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ORGANIC STRUCTURE DESIGN APPLICATIONS IN OPTICAL AND ELECTRONIC DEVICES editors Preben Maegaard Anna Krenz edited by Wolfgang Palz Tahsin J. Chow
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Preface xiii
1. Theoretic al Modeling for Electron Transfer in Organic Materials 1 Robert C. Snoeberger III, Bo-Chao Lin, and Chao-Ping Hsu 1.1 Introduction 1 1.1.1 Theoretical Modeling for Electron Transfer in Solar Cells 3 1.1.1.1 The in luence of energy gaps 3 1.1.1.2 Charge separation and recombination 4 f 1.1.2 Theoretical Modeling for Charge Transport 7 1.1.2.1 The hopping models 7 1.1.2.2 Polaron models 10 1.2 Electronic Coupling 11 1.2.1 Structural Models 12 1.2.1.1 Idealized models 12 1.2.1.2 Crystal structures 13 1.2.1.3 Simulated morphology 14 1.2.2 Calculation of Electronic Coupling 14 1.2.2.1 Energy gap 15 1.2.2.2 Direct coupling 17 1.2.2.3 The generalized Mulliken–Hush and fragment charge difference schemes 20 1.3 Conclusion 23 2. Organic Structure Design and Applications in Solution-Pr ocessed Organic Micro- and Nanomaterials 33 Ting Lei, Jie-Yu Wang, and Jian Pei 2.1 Introduction 34 2.2 Main Approaches to Organic Micro/Nanomaterials 35 vi Contents
2.2.1 Template Synthesis 36 2.2.2 Electrospinning 37 2.2.3 Lithography 39 2.2.4 Physical Vapor Transport 40 2.2.5 Solution Process 43 2.2.5.1 Vapor diffusion 43 2.2.5.2 Phase transfer 44 2.2.5.3 Rapid solution dispersion 44 2.2.5.4 Sol-gel process 45 2.3 Molecular Design Strategy for Solution Processed Organic Micro/Nanomaterials 45 2.3.1 p–p Interaction 45 2.3.2 Donor–Acceptor Interaction 48 2.3.3 Sulfur–Sulfur Interaction 49 2.3.4 Hydrophobic Interaction 51 2.3.5 Hydrogen-Bonding Interaction 52 2.4 Impact Factors and Growth Mechanism of Organic Micro/Nanomaterials 54 2.4.1 Alkyl Chain Effect and p–p Stacking 54 2.4.2 Isomeric Effect and Solvent Effect 56 2.4.3 Organic Microtwist and Temperature Effect 59 2.4.4 Organic Micro/Nanotube Formed by Etching 60 2.4.5 Organic Flowers Formed by Hierarchical Self-Assembly 61 2.4.6 “Oriented Attachment” Mechanism 64 2.5 Applications of Organic Micro/Nanomaterials 66 2.5.1 Organic Field-Effect Transistors 66 2.5.2 Organic Light-Emitting Diodes and Organic Photovoltaics 71 2.5.3 Photodetector 72 2.5.4 Photowaveguide 73 2.5.5 Gas and Explosive Detection 75 2.5.6 Superhydrophobic Material 76 2.6 Surface Modi ication of Organic Micro/ Nanomaterials 77 f 2.7 Summary and Perspectives 79 Contents vii
3. Synthesis, Structure, and Electronic and Photophysical Properties of Donor–Acceptor Cyclophanes 95 Masahiko Shibahara, Motonori Watanabe, Takaaki Miyazaki, Jun-ichi Fujishige, Yuki Matsunaga, Keisuke Tao, Zhang Hua, Kenta Goto, and Teruo Shinmyozu 3.1 Introduction 95 3.2 Structural, Photophysical, and Charge Transfer Interaction of Multilayered [3.3]Paracyclophanes 100 3.3 Synthesis, Structure, Electronic, and Photophysical and Properties of [2.2]Benzothiadiazolophane 115 3.4 Synthesis, Structure, Electronic and Photophysical Properties of Two- and Three-Layered [3.3]Paracyclophane-Based Donor–Acceptor Systems 122 3.5 Conclusion and Future Remarks 130
4. Light- and Electricity -Gated Internal Rotation of Molecular Rotors: Toward Artificial Molecular Machines 137 Jye-Shane Yang and Wei-Ting Sun 4.1 Introduction 137 4.2 cis-trans Photoisomerization 141 4.3 Chemicals-Gated Molecular Brakes 152 4.4 Light-Gated Molecular Brakes 160 4.5 Electricity -Gated Molecular Brakes 168 4.6 Concluding Remarks and Perspecti ves 172
5. Supramolecular Assemblies of Organogels Featuring p -Conjugated Framework with Long-Chain Dicarboxamides 185 M. Rajeswara Rao and Shih-Sheng Sun 5.1 Introduction 186 5.2 Classi ication of Gels 186 5.2.1 Organogelators Based on Elongated f Hydrocarbons, Fatty Acids, and Esters 188 5.2.2 Organogelators Based on Saccharides 189 5.2.3 Organogelators Based on Steroids 190 5.2.4 Organogelators Based on Aromatic Molecules 190 viii Contents
5.2.5 Binary Organogelators 191 5.2.6 Metal Complex Based Organogelators 193 5.2.7 Organogelators Based on Amino Acids and Ureas 194 5.3 Organogelators Based on Amides 196 5.4 Conclusions 220 6. Quinoxaline-Based Polycyclic Molecules Having Defined Shapes: From Orthocyclophanes to Polyazaacenes 229 Teh-Chang Chou 6.1 Introduction 229 6.2 Prologue 232 6.3 U- and Z-Shaped Multi-Bridged [n,n¢ ]Orthocyclophanes 235 6.3.1 The Quadruple-Bridged [5,5]Orthocyclophanes and [6,6]Orthocyclophanes 236 6.3.2 The U-Shaped Septuple-Bridged [7,7]Orthocyclophanes 242 6.3.3 The Z-Shaped [6,4]Orthocyclophanes 248 6.4 N-Shaped p,p -Stacking Molecules 252 6.5 Multi-Functionalized Polyazaacenes 259 6.5.1 Multi-Functionalized Chlorinated Polyacenoquinone Esters 261 6.5.2 Mechanistic Consideration for Fragmentation of Chlorinated Polyacenoquinone Esters 268 6.5.3 Amination of Chlorinated Polyacenoquinone Esters 272 6.6 Closing Remarks 275 7. Fluorogenic Sensors of Heavy Metal Ions Based on Calix[4]arenes Functionaliz ed by 1,3-Dipolar Cycloaddition Reactions 287 Wen-Sheng Chung 7.1 Introduction of Calixarenes 287 7.2 1,3-Dipolar Cycloaddition Reactions and Subsequent Ring-Opening Reactions 290 Contents ix
7.3 Calix[4]arenes with Upper- and Lower-Rim Isoxazolines and Isoxazoles 293 7.4 Fluorogenic Sensors of Calix[4]arenes with Lower-Rim 1,2,3-Triazoles 299 7.5 Metal Ion Sensing and Ditopic Sensing Based on Calix[4]arenes Functionalized by 1,3-Dipolar Cycloaddition Reactions 309 7.6 Summary and Perspecti ve 319
8. Electron Transport Materials in Organic Light-Emitting Diodes: Design Considerations and Structural Diversity 327
Samik Jhulki, Ishita Neogi, and Jarugu Narasimha Moorthy 8.1 Introduction 327 8.2 Metal Chelates 330 8.3 Six-Membered Heterocycles 332 8.3.1 Pyridines 333 8.3.2 Quinoxaline 336 8.3.3 Naphthyridines 337 8.3.4 Phenanthrolines 338 8.3.5 Pyrazines 340 8.3.6 Pyrimidines 341 8.3.7 Quinoxaline 342 8.3.8 Anthrazolines 344 8.3.9 Triazines 345 8.4 Five-Membered Heterocycles 346 8.4.1 Isobenzofurans 348 8.4.2 Oxazoles 349 8.4.3 Benzimidazoles 349 8.4.4 Benzothiazoles 351 8.4.5 Oxadiazoles 352 8.4.6 Triazoles 354 8.5 Per luorinated Compounds 356 8.6 Metalloles 359 f 8.7 Miscellaneous 361 Contents
8.8 Electron-Transporting Materials for Phosphorescent Organic Light-Emitting Diodes 366 8.9 Structural Determinants for Better ETMs: Our Perspecti ve 367 8.10 Conclusions and Outlook 389 9. Electrochemical Deposition of Carbazole and Triarylamine Derivativ es and Their Polymeric Optoelectronic Applications 399 Man-kit Leung 9.1 Introduction 399 9.2 Hole Mobility in Triarylamine-Based Materials 400 9.2.1 Hole-Mobility in Organic Glass 401 9.2.2 Hole-Mobility in Liquid Films 403 9.3 Electrochemical Deposition of Triarylamine-Based Materials 403 9.3.1 Electrochemical Polymerization of Carbazole 405 9.3.2 Electrochemical Polymerization of Dicarbazole, Polycarbazole, and Carbazole Dendrimers 408 9.3.3 Electrochemical Polymerization of Triphenylamine Derivati ves 416 9.3.4 Electrochemical Polymerization of Diphenylamine 419 9.3.5 Electrochemical Polymerization of 4-(1-Hydro-xyethyl)Triphenylamine 420 9.3.6 Electrochemical Polymerization of bis(Triphenylamine)s 422 9.3.7 Electrochemical Polymerization of Poly, Hyperbranched, and Dendritic Triphenylamines 428 9.4 Lithography and Nanopatt erning 436 9.4.1 Electrochemical Nanolithography 437 9.4.2 Colloidal Template Electropolymerization 438 9.4.3 Electropolymerization of Macromonomer Bearing Photolabile Linker for Imaging 440 Contents xi
10. Solution-Pr ocessed Acenes and Their Applications on Field-Eff ect Transistor 455 Motonori Watanabe and Tahsin J. Chow 10.1 Introduction 455 10.2 Synthesis of Acenes from Norbornadienone Type Precursors 457 10.2.1 Precursors of Pentacene 458 10.2.2 Precursors of Tetracene and Hexacene 462 10.3 Physical Properties of Acene Precursors 463 10.3.1 Pentacene Precursors 464 10.3.2 Tetracene and Hexacene 467 10.3.3 Solution-Pr ocessed Thin-Film Transistor 469 10.3.4 OFETs Made with Single Crystals of Acenes 470 10.4 Higher Acenes with Functional Substituents 472 10.4.1 Acenes Containing Trialkylsilylethynyl Substituents 472 10.4.2 Acenes Containing Thiolyl Substituents 476 10.5 Twisted Acenes from Lactam-Bridged Precursors 477 10.6 Summary 479 w11. Synthetic Ne Route to Acenes 487 Chih-Hsiu Lin 11.1 Introduction 487 11.2 New Methodology in Acene Synthesis 489 11.2.1 Novel Convenient One-Step Reduction of Acenoquinone to Acene Derivatives 489 11.2.2 One-Pot Syntheses of Tetracene Sulfoxide and Tetracene Sulfone Compounds via Cascade Cyclization 491 11.2.3 Synthesis of Ladder-Type Oligonaphthalene Derivatives via Cationic Cyclization 493 11.2.4 Iterative Synthesis of Acene Diesters and Acene Dinitriles 496 11.2.5 Synthesis of Hetero-Acene and Perylene Derivatives Conjugated Systems Using Elongation Protocol 503 xii Contents
11.2.6 Direct Derivatization of Acene Skeletons: Ether–Ether, Ether–Sul ide, Ether–Selenide Exchange Reactions 505 11.3 Summary and Conclusion f 510
Index 513 Preface
The application of organic materials to optical and electronic devices is a fast-growing research area nowadays. These new type of materials are expected to be the mainstream in the next generation of smart machines. They combine the classical electronic properties of metals, yet with the advantages of organic matters, such as softness, light weight, good solubility, and versatile, such as extended p-conjugate systems and macrocycles withhigh structural special shapes. flexibility. The The functional choice of properties molecular include structures light- is
Thisemitting book diodes provides (OLEDs), a revieworganic onphotovoltaics several top-notch (OPVs), field-effect topics in thesetransistors areas. (OFETs), The contents artificial are machines,mainly focused and on chemical the design sensors. and synthesis of organic functional molecules but also include related topics such as the model study on electron transfer phenomena and the fabrication technologies of organic nanostructures. The key features in this book may be grouped into the following categories: (1) molecular design and synthesis of functional compounds of organic and organometallic molecules in forms discrete and polymeric structures; (2) solution-processed fabrication technologies, nanostructure growth and crystallization, electro- polymerization of organic amines, supramolecular assemblies of organogels; (3) principles of long-range electron transfer through organic media, and for the design of donor–acceptor dipolar compounds; and (4) evaluation of device performance with respect to structural designs, for the application to light-emitting diodes, and chemical sensors. The following gives a brief outline of each chapter.organic solar cells, field-effect transistors, artificial machines, To make a single molecule work as a functional device, several active sites have to be implanted onto a common molecular backbone so that they can communicate effectively in a designated xiv Preface
manner. In these systems, electron and/or energy transfer
energy, the mechanism of electronic coupling must be realized. Theprocesses construction are involved. of theoretical In order models to control by the using flow computational of electron/ methods is reviewed in Chapter 1. An overall view on organic nano- and micro materials constructed by the solution process and their applications is described in Chapter 2. The design strategies, various growth mechanisms, and device performances of OFETs, OLEDs, OPVs, photo-detectors, and super-hydrophobic materials are summarized. In comparison with physical vapor deposition, solution processing
organic nano- and micro materials with various morphologies, provides a more convenient and cost-effective approach to obtain corresponding applications are discussed based on the general conceptsincluding of wires, supramolecular sheets, and chemistry. flowers. Their relationships with the The p p interaction in organic molecules plays a fundamental role in many processes, such as the self-assembly for supramolecular stacking, − light-harvesting antennae for photosynthesis, amyloid
DNA, and so on. The study of electron/energy transfer processes infibril organic formation structures in a variety depends of diseases, on the double-helix understanding structure of their of p p interactions. In Chapter 3, the electron movement in p-stacked linear arrays, namely the multilayered [3.3]paracyclophanes, is examined− by their transient absorption spectra of radical cation species. Their electron/charge transfer mechanism is regarded as analogous to that of the double strand DNA molecules. One of the ultimate goals in the development of functional
level. The mechanical work of ATP synthase involves the rotation ofmolecules a central is stalk to assemble that is an powered artificial by machine electrochemical at the molecular potential energy created by the concentration gradient of proton across the inner membrane of mitochondria. As inspired by ATP synthase and other biological molecular machines, the development of
nanoscience and nanotechnology. Chapter 4 reviews the progress ofartificial molecular molecular design machines and functions has beenof molecular an important machines, subject with of a special emphasis on molecular rotors. The rotation of the rotors is gated by light and/or electrical energy as energy sources. Preface xv
electrochemistry are discussed. TheSupramolecular principle and efficiency gels are semisolid of the related materials, photochemistry which can serve and a variety of purposes and appear ubiquitously in our daily lives in a variety of forms. Gels are prevalent in nature, within cells materials, including toothpaste, soap, shampoo, hair gel, contact lenses,and tissues and of gelbodies, pens. and These are also materials present in self-assemble a variety of artificial through the formation of non-covalent intermolecular interactions to form supramolecular networks that trap solvent within their matrices. Because of the non-covalent nature of the forces of self- assembly, the gelation process is typically thermally reversible. In Chapter 5, various types of organogelators, mainly including those grafted with amide functionalities, are reviewed. The pharmacological importance of quinoxalines and their utility as building blocks for preparing organic electronic materials have motivated a considerable number of studies in recent years. The synthetic approaches to multi-bridged U, N, and Z-shaped
mainlyartificial by compounds three fundamental embedded reactions, with i.e., quinoxaline Diels–Alder units reaction, are described in Chapter 6. The synthesis was executed efficiently oxidation with RuO4, and carbonyl-amine condensation reaction. as electron/energy transfer phenomena, host–guest complexation, andThese pharmaceutical compounds may applications. possess specific functions of interest, such
Calixarenes are [1n] metacyclophanes, which are derived from conditions. Gutsche coined the term “calix[4]arenes,” which is derivedthe condensation from Latin of “calyx,” phenols which and means formaldehyde vase, pointing in different out the presence of a cup-shape structure in these macrocycles when they assume the cone conformation, where all four aryl groups are oriented to the same direction. In Chapter 7, a series of calix[4]arene derivatives containing various bifunctional groups were prepared by using “click chemistry,” where metal ions can be encircled
The contents of Chapter 8 are focused on the electron- transportingmaking them useful materials fluorescent (ETMs) sensors. of OLEDs. The major objective is to glean a variety of structure types from a comprehensive survey of organic compounds that are exploited for application xvi Preface
as ETMs and identify key structural elements/motifs that allow design and development of newer materials with improved device performances. Although limited to ETMs, the insights that are developed in this chapter will apply equally to all other types of materials, viz., hole-transporting materials, emissive materials, etc., that are relevant to OLED device constructions.
an To electrode fabricate by thin-filmspin casting. organic It requires electronic good devices, solubility polymeric of the polymersmaterials in are an appropriate usually coated solvent. as uniform The low solubility thin films of directlyconjugated on polymers limits their use on optoelectronic applications. Direct
an option to bypass the solubility problem. In Chapter 9, some fundamentalfilm formation properties on ITO about glass bythe electro-polymerizationchemical reactivity of carbazole provides and triphenylamine derivatives toward electro-polymerization are reviewed.
Acenes exhibit a strong tendency to form highly ordered films arein various recognized substrates as promising under different materials growth for conditions. the application They oncan OFETs.display However, a high charge the low mobility solubility in an of electricpentacene field in and,most therefore, solvents is a major drawback that limited its utility through solution
pentacene precursors in order to go around this problem. Inprocesses. Chapters Many 10 and efforts 11, new have methods been attempted for the preparation to prepare of “soluble” acenes are summarized. Workable synthetic schemes are outlined, and can be used as practical guide for the preparation of similar poly-aromatic materials. The authorship of this book includes eleven renowned professors in the top-rated universities and institutions in Asia: Kyushu University (Teruo Shinmyozu), Peking University (Jian Pei), IIT Kanpur (J. N. Moorthy), National Taiwan University (Jye- Shane Yang and Man-kit Leung), National Chiao Tung University (Wen-Sheng Chung), National Chung Cheng University (Teh-Chang Chou), and Academia Sinica (Chao-Ping Hsu, Shih-Sheng Sun, Chih-Hsiu Lin, and Tahsin J. Chow). Many of the authors have developed long-term research collaboration among themselves.
results and integrated into the contents of this book. It provides They have made significant efforts to summarize their best Preface xvii
optoelectronicthe first-hand materials. reference information to the readers who are interested in the progresses of the emerging new field of organic
Chapter 1
Theoretical Modeling for Electron Transfer in Organic Materials
Robert C. Snoeberger III, Bo-Chao Lin, and Chao-Ping Hsu Institute of Chemistry, Academia Sinica, 128 Sec. 2, Academia Road, 115 Taipei, Taiwan [email protected]
1.1 Introduction
Research and development of organic materials have been potentials of organic electronic devices. Organic photovoltaics (OPV),developed organic vastly light-emitting and become andiodes exciting (OLED), field because of the versatile transistors are a few examples of the new technologies that have been developed based on organic, carbon-based,and organic materials field-effect (Braga and Horowitz, 2009; Dimitrakopoulos and Malenfant, 2002; Hoppe theirand Sariciftci, fabrication 2004; endows Hung them and with Chen, novel 2002). properties Organic that electronic enables thedevices innovation attract great of newinterest technologies. because the organic materials used for limitless possibilities for the synthesis ofThese new propertiesmaterials. In include order weight,for devices flexibility, and technologies low cost, ease based of manufacture, on organic and electronics the seemingly to be
978-981-4463-35-5 (eBook) www.panstanford.com Theoretical Modeling for Electron Transfer in Organic Materials
developed and improved, high performance organic materials need
modeling can withto be physical designed insight and synthesized, into the microscopic and this isprocesses, where computational enabling the rational designhelp. of improvedTheoretical organic and computational materials, and modeling therefore, helps has become increasingly important. Electron transfer is an important process in the organic devices.
The commonly seen electron transfer events in these molecular ( ) devices include those depicted in Fig. 1.1:
(2)1 CS (or exciton dissociation), where a pair of electron and hole are created from an (neutral) exciton (3) CR, where the electron or hole recombines back to the neutral ground state for the molecules . the charge transport (CT), in which an excess electron or hole is passed through an array of ground-state molecules
Figure 1.1
listed.Schematic sketch for two commonly studied devices: OPV and OLED. The processes that are of fundamental interested are the electrons or holes are drifted
As shown in Fig. 1.1, in OLED, across a film of organic or polymeric semiconducting material, generated.creating either In collecting an excited solar state energy or anwith accumulated an OPV cell, charge the optically at the interface, and further electrooptically useful results are then Introduction excited exciton migrates to the interface of an electron- and a hole- transporting material where charge separation (CS) takes place. The basic principle of dye-sensitized solar cells (DSSCs) is similar, where takes place at the dye-TiO2 the sunlight is absorbed by the dye, and subsequent charge injection interface. With a CS, the electrons and holes are generated, and subsequently transported in the electron- and hole-transporting materials. Charge recombination (CR) can andalso take exciton place migration at the interface, are mainly contributing the light to absorption, a loss of efficiency. emission, and Otherthe excitation processes, energy such astransfer exciton in generation, the condensed photon phase. generation, In this importantchapter, we as keepwell. our scope to those related to electron transfer, even though detailed studies on these excitation process are very 1.1.1 Theoretical Modeling for Electron Transfer in Solar Cells 1.1.1.1 The influence of energy gaps
hP
In solar cells,VJ the power conversion efficiency ( ) can be defined as , hP OCP SC i FF where V J (1.1) P is the intensity of the incident OC i SC light. is the open-circuit voltage, is the current density at short-circuit,In general, FF Vis the fill factor, and
OC is roughly the difference for the oxidation and reduction potentials. For bulk-heterojunctionq2 solar cells, it is simply qV max IP – EA – , 40 rr DA OC donor acceptor (1.2) where 0 r is the relative dielectric rDA the initial separation distance of is the the optically vacuum generated permittivity, hole and electron pair in the donor constantand acceptor of the layers bulk (Rand organic et material,al., 2007). and IP is for ionization potential, between V while EA stands for electron affinity. In practice, a linear relation OC and the redox potential difference is indeed found (Scharber et al., 2006). Therefore, computational determination for Theoretical Modeling for Electron Transfer in Organic Materials
the positions of IP and EA helps determining the V of OPV cells
OC V is determined by the redox potentials of the (Sini et al., 2011). In DSSC, OC excited dye and the TiO2 electrolyte and the counter electrode. The energy difference in the on the energy gap in the conductiondye itself recently band is (Olivares-Amayaessentially dissipated. et al., Theoretical work on different organic and inorganic dyes is focused J , a panchromatic 2011). The HOMO-LUMO energy difference determines the lowest SC theoreticalfrequency photon models the can dye help can absorb. predicting To increase the energy gaps and help screeningabsorption candidates with a high with extinction good potential coefficient of yielding is important. a high ProperJ (Le
SC 1.1.1.Bahers2 et Chargeal., 2013). separation and recombination
processes in organic devices, it is necessary to link the electron In order to theoretically predict the efficiencies of electron transfer
transfer rate to the molecular properties (Brédas et al., 2009). The Marcus theory of electron transfer (Marcus, 1956) is commonly used to predict the electron transfer rate, as 2 2 kET = | Vif | exp(–(DG + ) /(4 kB T )), 4k T 2 1 B (1.3) where Vif is the reorganization energy, and DG is the Gibbs free energy change. is the electronic coupling,
or in the dye-TiO2 CS and CR usually take place in the heterojunction of OPV, contact of DSSC. After a direct absorption of light (in DSSC) or with an exciton migrated to the interface (in OPV), an excited-state fragment may proceed with CS and generate an electron and a hole, leading to electric current from the cell. The subsequent CR reduces the electric current collected from the device. Both CS and CR are important electronic processesDG in the organic materials (Clarke and Durrant, IP 2010; and Faist et al., 2011). can be typically modeled as the difference of the the EA of the two molecules in contact (Rehm and Weller, 1970). When an exciton reaches the interface, the Coulomb attraction of unbalance electron population of the excited charge transfer Introduction state also DG in the and the contributes. Such an interfacial electron transfer leads to a CoulombHowever, energy in a solid-state that influences interface, the overallthere exist factorsCS that make CR processes (Clarke and Durrant, 2010). At the estimation of the site energies of the molecules complicated. the interface, the difference of the Fermi energies between the energydonor and level the alignment acceptor depends leads to ona charge the electronic redistribution properties so that of the an twoequilibrium components; is established it is also (Ishii highly et al., dependent 1999; Gao, on 2010). the The arrangement resulting of the molecules in the interface, and both are hard to model theoretically (Heimel et al., 2010). One way to accommodate this difficulty is to use experimental results to establish the free energy model (Gao, 2010; Kang et al., 2006). This way, other aspects of region.the CS and The CR processes electron-hole can still recombination be established. becomes dominated The CR is often discussed in the context of the Marcus’ inverted when the gap energy is as small as the molecular reorganization regionenergy happens (Tsutsumi when et al.,–D G 2012). > l, and CR in is this typically case, the exothermic, electron transfer due to the Coulomb interaction for the ion pair.DG becomes The Marcus’ more negative inverted rate from Marcus theory decreases as seen(Kuciauskas in solar etcells. al., 2000; Yi et al., 2009). The decreased CR rate helps reducing the loss of efficiency and it is one important character
The reorganization energy characterized the coupling of the electron transfer state energy to the fluctuation of the environments. arisesIt is often from divided solvent intopolarization. an inner The and inner an outer reorganization component. energy The outer reorganization energy is the dielectric response energy charge carrier. It can be obtained from the geometry change that refers to the geometry relaxation of the molecule supporting the accompanies the oxidation or reduction of the fragments, and therefore, can be simulated (Chang and Chao, 2010; Hutchison et al., the2005a; inner Lin reorganizationet al., 2003; Malagoli energy and can Brédas, be viewed 2000). asIn a a solid-state dominate parameterdevice where for surrounding determining media the activationare with a energylow dielectric of the constant, electron transfer reaction. A small reorganization energy is desirable in Theoretical Modeling for Electron Transfer in Organic Materials
order to lower the free energy barrier of electron transfer. Many computational studies have been devoted to elucidating the factors that determine the reorganization energy (Chang and Chao, 2010; Chen and Chao, 2005). The electronic coupling has become an important target in theoretical studies (Adams et al., 2003; Coropceanu et al., 2007; Hsu,- 2009; Newton, 1991; Prytkova et al., 2007). To properly calculate the electronic coupling for the CS or CR process, a proper determi Herenation wefor the note structure that both is necessary. the experimental We include determination a brief discussion (Al- on the structural aspects of modeling works below (Section 1.2.1).
Mahboob et al., 2009) and the theoretical simulation (Cantrell and Clancy, 2012; Liu et al., 2011; Maggio et al., 2012) for the interfacial structures are important. The computational methods for the electron transfer coupling are also discussed in Section 1.2.2. The calculation of CS or CR coupling typically involves an excited state, and therefore, computational schemes described in Section 1.2.2.3 are likely useful. , Theoretical modeling for the CS and CR for OPV has been an active area of research (Yi et al., 2009; Liu et al., 2011; Yen et al. 2008). However, theoretical modeling for CS in DSSC requires a different set of consideration. The CS (or charge injection) in DSSC longeris a sub-picosecond valid since the process.excited dye The does fast not charge have a injection chance thermally after the relax,photo andexcitation the electron indicates transfer that the may traditional be coherent, Marcus and theory therefore, is no new theoretical treatments have been developed (Liang et al., 2007;
Caruso and Troisi, 2012). Computational characterization for the distribution of excited state population for the dye molecules has also provided insights in DSSC dyes (Yen et al., 2008; Huang et al., 2 2008).it attaches A large to. electronic A positive population correlation shifted is observed towards the between anchoring the group of the dye increases the electronic coupling with the TiO
theoretically calculated charge shift at the electron-accepting group et(weighted al., 2008), by the which oscillator is consistent strength withto account the physical for the light-absorbing consideration. capacity) and the measured short-circuit current in DSSCs (Huang
Therefore, computational works for the electron transfer in solar cells are beginning to offer some insights. Introduction
1.1.2 Theoretical Modeling for Charge Transport
The relationship between the average velocity of charge carriers E is established with the carrier and the applied electric field mobility: = E
(1.4) E as well Mobility is often measured and reported in the new devices whichdeveloped, states and that it log varies m is withproportional the electric to | E field| , strengthhas been reported as the temperature. For example, the Poole–Frenkel1/2 relationship frequently, and a number of theoretical origins have been discussed (Bässler, 1993; Nagata and Lennartz, 2008; Pasveer et al., 2005). A number of theories have been proposed to model the charge mobility of organic films) (Bässler, 1993; Bouhassoune et al., 2009; Cheng and Silbey, 2008; Kenkre et al., 1989; Ortmann et al., 2009; Novikov et al., 1998 . Unfortunately, the predictive power of computational modeling is still limited. For example, to the best of our knowledge, there has not been a theory that can correctly describe the full mobility dependence at different temperatures and electric field strengths. Some of the simulated mobility may exhibit a very similar dependence to the observed one, but the mobility values are often quite different from the observed ones. While a complete computation of bulk properties is currently infeasible, theoretical modeling is still useful in providing system- dependent parameters, and in offering further physical insights. The ability to predict the electronic coupling factor, for example, allows researchers to interpret experimental results based on theoretical expressions with one less adjustable parameter. The electronic coupling factor plays an important role in almost all of 1.1.the theoretical2.1 The hopping models and models it will be discussed in Section 1.2.2. The charge hopping rates The charge carrier is able to move within
Each charge hopping event can be modeled as an electron transfer. an organic material by “hopping” from one molecule to another.
In recent years, there has been much effort to describe the conductivity of organic materials with the electron transfer theory Theoretical Modeling for Electron Transfer in Organic Materials
(Brédas et al., 2002; Coropceanu et al., 2007; Troisi, 2011). The MarcusAnother theory practical of electron way transferto model (Eq. the 1.3)rate (Marcus,of the hopping 1956) processis often used for charge hopping rate.
is with the Miller–Abrahams (MA) formalism (Miller and Abrahams, 1960): DDEEij+| ij | k= k exp –2 r 0 0 ij k T B (1.5) where Eij
is the energy difference between hopping sites, and distance rij. is used to model the exponential decay in terms of site-to-site
whenWe the note charge that hoppingthe MA rate is endothermic. has an exponential Otherwise, dependence it becomes on the energy difference, which is like a simple Arrhenius rate law
independent of energy difference. In the Marcus theory, there is a ofquadratic the environment, free energy as dependence. a strong system-bath The Marcus interaction theory is based has been on a classical statistics over a harmonic potential for the fluctuation
assumed. The exponential distance dependence in the MA rate is an effect arising from the electronic coupling, and it is modeled as an isotropic exponential attenuation. We note that, especially for organic molecules containing fragments with extended aromatic or conjugated bonded structures, an isotropic electron transfer rate is a poor approximation. Face-to-face molecular contact allows interaction through molecular orbitals (MOs); the electronic coupling for this type of orientation decays slower than edge-to- edge contact (You et al., 2006). A proper account for the electronicg) is coupling factor helps reduce the problem arising from the crude assumption,Estimating and mobilitythe value of from the theexponential diffusion attenuation coefficients rate If( the no longer an adjustable parameter. of an external force qE, the Einstein relationship give rise an charge carrier is modeled as a Brownian particle under the influenceD as qD expression= for, the mobility in terms of the diffusion coefficient k T B (1.6) Introduction
where q is the charge of the carrier, kB and T D depends on the is Boltzmann’s constant, is temperature. The diffusion coefficient material. TheD diffusion of the charge carriers is limited by the time required to hop between sites. For a periodic system, the diffusion coefficienthopping rate was (Deng estimated and Goddard, by a weighted 2004). sum The rateof hopping of the distance hopping squared, with each hopping path weighted by the corresponding process can be determined from either the MA formalism (Eq. 1.5) (Miller and Abrahams, 1960) or Marcus’ theory of electron transfer (Marcus, 1993). The Marcus’ electron transfer rate has been used to predict diffusion coefficients and charge mobilities for crystalline materials straightforward.(Chai et al., 2011; However, Wen et for al., a 2009). site with Predicting a high hopping mobilities rate from to diffusion coefficient has the advantage of being simple and one site but very slow hopping rates to all others, the high hopping rate Monte does not Carlo contribute simulations to a meaningful mobility. Therefore, the Monte Carlo simulation has been a commonly seen alternative. For more general problems, such as a disordered morphology other than crystals, the kinetic Monte Carlo (MC) simulation is a useful tool to model charge mobility inwithin the the details hopping of themodel hopping (Bässler, rates, 1993; and Kwiatkowski can generally et al., obtain2009; Nelson et al., 2009). It is not limited to the theoretical assumptions mobilities while include some details in the theoretical models. existsOne a good site with advantage very lowof the hopping MC simulation rates to is most its ability neighbors, to account the highfor the electron effect transferof network rates connectivity. to the remaining As mentioned few neighbors above, ifdo there not contribute to a meaningful mobility since it cannot pass the charge effectively. This effect can be properly accounted for with a proper Monte Carlo simulation (Nan et al., 2009; Vehoff et al., 2010) and it has been discussed with a more general and rigorous theoretical ground in the literature (Tessler et al., 2009). Since the charge mobility has been measured along different crystal directions, it is of great interest to reproduce such anisotropy theoretically. For example, the anisotropic charge mobilities in crystals have been studied extensively in the past. The anisotropy can be regarded as arising from different electron transfer rates in 10 Theoretical Modeling for Electron Transfer in Organic Materials
different directions, mainly due to the orientation dependence of the (Wenelectronic et al., coupling. In this regards, both the diffusion coefficients and Monte Carlo simulations have been reported in the literature 2009; Vehoff et al., 2010). A multi-dimensional effect was observed in Monte Carlo simulations of charge hopping (Bässler 1993; Vehoff et al., 2010). In such simulations, the mobility obtained orthogonalin different directions crystal directions helps facilitate was mobility not simply by avoiding proportional high- to the charge hopping rates in these directions. Charge motion along
network.energy sites (Bässler, 1993). We note that the overall anisotropy is a nontrivial result of the multi-dimensional charge transport the hopping rates, which can be properly addressed given the The electronic coupling is a major physical factor determining
structure of the material. We expect that an ab initio characterization for the coupling factor would facilitate the research work in this area, while the details of the modeling may be varied as the field develops. Thus, we include below a section summarizing the computational considerations1.1.2.2 Polaron for themodels couplings (Section 1.2.2).
Experimental findings indicate that in most cases the charge themobility charge increases hopping with rates. temperature, It was also which observed, agrees however, with the hoppingthat the model (see Eq. (1.3)), since thermal activation helps increasing
etmobility al., 2006; of carefullyPodzorov purified et al., 2004). oligoacene In these based crystals, crystals the decreasedelectronic with temperature (Jurchescu et al., 2004; Karl, 2003; Ostroverkhova
coupling can easily be similar to or even larger than the small modelreorganization are invalid. energy of these molecules. In this scenario, the hopping rate given by Marcus theory, Eq. (1.3), and the hopping
The inverse temperature dependence implies that the charge therefore,may not be disorder) localized increases. to a single Polaron molecule, models and are that developed delocalized for charge may become more localized when the temperature (and models have been developed to describe the charge transport such a delocalized charge transport. A, 2008;number Ortmann of different et al., 2009).polaron
(Kenkre et al., 1989; Cheng and Silbey Electronic Coupling 11
They are based on the Holstein (Holstein, 1959a,b) or Peierls (Peierls, 2001) model Hamiltonians. A tight-binding Hamiltonian is used as a starting point for the electronic part of the system, and it is coupled the charge carrier to a bath of harmonic oscillators for the effects of the nuclear fluctuation. In the Holstein model, the site energies (diagonal elements) are linearly coupled to the nuclear degrees of freedom. The reorganization energy of Marcus theory is equivalent to this electron-phonon coupling. The Peierls model goes one step further. The off-diagonal matrix elements of the tight-binding Hamiltonian, the electronic coupling terms, are coupled linearly to the phonon bath (Peierls, 2001). polaronA full models, quantum the dynamical Hamiltonian solution is transformed for either to the the Holstein polaron orbasis, the Peierls model is neither feasible nor practical (Xu et al., 2005). In position when a site is charged. The dynamical descriptions were which are obtained by shift the nuclei motion to their optimized then developed for the polaron basis (Cheng and Silbey, 2008; polaronKenkre et models. al., 1989; In theOrtmann Holstein-based et al., 2009). models, the electron or hole The electronic coupling is again an important factor in the model Hamiltonian. In the Peierls-based models, the dependence transfer coupling between nearest neighbor sites is constant in the originate mainly from the exponential distance dependence of the of electronic coupling strength on the nuclear displacements coupling strength. We expect that such a dependence would appear in low-frequency phonon modes that involve displacements of the molecules. The coupling strength, and its dependence on nuclear polaroncoordinates, or other can models be determined that describe by the ab initioelectron quantum transfer chemistrydynamics incalculations, organic materials. which offer a set of well-grounded parameters for the
1.2 Electronic Coupling
The electronic coupling factor is an important parameter in the theories described above. The magnitude of electronic coupling and its dependence on structure influence the electron transfer rates. Calculation of the electronic coupling factor is an important aspect of the emergent works in the literature that connect microscopic 12 Theoretical Modeling for Electron Transfer in Organic Materials
electronic and molecular structure to bulk properties. The starting point for any endeavor to compute the electronic coupling factor is a 1.reasonable2.1 Structural model of Modelsthe geometry of the molecules.
exponentialThe electronic decay coupling with distanceis highly betweendependent the on donor the inter-fragment and acceptor degrees of freedom. For example, the coupling strength exhibits an
fragments. The extent of molecular orbital contact has a direct effect on the electronic coupling magnitude. To compute the electronic coupling in a solid-state organic material, the choice of 1.the2 .1.1intermolecular Idealized structure models is paramount.
The early attempts to compute electronic coupling in organic materials focused on describing charge-transport behavior in simple dimer models (Cheng et al., 2003; Deng and Goddard, 2004; Hutchison et al., 2005b). The electronic coupling was calculated using reasonable guesses for the orientations between molecules. Low-bandgap molecules with planar -conjugated moieties are often good organic semiconducting materials. A large orbitals. electronic This typecoupling of alignment is obtained is also when favored these frommolecules the perspective are aligned of in maximizing a co-facial thearrangement – interaction. because it improves the overlap of the moieties stacked
Molecular pairs were studied with their face-to-face. Electronic couplings were reported as a function of the distance of one molecule shifted horizontally, revealing that the electronic coupling rapidly oscillates as the cofacial arrangement is slipped (Brédas et al., 2002; Kwon et al., 2004). The oscillations- closely resemble the nodal structure of the frontier molecular orbitals.On the These other results hand, showed we can that reasonably small intermolecular expect that in displace a solid- ments can create significant changes in the electronic coupling.
state device, the intermolecular structure is flexible enough to allow for a broad range of different electronic coupling values. Lin et al. (2011) showed that across several different crystal structure, even with a very similarly arranged molecular pairs, there still Electronic Coupling 13
exists about 5 Å in the distribution of relative positions, which is almost three cycles of oscillation in the coupling. Instead of fine- tuning for a peak in the oscillatory coupling strength, perhaps it is more practical to consider a coupling strength in a more coarse- grained scale. However, for tris(8-hydroxyquinolinato)aluminum (Alq) studied, the pyridyl part of the aromatic ligand has a larger lowest-unoccupied MO (LUMO) population, implying that a pyridyl– pyridyl contact may lead to a better electron transfer coupling. Therefore, it is still possible to take advantage of computational results for experimental design, but it is necessary to consider the uncertainty from the natural occurring disorder. Idealized models are also useful to test the methodologies thefor calculating donor and electron acceptor transfer orbitals coupling. where the The transferred electron electron transfer coupling is essentially an off-diagonal Fock matrix element between occupies. The molecular orbitals decay exponentially when the electron moves away from the molecule. Therefore, the electron transfer coupling decays exponentially with respect to donor- acceptor distance, as discussed in Section 1.1.2.1. In numerically calculating the electron transfer coupling, the exponential distance dependence is a desirable property to test for (You et al., 2004). Therefore, the artificially created structures are helpful in testing the computational1.2.1.2 Crystal schemes. structures obtained from crystallography, which is only available for crystalline The best source of structural information about a material is materials. Even though the structure in an amorphous film is different, studies on molecules arranged according to the crystal structure still provide important insights into the structure-function relationships. The electronic coupling has been computed using DFT for a number of organic materials with structural models obtained from the experimental crystal structures (Kim et al., 2007; Wen et al., 2009; Yang et al., 2008). The electronic coupling in Alq has– interactionbeen studied promotes based onstrong the geometry obtained from its crystal structure (Yang et al., 2006). In another study, it was found that the electronic coupling between the Alq LUMOs (electron transfer) but weak electronic coupling between the highest occupied MOs 14 Theoretical Modeling for Electron Transfer in Organic Materials
interaction (HOMOs) (hole transfer) (Lin et al., 2005). A survey across several mobilityavailable (Lin crystal et al., structures indicated that the CH– could play a role in biasing the material toward a higher electron 2011). This result interactions is consistent (Liao with etexperimental al., 2008). work that reported bipolar charge mobility for Alq derivatives that are incapable of forming CH– Therefore, it is possible to extract general insights from molecules with1.2.1. several3 Simulated different morphology crystal structures.
Simulations of morphology are necessary when experimental crystal structures are unavailable or when it is desirable to model the effect of disorder. In most solid-state organic devices, the films are amorphous. Disordered materials inherently have more disorder in structure, site energy, electronic coupling, and electron-- mentphonon in this coupling. area has Molecular been reviewed modeling (Nelson methods et al., have 2009). been use to simulate the morphology of organic films and the recent advance 1.2.2 Calculation of Electronic Coupling
A two-state model is often employed to model electron transfer
(Cave and Newton, 1996, 1997). However, multistate electron transfer effects have been studied both experimentally and theoretically by a number of groups (Bixon et al., 1994; Gould et al., expressed1994; Herbich as aand linear Kapturkiewicz, combination 1998; of two Rust eigenstates et al., 2002). (adiabatic In the states).two-state The model, Hamiltonian it is assumed H that the diabatic states can be
EVi if can be written in the diabatic basis: H diad = , VE if f (1.7)
diagonal matrix element. The Hamiltonian matrix in the diabatic representationwhere the electronic can be diagonalized coupling factorto yield is the located Hamiltonian on the matrix, off-
E 0 H ad = , 01 E 2 (1.8) Electronic Coupling 15
- in the adiabatic representation. In this representation,A the off energies, E and E2 . The adiabatic diagonal elements are zero and the diagonal values are the energies, E and E2, can be expressed in terms of the diabatic 1 , of the adiabatic wavefunctions Ψ 1
Hamiltonian matrix elements:2
EEEEi + f i – f 2 E = ± +V 2 2 1,2 if (1.9) The exact kinetic energy most cases. diabatic states, those that diagonalize the nuclear- operator, are overdetermined and cannot be found in Therefore, a number of different methods for determi ning approximate diabatic states and calculating the electronic thecoupling diabatic have states. been introduced into the literature. For electron transfer, charge localization is often used as the criteria to define In the following, a number of computational methods to compute the electronic coupling factor will be discussed. These methods can be classified into three groups: methods based on the energy gap (Newton, 1991; Yang and Hsu, 2006; You et al., 2004), direction coupling methods (Broo and Larsson, 1990; Farazdel et al., 1990; Ohta et al., 1986; Zhang et al., 1997), and methods similar to generalized Mulliken–Hush scheme that are based on defining localization with additional operators (Cave and1.2. 2Newton,.1 Energy 1996; gap Voityuk and Rösch, 2002).
In the special case that the diabatic energies Ei and Ef are resonant (i.e., Ei = Ef E = Ei ± Vif erence in electronic energies is ), the expression for the electronic energies, Eq. (1.9), E2 E = 2Vif, where the energy of the diabatic state has been simplifies to 1,2 . The diff − 1 eliminated by subtraction and we are left with an expression that gives the electronic coupling factor as one-half of the energy difference:|VEE |= | – | 2 if 1 2 1 This method has been referred to as the “energy gap” or “energy(1.10) 2004; splitting” method in the literature (Newton, 1991; You et al., Yang and Hsu, 2006). 16 Theoretical Modeling for Electron Transfer in Organic Materials
The energy gap requires simultaneous calculation of two state energies. A widely used approximation to the adiabatic energies is based on Koopmans’ Theorem (KT) (Koopmans, 1934). According to KT, the ionization potential (or electron affinity) is approximately the minus energy of the HOMO (or the LUMO), and so the energy ofdifference the neutral for system.two cationic In this (or way, anionic) the many-electron radical states energy equals gapto the is energy difference of the two corresponding HOMOs (or LUMOs)
reduced to a 1-electron energy difference. We note that KT works mainly for Hartree–Fock calculations. For the popular density-functional theory (DFT), KT is not valid for most of the commonly used functionals because of the self- interaction error in these functionals. On the other hand, if a DFT functional were exact, the negative energy of the Kohn–Sham HOMO is exactly the first IP (Perdew et al., 1982). Therefore, the quality of DFT-based coupling can be improved when the self-interaction error is corrected. The long-range correct DFT (Tawada et al., 2004; Yanai et al., 2004) has great potential in this regard. It has been pointed out that care must be taken when using the energy gap method because even dimers constructed from identical monomers may have different site energies (Valeev et al., 2006). If the structure of the dimer is not symmetric, the diabatic (i.e., Ei Ef energies are not guaranteed to be resonant. In off-resonant cases ), perturbations such as external electric fields can be introduced to force the diabatic states into resonance (Newton, 1991), and at resonance, the energy gap of the two eigenstates (adiabatic states) is minimized. The perturbation could be from a homogeneous electric field (Tong et al., 2002; Voityuk et al., 2000, 2001), external point charges (Larsson and Volosov, 1986), or diabaticeven artificial states energy are driven terms into added resonance, onto the it atom is a transition centered basisstate functions of the Hamiltonian (Daizadeh et al., 1997). When the
in an electron transfer process. In Marcus’ theory of electron transfer, a fluctuating dielectric environment offers such a driving force for electron transfer. Therefore, the search and computation modelfor a perturbed for the transition resonance state condition of the electron does not transfer only offer reaction. a feasible means for calculating electronic coupling, it also offers a proper
For electron transfer problems, the states often involve open- shell configurations where singly occupied orbitals exist, and the Electronic Coupling 17
importantdifferent singlyto treat occupied these orbitals orbitals in are a “balanced” often similar manner, in their and energies, in this whose energy difference directly influence the energy gap. It is way, the nondynamical correlation due to these nearly degenerate orbitals (and states) is minimized. For example, in the HF-KT scheme, the MO energy derived from the neutral system is used, even though the cationic or anionic state energies are desired. A computationally efficient procedure for this nearly degenerate issituation a high-spin is the state,spin-flip and scheme the two (Krylov, desired 2006; low-spin You et eigenstates al., 2004; Yang are and Hsu, 2006). In the spin-flip scheme, the reference determinate obtained as “excited states” in the calculation. Based on through- bond electron transfer couplings computed from spin-flip coupled- cluster methods, it is concluded that dynamical is an correlation attractive means is not sinceimportant in most cases (Yang and Hsu, 2006). chemistryCalculating program. coupling It is not from limited energy to gapthe theoretical level for the eigenstateit is not energies limited either, to the as functionality long as the of two a particular eigenenergies quantum are obtained in the same theoretical ground and the nondynamical asymmetriccorrelation systems. is properly accounted. It is also important to use an external electric field to bring the system to resonance for 1.2.2.2 Direct coupling
(Broo“Direct and coupling” Larsson refers to schemes that use wavefunctions calculated from quantum chemistry programs as diabatic states , 1990; Farazdel et al., 1990; Ohta et al., 1986; Zhang et al., 1997). For electron transfer reactions, charge-localized unrestricted Hartree–Fock (UHF), symmetry-broken solutions are often used (Broo and Larsson, 1990; Newton 1991; Ohta et al., 1986). d d diabatic states, referred to as the initial i f charge- localizedAssuming states weof an have electron charge-localized transfer reaction, states the representing matrix elements the and final of the diabatic Hamiltoniand|H | d can bed |computedH | d asHH d i i i f i if H() d d d d |H | |H | HH f i f f if f (1.11) 18 Theoretical Modeling for Electron Transfer in Organic Materials
H. In general, the charge- localized states may not be orthogonal. This can be addressed
with the full electronic Hamiltonian Horth = S HS , with the overlap matrix S −1/2 −1/2 by Löwdin symmetric orthogonalization (Löwdin, 1950): d d i|| f Sif S d d f|| i Sif 1 1 (1.12) 1 1
The electronicHHHS–( +coupling ) /2 obtained after orthogonalization is V if i f if . if S 2 if (1.13) 1– Vif = Hif, when We can see that the orthogonalized electronic coupling reduces the overlap term Sif to the off-diagonald Hamiltoniand matrix element, and f i is zero. For symmetric systems, the diabatic wavefunctions d d are symmetry-broken solutions. The combination of and f eigenfunctions cani be approximated by a symmetry-restored linear d d : i± f ± S if (1.14) 2(1± )
The expression in Eq. (1.13) is half of the energy gap of these two eigenstates, which are modeled as dual-configuration symmetry- restored solutions (Broo and Larsson, 1990; Yang and Hsu, 2006). In restoredother words, linear in combination order to account of several for the (in non-dynamical this case, two) correlationsymmetry- effect, the eigenfunctions can be described as the symmetry-
broken (charge localized) wavefunctions (Broo and Larsson, 1990; Newton, 1991; Ohta et al., 1986;d Yangd and Hsu, 2006). the charge-localized states and f There are several differenti approaches for the construction of and acceptor A fragments. The electron. We will transfer proceed between by assuming these fragments,the molecular in the system case can of an be excess partitioned electron into initially separate localized donor (D) on
fragmentD– + A(D), isD described+ A– by the following reaction:
(1.15) Electronic Coupling 19
–, A, and A–. The Fourd independentd electronic structure calculations provide a set initial Ψi Ψf of molecular orbitals for each fragment species: D, D and final wavefunctions are constructed by joining the moleculard = orbitalsAˆ { } of the fragments together: i D– A
d (1.16) =Aˆ {– }, f D A d ˆ (1.17) wheref= A { D A– },
antisymmetrizes the electronic wavefunctions. This way, we obtainedAnother waya direct to obtain coupling charge-localized from a set of states unrelaxed is from and a brokenstrictly charge-localized wavefunctions. symmetry self-consistent field (SCF) solution (Davidson and Borden, 1983). The donor and acceptor wavefunctions discussed above, othersince theyfragment. are obtainedIt is straightforward from independent to start fragmentwith strictly calculations, localized do not account for electronic relaxation in the presence of the wavefunctions as an initial guesses for a SCF relaxation. This procedure usually relaxes to a broken-symmetry UHF solution. In a previous test for several through-bridge electron transfer systems, the relaxed direct coupling results are often very close to those derived from coupled-cluster (EOMCCSD) energy gaps, which seems to indicates that this relaxed approach is perhaps the best “dual- configuration” approximation to eigenfunctions of the system (Yang and Hsu, 2006). In some cases, a simplified description of the electronic structure based on the 1-electron molecular orbitals is advantageous, especially when a great number of coupling calculations is required or the molecular system is large. The frontier molecular orbital (FMO) approach takes the 1-electronic off-diagonal Fock matrix element for the coupling. Assuming that all other MOs are frozen in the process of electron transfer, the off-diagonal Hamiltonian matrix element can be reduced to an off-diagonal Fock matrix element abetween “direction the two MOs whosevariant electronic of This occupancy approach is is changed in the to electron transfer process (Zhang et al., in 1997). Section FMO can beif the viewed orbital as energies are coupling”resonant. KT. equivalent the KT energy gap method discussed 1.2.2.1 20 Theoretical Modeling for Electron Transfer in Organic Materials
An efficient computational scheme has been developed by etemploying al., 2009) FMO and in semiempirical conjunction with Hamiltonians low-cost (Troisielectronic and structure Orlandi, methods, such as DFT (Valeev et al., 2006; Yang et al., 2008; Wen
2001, 2006; Troisi et al., 2004). Couplings from semiempirical Hamiltonians have a faster exponential decay in through-bridge electron transfer (Chen and Hsu, 2005), which is presumably from the parameters for the interatomic interactions. The quality of DFT- based FMO coupling is again highly dependent on the functionals chosen. The deficiency associated with the self-interaction errors seen in most DFT functionals may create problems. The 1-electron potential derived from most commonly used exchange-correlation functionals is wrong in the asymptotic region when the electronic coordinate is far removed from the nuclei (Casida and Salahub, 2000). The incorrect asymptotic potential may affect the quality of the wavefunction in the peripheral region, and the coupling values may be affected. As discussed above, the long-range corrected DFT (Tawada et al., 2004; Yanai et al., 2004) may be a good choice. We note that in calculating the off-diagonal Fock (or Kohn–Sham) matrix element, it is still necessary to specify the configurations (density) of all other electrons in the evaluation of the Coulomb and exchange (exchange-correlation) contributions. One way to proceed is to use the configurations (density) obtained from neutral donor and acceptor fragments (Yang et al., 2008), and again we note that this is very similar to the common usage of KT. The direct coupling scheme is useful mainly for electron transfer in the ground state. It is perhaps also the best and most commonly used method for modeling charge hopping in molecular semiconductors.1.2.2.3 The generalized Mulliken–Hush and fragment charge difference schemes
and Charge-localized states can be obtained by mixing the eigenstates additional from a set of calculations. Generalized Mulliken–Hush (GMH) to create its variant, the fragment charge difference (FCD), use charge- operators to define the necessary linear transformation charge-localized states. These methods are suitable for transfer reactions involving excited states. In particular, we expect Electronic Coupling 21
in organic solar cells. that they may be useful in cases where the CS and CR processes
The GMH scheme is a generalization from the Mulliken–Hush matrixexpression in the (Hush, adiabatic 1967, representation 1968), which is uses absorption spectra to determine coupling values for optical electron transfer. The dipole . 1 12 Diagonalization12 2 of the dipole matrix leads to the largest dipole(1.18) moment di transformation from the adiabatic to the diabatic representation. fference in the two states, which is used to define the
The corresponding linear combination gives the electronic coupling factor as the off-diagonal | DE Hamiltonian matrix element, V . if 2 –12 ) + 12 4 1 2 (1.19) operator Dq The FCD scheme is similar to GMH, except that a charge difference erence is between used to the define charge the of linear the donortransformation fragment (Voityuk (D) and andthe charge Rösch, of 2002). the acceptor The charge fragment difference (A operator is defined as the diff from the one-particle density, mm(r ). The charges are computed from the transition density, mm(r ), and corresponding quantity ): * r ()rNdrdr (,,,)(,,,) rr r rr r , mm 2 N m 2 N n 2 N where N is the of electrons in the system. With the densities,(1.20) the charge matrix elements can be as number difference computed Dq ()–().r dr r dr mn r D mn rA mn In practice, a (1.21) or Becke (Becke, analysis, is population analysis scheme, such as Mulliken dipole (Mulliken, 1955) 1988) population operatorused to compute the necessary a donorintegrals and in acceptor Eq. (1.21). fragment. Unlike The the charge operator used in the GMH scheme, the charge difference relies on the definition of difference matrix in the adiabatic representation is 22 Theoretical Modeling for Electron Transfer in Organic Materials
DDq q Dq . DDq11 q 12 12 22 (1.22)
transformation for the diabatic states, since they have the largest Diagonalization of the charge difference operator defines the q and q22 after transformation of the Hamiltonian difference in 11 . The electronic coupling factor is obtained Dq| D E V . if 2 Dq– D12 q ) +12 4 D q 1 2 (1.23)
In practice, a vertical excitation is calculated for the full system that includes the donor and the acceptor. The locally excited and the charge-transfer states are assigned, and the GMH or FCD scheme are used to find the diabatic states and the coupling values. They have been used to characterize the electron transfer in DNA (Voityuk et al., 2001; Voityuk and Rösch, 2002), and a number of electron transfer system (Lee et al., 2010; Orian et al., 2012; Yang et al., 2012). dipolesWe arisingnote that from the a GMH local couplingsexcitation arein the sometimes transition. overestimated Since a local (Chen and Hsu, 2005). This is due to an increasedq transition
excitation component does not contribute to , the FCD coupling is less likely be affected (Lee et al., 2010). It is possible to generalize GMH and FCD to include three or four theadiabatic or statesq (Rust et al., 2002; Lee et al., 2010). In this approach, two linear transformations are employed: first, the eigenstates for diagonalized. matrices are obtained; then, the Hamiltonian subspaces, which include states of a similar charge separation nature, are re-
the formationGMH and of FCD exciton are from suitable an electron for electron and a hole transfer in light-emitting problems involving excited states. We expect that CS and CR in solar cells, or are applicable to almost any model Hamiltonian for solving the excitedmaterials state, would as belong potential as transition problems dipoles for GMH (or transition and FCD. Sincedensities) both are available, it is expected that they will become even more widely
used in the future. References 23
1.3 Conclusion
reviewed,In this chapter, with thean emphasiscomputational on organic methods electronic and theoretical materials. models With used for the computation of electronic coupling are listed and hasthe vast become progress an achievable in computer goal. software In the area and hardware, of organic predicting materials forproperties electronic of large devices, molecules it is now and possible computer-aided to model molecular many important design parameters of electron transfer. However, in order to predict device performance, there are still several aspects that need attention.
We believe that the main difficulties are in correctly simulating therethe detailed is active intermolecular development instructure these areas and andenergy we canin the optimistically solid-state expectdevices, that both these inside problems the film or will at bethe overcome interface. asOn newthe other theoretical hand,
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