. 69, 373 (1983).35V. B. Koutecky, P. Fantucci, and J. Koutecky, Chem. Rev. 91, 1035(1991).

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Akira Takahashi and Shaul Mukamel- Anharmonic oscillator modeling of nonlinear susceptibilities and...

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