sergei tretiak and shaul mukamel- density matrix analysis and simulation of electronic excitations...

8/3/2019 Sergei Tretiak and Shaul Mukamel- Density Matrix Analysis and Simulation of Electronic Excitations in Conjugated a

1/42

Density Matrix Analysis and Simulation of Electronic Excitations inConjugated and Aggregated Molecules

Sergei Tretiak*

Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Shaul Mukamel*

Department of Chemistry and Department of Physics & Astronomy, University of Rochester, P. O. RC Box 270216,Rochester, New York 14627-0216

Received May 23, 2001

Contents

I. Introduction 3171II. The CEO Formalism 3175

A. Electronic Hamiltonian and Ground StateCalculations

3175

B. Computation of Electronic Oscillators 3177C. Real Space Analysis of Electronic Response. 3179

III. Electronic Coherence Sizes Underlying theOptical Response of Conjugated Molecules

3181

A. Linear Optical Excitations of Poly(p-phenylenevinylene) Oligomers

3181

B. Linear Optical Excitations ofAcceptor-Substituted Carotenoids

3182

C. Quantum Confinement and Size Scaling ofOff-Resonant Polarizabilities of Polyenes

3183

D. Origin, Scaling, and Saturation ofOff-Resonant Second Order Polarizabilities inDonor/Acceptor Polyenes

3184

E. Localized and Delocalized ElectronicExcitations in Bacteriochlorophylls

3185

IV. Optical Response of Chromophore Aggregates 3186A. Excitonic Couplings and Electronic

Coherence in Bridged Naphthalene Dimers3187

B. Electronic Excitations in StilbenoidAggregates

3188

C. Localized Electronic Excitations inPhenylacetylene Dendrimers

3189

D. Exciton-Coupling for the LH2 AntennaComplex of Purple Bacteria

3191

V. Discussion 3192VI. Acknowledgments 3194

VII. Appendix A: The TDHF Equations of Motion ofa Driven Molecule

3194

VIII. Appendix B: Algebra of Electronic Oscillators 3196IX. Appendix C: The IDSMA Algorithm 3197X. Appendix D: Lanczos Algorithms 3199

A. Lanczos Algorithm for Hermitian Matrices 3199B. Lanczos Algorithm for Non-Hermitian Matrices 3200

XI. Appendix E: Davidsons Algorithm 3202A. Davidsons Preconditioning 3202B. Davidsons Algorithm for Non-Hermitian

Matrices3202

XII. Appendix F: Frequency and Time DependentNonlinear Polarizabilities

3203

A. Equation of Motion for Electronic Oscillatorsand Anharmonicities

3203

B. Definition of Nonlinear Response Functions 3204C. Linear Response 3204D. Second-Order Response 3205E. Third-Order Response 3206

XIII. References 3207

I. Introduction

Predicting the electronic structure of extendedorganic molecules constitutes an important funda-mental task of modern chemistry. Studies of elec-tronic excitations, char ge-tra nsfer, ener gy-tra nsfer,and isomerization of conjugated systems form thebasis for our un dersta nding of th e photophysics andphotochemistry of complex molecules1-3 as well asorganic nanostructures a nd supramolecular assem-blies.4,5 Photosynthesis and other photochemical bio-

logical processes t ha t constitut e t he basis of life onEarth involve assemblies of conjugated chromophoressuch as porph yrins, chlorophylls, and carotenoids.6-8

Apart from the fundam ental interest, these stu diesare also closely connected to numerous importan ttechnological applications.9 Conjugated polymers areprima ry can didat es for new organ ic optical ma teria lswith large nonlinear polarizabilities.10 -19 Potentialapplications include electroluminescence, light em it-ting diodes, u ltra fast switches, photodetectors, bio-sensors, and optical limiting mat erials.20-27

Optical spectroscopy which allows chemists andphysicists to probe the dynamics of vibrations andelectronic excitations of molecules and solids is a

powerful tool for the study of molecular electronicstructure. The theoretical techniques used for de-scribing spectra of isolated small molecules areusually quite different from those of molecular crys-tals, and many intermediate size systems, such asclusters and polymers, are not readily described bythe methods developed for either of these limitingcases.28

* Correspondin g auth or. E-ma il: serg@lan l.gov (S.T.); mu ka [email protected] (S.M.).

3171Chem. Rev. 2002, 102, 31713212

10.1021/cr0101252 CCC: $39.75 2002 American Chemical SocietyPublished on Web 08/24/2002


2/42

Solving the many-electron problem required for theprediction a nd inter pret at ion of spectr oscopic signa lsinvolves an extensive numerical effort that growsvery fast with molecular size. Two broad classes oftechniqu es are gener ally employed in the calcula tionof molecular response functions. Off-resonant opticalpolarizabilities can be calculated most r eadily by avariational/perturbative tr eatment of the groundstate in the presence of a static electric field by

expan ding th e Sta rk en ergy in powers of electr ic field.The coupled perturbed Hartree-Fock (CPHF) pro-cedure computes the polarizabilities by evaluatingenergy derivatives of the m olecular Ha miltonian . Itusu ally involves expensive ab initio calculat ions withbasis sets including diffuse and polarized functions,that are substantially larger than those necessary forcomputing ground-stat e properties.14

The second approach starts with exact expressionsfor optical response functions derived using time-dependent per turbat ion theory, which relate theoptical response to the properties of the excitedstat es. It ap plies to resonant as well as off-resonantresponse. Its implement at ion in volves calculations of

both the ground state and excited-state wave func-tions and the transition dipole moments betweenthem.29,30 The configur at ion-intera ction/sum-over-states (CI/SOS) method15,31 is an example for thisclass of methods. Despite the straightforward imple-ment ation of the pr ocedure an d the int erpreta tion ofthe resul ts in terms of quantum states (which iscommon in quantum chemistry), special care needsto be tak en wh en choosing the right configur ations.In addition, this method is not size-consistent,32,33 andintrinsic interference effects resulting in a nearcancellation of very large cont ribu tions furt her limitits a ccura cy and complicat e th e an alysis of the size-scaling of the optical response. The SOS approach has

been widely applied using semiempirical Hamilto-nia ns (e.g., simple t ight-bindin g or H uckel, -electronPariser -Parr-Pople (PPP), valence effective Hamil-tonian s (VEH), complete neglect of differen tia l over-lap (CNDO), an d inter media te neglect of differen tia lOverlap (INDO) models).14,15,34-39 The global eigen-stat es carry too much informa tion on ma ny-electron

correlations, m aking it ha rd to use t hem effectivelyfor the interpretation of optical response and theprediction of various tren ds.

A completely different viewpoint is a dopted incalculat ions of infinite per iodic stru ctures (molecularcrystals, semicondu ctors, lar ge polymers). Band st ru c-t ur e approaches t ha t focus on t he dynamics of electron-hole pairs are th en used.40-44 Band theoriesmay not describe molecular systems with significantdisorder and deviations from periodicity, and becausethey are formulated in momentum (k) space they donot lend t hem selves very easily to real-space chem icalintu ition. The conn ection between t he molecula r an dthe ban d structur e pictures is an importa nt t heoreti-

cal challenge.45

To formulat e a un ified formulat ion th at br idges thegap between t he chemical and sem iconductor pointsof view, we must retain only r educed informat ionabout the many-electronic system necessary to cal-culat e th e optical r esponse. Cert ainly, the completeinformation on the optical response of a quantumsystem is contained in i ts set of many-electroneigenstates |, |, ... and en ergies , , ....29 Usingthe many-electron wave functions, it is possible tocalculat e all n-body quant ities an d corr elations. Mostof this informat ion is, however, ra rely used in thecalcula tion of common observables (ener gies, dipolemoments, spectra, etc.) which only depend on the

Sergei Tretiak is currently a Technical Staff Member at Los AlamosNational Laboratory (LANL). He received his M.Sc. (highest honors, 1994)from Moscow Institute of Physics and Technology (Russia) and his Ph.D.in 1998 from the University of Rochester where he worked with Prof.Shaul Mukamel. He was then a LANL Director-funded Postdoctoral Fellowin T-11/CNLS. His research interests include development of moderncomputational methods for molecular optical properties and establishingstructure/optical response relations in electronic materials, such as donoracceptor oligomers, photoluminescent polymers, porphyrins, semiconductornanoparticles, etc., promising for device applications. He is also developing

effective exciton Hamiltonian models for treating charge and energy transferphenomena in molecular superstructures such as biological antennacomplexes, dendrimer nanostructures, and semiconductor quantum dotsassemblies.

Shaul Mukamel, who is currently the C. E. Kenneth Mees Professor ofChemistry at the University of Rochester, received his Ph.D. in 1976 fromTel Aviv University, followed by postdoctoral appointments at MIT andthe University of California at Berkeley and faculty positions at theWeizmann Institute and at Rice University. He has been the recipient ofthe Sloan, Dreyfus, Guggenheim, and Alexander von Humboldt SeniorScientist awards. His research interests in theoretical chemical physicsand biophysics include: developing a density matrix Liouville-spaceapproach to femtosecond spectroscopy and to many body theory ofelectronic and vibrational excitations of molecules and semiconductors;multidimensional coherent spectroscopies of structure and folding dynamicsof proteins; nonlinear X-ray and single molecule spectroscopy; electrontransfer and energy funneling in photosynthetic complexes and Dendrimers.He is the author of over 400 publications in scientific journals and of thetextbook, Principles of Nonlinear Optical Spectroscopy (Oxford UniversityPress), 1995.

3172 Chemical Reviews, 2002, Vol. 102, No. 9 Tretiak and Mukamel


3/42

expecta tion va lues of a few (typically one- and two-)electron quantities. In addition, since even in practi-cal compu tat ions with a finite basis set, the nu mberof molecular many-electron states increases expo-nent ially with t he n um ber of electr ons, exact calcula-tions become prohibitively expensive even for fairlysmall molecules with a few atoms. A reduced descrip-t ion that only keeps a small amount of relevantinformation is called for. A remarkably successful

example of such a method is density-functionaltheory (DFT),46-51 which only retains the ground-state charge density profile. The charge density ofthe n th orbital is

where |g denotes the ground-state many-electronwave function a nd cn

(cn) are the Fermi annihilation(crea tion) opera tors for t he n th basis set orbital, whenthe overlap between basis set functions is neglected,th e molecular charge density depends on Fnn . Hohen-berg and Kohns theorem proves that the ground-state energy is a unique and a universal functionalof the charge densi ty,52,53 making i t possible inprinciple to compute self-consistently the chargedistribution and the ground-state energy.

The single-electron density mat rix54-60 given by

is a natu ra l genera lizat ion of th e ground-sta te cha rgedensity (eq 1.1). Here | and | represent globalelectronic states, whereas n and m denote the a tomicbasis functions. F i s t he redu ced si ngl e-electrondensity matrix of stat e . For * F is the density-mat rix associated with t he tra nsition between and

. These quantities carry much more informationt han Fjn n Fnngg (For brevity, the ground-stat e density

matrix Fgg will be donoted Fj thr oughout th is review),yet cons ider ably l ess t han t he comple t e set of eigenstates.51,61-66

Density functional theory has been extended toinclude cur ren t (in ad dition to char ge) dens ity.67 Th ecur rent density can be readily obtained from the n eardiagona l element s of th e density ma trix in rea l space.The current is thus r elated to short r ange coherence,whereas t he density mat rix includes short a s well aslong range coherence. The single electron densitymat rix is the lowest order in a system at ic hierar chy.Higher order density matrices (2 electron, etc.) have

been used as well in quantum chemistry. They retainsuccessively higher levels of inform at ion.68-73 Greenfunction techniques provide an alternat ive type ofreduced description.74,75

The wave function of a the system driven by anoptical field is a coherent superposition of states

and its density matrix is given by

Fnm are thus the building blocks for the time-de-

pendent single-electron density ma tr ix Fm n(t).The great ly reduced informa tion about t he global

eigensta tes cont ained in th e mat rices F is su fficientto compute the optical response. To illustrate this,let us consider the frequency-dependent linear po-larizability R() (see Appen dix F3).

where g g|| are the transition dipoles, and - g are the tran sition frequencies. is aphemenological dephasing rate which accounts forboth homogeneous (e.g., an interaction with bath) andinh omogeneous (e.g., sta tic distr ibution of moleculartra nsition frequencies) mecha nisms of line br oaden-ing (for a review see ref 76).

The molecular dipole is a single-electron operatorthat may be expanded in the form

We therefore h ave

The matrices Fg an d t he corr esponding frequencies th us contain all necessary informa tion for calcu-lating t he linear optical r esponse. Complete expres-sions for higher order polarizabilit ies up to thirdorder and other spectroscopic observables are givenin Appendix F.

Equat ion 1.2 apparent ly implies that one f i rs t

needs to calculate th e eigensta tes | and |g and thenuse them t o compu te the ma trix element s Fg. If tha twas the case, no computa tional saving is obtainedby using the density ma trix. However, its great poweris derived from the ability to compute the electronicrespons e directly, totally avoiding t he explicit calcu-lation of excited states: the time-dependent varia-tional principle (TDVP)64,65,77,78 and time-dependentdensity-functiona l theory (TDDFT)49,50,79,80 i n t h eKohn-Sham (KS) form 52,53 are two widely usedappr oaches of this type. In either case, one followsthe dyna mics of a certain r educed set of param etersrepresent ing the system driven by an extern al field.In th e TDVP, these para meters describe a tr ial many-

electron wave function, whereas in TDDFT th eyrepresent a set of KS orbitals. The time-dependentHartree-Fock (TDHF) equations are based on t heTDVP where th e trial wave function is assumed t obelong to the spa ce of single Slater det erm inan ts.77,81

Both TDHF and the TDDFT follow the dynamicsof a similar quantity: a single Slater determinantthat can be uniquely described by an idempotentsingle-electron density matrix F (with F2 ) F).62,63,77,78However, they yield different equations of motion forF(t), stemming from the different interpretation ofF(t). In the TDHF , F(t) is viewed as an app roxima tionfor th e a ctu al single-electr on density m atr ix,77 whereasin TDDFT F(t) is an auxiliary quan tity constra ined

R() )

2gg/

2 - ( + i)2 (1.5)

) n m

m ncn

cm (1.6)

g ) n m

m nFn mg (1.7)

Fjn n ) g|cn

cn|g (1.1)

Fn m |cn

cm | (1.2)

(t) )

a(t)| (1.3)

Fnm(t) (t)|cn

cm |(t) )

a/(t)a(t)Fn m

(1.4)

Density Matrix Analysis in Conjugated Molecules Chemical Reviews, 2002, Vol. 102, No. 9 3173


4/42

to merely reproduce the correct electronic chargedistribution at a ll times.52,53 TDDFT is forma lly exact.However, in practice it yields approximate resultssince exact expressions for the exchange-correlationenergy Exc[n(r)] an d th e corresponding potent ial vxc(r,[n ] ) in the KS scheme are not avai lable and areintroduced semiempirically. A close resemblancebetween TDHF an d TDDFT (especially its adiaba ticversion) may be establ ished by formulat ing KS

densi ty funct ional theory (DFT) in terms of thedensity matrix F rather than on the KS orbitals .78This formal similarity makes it possible to apply thesame algorithms for solving the equations for thematrices Fg (Abbreviated notat ion for thefamily of single-electr on den sity m at rices Fg will beused throughout this review) and frequencies ,directly avoiding th e tedious calculations of globaleigenstates in both cases.

This review focuses on the TDHF method,77,82-88

which combined with a semiempirical model Hamil-tonian provides a powerful tool for studying theoptical response of large conjugated molecules andchr omophore aggregates.81,89-96 The accuracy of this

combination is determined by t he approximationsinvolved in closing the TDHF equations and by thesemiempirical models. The TDDFT approach is onthe other ha nd usu ally based on th e ab initio Ham il-tonians,49,50,79,80,97,98 mak ing these compu tat ions sig-nificantly more expensive and limited to smallermolecular systems than TDHF/semiempirical tech-nique. F(t) compu ted in th e TDHF ap proach providesthe variation of electron charge distribution (diagonalelements) and the optically induced coherences, i.e.,changes in chemical bond orders, (off-diagonal ele-ments) caused by an external field. The latter areessential for understan ding optical properties ofconjugated molecules and for the first-principles

derivation of simple models for photoinduced dynam-ics in molecula r a ggregates (e.g., the F ren kel-excitonmodel).90

The TDHF equat ion of motion for the s ingle-electron density mat rix (eq A4 in Appendix A) wasfirst proposed by Dirac in 1930.99 This equation hasbeen introduced and explicitly applied in nuclearphysics by Ferrel.100 The TDHF description waswidely used in nuclear physics in the 50-60s.101-104,83,84The ran dom phase approximation (RPA) was firstintroduced into many-body theory by Pines andBohm.105 This approximation was shown to be equiva-lent to the TDHF for the linear optical response of

many-electron systems by Lindhard.106

(See, for ex-ample, Cha pter 8.5 in r ef 83. The electr onic modesare identical t o the tran sition densities of the RPAeigenvalue equ at ion.) The textbook of D. J. Th ouless82

cont ain s a good overview of Ha rt ree-Fock an d TDHFtheory.

The RPA approach was subsequently introducedinto molecular structure calculations and was exten-sively studied in 60th and 70th as an alternative toth e CI appr oach for solving man y-electr on problems.The RPA th eory was developed based on t he pa rt icle-hole propaga tors or two-electronic Green s fun ctionstechnique74 employing a direct decoupling of equa-tions of motion 107,108 or perturbative approach.109,110

In t his langua ge, the RPA procedure corr esponds t oth e summ at ion of ring diagra ms to infinite order.82,104

The RPA approach in combinat ion with the P ar iser-Par-Pople (PPP) Hamiltonian 111,112 was used to studylow-lying excited st at es of eth ylene an d form aldeh ydeby Dunn ing an d McKoy in 1967.113,114 This investiga-tion concluded that the RPA results are superior tosingle-electr on tra nsition approximation and are verysimilar to CI Singles (the latter coincides with the

Tamm -Dan coff approxima tion). Su bsequent compu-tat ions of small m olecules,107,108,115 -12 1 such as ben-zene,107 free radicals118 diatomics and triatomics,11 7

showed high promise of RPA for molecular excitationenergies. However, it was foun d t ha t t he first-orderRPA yields inaccura te r esults for t riplet sta tes113,119

and impractical for unstable HF ground state.12 2-12 6

This h appens when electronic correlations (doublesand higher orders) are significant for the ground-stat e wave function, and t he Ha rtr ee-Fock referencestate becomes a poor approximation for the trueground state wave funct ion. For example, largecontributions from doubly excited configurations leadto imaginary RPA energies of triplet states in bothethylene and formaldehyde.113,114 Several improvedschemes that take into account correlations beyondthe first-order RPA have been su ggested 120,127-133 toavoid these difficulties. Subsequently, RPA-basedmethods have been applied to calculate dynamicspolarizabilities of small molecules using an analyticalpropagator approach.134 -137 We refer r eaders to re-views104,74,138,75 for furt her deta ils of this ear ly devel-opment of RPA appr oaches.

Zern er an d co-workers h ad su bsequently attem ptedt o use RPA as an a l t e r na t i ve t o Si ngl es CI f orcompu ting molecula r electr onic spectra with ZINDOcode.139 -141 However, historically, these early RPA

advances did not develop into s tandard quantumchemical software. Modern computa tional pack-ages142 -145 usu ally offer extensive CI codes bu t notpropagator-based techniques for handling the elec-tr onic corr elations. However, curr ent stu dies of propa-gator techniques146,147 will be gradu ally incorporat edinto quan tu m-chemical softwar e.

Faster computers and development of better nu -merical algorithms have created the possibility toapply RPA in combination with semiempirical Hamil-tonian models to large m olecular systems. Sekino an dBartlett85,86,148,36 derived the TDHF expressions forfrequency-dependent off-resona nt optical polar izabili-

ties using a perturbative expansion of the HF equa-tion (eq 2.8) in powers of extern al field. This app roachwas further applied to conjugated polymer chains.The equations of motion for the time-dependentdens ity ma tr ix of a polyenic chain wer e first derivedand solved in refs 149 and 150. The TDHF approachbased on t he PPP Hamilt oni an 111,112 was subse-quently applied t o l inear and nonlinear optical r e-sponse of neutral polyenes (up to 40 repeat units)151,152

and P PV (up to 10 repeat un its).153 -155 The electronicoscillators (We shall refer to eigenmodes of thelinearized TDHF eq with eigenfrequencies aselectronic oscillators since they represent collectivemotions of electrons and holes (see Section II))



5/42

contributing t o the r esponse were ident ified, and t hesize-scaling of optical su sceptibilities wer e an alyzed.Furt her development of the classical TDHF repre-sent at ion a nd algebra of electr onic oscillat ors77,156-158

reduced the number of independent var iables toelectron-hole oscillators representing occupied-un occupied orbit al pa irs. These development s evolvedinto th e collective electronic oscillator (CEO) ap-proach for molecular electronic structure.

In t his ar ticle, we review the ba sic ideas an d recentdevelopments in th e CEO fram ework for computingthe optical excitations of large conjugated systemsand connecting them directly with the motions ofelectron-hole pairs in real spa ce. The CEO appr oachsolves the TDHF equations to genera te th e electronicnormal modes; quasiparticles which represent thedynamics of the optically driven reduced singleelectron density matr ix. Fast Krylov-space basedalgorithm s for th e requ ired diagonalizat ion of largeHa miltonian ma trices are u sed to calculat e excited-state structure of organic molecular systems withhundreds of heavy atoms with only moderate com-puta tional effort.

A real space an alysis of electronic norma l modes(transition densities) results in a systematic proce-dure for identifying the electronic coherence sizeswhich cont rol the scaling and satu ra tion of spectro-scopic observa bles with molecula r size. Localizat ionof these density m atr ices is furth er u sed to simplifythe descr ipt ion of the opt ical response of largemolecules by dissecting th em into coupled chromo-phores. Illust rat ive examples are p resented, includ-ing linear polyenes, donor/acceptor substitu ted oli-gomers, poly-phenylenevinylene (PPV) oligomers,chlorophylls, naph th alene an d PP V dimers, phen yl-acetylene dendrimers, and photosynthet ic l ight-harvesting antenna complexes.

In S ection II, we describe the CEO comput at iona lappr oach combined with semiempirical molecularHamiltonian. Section III presents a real space analy-sis of electronic excitations an d optical response ofdifferent conjugated molecules. In Section IV, wecompute int erchromophore inter actions to derive aneffective Frenkel exciton Hamiltonian for molecularaggregates. Finally, summary and discussion arepresent ed in Section V.

II. The CEO Formalism

The CEO computation of electronic structure81,89

starts with molecular geometry, optimized using

standard quantum chemical methods,142 -144

or ob-tained from experimental X-ray diffraction or NMRdata. For excited-state calculations, we usually usethe INDO/S semiempirical Hamiltonian model (Sec-tion IIA) generated by t he ZINDO code;145,159-163

however, other model Hamiltonians may be employedas well. The next step is to calculate t he H artr ee-Fock (HF) groun d sta te density ma trix. This densitymatr ix and th e Hamil tonian are the input into theCEO calculation. Solving t he TDHF equation ofmotion (Appen dix A) involves th e dia gona lizat ion ofthe Liouville operator (Section IIB) which is ef-ficiently performed using Krylov-space techniques:e.g., IDSMA (Appen dix C), Lan czos (Appen dix D), or

Davidsons (Appendix E) algorithms. A two-dimen-sional real space representat ion of the resul t ingtran sition density matr ices is convenient for anan alysis a nd visua lization of each electr onic tra nsi-tion and the molecular optical response in terms ofexcited-state charge distribution and motions ofelectrons a nd holes (Section IIC). Fina lly, th e com-puted vertical excitation energies and tran sitiondensities may be used to calculate molecular spec-

troscopic observables such as tran sition dipoles,oscillator strengths, linear absorption, and static andfrequency-dependent nonlinear response (AppendixF). The overall scaling of th ese compu ta tions does notexceed K3 in t ime and K2 in memory (K being thebasis set s ize) for both gr ound a nd excited-stat e (perstate) calculations. Typically, direct diagonalizationof the Liouville operator L or CI Singles ma trix Awithout invoking Krylov-space methods increases thecomputa tional cost to K6 i n t i m e a n d K4 inmemory for the excited states. The cost is even higher(K8-12) for methods taking into account higher orderelectronic correlations, such as higher order CI,coupled cluster and CAS-SCF.60

A. Electronic Hamiltonian and Ground StateCalculations

The general Hamiltonian of a molecule interactingwith an external field in second quantization formreads60

where the subscripts m , n , k, l ru n over known atomic

basis functions {n} and , label spin components.These at omic orbitals a re a ssum ed to be orth ogonal

cn (cn ) are the creat ion (annihilat ion) operators

which satisfy the Fermi anticommutation relations

and all other ant icommut ators ofc and c vanish.The first term in eq 2.1 is the core single-body

Ha miltonian describing the kinetic energy and nu clearatt raction of an electr on

where RA is the nuclear coordinate of atom A . Thesecond t wo-body t erm represent s electron-electronCoulomb intera ctions where

H ) m n

tm ncm

cn + m n k l

n m |k lcm

cn

ckcl -

E(t)m n

m ncm

cn , (2.1)

n |m dr 1nf(1)m (1) ) n m (2.2)

cm cn + cn

cm ) m n (2.3)

tn m ) n | - 1212 - AZA

|r 1 - RA||m

dr1nf(1) (-1212 - A

ZA

|r1 - R A|) m (1) (2.4)

n m |k l dr1dr2nf(1)m

f(2)1

r 12k(1)l(2) (2.5)



6/42

are the two-electron integrals. The interaction be-tween t he electr ons an d th e externa l electric field E(t)polarized along th e chosen z-axis is given by th e lastterm in eq 2.1, being th e dipole operat or

To simplify th e nota tion, we her eafter focus on closed-shell molecules and exclude spin variables assuming

that N electron pairs occupy K (Ne K) spat ial atomicorbitals. Generalization to the unrestricted opened-shell case and nonor t hogonal basi s set i s pos-sible.60 The ground state is obtained by solving theSchrodin ger equa tion H ) E for the ground-state assuming the s implest ant isymmetr ic wavefunct ion, i .e ., a s ingle Slater determinant )|1(1)2(2)...N(2N)60 (HF appr oximation). Here {R}are the molecular orbitals (MO). Following Roothaansprocedure,60,164 they a re expanded a s l inear combi-nations of localized atomic basis functions {n}

The HF appr oximat ion ma ps th e complex ma ny-bodyproblem onto an effective one-electron problem inwhich electron-electron repulsion is treated in anaverage (mean field) way. Even though the resultingground state is uncorrelated, this approximationworks reasonable well for ma jority of extended mo-lecular systems. However, the HF solution is notalways stable, in particular, for opened-shell12 4-126

and near degenerate cases (e.g., conical intersec-tions165,166).

The H F eigenvalue equation is derived by mini-mizing the ground-state energy with respect to thechoice of MOs

This equation may be recast using the density matrixin th e form

For closed-shells, the groun d-state d ensity ma trix isrelated to the MO expansion coefficients (eq 2.7) as

F(Fj) is the Fock matrix with matrix elements

and the matrix representation of the Coulomb elec-tronic operat or V in the atomic basis set {n} is

The H F eq 2.9 for Fj is nonlinear a nd ma y be readilysolved iteratively using the self-consistent field (SCF)procedure.60

In all computations presented below, we use asemiempirical (INDO/S) para metr ization of th e H amil-tonian (2.1) tha t wa s fitt ed to reproduce the spectraof simp le molecules a t t he sin gly excited CI level. TheINDO approximation 159 -163 l imits the basis set tovalence orbitals of Slater type. Excha nge ter ms in t hetwo-electron interaction are permitted only amongorbitals located on the same atom

where nA belongs to atom A and n

B to atom B. Thetetradic matrix nk|ml th us becomes block-diago-nal in two dimensions. Thus, this approximationl imits the number of computed Coulomb matr ixelements and allows the storing of all of them inmemory instead of recalculating them when n eededas is commonly done in ab init io computat ions,ma king semiempirical techniques significantly easierand faster.

The parameters of the INDO/S Hamiltonian are

given in refs 159-163. This widely used model firstintroduced by Pople159,160 and later carefully param-etr ized by Zerner a nd collabora tors to repr oduce UV-visible spectra of small organic chromophores at CIsingle level.161 -163,167-172 The INDO/S parameterswere init ially a vailable for the main group ele-ments161,162 a n d s u bs eq ue nt ly for t r a n sit ionmetals,163,168,173 -175 actinides,176 and lantha nides.177,178

Special a t tent ion was paid to reproduce t r ipletstates.167 INDO/CIS calculations have been success-fully ap plied to stu dies of electronically excited st at esin a wide variety of chromophores,179,180 and t h i smodel is currently widely used in optical responsecomputations.14,15,181 The ZIN DO code145 developed b y

Zerner and co-workers serves as a convenient plat-form for these calculations. In addition to the CIScalculations, t hey ha ve investigat ed how INDO workswith RPA approximation for molecular exci tedstates139 -141 using conventional diagonalization of theRPA matrix (see Section IIB). These studies con-cluded th at t he INDO/RPA excited-state ener gies ar eclose to INDO/CIS where both show some red-shiftscompa red t o experiment . However, RPA shows betteraccur acy for th e oscillator str engths an d for molec-ular system s with fine splittings in the spectrum suchas free base porphins.140 We also found that theTDHF (RPA) combined with the INDO/S Hamilto-nian works extr emely well for ma ny molecules with -

out furth er repara metrization a nd t hus provides analternative approach for computing their opticalproperties.182,183 Typically, this method reproducesvertical excita tion ener gies with a ccur acy of 0.1-0.3eV, whereas t ra nsition dipoles an d nonlinear polar-izabilities agree with experimenta l data within 10%and 20-30%, r espectively.182,183

Effects of th e sur roun ding media (e.g., solvent ) ma ybe readily incorporated using the self-consistentreaction field (SCRF) approach,184,185 whereby theintera ction energy between a solut e and th e solventis added to the HF energy of an isolated molecule,and the t otal energy of the system is th en minimizedself-consistently. The SCRF method is based on

nA k

B|m

A lB ) {

nA

kA

|mA

lA

A ) B ,n

A kB|n

A kBn m k l, A * B ,

(2.13)

n m ) n |z|m dr1nf(1)z1m (1 ) (2 .6 )

R ) i

K

CRii (2.7)

FC ) C (2.8)

[F(Fj), Fj] ) 0 (2.9)

Fjn m ) 2a

Nocc

Cn aCm af ) 2

a

N

Cn aCm af (2.10)

Fn m (Fj) ) tn m + Vn m (Fj) (2.11)

V(Fj)m n ) k,l

K

Fjkl[m l|n k -1

2m l|k n ] (2.12)



7/42

classical ideas originally introduced by Onsager 186

and Kirkwood.187

For electrically neutral solutes, only the dipolarintera ctions cont ribute t o th e solvation ener gy. In t heOnsagers spherical cavity model, the Fock operatorFm n is then modified by adding the response of adielectr ic medium, r esulting in

where Fm n0 is the isolated complex Fock operator, bg

is the ground-stat e dipole moment, is the dielectricconstant , and ao i s a cavity radius. The secondOnsager d ipolar t e r m in eq 2 .14 has been de-rived188,185 assuming that the solute is separated fromthe solvent by a sphere of radius ao. The Gaussian98 package142 provides a reasonable estimate for acavity ra dius.

Onsagers SCRF is the simplest method for takingdielectric m edium effects into a ccount and moreaccurate approaches have been developed such as

polarizable continuum modes,189,190

continuum di-electric solvation models,191,192 explicit-solvent dynamic-dielectric screening model,193,194 and conductor-likescreening model (COSMO).195 Extensive refinement sof th e SCRF meth od (spherical, elliptical,188 multi-cavity models) in conjunction with INDO/CIS wereintroduced by Zerner and co-workers 185,196-202 a swell.

The shape of the cavity has some effect on themolecular polarizabilities;203,204 however, th e meth odstaking into account real molecular shapes arecomputationally expensive and are most appropri-ately ut i lized with accurate ab init io or densi tyfun ctiona l th eory (DFT) app roaches.205,206 Even though

spherical cavity is a crude approximation for mostmolecules, the predicted trends usually agree wellwith experiment and with the results of much moresophisticated an d expensive methods.182,185,200

B. Computation of Electronic Oscillators

Using the ground-state density matr ix as an input,the CEO procedure81,89 comput es vertical tra nsitionenergies and the relevant t ransi t ion densi tymat rices (denoted electronic n orm al m odes ()m n )g|cm

cn|), which connect th e optical response with

the underlying electronic motions. Each electronictransi t ion between the ground state |g a n d a n

electr onically excited st at e | is described by a modewhich is represen ted by K K mat rix. These modesare computed directly as eigenmodes of the linearizedtime-dependent Har tree-Fock equations of motionfor t he den sity ma tr ix (eq A4) of th e molecule dr ivenby the optical field.

where L is a linear Liouville space operator (i.e.,superoperator) whose eigenvectors are the transitiondensities .81,89 The electr onic modes obey n ormal-ization conditions (see Section B)

The complete set of density ma tr ices (eq 1.2) may besubsequent ly calculated using th e eigenvectors.207

Only par ticle-hole and hole-par ticle component s of are computed in the restricted TDHF scheme 77

(Appendix A). Therefore, this non-Hermitian eigen-value pr oblem of dimension 2M 2M, M ) Nocc Nvir ) N (K - N) in the MO basis set representa-tion may be recast in the form 82,74

This is known as the first-order RPA eigenvalueequation,79,107,113,127,130,131,208 where X and Y are, re-spectively, t he par ticle-hole a nd hole-particle com-ponents of the transition density ) [Y

X] in t he MOrepresentation.77,79,80,113,120,208 In eq 2.18, the m atrixA is Hermitian an d identical to the CI Singles mat rix,

whereas the H ermitian matr ix B represents higherorder electronic correlations (double excitations) in-cluded in th e TDHF appr oximation. We recall, how-ever , t ha t t he TDHF uses t he HF gr ound s t a t e(Section IIA) as a reference state. If this state isunstable (e.g., saddle point) near curve crossings orconical intersections, or if the second-order electroniccorrelations are large (the ma gnitudes of mat rix Belements ar e compara ble to tha t of ma tr ix A), eq 2.18may have imaginary eigenvalues (frequencies). Inthis case, the first-order RPA breaks down,113,120 andhigher order RPA are called for.128 -131,133,209 We notethat the extended conjugated molecular systemsconsidered in this review have stable HF ground

stat e (closed sh ell), an d t he first-order RPA is wellsuit ed for comput ing th eir electronic excita tions. Wetherefore restrict our subsequent discussion to thisapproximation.

The form al pr operties of operator L eq 2.18 (knownas th e symplectic stru cture 77) allow th e intr oductionof a variational pr inciple eq D3,210 a scalar product(eq B1), and ultimately to reduce the original non-Her mitian eigenvalu e problem (eq 2.18) to the equ iva-lent Hermitian problem which may be solved usingstandard numerical algorithms (Appendices B-E).For example, L 2 is a Her mit ian operat or. Lowdinssymmetric orthogonalization procedure60,211,212 leadsto the H erm itian eigenva lue problem as well (eq E5),

which may be subsequently solved by Davidsonsalgorithm (Appendix E). The spectral tran sformLanczos m ethod developed by Ru he an d E ricsson213

is another example of such transformation.Direct diagona lization of the TDH F operat or L or

the CIS operator A in eq 2.18 is the computationalbottleneck, requiring computa tional effort whichscales as K6 i n t i m e a n d K4 in memory (forcomparison, SCF ground-state calculations scales asK3 in t ime and K2 in memory) because we areworking in the space of higher dimensionality (elec-tron-hole pairs). Direct diagona lizat ion of eq 2.18should give th e entire spectru m of excited sta tes. Thetraditional quantum-chemical approach addresses

Tr (Fj[R , ]) ) R (2.16)

Tr (Fj[R ,

]) ) Tr (Fj[R, ]) ) 0 (2.17)

(A B-B -A )[X

Y] ) [X

Y] (2.18)

Fm n ) Fm n0

-

- 1

2 + 1

bg bm n

ao3 (2.14)

L ) L ) -

) 1, ..., K2/2(2.15)



8/42

this problem by l imit ing the total basis set s izevariables K to a few MOs which are important forvisible-UV optical response. Indeed, most of the

electronic states obtained by diagonalization of eq2.18 lie in t he X-ray spectra l region an d correspondto at omic-core type tr an sitions. Visible-UV collectivemolecular excitations, on the other hand, could beadequa tely described by trun cat ing an active space,taking into account only few MOs close to HOMO-LUMO energy gap. Although this approach worksquite well and th e ZINDO code161,163,168 becam e verysuccessful, truncating the active space is a compli-cat ed and somewhat a rbitra ry procedure. In addition,even truncated CI calculations are usually signifi-cantly more expensive than ground-state computa-tions. The effects of size of the active space on t hecompu ted spectr a of sma ll molecules for CIS a nd RP A

approximations have been studied by Zerner andBaker.139 They showed that (i) even minimal con-figurational space (7 eV) provides qualitative de-scription for lowest electr onic t ra nsition, (ii) fairlylarge active space is required (10 eV) (The nu mberof molecular orbitals and s ubsequen tly the CI expan -sion size grows very rapidly with the active spacewindow size) to a ccoun t for all essent ial configura -tions, and (ii i) inaccuracy grows for higher lyingelectronic transitions. Figure 1 shows variation ofenergy of calculated 1E1u sta te of benzene (th e thirdtransition in electronic spectrum) as a function ofactive space size. These data ar e extracted from ref139. Both CI Singles an d RPA ener gies show consid-

era ble red-shift with increa sing th e active space size.It is interesting to note that CI Singles gives theclosest agreement with experiment for the smallactive space size used for parametrizing the INDO/Smodel. This points out th e need for a fut ur e repar am-etrization of the INDO/S Hamiltonian to account forthe entire active space.

An alter na tive solution to this problem is providedby fast Krylov-space algorithms.214,215 These tech-niques constr uct a sm all subspa ce of orth ogonal vec-tors which contains a good approximation to the trueeigenvector. This Krylov subspace S p{, L , L 2, ...,Lj}, j , M, span s th e sequence of vectors gener at edby the power meth od (th e mult iple action of th e RPA

operator L on some initial vector ). These m ethodsfind several eigenvalues and eigenvectors of a largematrix L using only mat rix-vector operat ions.214,215

Indeed, usu ally only a small fraction of eigensta tesof L (100) lie in the UV-visible region and are ofinterest for optical spectroscopy. In addition, theaction of th e TDHF opera tor L on an arbitr ary singleelectron matrix , which only contains particle-holeand hole-particle components is given by

This product ma y ther efore be calculated on th e flywithout constru cting an d st oring th e full matrix Lin memory.77,79-81,89,208 The a ction of the CIS opera torA on an arbi t rary matr ix can be a lso computeddirectly216,217 (e.g., using eq 2.19 by setting the hole-particle component of to zero). The cost of suchoperation in Hilbert (K K) space scales a s K3 intime and K2 in memory with system size. Comput -ing a single eigenvalue-eigenvector of matrix L whichcorresponds to molecular excited sta te t hu s requiresa computational effort comparable to that of the

ground state.I n Appendi ces D, E, and C, we out l i ne t hr eeKrylov-space bas ed a lgorith ms. Th e origina l Lan czosalgorith m compu tes effectively the lowest eigenvalueand the corresponding eigenvector of a large Hermi-tian matrix.214,218 Since the matrices L that n eed tobe diagonalized in the TDHF or adiabatic TDDFTapproaches are non-Hermitian, a modified nonstand-ard Lanczos algorithm should be used 219 -221 (Ap-pendix D). Similarly, Davidsons algorithm originallyformu lated for the diagona lizat ion of lar ge Herm itianCI matrices216 was furt her modified for th e TDHF 208,222

and adiabatic TDDFT49,50,79,80,97,217 methods. A thirdmethod for computing the lowest frequency eigen-

mode of a large Ham iltonian mat rix is based on th eiterative density mat rix spectra l moments algorithm(IDSMA).81,89 All thr ee algorith ms sh ow similar scal-ing of computational t ime, resulting from K Kmatrix multiplications. However, the scaling prefac-tors are different. The Davidson type algorithms,especially the recently improved versions,79,80,142 areextrem ely fast but I/O (inpu t/out put ) inten sive, sinceone needs to keep all th e previous iter at ions for t heeigenmodes thr oughout t he itera tion procedure. Con-sider, for example, the computa tion of t he lowesteigenmode of a mat rix using t he Da vidson iter ationin a 200 dimension Krylov space (default maximumdimension in the Gaussian 98). To improve th e

accura cy, we need to calculat e the 201st t ria l Krylovvector, which sh ould be ort hogona l to all oth ers. Th isrequires storing of all previous 200 vectors! On t heot her hand, t o comput e t he 201st vect or in t heLanczos procedure we only need the 200th and the199th vectors: by orth ogona lizing th e 201st to the200th a nd 199th , it au tomatically becomes orth ogonalto all previous vectors. The need to store only twovectors, rather than 200, constitutes a substantialimprovement in memory requirements of Lanczosover Davidsons. However, the Lanczos method usu-ally requires larger Krylov-space dimension t o obta inan approximate eigenvalue with the same accuracyas Da vidsons. The latt er t hu s converges faster a nd

F i gur e 1. Benzene 1E1u tra nsition as a function of activespace calculated with CI Singles and RPA methods com-bined with INDO/S model. Adapt ed from Bak er an d Zernerref 139.

L ) [F(Fj),] + [V(), Fj] (2.19)



9/42

generally involves lower computational effort com-pared to Lan czos. This fast convergence is ensur edby Davidsons preconditioning (Appendix E), whichassumes that the ma tr ices L (or A) are dominatedby their diagonal element s.216 In practice, the Lanczosis 2-4 times faster th an the IDSMA; however, theIDSMA ha s low memory requirement s an d allows tocompu te both exact an d effective eigenst at es. Thelatter may represent the overall contribution from

several electronic stat es t o the optical r esponse by asingle effective state,89,81 providing an a pproximationfor the spectrum in terms of very few variables.91

Ther e is no clear single met hod of choice and differen talgorithms may be preferable for specific applica-tions.

All algorithms converge to the lowest eigenmode,and higher eigenmodes can be successively obtained(Appendix D) by finding the lowest mode in thesubspace orthogonal to that spanned by the lowermodes a lready foun d. Th is orth ogonalization proce-dur e is not always sta ble, leading to the a ccumu lationof nu mer ical err or for t he h igher m odes. A deflectionprocedure214,219,223 that involves the antisymmetric

scalar product eq B1 may be alternatively used tosolve th is problem. There is a wh ole ars ena l of otherrelated algorithms, such as Chebyshevs polyno-mial224,225 and Arnoldis226 -228 which ma y be used aswell. These are included in sta nda rd pa ckages suchas Matlab.

These outlined numerical methods are commonlyused in quantum-chemical computations and becamea part of standard quantum-chemical packages.142-144

However, new developments in comput at iona l tech-niques may offer even faster and more dependablenumerical algorithms (such as the rational Krylovalgorithm for nonsymmetric eigenvalue problemsproposed by Ruh e229 -232) which will undoubtedly find

their place in the future quantum-chemical codes.

C. Real Space Analysis of Electronic Response

Each calculated transition density matrix Fg with the corresponding frequency ent er s t heTDHF equations of motion as an electronic oscillator.Density matr ices establ ish a natural connect ionbetween electronic structure and the molecular opti-cal response. The groun d-state density ma trix Fjm n g|cm

cn|g is widely used in the description of the

ground-state properties.54,55,233-235 Its diagonal ele-ments Fjn n are used in various types of populationanalysis56,59,233,234,236 to prescribe a char ge to specific

atoms and are commonly visualized using contourcharge density maps. The off-diagonal elements,m * n , known as bond orders repr esent t he bondingstructure associated with a pair of atomic orbitalsand ar e useful for interpr eting the chemical bondingpattern across the molecule.57-60,233,234

In complete an alogy with Fj, the diagonal element sof ()nn represent the net charge induced on the n t hatomic orbital wh en t he m olecule un dergoes th e g t ov electronic transition, whereas ()m n n * m i s thedynam ical bond -orderrepresenting the joint ampli-tude of finding a n extra electr on on orbita l m and ahole on orbita l n . The electronic modes thus directlyshow the flow of opt ical ly induced charges and

electronic coherences. To display these modes, weneed to coarse grain them over the various orbitalsbelonging to each atom. The INDO/S Hamiltonianuses from one to nine at omic orbita ls (s, p, and d type)for each atom. In pr actice, the hydr ogen at oms tha tweakly pa rticipat e in the delocalized electronic ex-citations (such as -type) are usually omitted. Forother atoms, we use the following contraction: thetotal induced charge on each at om A is given by th e

diagona l elements

whereas an average over all the off-diagonal elementsrepresents the effective electronic coherence betweenatoms A and B

Here the indices n A and m B run over al l a tomic

orbitals localized on atoms A and B, respectively. Thesize of the mat rix ()AB is now equal t o the nu mberof hea vy atoms. (For plan ar molecules it is su fficientto include th e -electr on contr ibutions perp endicularto the molecular plane to represent -excitationssince contributions are usually negligible.) Theresu lting two-dimensional repr esenta tion of th e elec-tronic modes ()AB is useful for interpreting andvisua lizing th ese collective electronic motions int e r ms of t he e lect r oni c dens it y mat r i x i n r ea lspace.81,90-92 This is i llustrated schematically byFigure 2: th e coordina te axes label at oms and indicesA and B of matrix ()AB run along the y and x axes,respectively.

Two types of cha ra cteristic size for t he degree oflocalizat ion of th e mode () ma y be clear ly iden tified.The diagonal size (L d) reflects the nu mber of atomsover which the optical excita tion is sprea d, i.e., th ewidth of the distribution of the electron hole paircenter of mass. The off-diagonal size L c measures thedegree of coheren ce between electrons an d h oles a tdifferent sites, an d cont rol th e scaling of molecularproperties with size. It reflects the size of electron -hole pair created upon optical excitation, (i.e., theconfinement of their relative motion). L d and L c canbe calculat ed qu an titat ively as follows.237 To intro-duce L d, we first define a normalized probabilitydistribution of the charge induced on the n th atom

L d is then defined as t he inverse participation ratioassociated with the distribution of populat ions:

For a localized excita tion on sit e k Pn ) nk and L d )1; For a delocalized excita tion Pn ) 1/L and L d ) L .

()A ) |n A

()n An A| (2.20)

()AB ) n Am B[()n Am B]2 (2.21)

Pn )|n n|

j

|jj|

(2.22)

L d [n

Pn2]-1 (2.23)



10/42

L c can be defined in term s of coherence part icipa-tion ratio.237,238 At first, we introduce a normalizedprobability distribution of the density matrix ele-ments

L c is then defined as follows:

For tightly bound e-h pairs Qn m ) nm/L d and L c )1; for loosely bound e-h pairs Qnm ) 1/L d2 and L c L . Both L c and L d thus vary between 1 and L , whereL i s the number of atoms. Un like L d, which only

depends on th e populat ions, L c measures th e degreeof coherence and is sensitive to the off-diagonalelements of the densi ty matr ix. Both L c, a n d L ddepend on the basis set.

Note, tha t L d and L c defined by eqs 2.23 and 2.25,respect ively, represent a total number of atomsinvolved int o electronic excita tion, wher eas coheren cesizes obta ined from t he t wo-dimens iona l plots reflectthe extent of the t ransition densities in real space.They may n ot be the sa me. For example, the excitoncorr esponding to th e band -gap excita tion in PP V (see

Section IIIA) is extended over 5 repeat units (40atoms) (Figure 3I). However, the coherence size L ccompu ted with eq 2.25 is only 26 atoms. This r eflectsuneven participation of phenyl and vinyl carbonatoms in the optical excitation. In remainder of thepaper, we will be using two-dimensional plots toobtain necessary coherence sizes relevant to t hedelocalization of the transition densities in real space.

The significance of the CEO oscillators may beexplained by drawing upon the analogy with thedescription of vibrational spectroscopy,239 wherebythe coherent motion of various atoms with well-defined amplitude and phase relations are repre-sented by collective nuclear coordinates; the normal

F i g u r e 2 . Two-dimensional representation and physicalsignifican ce of electronic modes. Ea ch m ode is an L Lmatrix, L being t he number of atoms. The contour plotprovides a direct rea l-space conn ection between th e opticalresponse and motions of charges in the molecule uponoptical excitation. The x axis represents an extra electronon site n , and the y axis describes an extra hole on site m .The incident light moves an electron from some occupiedto an unoccupied orbitals, creating a n electron-hole pair(or exciton). The state of this pair can be characterized bytwo lengthscales: first, th e distance between electron an dhole (i.e., how far t he electron can be separ at ed apa rt fromthe hole). This coherence size L c is the width of the

density ma tr ix along th e an tidiagonal direction. The secondlength L d describes the exciton center of mass position (i.e.,where t he optical excitat ion resides within the molecule).

L d is the width of the mode along the diagonal antidi-agonal direction. Charge-transfer processes can be char-acterized by the asymmetry of mode with respect to th ediagonal symmetrical mode atom. ()m n ()nm meansth at t her e is no prefera ble direction of motion for electrons(or holes), whereas ()m n > ()nm shows the transfer of electron from m t o n .

Qn m )|n m |

ij

|ij|

(2.24)

L c [L dm n

Qn m2 ]-1 (2.25)

F i g u r e 3 . (A) Geometry and atom labeling of PPV oligo-mers. Molecular str ucture was optimized using the Austinmodel 1 (AM1) semiempirical model492 in Gauss ian 98package;142 (B) Absorption spectrum of PPV(10). Dashedline: experimental a bsorption of a PP V thin film.243 Solidline: absorption line sh ape of PPV(10) obtained with 12effective modes DSMA calculat ion wit h line width ) 0.1eV; Contour plots of ground-sta te dens ity mat rix Fj an d fiveelectronic modes (I-V) which dominat e th e linear a bsorp-tion of PPV(10). The sizes of plotted matrices are 78 78(equa l to th e nu mber of carbon at oms in PP V(10)). The a xisare la beled by the r epeat u nits. The color m aps a re givenon the top of color plots. Reprinted with permission from

ref 91. Copyright 1997 American Association for theAdvancement of Science.



11/42

modes. The norma l modes provide a n atu ra l coordi-na te system an d a h ighly int uitive classical oscillatorreal-space interpr etat ion of infrared or Rama n spec-t r a ,240,241 which offers a n a lterna tive to the descrip-tion in terms of transitions among specific vibrationalstates. The normal modes of nuclear vibrations aresimply superpositions of the 3N nuclear displace-ments. In complete analogy, can be viewed ascollective coordinates which represent not t he indi-

vidual electrons but the displacements of the elec-tronic density ma trix elements from their ground-state values Fjnm .

The electronic modes provide a direct real-spacelink between the structure of complex molecules suchas organic oligomers with a delocalized -electronicsystem and their optical properties. They clearlyshow how specific variations in molecular design,such as chain length or donor/acceptor subst itut ions,can impact their optical response. In the remainderof the paper , we apply this approach to var iousclass es of molecules a nd to differen t t ypes of opticalresponse. The two-dimensional real space analysisof the transition densities (slices16 or two-dimensional

plots18 1

) provides an at t ract ive al ternat ive to thetra ditiona l molecular orbita l based quan tum -chemi-cal analysis of photoexcitation processes.

III. Electronic Coherence Sizes Underlying theOptical Response of Conjugated Molecules

A. Linear Optical Excitations of Poly(p-phenylenevinylene) Oligomers

In this section, we examine t he electronic excita-tions of poly(p-phenylene vinylene) (PPV) oligomer s(Figure 3A) and their scaling with molecular size.91,96,95

Understa nding the electronic structure and the over-

all electronic excitation processes in this photolumi-nescent polymer is needed to provide a consistentpicture for the numerous experimental242 -250 andtheoretical16,95,149,153,251 -25 3 studies of PP V.

The absorption spectrum of PPV(10) calculatedusing the CEO/DSMA algori thm combined withINDO/S H am iltonian (Figure 3B, solid line)91 is notinconsistent with t he experimenta l spectru m of PP Vthin film 243 (dashed line), which is typical for otherPPV derivatives.254,243,255 The experimental absorp-tion h as a fun damen ta l band at 2.5 eV (I), two weakpeaks a t 3.7 eV (II) and 4.8 eV (III), an d a strongban d at 6.0 eV (IV). Pea k II originat es from electr oncorrelations 247,253,255 an d is missed by HF calculat ions.

Before analyzing the tr an sition densities un derly-ing each absorpt ion peak , let us examin e the ground-state density matrix. A contour plot of the absolutevalue of th e mat rix element s ofFj of PPV(10) is shownin Figure 3 . The mat r ix s ize has been r educedaccording to contr action eqs 2.20 and 2.21. It is equalto the number of carbon atoms, and the axes arelabeled by r epeat u nits a long t he chain. Fj is domi-nated by the diagonal and near-diagonal elements,reflecting th e bonds between nea rest n eighbors. Thefive oscillators denoted I-V which dominate theoptical absorption ar e sh own as well. All tr an sitiondensities are almost symmetric with respect to thediagonal (m n nm ). This reflects the absence of

char ge separ at ion for t he lack of prefera ble directionof motion for electrons (or holes). Mode I is delocal-ized. The coherence size, L c, that is the width of thedensity mat rix along t he a ntidiagonal section, wherethe coheren ces decrease to


12/42

modes IV and V ar e not a ffected by size. These tr endsare consistent with the delocalized and localizedna tu re, resp ectively, of the t wo groups of modes. Oneimporta nt consequ ence of th is localizat ion of opticalexcitations is that the Frenkel exciton model formolecular aggregates may be applied to high fre-quency spectral region in PPV, even though thechromophores are not separated spatially (see SectionIV). Subsequent CI/INDO computations 16,253 which

used t he s lices of t ra nsit ion densi ty to s tudy th ecoherence sizes and formulated an essential-statesingle-chain model to model l inear and nonlinearresponse of PP V oligomers ar e in agreem ent with t hisanalysis.

B. Linear Optical Excitations ofAcceptor-Substituted Carotenoids

Substituted conjugated molecules have opticalproperties tha t reflect the interplay of the donor-acceptor strength an d the type an d the length of th econn ecting bridge.1,2,12,13,18,31,259-261 The electronic spec-tra of a family of un substitu ted, neut ra l (N(n )) (an dsubst i tuted with the s t rong acceptor polar) P(n )molecules shown in Figure 4A17,262 were calculatedusing th e CEO/INDO/S with IDSMA algorith m.89,91,81

Our an alysis shows the difficulties in disent anglingthe effects of donor acceptor and br idge in thespectroscopy of molecules wit h r elatively short bridges.To obtain a clear picture of the optical response ofacceptor-substituted molecules it is instructive tostud y th e size-dependence of optical properties, star t-ing with very long br idges, where th e effects of th eacceptor an d t he br idge regions can be clearly sepa-ra ted. Opt ical pr operties of acceptor-substitut ed mol-ecules with shorter bridges can then be attributedto quant um confinement, which is importan t whenth e bridge size becomes compa ra ble to the coher encelength L c.

We first consider the effect of the acceptor on theground state by ana lyzing t he bond-length alterna-tion (BLA) parameter and relevant charge distribu-t ions. The BLA lj is defined as the differencebetween the single (l2j) and the double (l2j-1) bondlengths in the jth repeat unit along the bridge:

The BLA is a signa tur e of the un even distribution ofthe densities over the bonds (Peierls distortion),which has a well-established relation to molecular

polarizabilities.260,263-

268 Figure 4B displays the BLAparam eter an d th e variation of the total char ge QAfrom t he a cceptor end

where Qacceptor ) 0.69e is th e tota l electr onic cha rgeon the acceptor and qa are t he at omic cha rges. Thesecalculations illustrate the roles of bridge and bound-ar y (end) effects in electronic stru ctur e of conjuga tedmolecules. The acceptor attracts electronic chargeand a t t empt s t o conver t t he cha i n s t r uct ur e t o

zwitterionic. In response, the -electronic systemscreens t he acceptor influence by indu cing a positivechar ge at th e acceptor end . The electrons complet elyscreen t he a cceptor over an effective length of about10 double bonds leading t o a sa t ur a t ion of t heground-sta te dipole moment at th is molecular size.Other parts of the molecule are unaffected by theacceptor. lj and QA deviate a gain from th eir bulk

values near th e neut ra l end of the m olecule (Figure4B) due to boundary condition effects imposed bystructure on the right molecular end.

This effect of t he acceptor substitution furt herstrongly affects the absorption spectra:17,81,262,269 Th espectrum of the unsubstituted molecule N(20) isdomina ted by a single peak a, whereas in t he accep-tor molecule P(20) this resonance is red shifted anda second, weak er, peak b appears. These trends maybe accounted for by inspecting th e relevant tr an sitiondensities. The electronic modes ofN molecule (panelsa a n d b in Figure 4) are almost symmetric withrespect to the diagonal (m n n m ). This means thatthere is no preferable direction for the motion of

lj ) l2j - l2j-1, j ) 1, ..., n (3.1)

QA ) Qacceptor + a)1

A

qa (3.2)

F i g u r e 4 . (A) Structures of the neut ral N(n) and polarP(n) (substituted by the strongest acceptor) carotenoids.Molecular geomet ries were optim ized using AM1 model492in Gaussian 98 package.14 2 Calculations were done forchain lengths ofn ) 5, 10, 20, an d 40 double bonds; (B)Variation of the bond-length alternation (top) an d totalcharge QA (bottom) along the chain in polar P(40) molecule;(C) Linear absorption spectra calculated with line width) 0.2 eV of the N (20) (dash ed lines) an d P(20) (solid lines)molecules; contour plots of electronic modes which domi-nat e th e absorption spectra of N(20) and P(20). Reprintedwith permission from ref 81. Copyright 1997 AmericanChemical Society.



13/42

electrons (or holes). a is a bulk mode similar to thebulk t ra nsition in P PV (Figure 3(I)) with coheren cesize L c 12 double bonds.81 The second oscillat or bhas a n onuniform diagonal spa tial distribution, withthree distinct contributions to th e dipole moment,mak ing a weak contr ibution t o the linear a bsorption.The lowest feature (a) in P(20) Figure 4) is a chargetransfer mode with diagonal size of L d 17 andcoherence size L c 12 double bonds,81 completely

localized at the acceptor end. Its dipole moment islarge and local ized. This mode carr ies a s t rongoscillator strength in the optical response of smallchains, which saturates in larger molecules (n > 17).The second mode (b) resembles th e bulk mode of th eneutral molecule (compared to a ). Its oscillat or-str ength for molecules with J 12 grows linea rly withn . The absorption spectra of sma ll cha ins ar e th ere-fore dominated by the charge-t ransfer mode (a)wherea s th e bulk mode (b) ta kes over with increas ingmolecular size. The distin ct cha ra cter of th ese modesis less apparent in chains shorter than the effectivecoher ence size of 12 double bonds .81

The optical r esponse of long d onor/acceptor subst i-tut ed molecules can th us be inter preted by dividingth em into th ree effective regions: th e acceptor (I) an dthe donor (III) boundar y regions at the molecularends, connected by the bridge (middle) region (II).The absence of electr onic delocalization betweenthese r egions implies tha t th e optical properties areadditive and can be described in the same way asth ose of molecular aggregat es.270,271 Region II ha s thesame properties as neutral molecule; it only showsodd order responses which scale linearly with size,whereas regions I and III have a fixed size deter-mined by th e screening length of the su bstituents.The ground and the excited states are zwitterionic.

These effective regions are solely responsible foreven-order optical responses. The odd-order re-sponses for long chains are dominated by the contri-bution of region II, which is proportional to the size,81

but regions I and III affect the response as well.

These acceptor substitution effects on the nonlinearresponse of car otenoid h ave been stu died extensively.Experimental investigations17,262 reveal tha t th e sub-stitution resulted in third-harmonic generation val-ues up t o 35 times higher t han in -carotene whichcorresponds to N(11) molecule. Subsequent CI/SOSquantum-chemical calculations269 rationalized theorigin of this enhancement and assessed the ap-

plicability of simple models t o describe th e evolutionof the molecular polar izabilities. In par ticular , th isstudy shows a steplike increase of the longitudinalcomponent of the dipole moment with the appliedexterna l field, cau sed by cha rge-tra nsfer toward t heacceptor end leading to an enhan ced nonlinear re-sponse.

C. Quantum Confinement and Size Scaling ofOff-Resonant Polarizabilities of Polyenes

Conjugated polymers have large polarizabilitiesattributed to the delocalized nature of electronicexcitat ions. Num erous experimental an d th eoretical

studies have forged a pretty good understanding oftheir electronic and optical cha racteristics.15,31 Pio-neering theoretical investigations of NLO propertiesof polymer s usin g solid-sta te ph ysical concepts h avebeen carried out by Andre, Champagne, and co-workers27 2-27 6 These investigations u tilized the su mover states273 and the polarization propagator tech-nique.274,276 A similar study has been done by usinga variational method for the time-dependent wave

function.277,278

Ab initio approach combined CPHFmethod has been applied to polyene oligomers ofmoderate sizes,34,35 extrapolated to the infinite sys-tems using the periodic boundar y conditions,279,280

and extended into finite frequency off-resonantregime.28 0-28 2 I t has been shown that vibrat ionalcontributions to t he polarizability may be as impor-tant as their electronic counterparts.283 -289 Thesenu clear effects ar ise from geometr y deforma tionsinduced by th e externa l field -electron delocalizationand polymer n onrigid energy potential su rface str onglyenhances the vibrational contribution.

The va ria tion of off-resona nt optical polar izabilitiesof polyenes w ith molecula r size ma y be described by

the scaling law n b, n being t he n umber of repeatuni ts and b is a scaling exponent. In first (R) andthird () order responses th e scaling exponents b varyconsidera bly for short molecules: 1 < bR < 2 a n d2 < b < 8.10,15,32,33,289-29 7 For elongated chains, theexponent b atta ins th e limiting value 1, indicatingth at th e polarizabilities become exten sive propert ies.Recent theoretical studies suggest that this sets inat about 30-50 repeat uni ts . An unusual ly largesaturation length was reported experimentally in onecase,298 which was then corrected to yield a value of60 repeat units.299 )

Static electronic polarizabilities up to seventh orderfor polyacetylene oligomers with up to 300 carbonatoms were computed using the PPP Hamiltoniancombined with the DSMA.89,300 The polarizabilitiesar e obta ined by adding respective contribu tions fromeffective electronic modes calculated in th e DSMAprocedure.89,300 These modes manifest themselvesin the response with different effective oscillatorstren gths at each order. Typically, higher frequencymodes make more significant contributions to thehigher order responses. The ground state densitymatrix Fj (a) as well as the five dominant modeslabeled b-f are depicted in Figure 5 for N ) 30 (toptwo rows) an d N ) 100 (bott om two r ows). As notedear lier, the delocalization of th e off diagona l element s

represents electron ic coherence between differentatoms. Figure 5 clearly shows how electronic coher-ence which is very limited for the almost diagonal Fjincreases very ra pidly for the higher modes in thecase of longer oligomer (N ) 100), wher eas finite s ize(quantum confinement) effects are i l lustrated forN ) 30. We note that modes a and b are hardlyaffected by redu cing th e size from 100 to 30. H ow-ever, the more delocalized, higher modes, showsignificant confinemen t effects.

This coherence size directly controls the size-scaling beha vior of nonlinear optical response. Thecalculated first- (R), th ird- () an d fifth -ord er () sta ticpolarizabilities of polyacetylene cha ins with up t o 200



14/42

carbon atoms ar e shown in th e Figure 6A. Pan el Bshows their scal ing exponents . We note that thevariation with size is very ra pid at sm all sizes but beventually saturates, and attains the bulk value of1. In general, higher frequency modes contributemore to the higher n onlinea r response. Since the sizeof the mode grows with mode frequency (Figure 5),the crossover (coherence) size increases for higher

orders nonlinearities (Figure 6A). The scaling andsat ur at ion sizes of sta tic nonlinear polar izabilities inpolyenes and other polyconjugated oligomers havebeen studied in detail using the CE O appr oach.297,238

Simple analytical expressions for size and bond-length alternation dependence of off-resonant polar-izabilities were derived297 using a single-oscillatorapproximation. The relations between the magnitudeof the saturation size L c ha ve been investigated forseveral families of molecules in ref 238. The size-scaling behavior of the second-order nonlinear re-sponse in conjugated oligomers substituted by donorand acceptor groups will be ana lyzed in SectionIIID.301,302

D. Origin, Scaling, and Saturation ofOff-Resonant Second Order Polarizabilities inDonor/Acceptor Polyenes

Donor/bridge/acceptor t ype molecules a re n ot cen-trosymmetr ic and t herefore possess even-order non-linear polarizabilities. Experimental12,13,18,19,303 andtheoretical14,304-310 studies have thoroughly investi-gated the variation of polarizabilit ies magnitudeswith donor and acceptor strength, length, and type

of the congugation bridge, and molecular conforma-tions. A common approach for computing nonlinearpolarizabilit ies is to use a perturbative expansioninvolving a summation over all molecular states. Byrestr ict ing the summation to a s ingle low-lyingexcited state and assuming that the charge-transfertra nsition is unidirectional, Oudar an d Chemla 311,312

obtained th e two-level expression comm only used forestimat ing t he second-order polarizability

where gg and ee are the ground and excited-statedipole moments, ge is the t ran sition dipole, and Egeis th e t ran sition frequency. A superficial look at eq3.3 suggests a rapid nonlinear scaling with n sincethe permanent dipole m oments gg, ee a n d t h etra nsition dipole ee ar e expected to grow with n . Itis not clear from eq 3.3 precisely how should scalewith molecular size. Establishing the precise scalinglaw of and its crossover to the bulk is an importa ntissue. E xperimental studies restricted by syntheticlimitations to chain length of 15-20 repeat uni tsshow 1.4 < b < 3.2,14,18,13,12,19,313 wherea s calculat ionsperformed with up to 22 repeat units yield 1.5 0 is found, one can work in the orthogonalsubspace by choosing initial vectors p (2) and q(2)

ort hogonal to p1 and q1, respectively. All subsequ entLanczos expansion vectors {pm

(2), qm(2)} will remain

orthogonal to {p1, q1} as follows from eq D4 (theoblique projection may be used to correct the loss oforthogonality at large M.214,220,221) The Lanczos al-gorith m will thu s converge t o the second-lowest RP A

eigenvalue. Alterna tively, th e deflection pr ocedur e214,219

could be used for t he sa me pu rpose. Suppose we havefound the j lowest eigenmodes (1, (2, ..., (j. Weintroduce t he deflected operat or L def:

where th e modes are normalized:

| ) 1 for > 0. The operat or L def has t he same eigenmodes asL ; however, the eigenvalues of( for v ) 1, ..., j ar eshifted: (

(d) ) (( + ). The next pair of eigen-modes ofL , ((j+1), thu s corr esponds to the lowest pairofL def, provided is large enough. Ort hogona lizat ion(or deflection) procedur es th us allow one t o find RP Aeigenproblem (eqs 2.15 and 2.18) solutions one byone.

I t i s i l lust rat ive to compare the resul ts of theIDSMA algorithm, which provide an approximatespectru m, and Lanczos algorithm, which provideaccur ate eigensta tes. The PP V-4 oligomer ha s been

compu ted using oblique Lan czos algorith m (50 m odes)an d IDSMA (9 modes for each of the th ree polariza-tion dir ections).219 The results shown in the top andbottom pa nel of Figu re 20, res pectively, ar e very closefor both algorithms for low-energy spectrum (below4 eV) where th e peaks are well separated energeti-cally. On t he other h an d, the h igher energy (5-6 eV)spectrum has many closely lying modes resolved bythe Lanczos algorithm. IDSMA approximates thesepeaks by a single effective oscillator. Lanczos andIDSMA are th us complement ar y since th ey providehighand lowresolution spectra. In particular, theDSMA algorithm is extremely u seful for computingoff-resonant response becau se it allows one t o tak e

F i g u r e 1 9 . Convergence of the Lanczos algorithm forPPV-4 oligomer. Reprinted with permission from ref 219.Copyright 1996 American Institu te of Physics.

K cm ) m dm T dm ) m cm ) 1, ..., M (D7)

F i gur e 20. Linear absorption spectrum for PPV-4 oligo-

mer calculated us ing th e DSMA (top panel) and theLanczos algorithm (bottom panel). Reprinted with permis-sion from ref 219. Copyright 1996 American Institute ofPhysics.

L de f L + )1

j

{, - ,

} (D8)



32/42

into account integral effective oscillator contributionsfrom the ent ire spectrum. In comparison, otheralgorithm s, such as La nczos an d Davidson, ar e ableto calculat e contr ibutions to th e r esponse only froma narrow spectral region.

XI. Appendix E: Davidsons Algorithm

A. Davidsons Preconditioning

The Hermit ian Lanczos algori thm is t he bestmethod for approximating extreme eigenvalues, whenno extra information about the matrix H is givenbesides the prescription for computing the matrix-vector pr oducts. In some problems, ther e exists someuseful informa tion a bout t he intern al stru ctur e ofH,and preconditioning techniques can speed up theconvergence. One of the most widely used meth odsof th is class is th e Davidson algorithm 216 that utilizesthe information about the diagonal elements of H(e.g., CI Singles matrix H ) A in eq 2.18) , andrequires fewer iterat ions when t he diagona l elementsof H are dominant. Davidson derived his method

though per turbat ion analysis for large scale CIcalculations.216 The idea of Davidsons p recond ition-ing is s imple. As in the Lanczos algori thm, theeigenvalue problem is solved by projecting the matrixonto a certain subspace KM that expands with thenumber of iterations. In the Lanczos algorithm, thespace KM is augment ed at each iterat ion step by theresidual vector

where M and vM are approximations for an exacteigenvalue and eigenvector, respectively, in the space

KM. In contrast, the Davidson algorithm augmentsthe subspace KM by

where D is the diagonal pa rt ofH (e.g., for H ) A ,Dij ) i - j where i and j are the energies of unoccupied a nd occupied MOs in eq 2.8). In eqs E 1and E2 vectors rM and rM, respectively, are intendedto be a correction to vM.

To ra tiona lize t he merits of Davidsons precondi-tioning, we recall that the rate of convergence is

approximately exponential in the gap ratio48 9

where 1, 2, and M are t he sma llest, second, an d th elargest eigenvalues of H. The convergence thusdecreases if the desired eigenvalues are not wellsepara ted from t he rest of the spectru m. To impr oveconvergence, Lanczos algorithms with preconditionedconjugate gradient method has been developed.490 Inthe Davidson expansion eq E2, 1/(D - MI) can beviewed as appr oximate inverse of (H - MI) if H is

domina ted by its diagonal elements. Event ua lly, Mapproaches a true eigenvalue and, therefore, th edistribution of the eigenvalues of ) (H - I)/(D -I) contr ols t he a symp totic conver gence of Davidson smethod. We can easily see that the smallest eigen-value of is 0 and the other eigenvalues have theten dency to be compressed ar ound 1, mak ing the gapratio eq E3 large and the Davidsons method sub-stantially more efficient than Lanczos when H is

dominated by its diagonal elements. We also notetha t th e Davidson algorithm requires th e knowledgeof th e entir e basis of th e subspa ce KM which imposesheavier mem ory requirem ents compar ed to Lanczosalgorithm which only keeps thr ee vectors from KM.

B. Davidsons Algorithm for Non-HermitianMatrices

Similarly t o Lanczos m ethod for solving t he RPAproblem eq 2.15, Davidsons algorith m needs to bemodified to take into account the block paired struc-ture of eq 2.18 and scalar product eq B1. The firstRPA algorith m ha s been developed by Rettru p208 andlater improved by Olsen.222 The method has beenfurther refined in ref 79, combined with TDDFTtechnique, and incorporat ed into Gaussian 98 pa ck-age.14 2 We will follow r ef 79 to describe t his met hod.

We f i rs t note that in the space of coordinate-momentum variables q and p (eq B10), the RPAMM eigenvalue problem (eq 2.18) can be presentedas

where T and K are the stiffness and kinetic energymatrices, respectively. The right and left eigenvectorsof this n on-Hermitian equation are q and p elec-tronic modes which sat isfy eq B11 with (p, q) ) 1

normalization condition (eq B15). Alternatively eq2.18 can be presented in the form of Hermit ianeigenvalue problem:

where [q] ) K-1/2[q]. Similarly to the Lanczos pro-cedure, the Davidsons algorithm constru cts thereduced analogue of eqs E4 (or E5) in KM subspacewith M , M.

To calculate the first k eigenvectors of L , t h ealgorithm starts from selected trial vectors in theorthonormal subspace b1, ..., bM, M > k. We n extgenerat e configurat ion space vectors Kbm and T bm ,

m ) 1, ..., M using eq 2.19 (th e most inten sive CPUstep), and form mat rices M m n

+ ) (bm , T bn) and M m n- )

(bm , Kb n) (m , n ) 1, ..., M). The reduced analoguesof eqs E4 and E5 are constructed by computing,respectively,

Diagonalizing matrix M(1) (or M(2)) we obtain thereduced eigenvalues which ar e the appr oximations

rM ) (H - MI)vM (E1)

rM )1

D - MI(H - MI)vM (E2)

)2 - 1

M - 2(E3)

KT[q] ) 2[q] (E4)

K1/2

T K1/2[q] ) 2[q] (E5)

M m n(1) )

k

M m k-

M k n+ (E6)

M m n(2) )

jk

(M )m j-1/2(M jk

+)(M k n- )1/2 (E7)



33/42

for eigenvalues ofL . The appr oximate eigenvectorsof our RPA pr oblem p and q are th en computed as

where L m and R m ar e the left end r ight eigenvectorsof m atrix M(1) (or M(2)), respectively. It was found

numerical ly in ref 79 that eq E7 provides fasterconvergence tha n eq E6.To improve the appr oximation, t he dimen siona lity

ofKM needs to be extended. Following ref 79, wedefine 2k residual vectors

and a set of pertur bed vectors u sing Davidsonspreconditioning:21 6

where Dij ) i - j (i and j are the energies of un occupied an d occupied MOs in eq 2.8) and in dicesi and j ru n over the pa rticle an d hole var iables in Mspace. Finally, we orthogonalize the W vectorsamong themselves and with respect to the previousexpansion vectors b1, ..., bM, and add t hem t o t heexpansion set: b1, ..., bM+2k expanding M t o M +2k. We then sta rt with new expansion set and findnew approximations for eigenvalues and eigenvectorsofL and so on. This procedure is repeated until thedesired convergence criteria ar e sa tisfied.

XII. Appendix F: Frequency and Time DependentNonlinear Polarizabilities

A. Equation of Motion for Electronic Oscillatorsand Anharmonicities

We star t with the equat ion of motion for theinterban d component of the density ma trix (eq A11)

where L is a Liouville operator (eq 2.19), V is aCoulomb operator (eq 2.5), and T() (eq A8) areinterband an d intraban d part s of the time-dependentsingle-electr on density ma tr ix F(t) ) Fj + (t) + T((t)),respectively. T can be expanded in a Taylor serieswhich contains only even powers of (eqs A9 andA10). For optical signa ls not higher th an t hird order ,it is sufficient to retain only the lowest (second order)term:

We next expand (t) in terms of modes R (eq B16)

Ea ch oscillator R is described by two conjugatedmodes R and R

. Adopting the notation of refs 77,207, and 491, we define -R ) R

and -R ) - R, so

that equation L R ) RR would hold for R ) -M ...,M. zR an d its complex conjugate z-R ) zR

/ constitutethe complex oscillator amplitudes. Inserting theexpansion eq F3 into eq F1 and using eq F2 givesthe following equations for the complex amplitudes,

The a mplitudes for the adjoint (negative frequency)variables are simply the complex conjugates. Thisnonlinear equa tion may be solved by expanding z(t)(z*(t)) in powers of the external field E(t):

Similarly, using eqs A13 an d F2 we obtain th e opticalpolarization

I n eqs F4 and F6, we onl y r e t a i ned t e r ms t ha tcontribute to the th ird-order optical r esponse; R ) 1,..., M, , , ) - M, ..., M, an d th e coefficients in th erh s could be expressed u sing identit ies (B18) - (B23)in th e form

p ) m

M

L m bm , q ) m

M

R m bm , ) 1, ..., k (E8)

rp ) T[q] - p r

q ) K[p] - q ) 1, ..., k(E9)

(W)

ij)

1

I - Dij(r

)

ij ) 1, ..., 2k (E10)

i

t) L - E(t)[, Fj] - E(t)[, ] - E(t)[, T()] +

[V(), ] + [V(), T()] + [V(T()), ] + [V(T()), Fj](F1)

T() )12

[[, Fj], ] ) (I - 2Fj)2 (F2)

(t) ) R>0

(RzR(t) + R

zR/(t)) R ) 1, ..., M (F3)

izR

t) RzR - E(t)-R - E(t)

-R,z -

E(t)

-R,zz +

V-R,zz +

V-R,zzz

(F4)

z(t) ) z (1)(t) + z(2)(t) + z (3)(t) + .... (F 5)

P(t) )

z +1

2

zz. (F6)

R ) Tr([Fj, R][, Fj]) ) Tr (R) (F 7)

R ) Tr([Fj, R][, ]) ) Tr ((I - 2Fj)(R + R))

(F8)

R, ) Tr ([Fj, R][, 12[[, Fj], ]) )-

12

Tr((R + R)( + )) (F9)

VR, )1

2!

perm

(Tr([Fj, R][V(), ]) +

Tr ([Fj, R][V(1

2[[, Fj], ]), Fj])) )

1

2Tr((I - 2Fj)

(( + )V(R) + (R + R)V() +

(R + R)V())) (F10)



34/42

Here VR, and VR, have been symmetrized withrespect to all perm ut ations of indices , and , ,, respectively. These anharmonicities describe cou-pling among electronic oscillators mediated by Cou-lomb V and dipole interact ions (Note that theindices R, , , and run over positive an d negat ivemodes). describes optical tr an sitions between oscil-lators whereas V describes scattering between oscil-lators induced by the many-body Coulomb interac-tion. It is importan t t o note th at all the anh armoniccoefficients can be calculat ed usin g the groun d-stat edensity ma trix Fj as well as t he eigenmodes of th elinearized TDHF equation. Equations F6 and F4 mapthe task of computing the optical response of the

original m an y-electr on system onto finding t he oscil-lators and the nonlinear couplings and V. Wefurther note that the expressions for anharmonicitiesinvolving mu ltiplicat ions of electr onic modes m at ricesare better suited for numerical computations thanthose involving commuta tors.

B. Definition of Nonlinear Response Functions

Optical polarizabilities are induced by the deviationof the reduced density matrix from its equilibriumvalue Fj expan ded in powers of th e extern al field E(t).Following refs 76 and 77, we define time domainresponse functions R (j)(t, 1, ..., j) up t o th e third order

(j ) 1,2,3):

The corr esponding frequen cy domain p olar izabilitiesR (j)(-s;1,...,j) (j ) 1,2,3) are defined by

Here E() is the Fourier t ransform of the t ime-dependent externa l field E(t) defined as

The relations between response functions and polar-izabilities are obtained by comparing eqs F12-F14with eqs F15-F17 and using the Fourier tran sformeq F18:

The linear, second, and third

sergei tretiak and shaul mukamel- density matrix analysis and simulation of electronic excitations...

Documents