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Density Matrix Analysis and Simulation of Electronic Excitations in Conjugated and Aggregated Molecules

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Density Matrix Analysis and Simulation of Electronic Excitations in Conjugated and Aggregated Molecules 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 and Anharmonicities I. Introduction 3171 B. Definition of Nonlinear Response Functions 3204 II. The CEO Formalism 3175 C. Linear Response 3204 A. Electronic Hamiltonian and Ground State 3175 D. Second-Order Response 3205 Calculations E. Third-Order Response 3206 B. Computation of Electronic Oscillators 3177 XIII. References 3207 C. Real Space Analysis of Electronic Response. 3179 III. Electronic Coherence Sizes Underlying the 3181 Optical Response of Conjugated Molecules A. Linear Optical Excitations of Poly(p-phenylene 3181 I. Introduction 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 photochemistry of complex molecules1-3 as well as Donor/Acceptor Polyenes organic nanostructures and supramolecular assem- E. Localized and Delocalized Electronic 3185 blies.4,5 Photosynthesis and other photochemical bio- Excitations in Bacteriochlorophylls logical processes that constitute the basis of life on IV. Optical Response of Chromophore Aggregates 3186 Earth involve assemblies of conjugated chromophores A. Excitonic Couplings and Electronic 3187 such as porphyrins, chlorophylls, and carotenoids.6-8 Coherence in Bridged Naphthalene Dimers Apart from the fundamental interest, these studies B. Electronic Excitations in Stilbenoid 3188 are also closely connected to numerous important Aggregates technological applications.9 Conjugated polymers are C. Localized Electronic Excitations in 3189 primary candidates for new organic optical materials Phenylacetylene Dendrimers with large nonlinear polarizabilities.10-19 Potential D. Exciton-Coupling for the LH2 Antenna 3191 applications include electroluminescence, light emit- Complex of Purple Bacteria ting diodes, ultrafast switches, photodetectors, bio- V. Discussion 3192 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 * Corresponding author. E-mail: [email protected] (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 Solving the many-electron problem required for the Press), 1995. prediction and interpretation of spectroscopic signals involves an extensive numerical effort that grows been widely applied using semiempirical Hamilto- very fast with molecular size. Two broad classes of nians (e.g., simple tight-binding or Hu¨ ckel, π-electron techniques are generally employed in the calculation Pariser-Parr-Pople (PPP), valence effective Hamil- of molecular response functions. Off-resonant optical tonians (VEH), complete neglect of differential over- polarizabilities can be calculated most readily by a lap (CNDO), and intermediate neglect of differential variational/perturbative treatment of the ground Overlap (INDO) models).14,15,34-39 The global eigen- state in the presence of a static electric field by states carry too much information on many-electron expanding the Stark energy in powers of electric field. correlations, making it hard to use them effectively The coupled perturbed Hartree-Fock (CPHF) pro- for the interpretation of optical response and the cedure computes the polarizabilities by evaluating prediction of various trends. energy derivatives of the molecular Hamiltonian. It A completely different viewpoint is adopted in usually involves expensive ab initio calculations with calculations of infinite periodic structures (molecular basis sets including diffuse and polarized functions, crystals, semiconductors, large polymers). Band struc- that are substantially larger than those necessary for ture approaches that focus on the dynamics of computing ground-state properties.14 electron-hole pairs are then used.40-44 Band theories The second approach starts with exact expressions may not describe molecular systems with significant 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 not lend themselves very easily to real-space chemical 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- both the ground state and excited-state wave func- cal challenge.45 tions and the transition dipole moments between To formulate a unified formulation that bridges the them.29,30 The configuration-interaction/sum-over- gap between the chemical and semiconductor points states (CI/SOS) method15,31 is an example for this of view, we must retain only reduced information class of methods. Despite the straightforward imple- about the many-electronic system necessary to cal- mentation of the procedure and the interpretation of 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 29 to be taken when choosing the right configurations. eigenstates |ν〉, |η〉, ... and energies ν, η, .... Using In addition, this method is not size-consistent,32,33 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
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