article electronic coherence and collective...

Electronic Coherence and Collective OpticalExcitations of Conjugated MoleculesShaul Mukamel,* Sergei Tretiak, Thomas Wagersreiter, Vladimir Chernyak

Optical spectroscopy of conjugated molecules is described by using collective electroniccoordinates, which represent the joint dynamics of electron-hole pairs. The approachrelates the optical signals directly to the dynamics of charges and bond orders (electroniccoherences) induced by the radiation field and uses only ground-state information, thusavoiding the explicit calculation of excited molecular states. The resulting real-spacepicture is reminiscent of the normal-mode analysis of molecular vibrations and offers aunified framework for the treatment of other types of systems including semiconductornanostructures and biological complexes. Spatial coherence displayed in two-dimen-sional plots of the five electronic normal modes that dominate the optical response ofpoly( p-phenylene vinylene) oligomers with up to 50 repeat units (398 carbon atoms) inthe 1.5- to 8-electronvolt frequency range suggests a saturation to bulk behavior at aboutfive repeat units.

Spectroscopy allows chemists and physiciststo probe of the dynamics of vibrations andelectronic excitations within molecules andsolids. The theoretical models used for inter-preting molecular spectra compared withthose for extended solids are usually quitedifferent, and certain systems, such as clus-ters and polymers, are not readily treated byeither of these limiting cases. There is par-ticular interest to understand the opticalspectra of large polymers, which are extend-ed conjugated molecules such as poly(p-phe-nylene vinylene) (PPV) oligomers (Fig. 1)that have interesting optical applications.

Electronic and optical properties ofsmall conjugated chains can be interpretedmolecularly in terms of their global many-electron eigenstates obtained from quan-tum-chemistry methods (1, 2). Large poly-mers also can be analyzed with semi-conductor band theories that focus on thedynamics of electron-hole pairs (3). Thesize-scaling of the optical response and thetransition between these two regimes hasnot been fully explored for the lack of ade-quate theoretical methods. It is very hard toobtain the complete set of eigenstates,which carry considerably more informationthan necessary for the calculation ofspectra, for large molecules with strongelectron correlations (as occurs in conjugat-ed chains). Band theories, however, neglectelectronic correlation effects, and becausethey are formulated in momentum (k)space, they do not lend themselves easily toreal-space chemical intuition.

The collective-electronic oscillator(CEO) representation (4, 5) provides a hy-

brid formulation that bridges the gap be-tween the chemical and semiconductorpoints of view. This model uses an electron-hole picture in real space, overcomes manyof the difficulties associated with the formerapproaches, and provides a physically intu-itive link between electronic structure andoptical properties, that is, the optical prop-erties are related directly to the motions ofcharges and electronic coherences, thusavoiding the need to calculate the global(many-electron) eigenstates. The electronicoscillators, unlike the electronic orbitals,are directly observable spectroscopically(4–6). Despite the quantum nature of elec-tronic motions, the collective oscillators areclassical (7, 8), which relates well withchemical intuition. Typically only a fewoscillators dominate the response, greatlysimplifying the theoretical description. Areal-space picture of linear absorption thatpinpoints the origin of each optical transi-tion is obtained by two-dimensional (2D)display of the electronic mode matrices.Our results provide a unified description ofthe optical response of small and large mol-ecules as well as bulk materials.

The CEO Approach

The oscillator picture that we use here ismore familiar in the analysis of vibrationalspectroscopy (9), in which the coherent mo-tions of various atoms with well-defined am-plitude and phase relations are representedby collective nuclear coordinates—the nor-mal modes. The normal modes provide anatural coordinate system and allow an al-ternative classical real-space interpretationof infrared or Raman spectra (9, 10) insteadof a description in terms of transitions amongspecific vibrational states. The normalmodes of nuclear vibrations are simply super-

positions of the 3N nuclear displacements.Extending this concept to electronic mo-tions is not straightforward, however, be-cause spectroscopic observables are highlyaveraged, and following the complete many-electron dynamics is neither feasible nordesirable. For this reason, the oscillatorpicture is normally not used for electronicspectroscopy.

In this article we show how a CEOpicture can be rigorously established for op-tical excitations of conjugated molecules.We demonstrate how a natural set of elec-tronic coordinates can be constructed byusing the reduced single-electron densitymatrix (4, 5, 11) and how it offers tremen-dous conceptual as well as computationaladvantages. Consider a conjugated mole-cule described by a basis set fn of Natomic orbitals. [For simplicity we use in thecalculations presented below the Pariser-Parr-Pople (PPP) Hamiltonian where eachcarbon atom has a single p orbital (6, 12).N then coincides with the number of car-bon atoms.] The system can be described bythe Fermi operators cn

1 and cn representingthe creation and annihilation, respectively,of an electron in fn. The complete many-electron wave function representing thesystem’s ground state will be denoted cg(x),x being the complete set of electronic co-ordinates. The reduced single-electronground-state density matrix is then definedas the expectation value (13)

#rnm[^cgucn1cmucg& (1)

(Spin indices have been omitted for brevi-ty.) The physical significance of #r has beenrecognized since the early days of quantumchemistry (14). We first note that #r is anN 3 N matrix. The density matrix carriesconsiderably less information than the com-plete many-electron wave function (makingit much easier to calculate); however, thisinformation is sufficient to calculate all op-

The authors are in the Department of Chemistry and theRochester Theory Center for Optical Science and Engi-neering, University of Rochester, Rochester, NY 14627,USA.

*To whom correspondence should be addressed.

Fig. 1. Geometry and atom labeling of PPV oligo-mers. Bond angles are 120°, except a(r6,7, r7,8) 5128°, and the distances are r1,2 5 r2,3 5 r3,4 5r4,5 5 r5,6 5 r6,1 5 1.39 Å, r6,7 5 1.44 Å, andr7,8 5 1.33 Å.

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tical properties and develop an intuitivephysical picture of the optical response.[cg(x) allows us to calculate the expectationvalues of products of arbitrary numbers of cnat cn

1, whereas #r only gives the binaryproducts, which represent operators that de-pend on a single electron, hence its name.]#r is thus the quantum analog of the single-particle distribution in classical statisticalmechanics (15). The ground-state densitymatrix #rnm may be obtained by using stan-dard quantum chemistry packages. Its diag-onal elements (n 5 m) represent the chargeat the mth atom, whereas the off-diagonalelements (n Þ m) reflect the strength ofchemical bonding between each pair of at-oms and are known as the bond orders.Bond order is thus associated with a phaserelation (electronic coherence) between or-bitals. The eigenvectors of #r known as thenatural orbitals provide a convenient basisset for performing configuration interactioncalculations and for interpreting chemicalreactivity (16, 17).

When the molecule is driven by an ex-ternal electric field (such as provided by aphoton in a spectroscopic measurement), its

wave function (and consequently, the re-duced density matrix) becomes time-depen-dent, such that r(t) 5 #r 1 dr(t). Thematrix elements drnm(t) represent thechanges induced in the density matrix bythe electric field. drnn(t) is the net chargeinduced on the nth atom, whereas drnm(t),n Þ m, is a dynamical bond order represent-ing the joint amplitude of finding an elec-tron on atom m and a hole on atom n.

Quantum chemistry techniques that cal-culate properties such as polarizabilities byusing the many-body wave functions rapidlybecome more complicated with molecularsize and are therefore limited to small mol-ecules. Furthermore, in most practicalchemical applications we need much lessinformation than is carried by the completeeigenstates. This makes it hard to develop asimple intuitive understanding of varioustrends. A time-dependent procedure for cal-culating dr(t) directly for a molecule inter-acting with an external electric field ε(t)can be obtained by starting with the Heisen-berg equation of motion for cn

1cm. Thisequation is not closed, because higher orderproducts will show up when the time deriv-

ative is calculated. Writing equations of mo-tion for these higher order products willyield increasingly higher order products.This is the famous hierarchy of many-body(classical and quantum) dynamics. To over-come this difficulty we need a truncationprocedure. The simplest procedure assumesthat the many-body wave function is givenby a single Slater determinant at all timesand yields the time-dependent Hartree-Fock(TDHF) equations of motion (18):

dr 5 Adr 1 Bdrdr 2 ε~t!mdr (2)

The coefficients (A, B, and m) in theseequations are readily available and dependon the original Hamiltonian and on #r(which is the essential input in the presentapproach). Figure 2 displays #r of a linear30-atom polyacetylene chain as well as theinduced density matrix dr(v) [the Fouriertransform of dr(t)] to first order in theexternal field [ε(t)], for three frequenciescorresponding to the lowest peaks in theoptical absorption. #r is almost diagonal;only nearest neighbors have significant off-diagonal elements. This result is in agree-ment with our elementary picture of chem-ical bonding. Optical excitations, however,induce electronic coherences between at-oms that are much farther apart, as is clearlyseen in Fig. 2, B, C, and D.

With Eq. 2 the time-dependent densitymatrix (and optical excitations) can be cal-culated directly from #r. By understandingthe mechanism for the creation of the co-herence, we can develop a new type ofchemical intuition and relate the opticalresponse directly to the motions of chargesand bond orders. This is an attractivealternative to the conventional descrip-tion of spectra in terms of transitionsamong eigenstates. Both pictures are cor-rect, and for historical reasons, chemicalintuition is traditionally based on eigen-states. However, we argue that the real-space picture is much more natural, intu-itive, and easier to implement once theproper terminology is developed. To illus-trate this point, consider the variation ofoptical properties such as the electronicband gap and its oscillator strength or themagnitude of the off-resonant polarizabil-ity with molecular size. These propertiesstrongly depend on size for short chainsand level off at about 20 to 30 doublebonds, where they attain the bulk values(19). It is impossible to visualize this co-herence size by examining the molecularorbitals. These orbitals are completely de-localized, change gradually and continu-ously with chain length, and contain nosignatures of this coherence size. In con-trast, by looking at the induced densitymatrix one can immediately note the co-herence size associated with its off-diago-

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Fig. 2. Contour plots of density matrices of a 30-carbon polyacetylene chain. (A) Ground-state densitymatrix #r; (B, C, and D) frequency-dependent density matrices dr(v) at v 5 2.5, 3.5, and 4.7 eV (496,354, and 264 nm) corresponding to the lowest three dominant peaks in the absorption spectrum. Theaxes represent the individual carbon atoms, and the color code is shown in Fig. 4.

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nal section. This coherence size, whichmeasures how far apart different atomscommunicate, controls the scaling of op-tical properties with size, as will be dem-onstrated below.

At this point, we return to the analogywith classical molecular vibrations. Thedisplacements of nuclear positions fromtheir equilibrium values satisfy nonlinearequations of motion resulting from theanharmonic force fields. Infrared and Ra-man spectra are usually interpreted by us-ing normal modes obtained by diagonaliz-ing the linear (harmonic) part of theseequations of motion. Normal modes arenatural collective coordinates for atomicdisplacements. Nonlinear (anharmonic)effects can be treated as perturbations. Incomplete analogy, drnm(t) represent thedisplacements of the electronic densitymatrix elements from their equilibrium(ground state) values #rnm. The nonlinearTDHF equations are the electronic coun-terpart of the classical Newtons’ equationsof motion of nuclear displacements. Bydiagonalizing the linearized TDHF equa-tions, we obtain a set of collective elec-tronic normal modes. The induced densitymatrix can then be expanded as a super-position of these collective modes in thesame way that an atomic nuclear displace-ment is a superposition of the vibrationalnormal modes. Each normal mode (elec-tronic oscillator) with frequency Vn is de-scribed by a coordinate Qn and momentumPn. Qn and Pn are also N 3 N matrices (5).These N2/4 coordinates and momenta al-low us to represent the time-dependentdensity matrix in the form (4)

dr~t! 5 On51

N2/4

an~t!Qn 1 bn~t!Pn (3)

The time-dependent coefficients an andbn are obtained by solving the TDHFequations.

The optical polarization is related to thecharge distribution and may be expressed interms of the diagonal elements of dr(t). Thepolarization along the z axis is given by

^3~t!& 5 Oezn drnn~t! (4)

where zn is the z coordinate of the nth atom,and e is the electronic charge.

In the CEO method (4, 5), the N2 ma-trix elements of dr(t) are obtained by solv-ing the closed nonlinear TDHF equationsof motion (4). These equations map thecalculation of the optical response onto thedynamics of coupled electronic oscillators(analogous to calculating molecular vibra-tions), thus avoiding the tedious calculationof the global (many-electron) wave func-tions. Infrared and Raman spectra are great-ly simplified by selection rules that allow us

to include only a few modes in the calcula-tion. The same is true for the electronicnormal modes: Only a few dominant modestypically determine the spectra, thus greatlysimplifying the physical picture and reduc-ing the computational effort.

2D Real-Space Analysis ofOptical Responses

We investigated the electronic excitationsof PPV oligomers (Fig. 1) (6, 20–27) andanalyzed their scaling with size. Recent in-terest in PPV is connected with its possibleuse as a photoconductor (28, 29), as acandidate for electroluminescent devices, orfor optical switches.

The p molecular orbitals of PPV havebeen classified as either localized (l) ordelocalized (d) (26, 28). The former havean electron density on carbon atoms 1, 2,4, and 5 (Fig. 1), whereas the latter aredelocalized over all carbon atoms. Theexperimental absorption spectrum of PPVthin film (21) shown in Fig. 3A (dashedline) is typical for other PPV derivatives(21, 25, 28). It has a fundamental (d 3d*) band at 2.5 eV (496 nm) (I), two weakpeaks at 3.7 eV (335 nm) (d 3 d*) (II)and 4.8 eV (258 nm) (l3 d* and d3 l*)(III), and a strong (l3 l*) band at 6.0 eV(207 nm) (IV). Peak II originates fromelectron correlations (26, 28) and ismissed by HF calculations. The calculatedspectrum of PPV(10) shown in Fig. 3A (solidline) closely resembles the experimentalspectrum and has similar features at 2.83 (I),3.3 (II), 4.5 (III), and 5.6 eV (IV) (438, 376,276, and 221 nm). In addition, it shows afifth band centered at 7.0 eV (177 nm) (V).The oscillator strengths fn of PPV(10) areshown in Fig. 3A.

By displaying the dominant oscillators inthe site representation, we obtain a newpicture that relates the optical propertiesdirectly to motions of charges in the system,without introducing electronic eigenstates.The extent of spatial coherence then pro-vides a view of the underlying coherencesizes. A 2D plot of #r of PPV(10) is shown inFig. 4A. The coordinate axes represent re-peat units along the chain, and the absolutevalues of matrix elements are depicted bydifferent colors. Similar to Fig. 2, #r is dom-inated by the diagonal and near-diagonalelements, reflecting the bonds between near-est neighbors. A single unit of Fig. 4A on anexpanded scale is shown in Fig. 4B with theatom labeling given in Fig. 1. It reflectsbond-strength distribution over the benzenering (elements 1 to 6), strong double bond(elements 7 and 8), and weaker single bond(elements 6 and 7) of the vinylene group.This bonding pattern is to be expected fromthe molecular structure.

We next examine the coordinates Qn

and momenta Pn of the dominant elec-tronic oscillators. Vibrational normalmodes represent coherent displacementsof various atoms, and these electronicmodes represent the displacements of theelectronic density matrix with respect to#r. The diagonal elements reflect inducedcharges on various atoms, whereas the off-diagonal elements represent dynamicalfluctuations of interatomic chemicalbonding (4–6). Our calculations showthat the absorption is dominated by fiveoscillators denoted I to V. The coordinateand momentum eigenvectors of the oscil-lator responsible for the lowest absorptionpeak I of PPV(10) are shown in Fig. 4, Cand D. The same quantities for the secondoscillator corresponding to peak II areshown in Fig. 4, F and G. Despite thedifferent structures of these electronicmodes, the delocalization pattern of theoff-diagonal elements representing elec-tronic coherence between different atomsis similar. Both modes are delocalized andcan be viewed as d 3 d* transitions. Qn

and Pn clearly show that the weak coher-ences between the phenylene ring of theith repeat unit, and the vinylene group ofthe i 1 1-st repeat unit are enhanced byoptical excitation. In addition, a weakdynamical coherence develops betweenthe ith and the i 1 2-nd repeat units.These figures illustrate that finite size ef-fects are limited to the terminating repeat

Fig. 3. (A) Absorption spectrum of PPV(10) (theimaginary part of a Eq. 1. Dashed line, experimen-tal absorption of a PPV thin film (24); solid line,absorption lineshape of PPV(10) obtained with 12effective modes calculation with linewidth Gn 50.1 eV. The vertical lines represent oscillatorstrengths f n, n 5 1, K 2/4 of PPV(10) obtained bythe full TDHF. (B) The frequency-dependent in-verse participation ratio of the induced densitymatrix.

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units and that the momenta are more delo-calized than the coordinates for a singleunit. The coherence size, that is, the “width”of the momentum density matrix along thecoordinate axes, where the coherences de-crease to 10% of their maximum values, isfive repeat units. The same modes for alonger chain [PPV(20)] displayed in Fig. 4, Eand H, are virtually identical to those ofPPV(10). Therefore, 10 repeat units alreadyresemble the infinite chain as far as the

optical spectrum is concerned.The coordinate and momentum of peak

III of PPV(10) are shown in Fig. 5, A and B.This mode is delocalized with a coherencesize similar to modes I and II; however, itsstructure along the oligomer chain is verydifferent—bonding is weak at the centerand strong toward the edges. The electronicmodes are most suitable for investigatingcharge transfer processes and photoconduc-tivity (28, 29). The strong local optical

dipoles along the chain can affect chargetransfer and electron hopping. OscillatorIII, which has the strongest optical coher-ences induced at the chain ends (see Figs.5A and 4B), should play an important rolein effects involving charge separation.

The coordinates and momenta of thehigh-frequency peaks IV and V of PPV(10)(Fig. 5, D, E, G, and H) are completelylocalized on a single-repeat unit. This be-havior is markedly different from polyacety-

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Fig. 4. Contour plots of density matrices. (A) #r of PPV(10); (B) magnifiedregion of (A) representing the single unit of polymer chain and the colormaps; (C) momentum and (D) coordinate of PPV(10), and (E) coordinate ofPPV(20) of the lowest absorption peak I, (F, G, and H) are the same quan-

tities as in (C) to (E), but for the second absorption peak (II). The axis labelsrepresent the repeat units, except in (B) where the axes represent theindividual carbon atoms as numbered in Fig. 1.


lene, where the electronic coherence sizeincreases monotonically for the higher fre-quency modes (Fig. 2) (5). The coordinatesof these modes for a single PPV unit on anexpanded scale are shown in Fig. 5, F and I.For peak IV the optically induced coher-ences only involve the phenylene ring car-bon atoms 1, 2, 4, and 5 (Fig. 1), in agree-ment with the results obtained in (26, 28).The oscillator responsible for peak IV rep-resents several nearly degenerate localizedoscillators (Fig. 3A). The high-frequency

peak V predicted by our calculations liesbeyond the experimentally studied frequen-cy range. It corresponds to localized andweakly delocalized transitions involving thevinylene group atoms 7 and 8, and thephenylene ring atoms 3 and 6. A weakcoherence between the vinylene groups ofneighboring repeat units is observed as well.

Even though the CEO approach is ei-genstate-free, it is instructive to establish itsconnection to the more traditional eigen-state representation. The nth oscillator rep-

resents the optical transition between theground state cg and the nth excited statecn. The matrices representing the coordi-nate Qn and momentum Pn are given by(Qn)mn 5 ^cnucm

1cnucg& 1 ^cgucm1cnucn&,

(Pn)mn 5 ^cnucm1cnucg& 2 ^cgucm

1cnucn&. Qn

and Pn thus carry considerably reduced in-formation about the global eigenstates ucn&.A different perspective on these modes isobtained by expanding them in the molec-ular orbital representation using a basis setof pair molecular orbital pairs. Let us denote

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MOΩ = 4.5 eV

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Fig. 5. Contour plots of density matrices. (A) Momentum and (B) coordinatein the real-space and (C) coordinate in the molecular orbital (MO) represen-tation of peak III of PPV(10). (D) Momentum and (E) coordinate of PPV(10) forabsorption peak IV, and (F ) magnified area of (E) representing the single unit

of polymer chain. (G, H, and I) The same quantities as in (B) to (F ) but forabsorption peak V. The axes of (A), (B), (D), (E), (G), and (H) represent therepeat units of polymer chain. The axes of (C) denote the molecular orbitals.The axes in (F ) and (I) represent the number of carbon atoms.

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the creation and annihilation operator forthe ith molecular orbital as ci

1 and ci, re-spectively. We then have

Qn 5 Oi,j

N2/4

ai,jn ~ci

1cj 1 cj1ci! (5)

where i runs over initially unoccupied or-bitals (particles) and j denotes occupiedorbitals (holes) (Fig. 6A). The coefficients,a normalized as •i,jai,j

n 5 1, represent thecontribution of the j 3 i transition to thenth oscillator. Note that the indices n, mused earlier represent localized atomic or-bitals whereas i, j denote delocalized molec-ular orbitals. To illustrate how various mo-

lecular orbitals contribute to our five dom-inant electronic modes, we have introducedthe following two quantities

Rn~ j ! 5 Oi

@ai,jn #2 and Pn 5

1

Oi,j@ai,jn #2

(6)

where i, j 5 1, . . . , N/2, and n 5 I, II, . . . ,V. Rn( j) represents the total contribution ofthe jth molecular orbital to all orbital pairsappearing in the nth oscillator. Rn( j) for thefive dominant oscillators in PPV(10) are dis-played in Fig. 6B. RI( j) is relatively localizedin the vicinity of the HOMO-LUMO tran-sition (between the highest occupied andlowest unoccupied orbitals), whereas addi-tional pairs of orbitals contribute to thehigher modes. The inverse participation ra-tio Pn measures the number of orbital pairsthat contribute significantly to the nth oscil-lator. In the absence of electronic correla-tions, each oscillator represents a single tran-sition between an occupied and an unoccu-pied orbital (in the quantum chemistry ter-minology) or a single particle-hole pair (inthe semiconductor terminology) and Pn 5 1.In this case, the oscillator and molecularorbital pair descriptions coincide. In a corre-lated electronic structure, each mode be-comes a linear combination (that is, a wavepacket) of orbital pairs as represented by Eq.5, and Pn increases. Pn is thus a useful mea-sure of electronic correlations. The values ofPn given in Fig. 6B show that the higheroscillators are more collective and containgradually increasing numbers of electron-hole pair states. The oscillators III and Vcorresponding to d3 l* transitions have themost collective character. Such strongly cor-related excitations require extensive config-uration-interaction calculations in an eigen-states approach. Here they appear naturallythrough the modes. The CEO is most attrac-tive when Pn is large because in a very effi-cient way it lumps the important effects ofcorrelations directly into the observables.The collective nature of optical excitationsat different frequencies can be analyzed byexpanding the induced density matrix inmolecular orbitals dr(v) 5 •i,j ai,j(v)(ci

1cj1 cj

1ci). We can then define a frequency-dependent participation ratio P(v) by re-placing ai,j

n with ai,j(v) in Eq. 6. (A normal-ization •i,juai,j(v)u 5 1 is assumed). P(v)displayed in Fig. 2B is a weighted average ofthe participation ratios Pn of the contribut-ing electronic oscillators.

We have used the molecular orbital rep-resentation to analyze the nature of modeIII. Its coordinate in the molecular orbitalrepresentation is shown in Fig. 4C. Thefigure clearly shows that only a few molec-ular orbitals close to HOMO-LUMO con-

tribute to this transition. The strongest or-bitals can be identified as either delocalizedor localized, and mode III corresponds to l3 d* and d 3 1* transitions. Our calcula-tions further show that the frequencies ofmodes I, II, and III are red-shifted andgradually saturate with increasing chainlength, whereas the frequencies of modes IVand V are not affected by size. These find-ings are consistent with the delocalized andlocalized nature of the two groups of modesrespectively as displayed in Figs. 4 and 5.

Discussion and OtherApplications

The main reason for the success of the CEOrepresentation is the following. An opticalexcitation moves an electron from someoccupied orbital to an unoccupied orbital,thereby creating an electron-hole pair. Thenatural description of the optical responseshould therefore be based on following thesimultaneous and coupled dynamics of thispair; the two indices of the density matrixcarry precisely this information. Moleculareigenstates, however, use a single-particlebasis set. Correlations are incorporatedthrough an extensive configuration interac-tion calculation. By working in a space ofhigher dimensionality (the pair) we capturethe essential physics of the system, and eventhe simplest (TDHF) factorization yields anadequate description. In a single-particlebasis, a much more extensive numericaleffort is needed. A real-space analysis oflinear absorption that pinpoints the originof each optical transition is obtained bydisplaying the electronic mode matricesgraphically. The fact that only a few oscil-lators typically dominate the responsegreatly simplifies the theoretical descrip-tion. The weak anharmonicities that justifythe harmonic picture may be attributed tothe large delocalization size. On the otherhand, in atoms collective excitations havebeen found to converge to local modes rath-er than to normal modes (30). In semicon-ductors, the electron-hole pairs are looselybound and form Wannier excitons (3). Inmolecular aggregates, each pair is tightlybound and can be considered as a singleparticle (Frenkel exciton) (31, 32). Conju-gated polymers are intermediate betweenthese two extremes, and the collective os-cillators in conjugated polymers can beviewed as charge-transfer excitons. TheCEO thus offers a unified description ofdifferent materials and allows a direct com-parison of their optical properties (33).Also, one can go beyond the PPP Hamilto-nian and the TDHF approximation andinclude additional variables and use a dif-ferent ansatz for the wave function (34).Technically the calculation of optical prop-

Nh

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Fig. 6. (A) Origin of the collective electronic oscil-lators. Each transition between an occupied andan unoccupied orbital represents an electron-holeoscillator. In a molecule with Ne occupied (elec-tron) and Nh unoccupied (hole) orbitals we havealtogether Ne 3 Nh, oscillators. For a system witha filled valence and empty conduction band de-scribed by a “minimal basis set,” Ne 5 Nh 5 N/2,and the number of oscillators is N 2/4. The collec-tive oscillators Qn can be represented as super-positions of the electron-hole oscillators (see Eq.7). (B) The molecular orbital contributions and theinverse participation ratios of orbital pairs corre-sponding to the five dominant modes of PPV(10)absorption. The inverse participation ratio P n

measures the effective number of electron-holepairs contributing to a given collective oscillator.


erties with summation over states is alsounified and universal. However, very differ-ent approximate schemes and terminologiesare usually used in the calculation of theeigenstates of various systems, prohibiting aclear comparison and obscuring the originof differences. The electronic oscillator pic-ture applies to all materials by simplychanging parameters (such as the electronhole mass, the Coulomb interaction, andthe hopping matrix elements) (35).

We next review the computational advan-tages of the electronic oscillator approach.The sum-only states method becomes rapidlymore expensive with molecular size. Both cal-culating the eigenstates and performing thenecessary summations over them are intracta-ble for large systems. Knowing the completeset of eigenstates allows the calculation of anyoptical response including to strong fields.This is therefore an “all or nothing” approach.The oscillator approach carries less informa-tion but for considerably less effort. Compu-tational time of configuration interaction cal-culations scales as N6; the CEO procedurescales only as N2. Our results allow the inter-pretation of the most interesting crossoverregion toward the bulk.

The significance of the oscillator pic-ture is even more pronounced when non-linear optical properties are calculated (4,5). Interference effects in the sum-over-states approach result in an almost com-plete cancellation of large positive andnegative contributions to optical suscepti-bilities (36, 37), which limits the accuracyand makes approximate calculations dan-gerous (because innocent approximationsmay lead to huge errors). One conse-quence of this is that individual terms donot have the correct scaling with size. Thelatter is only obtained once all of theterms are carefully combined. In the oscil-lator picture these cancellations are builtin from the start and each separate con-tribution to the susceptibility scales prop-erly. The present discussion focuses on theresonant response. However, the real-space approach has been shown to providean adequate description of the scaling andsaturation of off-resonant linear and non-linear polarizabilities (4, 5).

We further note that by treating theelectronic degrees of freedom as oscillatorswe can couple them more naturally to nu-clear degrees of freedom, which constituteanother set of oscillators. The incorporationof nuclear notions thus becomes much morestraightforward compared with the eigen-state representation and lends itself moreeasily to semiclassical approximations.

The oscillator approach allows us to de-velop a natural framework for the interpre-tation and the design of molecules withspecific properties. Instead of asking which

of the many-electron states are most rele-vant, we can explore how different regionsof the molecule couple and affect each oth-er. We can translate dr(t) into a nonlocalresponse function anm(t), which shows howthe interaction with a field at point n affectsthe polarization at point m (38). The totalpolarizability is given by summing thisquantity over n and m a(t) 5 •nm anm(t).The nonlocal character of the response isintimately connected with the electroniccoherence of the induced density matrix.One can then address directly the effects ofdonor-acceptor substitutions and geometry.

Electronic motions may now be probeddirectly on the femtosecond time scale andthe nanometer length scale by nonlinearspectroscopic techniques. This has been re-cently demonstrated in semiconductorquantum wells (39), single-molecule spec-troscopy (40–42), and Rydberg atoms (43).The CEO approach should allow us to an-alyze the temporal and spatial microscopicdynamics underlying energy and electrontransfer processes in substituted conjugatedmolecules by using real-space wave packetsrepresenting the single-electron density ma-trix. A physical picture for coherent versusincoherent electron transfer processes canthen be developed in terms of off-diagonalor diagonal pathways, respectively, of theelectronic density matrix.

The CEO is conceptually similar to den-sity functional theory, which aims at calcu-lating the ground state with an energy func-tion that depends only on the charge den-sity, that is, the diagonal elements of thedensity matrix in a localized basis (44). TheCEO is a natural extension of density func-tional theory that includes the electroniccoherences contained in off-diagonal ele-ments. These carry the key informationabout electronic excitations and allow thecalculation of spectra with only ground-state information.

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45. Supported by the Air Force Office of Scientific Re-search, NSF, and the NSF Center for PhotoinducedCharge Transfer. T.W. acknowledges the support ofthe Fonds zur Forderung der WissenschaftlichenForschung, Austria, in the form of an E. Schrodingerstipendium (grant number J-01023-PHY ). Com-ments of R. S. Berry are appreciated. The calcula-tions were conducted with the resources of the Cor-nell Theory Center, which receives major fundingfrom NSF and New York State.

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