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Chem. Rev. 2002, 102, 3171−3212 3171
Density Matrix Analysis and Simulation of Electronic Excitations in
Conjugated 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 A. Equation of Motion for Electronic Oscillators 3203
I. Introduction 3171 and Anharmonicities
II. The CEO Formalism 3175 B. Definition of Nonlinear Response Functions 3204
A. Electronic Hamiltonian and Ground State 3175 C. Linear Response 3204
Calculations D. Second-Order Response 3205
B. Computation of Electronic Oscillators 3177 E. Third-Order Response 3206
C. Real Space Analysis of Electronic Response. 3179 XIII. References 3207
III. Electronic Coherence Sizes Underlying the 3181
Optical Response of Conjugated Molecules I. Introduction
A. Linear Optical Excitations of Poly(p-phenylene 3181
vinylene) Oligomers Predicting the electronic structure of extended
B. Linear Optical Excitations of 3182 organic molecules constitutes an important funda-
Acceptor-Substituted Carotenoids mental task of modern chemistry. Studies of elec-
C. Quantum Confinement and Size Scaling of 3183 tronic excitations, charge-transfer, energy-transfer,
Off-Resonant Polarizabilities of Polyenes and isomerization of conjugated systems form the
D. Origin, Scaling, and Saturation of 3184 basis for our understanding of the photophysics and
Off-Resonant Second Order Polarizabilities in 1-3
Donor/Acceptor Polyenes photochemistry of complex molecules as well as
E. Localized and Delocalized Electronic 3185 organic nanostructures and supramolecular assem-
4,5
Excitations in Bacteriochlorophylls blies. Photosynthesis and other photochemical bio-
IV. Optical Response of Chromophore Aggregates 3186 logical processes that constitute the basis of life on
A. Excitonic Couplings and Electronic 3187 Earthinvolveassembliesofconjugatedchromophores
Coherence in Bridged Naphthalene Dimers suchasporphyrins,chlorophylls,andcarotenoids.6-8
B. Electronic Excitations in Stilbenoid 3188 Apart from the fundamental interest, these studies
Aggregates are also closely connected to numerous important
technological applications.9 Conjugated polymers are
C. Localized Electronic Excitations in 3189 primarycandidatesforneworganicopticalmaterials
Phenylacetylene Dendrimers 10-19
D. Exciton-Coupling for the LH2 Antenna 3191 with large nonlinear polarizabilities. Potential
Complex of Purple Bacteria applications include electroluminescence, light emit-
V. Discussion 3192 ting diodes, ultrafast switches, photodetectors, bio-
sensors, and optical limiting materials.20-27
VI. Acknowledgments 3194 Optical spectroscopy which allows chemists and
VII. Appendix A: The TDHF Equations of Motion of 3194 physicists to probe the dynamics of vibrations and
a Driven Molecule electronic excitations of molecules and solids is a
VIII. Appendix B: Algebra of Electronic Oscillators 3196 powerful tool for the study of molecular electronic
IX. Appendix C: The IDSMA Algorithm 3197 structure. The theoretical techniques used for de-
X. Appendix D: Lanczos Algorithms 3199 scribing spectra of isolated small molecules are
A. Lanczos Algorithm for Hermitian Matrices 3199 usually quite different from those of molecular crys-
B. Lanczos Algorithm for Non-Hermitian Matrices 3200 tals, and many intermediate size systems, such as
XI. Appendix E: Davidson’s Algorithm 3202 clusters and polymers, are not readily described by
A. Davidson’s Preconditioning 3202 the methods developed for either of these limiting
28
B. Davidson’s Algorithm for Non-Hermitian 3202 cases.
Matrices
XII. Appendix F: Frequency and Time Dependent 3203 * Correspondingauthor.E-mail: serg@lanl.gov(S.T.);mukamel@
Nonlinear Polarizabilities chem.rochester.edu (S.M.).
10.1021/cr0101252 CCC: $39.75 © 2002 American Chemical Society
Published on Web 08/24/2002
3172 Chemical Reviews, 2002, Vol. 102, No. 9 Tretiak and Mukamel
Sergei Tretiak is currently a Technical Staff Member at Los Alamos Shaul Mukamel, who is currently the C. E. Kenneth Mees Professor of
National Laboratory (LANL). He received his M.Sc. (highest honors, 1994) Chemistry at the University of Rochester, received his Ph.D. in 1976 from
from Moscow Institute of Physics and Technology (Russia) and his Ph.D. Tel Aviv University, followed by postdoctoral appointments at MIT and
in 1998 from the University of Rochester where he worked with Prof. the University of California at Berkeley and faculty positions at the
Shaul Mukamel. He was then a LANL Director-funded Postdoctoral Fellow Weizmann Institute and at Rice University. He has been the recipient of
in T-11/CNLS. His research interests include development of modern the Sloan, Dreyfus, Guggenheim, and Alexander von Humboldt Senior
computational methods for molecular optical properties and establishing Scientist awards. His research interests in theoretical chemical physics
structure/optical response relations in electronic materials, such as donor− and biophysics include: developing a density matrix Liouville-space
acceptor oligomers, photoluminescent polymers, porphyrins, semiconductor approach to femtosecond spectroscopy and to many body theory of
nanoparticles, etc., promising for device applications. He is also developing electronic and vibrational excitations of molecules and semiconductors;
effective exciton Hamiltonian models for treating charge and energy transfer multidimensional coherent spectroscopies of structure and folding dynamics
phenomena in molecular superstructures such as biological antenna of proteins; nonlinear X-ray and single molecule spectroscopy; electron
complexes, dendrimer nanostructures, and semiconductor quantum dots transfer and energy funneling in photosynthetic complexes and Dendrimers.
assemblies. He is the author of over 400 publications in scientific journals and of the
textbook, Principles of Nonlinear Optical Spectroscopy (Oxford University
Solvingthemany-electronproblemrequiredforthe Press), 1995.
prediction and interpretation of spectroscopic signals been widely applied using semiempirical Hamilto-
involves an extensive numerical effort that grows nians(e.g., simple tight-binding or Hu¨ckel, π-electron
very fast with molecular size. Two broad classes of Pariser-Parr-Pople (PPP), valence effective Hamil-
techniquesaregenerallyemployedinthecalculation tonians (VEH), complete neglect of differential over-
of molecular response functions. Off-resonant optical lap (CNDO),andintermediateneglectofdifferential
polarizabilities can be calculated most readily by a Overlap (INDO) models).14,15,34-39 The global eigen-
variational/perturbative treatment of the ground states carry too much information on many-electron
state in the presence of a static electric field by correlations, making it hard to use them effectively
expandingtheStarkenergyinpowersofelectricfield. for the interpretation of optical response and the
The coupled perturbed Hartree-Fock (CPHF) pro- prediction of various trends.
cedure computes the polarizabilities by evaluating A completely different viewpoint is adopted in
energy derivatives of the molecular Hamiltonian. It calculations of infinite periodic structures (molecular
usually involves expensive ab initio calculations with crystals, semiconductors, large polymers). Band struc-
basis sets including diffuse and polarized functions, ture approaches that focus on the dynamics of
that are substantially larger than those necessary for
14 40-44
computing ground-state properties. electron-holepairsarethenused. Bandtheories
Thesecondapproachstartswithexactexpressions maynotdescribemolecularsystemswithsignificant
for optical response functions derived using time- disorder and deviations from periodicity, and because
dependent perturbation theory, which relate the they are formulated in momentum (k) space they do
optical response to the properties of the excited notlendthemselvesveryeasilytoreal-spacechemical
states. It applies to resonant as well as off-resonant intuition. The connection between the molecular and
response. Its implementation involves calculations of the band structure pictures is an important theoreti-
45
both the ground state and excited-state wave func- cal challenge.
tions and the transition dipole moments between Toformulateaunifiedformulationthatbridgesthe
29,30 gapbetweenthechemicalandsemiconductorpoints
them. The configuration-interaction/sum-over-
15,31 of view, we must retain only reduced information
states (CI/SOS) method is an example for this
class of methods. Despite the straightforward imple- about the many-electronic system necessary to cal-
mentationoftheprocedureandtheinterpretationof culate the optical response. Certainly, the complete
the results in terms of quantum states (which is information on the optical response of a quantum
common in quantum chemistry), special care needs system is contained in its set of many-electron
to be taken when choosing the right configurations. eigenstates |ν〉, |η〉, ... and energies ǫ , ǫ , ....29 Using
32,33 ν η
Inaddition,thismethodisnotsize-consistent, and the many-electron wave functions, it is possible to
intrinsic interference effects resulting in a near calculate all n-body quantities and correlations. Most
cancellation of very large contributions further limit of this information is, however, rarely used in the
its accuracy and complicate the analysis of the size- calculation of common observables (energies, dipole
scaling of the optical response. The SOS approach has moments, spectra, etc.) which only depend on the
Density Matrix Analysis in Conjugated Molecules Chemical Reviews, 2002, Vol. 102, No. 9 3173
expectation values of a few (typically one- and two-) Fνη are thus the building blocks for the time-de-
electron quantities. In addition, since even in practi- nm
cal computations with a finite basis set, the number pendent single-electron density matrix Fmn(t).
The greatly reduced information about the global
of molecular many-electron states increases expo- νη
nentially with the number of electrons, exact calcula- eigenstates contained in the matrices F is sufficient
tions become prohibitively expensive even for fairly to compute the optical response. To illustrate this,
small molecules with a few atoms. A reduced descrip- let us consider the frequency-dependent linear po-
tion that only keeps a small amount of relevant larizability R(ω) (see Appendix F3).
information is called for. A remarkably successful 2Ωµ µ/
example of such a method is density-functional R(ω) ) ∑ ν gν gν (1.5)
theory (DFT),46-51 which only retains the ground- ν 2 2
Ω -(ω+iΓ)
state charge density profile. The charge density of ν
the nth orbital is where µgν ≡ 〈g|µ|ν〉 are the transition dipoles, and
Fj )〈g|c†c |g〉 (1.1) Ων ≡ ǫν - ǫg are the transition frequencies. Γ is a
nn n n phemenological dephasing rate which accounts for
where |g〉 denotes the ground-state many-electron bothhomogeneous(e.g.,aninteractionwithbath)and
inhomogeneous(e.g.,static distribution of molecular
† transition frequencies) mechanisms of line broaden-
wavefunction and c (c ) are the Fermi annihilation
n n ing (for a review see ref 76).
(creation) operators for the nth basis set orbital, when Themoleculardipoleµisasingle-electronoperator
the overlap between basis set functions is neglected, that may be expanded in the form
the molecular charge density depends on F . Hohen-
nn
berg and Kohn’s theorem proves that the ground- µ) µ c†c (1.6)
state energy is a unique and a universal functional ∑ mn n m
52,53 nm
of the charge density, making it possible in
principle to compute self-consistently the charge Wetherefore have
distribution and the ground-state energy.
The single-electron density matrix54-60 given by µ ) µ Fgν (1.7)
gν ∑ mn nm
νη † nm
F ≡〈ν|c c |η〉 (1.2)
nm n m
The matrices Fgν and the corresponding frequencies
is a natural generalization of the ground-state charge Ων thus contain all necessary information for calcu-
density (eq 1.1). Here |ν〉 and |η〉 represent global lating the linear optical response. Complete expres-
electronic states, whereas n and m denote the atomic sions for higher order polarizabilities up to third
basis functions. Fνν is the reduced single-electron order and other spectroscopic observables are given
density matrix of state ν. For ν * η Fνη is the density- in Appendix F.
matrix associated with the transition between ν and Equation 1.2 apparently implies that one first
η. These quantities carry much more information needstocalculatetheeigenstates|ν〉and|g〉andthen
thanFj ≡Fgg (For brevity, the ground-state density usethemtocomputethematrixelementsFgν
nn nn . If that
gg
matrixF willbedonotedFj throughoutthisreview), was the case, no computational saving is obtained
yet considerably less than the complete set of byusingthedensitymatrix.However,itsgreatpower
eigenstates.51,61-66 is derived from the ability to compute the electronic
Density functional theory has been extended to response directly, totally avoiding the explicit calcu-
67
include current (in addition to charge) density. The lation of excited states: the time-dependent varia-
currentdensitycanbereadilyobtainedfromthenear 64,65,77,78
tional principle (TDVP) and time-dependent
diagonalelementsofthedensitymatrixinrealspace. density-functional theory (TDDFT)49,50,79,80 in the
Thecurrentisthusrelatedtoshortrangecoherence, 52,53
Kohn-Sham (KS) form are two widely used
whereasthedensitymatrixincludesshortaswellas approaches of this type. In either case, one follows
long range coherence. The single electron density the dynamics of a certain reduced set of parameters
matrixisthelowestorderinasystematichierarchy. representing the system driven by an external field.
Higher order density matrices (2 electron, etc.) have IntheTDVP,theseparametersdescribeatrialmany-
beenusedaswellinquantumchemistry.Theyretain electron wave function, whereas in TDDFT they
successively higher levels of information.68-73 Green represent a set of KS orbitals. The time-dependent
function techniques provide an alternative type of Hartree-Fock (TDHF) equations are based on the
reduced description.74,75 TDVP where the trial wave function is assumed to
The wave function of a the system driven by an 77,81
belongtothespaceofsingleSlaterdeterminants.
optical field is a coherent superposition of states Both TDHF and the TDDFT follow the dynamics
of a similar quantity: a single Slater determinant
Ψ(t) ) a (t)|ν〉 (1.3) that can be uniquely described by an idempotent
∑ ν 2 62,63,77,78
ν single-electron density matrix F (with F )F).
and its density matrix is given by However,theyyielddifferentequationsofmotionfor
F(t), stemming from the different interpretation of
† / νη F(t). In the TDHF, F(t) is viewed as an approximation
F (t) ≡ 〈Ψ(t)|c c |Ψ(t)〉 ) ∑a (t)a (t)F (1.4) for the actual single-electron density matrix,77 whereas
nm n m ν η nm
νη in TDDFT F(t) is an auxiliary quantity constrained
3174 Chemical Reviews, 2002, Vol. 102, No. 9 Tretiak and Mukamel
to merely reproduce the correct electronic charge In this language, the RPA procedure corresponds to
52,53 82,104
distribution at all times. TDDFTisformallyexact. thesummationofringdiagramstoinfiniteorder.
However, in practice it yields approximate results TheRPAapproachincombinationwiththePariser-
since exact expressions for the exchange-correlation Par-Pople(PPP)Hamiltonian111,112wasusedtostudy
energyE [n(r)]andthecorrespondingpotentialv (r, low-lying excited states of ethylene and formaldehyde
xc xc
[n]) in the KS scheme are not available and are 113,114
introduced semiempirically. A close resemblance by Dunning and McKoy in 1967. This investiga-
between TDHFandTDDFT(especiallyitsadiabatic tion concluded that the RPA results are superior to
version) may be established by formulating KS single-electron transition approximation and are very
density functional theory (DFT) in terms of the similar to CI Singles (the latter coincides with the
78 Tamm-Dancoff approximation). Subsequent compu-
density matrix F rather than on the KS orbitals. tations of small molecules,107,108,115-121 such as ben-
Thisformalsimilarity makesitpossible to apply the zene,107 118 117
same algorithms for solving the equations for the free radicals diatomics and triatomics,
gν showedhighpromiseofRPAformolecularexcitation
matrices F ≡ ê (Abbreviated notation ê for the
ν ν energies. However, it was found that the first-order
family of single-electron density matrices Fgν will be RPAyields inaccurate results for triplet states113,119
used throughout this review) and frequencies Ω , 122-126
ν andimpractical for unstable HF ground state.
directly avoiding the tedious calculations of global This happens when electronic correlations (doubles
eigenstates in both cases. and higher orders) are significant for the ground-
77,82-88
This review focuses on the TDHF method, state wavefunction, and the Hartree-Fockreference
which combined with a semiempirical model Hamil- state becomes a poor approximation for the true
tonian provides a powerful tool for studying the ground state wave function. For example, large
optical response of large conjugated molecules and contributions from doubly excited configurations lead
chromophore aggregates.81,89-96 The accuracy of this to imaginary RPA energies of triplet states in both
combination is determined by the approximations 113,114
involved in closing the TDHF equations and by the ethylene and formaldehyde. Several improved
schemes that take into account correlations beyond
semiempirical models. The TDDFT approach is on 120,127-133
the other hand usually based on the ab initio Hamil- the first-order RPA have been suggested to
tonians,49,50,79,80,97,98 making these computations sig- avoid these difficulties. Subsequently, RPA-based
nificantly more expensive and limited to smaller methods have been applied to calculate dynamics
molecular systems than TDHF/semiempirical tech- polarizabilities of small molecules using an analytical
nique. F(t) computed in the TDHF approach provides propagator approach.134-137 We refer readers to re-
thevariationofelectronchargedistribution(diagonal views104,74,138,75 for further details of this early devel-
elements) and the optically induced coherences, i.e., opment of RPA approaches.
changes in chemical bond orders, (off-diagonal ele- Zernerandco-workershadsubsequentlyattempted
ments) caused by an external field. The latter are to use RPA as an alternative to Singles CI for
essential for understanding optical properties of computingmolecularelectronicspectrawithZINDO
conjugated molecules and for the first-principles code.139-141 However, historically, these early RPA
derivation of simple models for photoinduced dynam- advances did not develop into standard quantum
ics in molecular aggregates (e.g., the Frenkel-exciton chemical software. Modern computational pack-
model).90 ages142-145 usually offer extensive CI codes but not
The TDHF equation of motion for the single- propagator-based techniques for handling the elec-
electron density matrix (eq A4 in Appendix A) was tronic correlations. However, current studies of propa-
first proposed by Dirac in 1930.99 This equation has gator techniques146,147 will be gradually incorporated
been introduced and explicitly applied in nuclear into quantum-chemical software.
100 Faster computers and development of better nu-
physics by Ferrel. The TDHF description was
101-104,83,84 merical algorithms have created the possibility to
widely used in nuclear physics in the 50-60s. apply RPAincombinationwithsemiempirical Hamil-
The random phase approximation (RPA) was first tonianmodelstolargemolecularsystems.Sekinoand
introduced into many-body theory by Pines and
Bohm.105 This approximation was shown to be equiva- Bartlett85,86,148,36 derived the TDHF expressions for
lent to the TDHF for the linear optical response of frequency-dependent off-resonant optical polarizabili-
many-electron systems by Lindhard.106 (See, for ex- ties using a perturbative expansion of the HF equa-
ample, Chapter 8.5 in ref 83. The electronic modes tion (eq 2.8) in powers of external field. This approach
are identical to the transition densities of the RPA was further applied to conjugated polymer chains.
eigenvalueequation.)ThetextbookofD.J.Thouless82 The equations of motion for the time-dependent
contains a good overview of Hartree-Fock and TDHF density matrix of a polyenic chain were first derived
theory. andsolvedinrefs149and150.TheTDHFapproach
111,112
The RPA approach was subsequently introduced based on the PPP Hamiltonian was subse-
into molecular structure calculations and was exten- quently applied to linear and nonlinear optical re-
151,152
sively studied in 60th and 70th as an alternative to sponse of neutral polyenes (up to 40 repeat units)
153-155
theCIapproachforsolvingmany-electronproblems. andPPV(upto10repeatunits). Theelectronic
TheRPAtheorywasdevelopedbasedontheparticle- oscillators (We shall refer to eigenmodes of the
hole propagators or two-electronic Green’s functions linearized TDHF eq ê with eigenfrequencies Ω as
ν ν
technique74 employing a direct decoupling of equa- electronic oscillators since they represent collective
107,108 109,110
tions of motion or perturbative approach. motions of electrons and holes (see Section II))
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