NORTHWESTERN UNIVERSITY Singlet Exciton Fission

NORTHWESTERN UNIVERSITY

Singlet Exciton Fission: A Discussion of the Mechanism and What It Means For Dye Sensitized Solar Cells

A DISSERTATION

SUBMITTED TO THE GRADUATE SCHOOL

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

for the degree

DOCTOR OF PHILOSOPHY

Field of Chemistry

Eric Chapman Greyson

EVANSTON, IL

December 2007 2

A Theoretical Analysis of Singlet Exciton Fission for Solar Cell Applications: Elucidating the Mechanism and Molecular Design

Eric Chapman Greyson

This thesis investigates singlet exciton fission, a physical process that converts one singlet exciton to a pair of triplet excitons. Singlet fission was discovered nearly forty years ago, but the mechanism for this process is still not well understood. Recent work has suggested that singlet fission may be capable of enhancing the performance of dye-sensitized solar cells. This thesis proposes the first complete mechanism for singlet fission and examines what type of molecules will undergo efficient singlet fission.

We begin by reviewing the literature of singlet fission and by examining individual chromophores that have been identified as promising for singlet fission. Electronic structure methods, such as density functional theory, are used to examine how the properties of individual chromophores are affected by combining them to create a coupled chromophore pair (CCP). We focus on how the energy levels and electronic coupling depend on the specific geometry of these

CCPs, and how those parameters are expected to affect singlet fission.

We propose a mechanism for singlet fission whereby the initial singlet excitation undergoes one electron transfer event to reach a charge transfer intermediate, followed by a second electron transfer to produce the triplet pair state. A ten state model is developed in order to analyze the dynamics and efficacy of singlet fission in molecular systems. We examine the dynamics of several molecules, both real and simulated, within this model in two different regimes of electron transfer. Density matrix formalism is used first, to examine how singlet 4 fission proceeds when the electron transfer steps are fast and coherent. We then use Marcus theory and classical kinetics to investigate singlet fission in the regime of slow electron transfer steps, where molecular rearrangement and vibrations play a critical role in electron transfer. We use each set of simulations to predict principles of molecular design for ideal CCPs for singlet fission.

5 ACKNOWLEDGEMENTS

This thesis would not have been written without the help and support of a great many individuals I have been fortunate to work with, live with, and know over the past five years. On an academic level, I have been blessed with many wonderful interactions with professors and coworkers. The opportunity to study this fascinating problem was provided by Professor Mark

Ratner, who has been a tremendous advisor, and a great encouragement throughout graduate school. This project, and theoretical chemistry in general, provided a daunting challenge for me, and it is because of Mark and the incredible community of researchers he has helped gather on the fourth floor of Ryan Hall that I was able to have the success I did. I also owe thanks to many other professors for their support and education over the years. Chiefly, I owe Teri Odom for providing me with the opportunity to investigate various problems with the realm of nanoscience, and my committee members, Tobin Marks, Michael Wasielewski, and Jim Ibers, all of whom provided insight and guidance along the way. In addition, I’ve enjoyed and learned from several conversations with Vladimiro Mujica, Abe Nitzan, Josef Michl, Troy Van Voorhis,

Franz Geiger, Ken Poeppelmeier, and Karl Scheidt.

I am constantly amazed at how much help I have received from several coworkers, and also at how many of my coworkers have become my best friends. In my life as a theoretical chemist, Sina Yeganeh has acted as an all-purpose resource, available for the discussion of anything theoretical, as long as I don’t start talking about real molecules. Irina Paci helped me begin this adventure in singlet fission and has remained helpful and encouraging even after she left. Chad Risko, Eric Brown, Thorsten Hansen, Christine Aikens and Tony Dutoi have all helped me gain a greater understanding of electronic structure theory and how it can be used 6 properly and improperly. I owe Josh Vura-Weis for his work implementing several codes, as

well as for several helpful discussions. I also owe thanks to several wonderful officemates who

have made it a pleasure to come in to work every day, Hyonseok Hwang, Sharon Koh, Marcel

Fallet, and Jen Roden. My previous lifetime as an experimental chemist was full of just as many

helpful and kind people, most notably Liza Babayan, Jeremy Barton, Chris Stender, Joel Henzie

and Perumal Sekar. Beyond this list, an incredible number of other students and friends have

helped me out in innumerable ways. Every person on this list has been not only a great help

academically, but a good friend.

Outside of work, I have had the pleasure to live with a parade of chemists and a pair of

non-chemists, all of whom have provided support and a feeling of home during the transient time

that is grad school. Several of these people could be mentioned elsewhere in these

acknowledgements, but I now acknowledge Randall Goldsmith, Chris Konek, Rick Kelley,

Khalid Salaita, James Cameron, Mark Witschi, Brandon Rodriguez, Julia Chamberlain, Remy

Zebrowski and Thomas Albrecht.

I also want to thank my family for their love and support. My parents and sister

provided me with wonderful guidance throughout and childhood, and continue to do so in my burgeoning adulthood. Obviously I would not be who I am, or where I am, without everything they have given me.

Finally, I’d like to thank my future family, Andrea, Diana, Barbara and David Voges.

Andrea has been a better friend and companion than I could have imagined possible throughout graduate school. I look forward to spending more time with her, and with the entire Voges clan, who have provided a wonderful second home for me. 7 Table of Contents

ABSTRACT...... 3

ACKNOWLEDGEMENTS ...... 5

Table of Contents ...... 7

List of Figures...... 10

List of Tables ...... 13

Chapter 1 Introduction...... 14

1.1 Context and Motivation...... 15

1.2 Dye Sensitized Solar Cells ...... 15

1.3 Experimental Observation of Singlet Fission ...... 16

1.4 Design of Molecular Systems for Singlet Fission ...... 18

1.5 Outline of Thesis ...... 18

Chapter 2 Investigating the Effects of Coupling Chromophores on Electronic Structure: How does Coupled Chromophore Pair Geometry Control Singlet Fission?...... 21

2.1 Introduction...... 22

2.2 Methodology ...... 25

2.3 Results and Discussion ...... 28

2.3.1 DPIBF CCPs ...... 28 2.3.2 BQD CCPs...... 34 2.3.3 Tetracene CCPs...... 38 2.3.4 Hetero-CCPs ...... 42 2.3.5 Trade-Off of Transfer Integrals with Thermodynamics ...... 45 2.3.6 Comparison to Experimental Data...... 46

2.4 Summary and Conclusion ...... 48 8 Chapter 3 Development of a Mechanism for Singlet Fission and Analysis of Fission Dynamics and Yield In The Regime of Coherent Electron Transfer...... 49

3.1 Introduction...... 50

3.2 Methodology ...... 50

3.2.1 Creation of a Model System ...... 51 3.2.2 Electronic Structure Computations...... 56 3.2.3 Time Evolution of Excitation...... 59

3.3 Results and Discussion ...... 61

3.3.1 Electronic Structure ...... 61 3.3.2 Time Evolution of Model Systems ...... 65 3.3.2.1 Basic Model ...... 65 3.3.2.2 Effect of Decay Rates on Singlet Fission ...... 68 3.3.2.3 Effect of Matrix Elements on Singlet Fission...... 71 3.3.2.4 Effect of Non-degenerate State Energies on Singlet Fission...... 75 3.3.2.5 Lessons for Simulations...... 79 3.3.3 Time Evolution of Real Molecules...... 79

3.4 Summary and Conclusion ...... 81

Chapter 4 Singlet Fission Yield and Dynamics In the Regime of Incoherent Electron Transfer ...... 83

4.1 Introduction...... 84

4.2 Methodology ...... 84

4.2.1 Electronic Structure Computations...... 85 4.2.2 Time Evolution of the Excitation...... 86

4.3 Results and Discussion ...... 89

4.3.1 Electronic Structure Computations...... 89 4.3.2 Time Evolution of Model Systems ...... 93 4.3.2.1 Basic System...... 93 4.3.2.2 Effect of Changing Matrix Elements on Singlet Fission ...... 96 4.3.2.3 Non-Isoenergetic State Schemes...... 97 9 4.3.2.4 Conclusions from Model Simulations ...... 103 4.3.3 Time Evolution of Real Molecules...... 103

4.4 Design Principles for Singlet Fission in DSSCs...... 105

4.5 Limits of the Model ...... 107

4.6 Perspective for Experimentalists ...... 108

4.7 Summary and Conclusion ...... 112

References...... 113

Appendix A Fortran Codes...... 120

A-1 Fortran (f90) Code To Compute Orbital Overlap ...... 121

A-2 Fotran (f90) Code for Coherent Dynamics...... 125

A-3 Fortran (f90) Code for Marcus Theory Dynamics...... 139 10 List of Figures

Figure 1 HOMO, LUMO and chemical structure (Left to Right) of DPIBF, BQD and tetracene (Top to Bottom)...... 24

Figure 2 The electron transfer integrals, t, for the HOMO and LUMO are half the splitting of the CCP orbitals in a homo-CCP...... 26

Figure 3 Chemical Structures of DPIBF CCPs...... 31

Figure 4 Frontier molecular orbitals (Left to Right, HOMO-1, HOMO, LUMO, LUMO+1) for both strongly (D20, Top) and weakly (D2, Bottom) coupled DPIBF CCPs look like a weakly perturbed sum and difference of IC orbitals. The HOMO-1and LUMO have bonding interactions where the ICs join, while the HOMO and LUMO+1 have antibonding interactions at the coupling...... 33

Figure 5 Chemical Structures of BQD CCPs...... 35

Figure 6 The frontier molecular orbitals of B9 (Left to Right, HOMO-1, HOMO, LUMO, LUMO+1) are linear combinations of the isolated BQD orbitals. Because BQD has more electron density on the bridging carbons in its HOMO than its LUMO, th is much greater than tl...... 37

Figure 7 Chemical Structures of tetracene CCPs...... 39

Figure 8 The Frontier orbitals of T4 are linear combinations of the frontier orbitals of two isolated chromophores. (Left to Right, HOMO-1, HOMO, LUMO, LUMO+1)...... 41

Figure 9 Chemical structures of Hetero-CCPs...... 43

Figure 10 Frontier orbitals for the hetero-CCP DT2 are localized, in contrast to the delocalized orbitals seen in homo-CCPs. (Left to Right, Top row then Bottom HOMO-1, HOMO, LUMO, LUMO+1)...... 44

Figure 11 The S 1 excitation for a hetero-CCP (Center) is less energetic than either IC’s S 1 excitation (Left and Right), making fission more endothermic in hetero-CCPs than in homo-CCPs with comparable coupling...... 45

Figure 12 The free energy of singlet fission (Eqn 2.5, Table 1) becomes more endothermic nearly linearly as the coupling between two DPIBF chromophores is increased ...... 46

Figure 13 There are 70 possible electron configurations in a 4 electron 4 orbital basis comprised of a pair of chromophore HOMOs and LUMOs. 36 states have two neutral chromophores, 32 are a single electron transfer charge transfer pair, and 2 states put all 11 four electrons on one chromophore. Highlighted states are those that are most energetically accessible...... 52

Figure 14 Ten state system for singlet fission with all allowed electron transfers shown. States have been renumbered...... 54

Figure 15 Structures of three promising CCPs. We examine these three CCPs, as well as some variants that have amino (a), or nitro (n) ‘R’ groups...... 62

Figure 17 (Left) Doubled triplet-triplet decay rates lead to increased fission yield, while (Right) doubled singlet decay rates decrease fission yield...... 67

Figure 18 (Left) Even with no possible singlet decay route, only 25% of the excitation undergoes fission because of destructive interference if all states have equal energy and all matrix elements are equal, (Center) If states 1 and 4 are artificially removed from the system, the fission yield is 50% under these conditions, (Right) With only one S 1, one CT, and one TT state, the fission yield is 100%...... 69

Figure 19 Doubling all of the matrix elements in an isoenergetic system with slow decay rates does not affect fission yield, although it does increase the frequency of population oscillation ...... 72

Figure 20 If the symmetry of the system is broken by raising some matrix elements and not others, the fission yield increases. (Left) If TL is doubled, the fission yield increases to 19%, (Center) If TD1 is doubled the fission yield increases to 23%, (Right) If both are doubled, fission yield increases to 27%...... 73

Figure 21 A moderate CT barrier does not affect fission yield if the S 1 and TT states are isoenergetic, although the period of oscillation increases with larger barriers (Left) 0 .081 eV barrier, and (Right) 0.27 eV...... 75

Figure 22 If the S 1 and TT states are not isoenergetic, the fission yield is decreased. If there is a G, then the presence of a moderate CT barrier does hurt fission yield. (Left) G =0.027 eV. (Center) G =0.027 eV and CT=0.081 eV (Right) G =0.027 and CT=0.27 eV ...... 77

Figure 23 Hetero-CCPs show promise because the different energy levels of the left and right localized S 1 and CT states lead to higher fission yields. (Left) States 1,4 are raised 12 0.027 eV above all others. (Center) States 5,8 are raised 0.027 eV. (Right) States 1,4,5,8 are all raised 0.027 eV...... 78

Figure 24 If all states are isoenergetic, at equilibrium each of the ten states is evenly populated. This leads to 33% fission yield, if the singlet and triplet decay rates are equal, because twice as much population resides in the four S 1 states as in the two TT states. Equilibrium is reached very quickly when all states are isoenergetic...... 94

Figure 25 Early time profiles show that larger electronic coupling matrix elements lead to quicker equilibration between states. (Left) All coupling elements .027 eV (Center) TH and TL are .054 eV so S 1 and CT equilibrate faster (right) All coupling elements are .054 eV, so all states equilibrate faster ...... 97

Figure 26 When not all states are isoenergetic their relative populations change. (Left) 0.054 eV CT state with no decay leads to each CT state having less population than the S 1 and TT states, (Center) With decay, the fission yield is only slightly depressed for this small barrier value, (Right) but a CT barrier of 0.11 eV significantly impedes fission...... 99

Figure 27 (Left) If the TT states are higher (0.027 eV) in energy than the S 1 states, less population goes to the TT states and the fission yield decreases (Center) while a low energy (-0.027 eV) TT state leads to more TT population, and higher fission rate. (Right) Adding a CT barrier (0.11 eV) does not affect the equilibrium population ratio of S 1 and TT states, but it causes more population decay out of the S 1 states before equilibrium is reached, lowering fission yield ...... 100

Figure 28 (Left) In a model hetero-CCP with the S 1 states at 0 and 0.081 eV, the CT states at - 0.027 and 0.135 eV and the TT states at -0.027 eV the fission yield is very high. (Right) If the low energy localized singlet cannot inject into the electrode, the fission yield jumps to nearly 100%...... 102

Figure 29 (Left) anPOLY has a two CT states lower in energy than its S 1 states, and has a TT state that is even lower in energy, providing excellent fission yield. (Center) anPENT also has two CT states that are lower in energy than its S 1 states, but its TT states are also above these low CT states. If there is no decay route directly from the CT states, this will eventually provide a high fission yield, (Right) but even a very slow CT decay route makes this energy scheme poor, and results in ...... 105

Figure 30 Flowchart for designing promising molecules for singlet fission. The important molecular properties are displayed at right, with possible means of achieveing those properties at left...... 109 13 List of Tables

Table 1 Electronic Matrix Elements and Free Energies for DPIBF CCPs ...... 29

Table 2 Electronic Matrix Elements and Free Energies for CCPs of BQD...... 34

Table 3 Electronic Matrix Elements and Free Energies for tetracene CCPs ...... 40

Table 4 Electronic Matrix Elements and Free Energies for Hetero-CCPs ...... 43

Table 5 Electronic Matrix Elements for Several Bare and Functionalized CCPs. The prefixes aa, an, and nn are diamino, aminonitro, and dinitro functionalized variants...... 63

Table 6 Energies of Electronic States for several CCPs (in eV)...... 64

Table 7 Summary of several simulations with various parameters for coherent dynamics. Each row is a different simulation, and the figure in which more details can be found is listed. The electronic coupling matrix elements are listed first, then the decay rate constants, followed by the energies of the states (E14 is states 1 and 4), and finally the fission percent is listed. More detail on every simulation is in the text...... 68

Table 8 Fission Yield of Real CCPs with State Energies Provided (in eV) ...... 80

Table 9 Energy of 16 ICs at every Relevant Electronic State and Geometry. Molecules are labeled by IC type (PO=polyene, DP=DPIBF, PE=pentacene) and functional group (a=amino, h=hydroxy, n=nitro). All values are in eV...... 91

Table 11 Summary of several simulations with various parameters for incoherent dynamics. Each row is a different simulation, and the figure in which more details can be found is listed. The electronic coupling matrix elements are listed first, then the decay rate constants, followed by the energies of the states (E14 is states 1 and 4), and finally the fission percent is listed. More detail on every simulation is in the text ...... 95

Chapter 1

Introduction

15 1.1 Context and Motivation

One of the greatest challenges of the current century is sustainability. One aspect of sustainability is global energy use, which has rapidly grown in recent years, and promises to continue its rapid ascent. Various scenarios predict a global energy use of 26.4 to 32.9 terawatts

(TW) by 2050, up from only 12.7 TW in 1998.1 While a majority of energy currently comes from fossil fuels, there are several reasons to look for alternate energy sources, ranging from possibly dire environmental consequences from climate change and pollution, to the economic cost of supplying and distributing a rapidly increasing amount of fuel from a global supply that is, if anything, becoming more difficult to extract. 2, 3 The amount of energy that strikes the Earth

from sunlight in one hour is more than the amount of energy used by the global population in a

whole year. The sun clearly provides enough energy, if we are only able to harness it efficiently

and economically. This abundant supply makes solar energy one of the most promising

alternative energy sources. 4-6

1.2 Dye Sensitized Solar Cells

One of the most promising solar cells being studied is the dye sensitized solar cell

(DSSC), also known as the Grätzel cell. DSSCs are promising because they combine low cost

fabrication with potentially high energetic yield. 7, 8 A DSSC is composed of a nanoscale porous metal oxide semiconductor, coated with an organic dye, and embedded in a redox electrolyte.

The organic dye is excited by photon of light. These excited state dye molecules then each inject

(ideally) one electron into the conduction band of the metal oxide. The electrons quickly thermalize to the bottom of the conduction band as they migrate through the metal oxide to an 16 electrode. The electrons continue through the external circuit, providing power, and emerge

from another electrode at lower potential. These lower potential electrons are then recombined

with the dye molecule to reform the ground state of the molecule by a redox shuttle. The highest

reported yield for a Grätzel cell is ~11%,8-10 and this value has not increased for several years despite multiple innovative attempts. 9, 11-13 These cells suffer from many avenues of loss, but one of the most significant is the loss that comes from using a dye sensitizer that can, at best, yield one electron of current per photon absorbed. Based on the solar spectrum and a single absorber such as this, the thermodynamic limit for solar cell efficiency is 32%. 14 One possible way to improve the energetic yield of a DSSC is to use an absorber that is capable of injecting multiple electrons per absorbed photon. One process that would enable this is multiple exciton generation, which has been observed in various inorganic nanocrystals. 15-18 A molecular analog of this process is singlet fission, a physical process that converts one singlet excitation to two triplet excitations. It has been pointed out that a DSSC using a dye capable of efficient singlet fission could provide a 46% energetic yield by injecting two triplets instead of one singlet excitation for all green, blue and ultraviolet light. 14

1.3 Experimental Observation of Singlet Fission

Singlet exciton fission was first discovered in solid state polyacenes in 1965.19 It was determined that the main nonradiative decay in tetracene was triplet formation via singlet fission.

Subsequent studies examined the rate constant for singlet fission in a few solid state polyacene systems 20-29 and explored the unique magnetic field dependence of fission. 30-36 This work also led to the discovery that fission primarily occurs when the energy of the first excited singlet is 17 37-39 greater than twice the energy of the first excited triplet ( E(S 1) ≥ 2 E(T 1)). In the last

dozen years, high triplet yields attributed to singlet fission have been observed in several

crystalline systems, such as benzophenone 40 and p-sexiphenyl. 41 Singlet fission has also been

suggested in a number of polymeric systems such as polydiacetylene, poly( p-

phenylenevinylene), and a ladder-type poly( p-phenylene). 42-49 Recent studies have suggested

singlet fission in molecular systems consisting of weakly coupled chromophores such as

carotenoids, tetracene and diphenylisobenzofurans (DPIBF). 50-54

One impediment to these studies has been the difficulty in confirming whether singlet

fission occurred. Delayed fluorescence is sometimes used as evidence that singlet fission has

occurred and then reversed course, but other physical processes can also lead to delayed

fluorescence. High triplet yield is also a possible indicator for singlet fission, but unless the

triplet yield is greater than 100%, simple observation of triplet population does not rule out other

mechanisms such as intersystem crossing. As mentioned earlier, singlet fission has a unique

magnetic field angle dependence in single crystal polyacenes owing to a level crossing resonance

between pairs of triplet excitons. No group has conducted a molecular analog to this experiment

yet, which could be imagined by using polarized light to selectively excite molecules in certain

orientations, or by poling molecules to align them, for example in a liquid crystal. 55 It is also possible that an electron paramagnetic resonance experiment could definitively identify triplet fission based on the relative populations of the various triplet sublevels. 56 As of now, other than reports in single crystal polyacenes, there have been no indisputable experimental reports of singlet fission.

18 1.4 Design of Molecular Systems for Singlet Fission

The studies of polyacenes revealed that singlet fission was observable when the energy of a singlet excitation was greater than or equal to the energy of two triplet excitations. One recent paper considered what sort of molecular motif would lead to promising chromophores for singlet fission in dye sensitized solar cells. 57 The optimal singlet energy for fission with solar radiation

and common DSSC elements would be ~ 2.2 eV, with a triplet roughly half that energy. It was

also determined that the second excited triplet energy level would ideally be above the first

singlet to prevent intersystem crossing, as many chromophores with a low energy first triplet also

have other low lying triplet levels. In a simple Hückel model, the S 1 and T 1 levels are

isoenergetic. Using a simple self consistent field approximation, the T1 energy level drops below

the S 1 energy level by twice the exchange integral. It was concluded that by maximizing this

exchange integral, by creating molecules with similar electron densities across the molecules in

both the HOMO and LUMO, one is able to generate several chromophores with low triplet

energy levels. Three molecules were chosen as a result of this work as being most promising for

singlet fission: pentacene, benzoquinodimethanes (BQD), and diphenylisobenzofuran (DPIBF). 57

1.5 Outline of Thesis

The objective of this thesis is to build up a theoretical understanding of singlet fission, so

that we can work more efficiently to design molecules and systems that are capable of efficient

fission yield. We approach the problems using the tools of theoretical chemistry and aim to

provide physical insight to the process as well as to guide future work in the field. 19 In Chapter 2 we examine the effects of coupling two isolated chromophores (ICs) to form a coupled chromophore pair (CCP). We outline a framework for understanding the relation between strong electronic coupling between chromophores, which often leads to efficient electron transfer, and reaction driving force, which plays a key role in determining whether fission will occur. Electronic structure techniques are used to scan through a variety of CCP geometries for three different ICs and to pick out the most promising CCPs for molecular singlet fission in solar cells

In Chapter 3 we consider singlet fission in a weakly coupled CCP as a four electron four orbital system, using the HOMOs and LUMOs of the ICs as our orbitals, and we derive a mechanism for singlet fission after considering all seventy possible electron configurations in this basis. The derived mechanism involves two subsequent electron transfer events, going from a localized singlet exciton to a charge transfer intermediate and ending in a pair of localized triplets. We assume the electron transfer steps are fast, and coherent in this chapter, and examine how various molecular properties affect singlet fission yield in this regime. Electronic structure methods are used to compute properties of real molecules, and the density matrix formalism is used to examine the time evolution of the initial singlet excitation. We use these simulations to derive design rules for singlet fission in a system governed by coherent electron transfer.

In Chapter 4, we re-examine the two step electron transfer mechanism in the context of

Marcus theory, where the electron transfer steps are incoherent, and their rates are mediated by fluctuations of the molecules and of their environment. We calculate energy levels and electronic coupling matrix elements for several promising molecules, and examine the time evolution of the initial excitation. We consider how molecular properties as well as the 20 properties of the environment affect singlet fission yield and derive guidelines for an optimal molecular system for singlet fission within this regime.

In summary, this work begins by considering individual chromophores, the most basic building blocks for a singlet fission enhanced dye-sensitized solar cell. We examine how the individual chromophores interact, and what control we can have over their molecular interactions. A complete mechanism for singlet fission is derived for weakly coupled chromophore pairs and we examine how the initial singlet excitation evolves and decays in both the coherent and the incoherent regimes of electron transfer. These studies are used to propose target molecules to realize high singlet fission yield in molecular systems and to move towards improved solar cell efficiency by means of singlet fission.

Chapter 2

Investigating the Effects of Coupling Chromophores on Electronic

Structure: How does Coupled Chromophore Pair Geometry Control

Singlet Fission?

22 2.1 Introduction

Recent experiments have examined the effects of chemically connecting two promising singlet fission chromophores in order to encourage energy transfer, and hence singlet fission.

The highest reported triplet yields in these coupled chromophore pairs (CCPs) are under 10%, as opposed to an ideal of 200% triplet yield. 50-53, 58 The exact nature of the connection, and therefore interaction, between coupled chromophores should control the efficiency of singlet fission. A CCP can be viewed as a perturbation of the electronic structure of two isolated chromophores (ICs) to a combined system, much like excimer formation in solution. Unlike excimer formation, one can control the intermolecular interaction by controlling the bonding between the two chromophores. The exact arrangement of the chromophores in space is critical: while single crystal tetracene has a very high triplet yield from singlet fission (room temperature fluorescence yield is less than 0.2%), solutions of molecular tetracene show no triplets, and the three known CCPs of tetracene to have been studied have delayed fluorescence yields from 0 to

3%, which could indicate weak fission.53, 54 Similarly, three CCPs of DPIBF have been studied, and have triplet yields that vary from 0% to 9% depending on the CCP geometry. 58

The specific chemical connection between two monomers will have two major effects.

First, it will affect the overall energetic balance of singlet fission, which is critical because fission yields will be low or zero if the process is significantly endothermic. One would expect that CCP formation would lead to some delocalization of the initial singlet exciton, and thus a stabilization of the initial singlet exciton state. The final pair of triplets state is not stabilized, however, as the two triplets are each localized on separate halves of the system. Stronger electronic communication between the two halves would be expected to result in greater 23 delocalization, hence more stabilization of the initial excited singlet, and a less energetically favorable singlet fission process. Because it is difficult to find ICs where singlet fission is significantly exothermic, it is important to monitor this property when creating a CCP. Second, the degree of electronic communication between the two halves is determined by the connection.

The figure of merit for electronic coupling is the electron transfer integral (or electronic coupling matrix element), t, and is an important factor in energy and electron transfer, which often (for small t) have rates proportional to | t|2.59-61 It is clear that the ideal CCP will have to balance boosting the kinetic driving force with harming the energetic balance of fission.

In this chapter we design and examine a series of CCPs based on tetracene, BQD and

DPIBF.(Figure 1) We investigate how various modes of chemical connectivity affect the electronic structure of the system, focusing on important parameters for singlet fission. In particular, we examine the electron transfer integral (i.e. electronic coupling), a key parameter in energy and electron transfer. We also study the free energy of singlet fission - whether singlet fission is endothermic or exothermic. Finally, we use these computations to predict specific

CCPs that are promising for singlet fission in solar cells.

N N

Figure 1 HOMO, LUMO and chemical structure (Left to Right) of DPIBF, BQD and tetracene (Top to Bottom)

The three ICs we consider are large aromatic systems, a structural motif which is known to lead to low triplet energies because of large exchange stabilizations. 57 Each of the three chromophores was coupled with a partner in multiple ways to see how the electron transfer integral and free energy of singlet fission would vary based upon CCP geometry. We use the frontier molecular orbitals of the ICs to guide our study, as they provide a simple tool for predicting how strongly perturbed a CCP will be from two non-interacting chromophores. The perturbation is roughly proportional to the amount of overlap between the frontier orbitals of the two monomers. In some cases, the IC structure was altered slightly to create a larger variety of

CCP structures and parameter values.

2.2 Methodology

We focus on two important parameters for singlet fission, the electron transfer integral, t, and the thermodynamic balance of singlet fission, Gf, which is the energy of the final state

minus the energy of the intial state. We derive both parameters from electronic structure

computations. Electronic structure calculations for both ICs and CCPs are done at the Density

Functional Theory (DFT) level with the B3LYP functional, using the 6-31G** basis set. Time

dependent DFT (TDDFT) calculations were used to evaluate the excited singlet states for all

systems. Hartree-Fock theory and configuration interaction (HF-CIS) were used to confirm DFT

and TDDFT results. Both the ground and first excited singlet were evaluated at the optimized

geometry of the ground state singlet, while the first excited triplet state was evaluated at its own

minimum energy geometry. All calculations were performed with either QChem 3.0 62 or

NWChem 5.0.63, 64

The electron transfer integrals were approximated by invoking Koopmans’ theorem and

assuming that only the two monomer LUMOs (HOMOs) mix to form the CCP LUMO and

LUMO+1 (HOMO and HOMO-1).(Figure 2) 65 We believe this is a safe approximation because

the energetic gaps between the HOMO-1 and HOMO and LUMO and LUMO+1 are more than 1

eV for all three chromophores.

LUMO 1 LUMO 2 2 tl ENERGY

HOMO 1 HOMO 2 2 th

Figure 2 The electron transfer integrals, t, for the HOMO and LUMO are half the splitting of the CCP orbitals in a homo-CCP

Specifically, the 2x2 Hamiltonian:

E1 t  H =   (2.1) t E 2 

where E1 and E2 are the energies of the HOMOs (LUMOs) for the two uncoupled chromophores, gives rise to eigenstates which are the HOMO and HOMO-1 (LUMO and

LUMO+1) of the CCP. If we take the orbital overlap to be very small, the electron transfer integral, t, can be approximated as:

2t= 4'( t2 − G ) 2 (2.2) 27

where G is the difference in the HOMO (LUMO) energies of the two ICs (E 1 and E 2)

that are mixing, and 2 t ' is the difference in the energy of the HOMO and HOMO-1 (LUMO,

LUMO+1) of the CCP. When both chromophores are the same, E1 = E2, the equation simplifies

so that t is the amount the CCP HOMO and HOMO-1 are above and below the chromophore

HOMO.

E+/ − = E ± t (2.3)

To evaluate the energetic balance of singlet fission, Gf, we assume the initial state is the lowest energy singlet exciton of the CCP. The CCP LUMO+1 orbital could be occupied after the molecule absorbs light, but the excess energy will quickly thermalize in most cases. The initial state energy is thus E , the CCP excited state as calculated by TDDFT, minus E , the S1 S0

CCP ground state as computed by regular DFT. There is no good way to compute the energy of

the pair of triplets state for the CCP. Because of this, we approximate the energy of the pair of

triplets as being twice the triplet-ground state energy gap for an IC, as both of these energies can

be solved with DFT. This leads to the equation:

=G2( EE − )( − EE − ) (2.4) f T1 GS S 1 S 0

28 where E and E are the IC triplet and ground state energies; the change in entropy is T1 GS

considered to be negligible and not included.

Finally, we investigate the inherent trade-off between wanting an exothermic free energy

of singlet fission, and wanting large electronic matrix elements. If we approximate a first excited

singlet as being a one electron HOMO to LUMO transition, the energy of the first excited singlet

for a CCP can be solved for based on the singlet energy gap of the IC minus the th and tl, the

amounts the HOMO and LUMO are raised and lowered by creation of the CCP. This

approximation can thus be written as:

=Gf2( EE T1 − GS )( − EEtt S * −−− GShl ) (2.5)

and clearly shows why the two ideal parameters, exothermic free energy of fission

and high electronic matrix elements, compete with each other. We examine the legitimacy of

this one electron approximation

2.3 Results and Discussion

2.3.1 DPIBF CCPs

The frontier molecular orbitals (HOMO and LUMO) of DPIBF differ noticeably in their nodal structure, but are both extended π structures with high electron density everywhere except the meta positions of the two phenyl rings. Because the two phenyl rings are only ~30 degrees out of the plane of the isobenzofuran core, the whole molecule acts as one extended π system. 29 Most regions of the chromophore have a similar electron density in the HOMO as they do in

the LUMO, which means most CCPs should have similar overlap, hence coupling, between the

two HOMOs and between the two LUMOs ( th and tl). The free energy of singlet fission, Gf, for one isolated DPIBF chromophore is listed in Table 1, based on the first excited singlet energy and the first triplet energy. The localized pair of triplets final state has the same energy for all CCPs of DPIBF, but each CCP has a different Gf owing to different delocalized singlet excitation energies.

Table 1 Electronic Matrix Elements and Free Energies for DPIBF CCPs

DPIBF th tl S1-S0 ∆Gf (eV) (eV) (eV) (eV) monomer N/A N/A 2.917 -0.056 D1 0.027 0.014 2.835 0.026 D2 0.027 0.027 2.770 0.092 D3 0.203 0.133 2.356 0.505 D4 0.190 0.200 2.152 0.710 D5 0.122 0.136 2.441 0.420 D6 0.122 0.122 2.525 0.337 D7 0.041 0.054 2.715 0.147 D8 0.027 0.027 2.772 0.089 D9 0.054 0.068 2.693 0.168 D10 0.004 0.009 2.798 0.064 D10@70 o 0.068 0.068 2.659 0.203 D11 0.002 0.006 2.805 0.056 D12 0.014 0.014 2.789 0.073 D13 0.014 0.014 2.797 0.064 D14 0.019 0.006 2.822 0.039 D15 0.014 0.014 2.822 0.040 D16 0.176 0.216 2.269 0.593 D17 0.036 0.054 2.713 0.149 D18 0.035 0.025 2.766 0.095 D19 0.079 0.105 2.629 0.232 D20 0.118 0.155 2.523 0.339 D21 0.085 0.082 2.611 0.251 D22 0.027 0.013 2.733 0.129 D23 0.064 0.073 2.583 0.278 D24 0.094 0.082 2.448 0.414

30 We examined CCPs of DPIBF that were bound through both the meta and para positions of the phenyl rings, as well as the two unique positions on the back of the isobenzofuran unit (Figure 3). The ortho position of the phenyl rings was not used, as attachment there would force the phenyl rings out of the plane of the isobenzofuran core, significantly changing the electronic structure of the IC. This dramatic geometric change to the

IC would localize the excitation on the core, and perturb the required energy relation, E S1 ≥ 2E T1 .

The majority of CCPs are designed based on creating a weak side to side π – π interaction between the two chromophores, while three CCPs (D4, D23, D24) force π stacking interactions.

N O O O O O O O D4 D1 D2 D3 O

O O O O D5 O D6 O O D7 O D8

O O O D12 O D9 D10 D11 o O O O

O O D13 O O D16 O D15 O O D14 O

O O O D17 D20 O D18 O D19 O O O

O O O O

O O O O D21 D22 D23 D24

Figure 3 Chemical Structures of DPIBF CCPs 32

In general, we see that preserving conjugation across the CCP, as in D16, leads to very strong coupling, as does any direct interaction of the π orbitals, such as connection at the para position of the phenyl rings on the two monomers in D5. CCPs that are connected by a methylene bridge (D2), or directly bonded through the meta position of a phenyl ring (D11) all have very low coupling, owing to little orbital overlap. The three π-stacked CCPs, (D4, D23,

D24) have electron transfer elements that are roughly proportional to the overlap of their π systems, which is altered by changing the location of the propylene tethers, from meta (D23) to para (D24), or by adding a third tether (D4). The electron transfer elements th and tl are similar, within error, for almost all of the CCPs owing to the similar electron density in the HOMO and

LUMO of the IC. By adding a heteroatom to the linker, such as in D3, one can form CCPs of

DPIBF that have different amounts of coupling between the HOMOs and LUMOs. One interesting series of CCPs is D5-D10, all of which are directly connected through the para bond of the phenyl rings, but which have varied twist angles (from 31.1 to 88.7 degrees) based on steric methyl groups. While all of these CCPs are connected the same way, the steric groups force the two phenyl rings out of plane, reducing the pi overlap dramatically.

All computations are done at the optimized geometry for all CCPs, and for many molecules a quick search of the potential energy surface reveals no geometries with significantly different coupling or energetics. This means that the electronic coupling is not likely to change as a function of temperature owing to vibrations and geometric changes. The notable exceptions to this lack of temperature dependence are molecules where the primary overlap is controlled by the torsion angle about a single bond, such as the D5-D10 series. To illustrate the importance of finite temperature on CCPs with torsion dependent coupling we performed a potential energy 33 surface scan for D10 and found that the central bond could twist from 88.7 degrees in the

optimized structure to 70 degrees at room temperature. The coupling of this twisted D10

(D10@70˚) is roughly an order of magnitude above the coupling in the optimized geometry. As

one would expect, this increasing planarity leads to greater orbital overlap and stronger

electronic coupling. Other DPIBF CCPs expected to have similar behavior include D19-D22.

The frontier orbitals of all DPIBF CCPs examined are visually similar to a sum or difference of the orbitals from two ICs. In both a strongly (D20) and a weakly (D2) coupled

CCP the HOMO-1 and HOMO are similar, and look very much like the sum of two HOMOs on two ICs (Figure 4). In both CCPs the LUMO and LUMO+1 orbitals look similar (except for the relative fragment signs), and appear to be the sum and difference of two IC LUMOs. Even in the strongly coupled D20 CCP there is only a small perturbation from the IC frontier orbitals, which is most noticeable in the LUMO, where significant electron density bridges the two halves of the 34 CCP. This verifies our model that each CCP is formed from two weakly coupled ICs with the

IC HOMOs interacting to form the CCP HOMO-1 and HOMO, and the IC LUMOs interacting to form CCP LUMO and LUMO+1.

2.3.2 BQD CCPs

Each isolated BQD chromophore is a roughly planar system, with both frontier orbitals composed primarily of an extended π-electron system. Unlike the DPIBF chromophore, the

HOMO and LUMO do not have the same electron density as each other on all regions of the molecule, owing to the asymmetry across the quinone ring. We have created CCPs by connecting BQD chromophores through both unique positions on the quinone ring as well as through both the nitrogen and carbon positions on the imidazole ring (Table 2, Figure 5).

Table 2 Electronic Matrix Elements and Free Energies for CCPs of BQD

TDDFTHF-CIS CORRECTED BQD th tl S1-S0 ∆Gf S1-S0 ∆Gf S1-S0 ∆Gf (eV) (eV) (eV) (eV) (eV) (eV) (eV) (eV) monomer N/A N/A 3.161 0.092 3.757 -1.598 3.161 0.092 B1 0.082 0.109 2.281 0.972 3.679 -1.520 3.082 0.170 B2 0.014 0.027 2.539 0.714 3.944 -1.785 3.347 -0.095 B3 0.020 0.029 2.529 0.724 3.615 -1.456 3.019 0.234 B4 0.054 0.104 2.254 0.999 3.386 -1.228 2.790 0.462 B5 0.008 0.007 2.543 0.709 3.575 -1.416 2.979 0.274 B6 0.054 0.041 2.489 0.764 3.757 -1.598 3.161 0.092 B7 0.068 0.014 2.471 0.782 3.766 -1.607 3.170 0.082 B8 0.328 0.306 1.683 1.569 3.141 -0.982 2.545 0.708 B9 0.122 0.041 2.266 0.987 3.742 -1.583 3.146 0.107 B10 0.041 0.177 2.234 1.018 3.698 -1.540 3.102 0.150 B11 0.172 0.122 2.292 0.960 3.308 -1.150 2.712 0.540 B12 0.069 0.091 2.289 0.964 3.474 -1.315 2.878 0.375 B13 0.188 0.751 1.741 1.512 2.518 -0.359 1.922 1.331 B14 0.193 0.720 1.907 1.346 2.560 -0.401 1.963 1.289 B15 0.200 0.737 1.478 1.774 2.413 -0.254 1.817 1.436 B16 0.163 0.658 1.866 1.387 2.699 -0.540 2.103 1.150 B17 0.395 0.516 2.208 1.044 2.930 -0.771 2.334 0.918 35

N N N N N N N N

B1 B2 B3 HN N N NH HN N HN NH N HN NH B4 N HN NH N N N H HN N N N N N H

N N HN NH N N N NN B5 B6 R R N N N N N N HN NH R R N N B7 B8 N N N N N N N N R NN R N N R R NN N N

N N N N R N N R B12 N N HN NH N N R N N R B11 N N N R NN R N N N N HN R NN R N B9 N B10 N H

N N N N N N N N N N

B13 NH B15 B16 N H B14 H N NH HN N H HN B17 N N N H

N N N N N N N N N N Figure 5 Chemical Structures of BQD CCPs 36

The asymmetry across the quinone ring stabilizes a zwitterionic electron configuration for BQD. TDDFT has difficulty properly evaluating energies for molecules of this sort, because of their multi-reference nature, and because DFT has a tendency to over-delocalize electron density. 66-71 The third and fourth columns of Table 2 show that TDDFT predicts that any CCP, regardless of the amount of coupling, has a significantly different first singlet excitation energy from the IC because of this problem. Even two BQD molecules with no connection, placed 20 angstroms apart are predicted to have a wildly different S 1 energies (2.4 eV instead of 3.2 eV) than a single BQD according to TDDFT. We correct for this problem by using HF-CIS computations, which do not predict the absolute magnitude of excited states as well as TDDFT, but which do not suffer from the same problem evaluating singlet exciton energies for BQD

CCPs as TDDFT. The last two columns of Table 2 present our best guess as to the actual molecular values, which are found by using the energy of the isolated BQD S1 from TDDFT, and using the differences in HF-CIS energies for BQD and the various CCPs of BQD. These tend to agree with the simple approximation that the first singlet excitation decreases in energy proportional to the increase in the electron transfer integrals. We believe these numbers illustrate the proper behavior of BQD and BQD CCPs, but note that a set of far more computationally intensive multi-reference calculations could be done to confirm these figures.

The molecular asymmetry of BQD means that it is possible to make CCPs of BQD that have very different th and tl values such as B9 and B10, because of the disparate electron densities in the two positions of the quinone ring in the HOMO and LUMO. B9 has a large th because of the large electron density in the HOMO, while B10 has a larger tl owing to the larger electron density on the imidazole side of the quinone in the LUMO. CCPs with connections 37 through the carbon of the imidazole ring have a small coupling (B3, B5) because the π-system does not extend onto the carbon atoms of BQD. The BQD IC can be changed such that there is

π-electron density on the carbons by unsaturating the terminal C-C bond, resulting in an IC

57 predicted to have approximately the same Gf as BQD. B4 is an example of a CCP of unsaturated BQD that has a large coupling owing to a direct bond, while B12 maintains this large coupling despite the methylene spacer because the two BQD systems are closer to planar than in the twisted B4. All CCPs of BQD that are connected through the nitrogen of the imidazole ring

(B1, B2) have small coupling because of alkyl spacers, as we avoided forming an unstable N-N single bond. We also examined CCPs of slightly modified BQD chromophores, such as B13-

B17. These CCPs all exhibit large electron transfer integrals, but fission is very endothermic because they have low energy singlet excitations owing to their continuous π systems. As was the case for DPIBF, all frontier orbitals of BQD CCPs appear to be the weakly perturbed sum or difference of frontier orbitals for two ICs (Figure 6).

Overall, there are fewer viable CCPs of BQD than DPIBF owing to fewer attractive connection points, and a worse IC free energy of fission. The BQD chromophore, however, provides more options for creating CCPs with disparate th and tl owing to the molecular asymmetry. If the mechanism of singlet fission is found to be dominated by either hole or electron transfer, for example, maximizing one electronic matrix element while keeping the other small could result in both good kinetics and a good free energy of fission.

2.3.3 Tetracene CCPs

Polyacenes are promising chromophores for singlet fission in CCPs because they have very low triplet levels which lead to a good Gf (Table 3), and because there is ample evidence for efficient singlet fission in polyacene crystals. We examine tetracene because of the recent experimental work, 54 but emphasize that these results should generalize well to pentacene, which we will discuss in more detail in future work, and which we believe is more promising because of its lower triplet relative to the first singlet excitation. Similar to the other chromophores we have studied, tetracene is a planar molecule with frontier orbitals dominated by a large π-electron system. The high symmetry of tetracene means that there are only three unique locations from which to couple the chromophore to a twin. We examine CCPs of tetracene from all three unique locations, as well as one π-stacked CCP (Figure 7). 39

T1 T2 T3

T4 T5

T7 T8

Figure 7 Chemical Structures of tetracene CCPs 40 Table 3 Electronic Matrix Elements and Free Energies for tetracene CCPs

Tetracene th tl S1-S0 ∆Gf (eV) (eV) (eV) (eV) monomer N/A N/A 2.494 -0.088 T1 0.302 0.308 1.658 0.748 T2 0.014 0.014 2.504 -0.044 T2@70 o 0.068 0.082 2.385 0.021 T3 0.211 0.189 1.934 0.472 T4 0.041 0.014 2.326 0.080 T5 0.012 0.010 2.419 -0.012 T6 0.010 0.010 2.425 -0.019 T7 0.034 0.026 2.383 0.024 T8 0.031 0.024 2.393 0.013 T9 0.190 0.159 2.033 0.373

When tetracene molecules are π-stacked, as in T1, there is a large overlap in both the

HOMO and LUMO, leading to large electronic matrix elements. Unfortunately this large coupling also makes singlet fission significantly endothermic. A more moderate degree of coupling can be obtained by using methylene linkers to align the two chromophores in a sort of herringbone structure (T9), similar to the interaction in a tetracene single crystal. Despite the methylene spacers, the two chromophores maintain a very large orbital overlap between the bottom lobes of their π systems. Inserting a phenyl bridge between the chromophores, as in T5-

T8, usually results in very low coupling because the hydrogen atoms on the bridge and chromophores repel, forcing the phenyl group more than 89 degrees out of plane from both polyacene monomers. An ethynylene bridge between the chromophores leads to large coupling

(T3), but, as in the case with T1, singlet fission becomes less thermodynamically favorable with this strong coupling. Two options for pairing tetracene chromophores to a more moderate degree are illustrated by T4, which is shaped like a ‘V’ owing to steric repulsions, and T2, where the two tetracene chromophores have nearly orthogonal π systems. These connections are very 41 similar to those for CCPs of DPIBF (D5-D10), and methyl groups can be added or removed to tune the interaction. Room temperature thermal vibrations allow T2 and T4 to achieve higher coupling (up to 70˚ for T2) by rotating about the linking bond, similar to the DPIBF cases D5-

D10. The steric interactions in T5-T8 provide a dramatically sharper well in the potential energy surface, however, and the coupling will not increase appreciably at room temperature. It would, of course, be possible to make all of these CCPs with pentacene instead of tetracene. The electronic matrix elements would be largely the same, and the free energy of fission would be close to 0.5 eV more exothermic for each CCP listed, owing to the IC singlet and triplet energy levels. As is the case for the previous two sets of CCPs, the frontier orbitals of tetracene CCPs, such as T4, look very much like the sum and difference of frontier orbitals from two isolated tetracene chromophores (Figure 8).

Figure 8 The Frontier orbitals of T4 are linear combinations of the frontier orbitals of two isolated chromophores. (Left to Right, HOMO-1, HOMO, LUMO, LUMO+1) 42

2.3.4 Hetero-CCPs

It is also possible to combine two different ICs into a CCP to form a hetero-CCP (as

opposed to previously discussed homo-CCPs). This could be beneficial if desired properties of

two ICs could be combined, for example, a specific ratio of HOMO and LUMO wave functions

to lead to a specific th to tl ratio provided by one IC and a very low triplet level provided by a

different IC. The disadvantage to making hetero-CCPs is that from a simple one-electron picture

they will have a more endothermic free energy of fission than homo-CCPs with equal amounts of

coupling. If we approximate the first singlet state as being a singlet electron transition from the

HOMO to the LUMO, the first singlet exciton in a hetero-CCP is a transition from a CCP orbital

similar to the HOMO of the IC with higher energy to a CCP orbital similar to the LUMO of the

IC with lower energy (Figure 11). The final state of a pair of local triplets is the sum of the

triplet energies for each IC. This means the energy of the triplet pair state for a hetero-CCP is an

average of the triplet pair states for each homo-CCP, whereas the S 1 state for a hetero-CCP is lower in energy than the smaller of the two homo-CCP singlet excitons. While all CCPs have slightly smaller S 1 excitation energies than the ICs owing to the splitting of the frontier molecular orbitals, hetero-CCPs also have the S 1 excitation energy reduced by whatever the offset in the IC orbital energies is. This flaw of hetero-CCPs will be more noticeable when the two ICs that are combined have more disparate HOMO and LUMO levels. Because the three

ICs we examine all have similar HOMO and LUMO energies, they are expected to be only slightly worse than the homo-CCPs. DT1 and DT2 (Figure 9, Table 4) have higher free energies of fission than D20 and D6, but weaker coupling. The tetracene homo-CCP T9 has similar free energy to DT1 and DT2, but almost 50% more coupling. BD1 and BD2 both have the sum of th 43 and t l equal to approximately 0.4 eV, which is comparable to the coupling in D16 or B11, which are both roughly .3 eV less endothermic for singlet fission. Based on these results, we would not absolutely rule out hetero-CCPs for singlet fission, but note that the decreased singlet exciton energy makes them less attractive than homo-CCPs.

O DT2 DT1 O

N N N N N O

O R O N R N N R N R BD1 N R N BD2 BD3 N R

Figure 9 Chemical structures of Hetero-CCPs

Table 4 Electronic Matrix Elements and Free Energies for Hetero-CCPs

Mixed t_h t_l S1-S0 ∆Gf Dimers (eV) (eV) (eV) (eV) DT1 0.131 0.090 2.279 0.355 DT2 0.113 0.123 2.252 0.382 BD1 0.143 0.257 2.187 0.871 BD2 0.096 0.264 2.138 0.920 BD3 0.104 0.097 2.369 0.688 44

Figure 10 Frontier orbitals for the hetero-CCP DT2 are localized, in contrast to the delocalized orbitals seen in homo-CCPs. (Left to Right, Top row then Bottom HOMO-1, HOMO, LUMO, LUMO+1)

Visual inspection of the frontier molecular orbitals of a hetero-CCP illustrates the energetic offset of the HOMOs and LUMOs (Figure 10). Unlike all three homo-CCP examples, the frontier orbitals on the hetero-CCPs are very localized. When the HOMOs and LUMOs of

DPIBF and tetracene split in DT2, the IC orbitals are not isoenergetic, as they are with homo-

CCPs, and do not contribute equally to the HOMO and HOMO-1 (LUMO and LUMO+1). The

HOMO-LUMO gap in DT2 looks a lot like the gap between a DPIBF (higher energy ) HOMO and tetracene (lower energy) LUMO, as predicted by the one-electron description. 45

LUMO 2

LUMO 1

CCP IC 2 IC 1 S1 S1 ENERGY S1

HOMO 2

HOMO 1

2.3.5 Trade-Off of Transfer Integrals with Thermodynamics

We now evaluate the idea that there will necessarily be a trade-off between thermodynamic driving force and the transfer integral. Equation 2.5 illustrates this trade-off given a one electron picture of the ground and excited states. In this approximation the driving force decreases linearly with the sum of the transfer elements. This is not far from what we see in doing full TDDFT, as shown by Figure 12. For a series of DPIBF CCPs there is a strong linear correlation (R 2=.94) between increased transfer integrals and unfavorable thermodynamic driving force. The slope of this line is 1.54, meaning that the driving force, and thus the actual singlet excitation, is more strongly affected by coupling than this simple approximation predicts. 46 The error in this simple model can be attributed to its exclusion of multi-configuration

electronic states contributing to the excited state character. The three worst outliers to the linear

regression are the three pi conjugated systems, which is not a surprise as the method we used

(DFT B3LYP) are known to have difficulty accurately assessing π-stacking interactions.

Overall, the data show that the simple one electron picture can be used to approximately predict the singlet excitation, and hence free energy of fission, without the need for TDDFT.

0.8

0.7

Gf 0.6 ∆ 0.5

0.4

(eV) 0.3

0.2

0.1 Free Energy of Fission, Energy of Free 0 0 0.1 0.2 0.3 0.4 -0.1 Sum of Transfer Integrals (eV)

Figure 12 The free energy of singlet fission (Eqn 2.5, Table 1) becomes more endothermic nearly linearly as the coupling between two DPIBF chromophores is increased

2.3.6 Comparison to Experimental Data

Exciton fission yields have been experimentally studied for three DPIBF CCPs, D2, D5, and D10. D5 showed no triplet formation at all, which could be expected from the high 47 coupling, and thus extreme endothermicity of singlet fission. D2 evinced a small amount of

triplet, while D10 had a higher triplet yield. 58 At first glance one could take this to mean that the

coupling of D2 is too large and the thermodynamic cost makes this coupling disadvantageous. If

one accounts for thermal motion of the CCPs, however, D10 can rotate such that the two IC π-

systems are only 70 degrees out of plane (D10@70) at room temperature, yielding a significantly

higher coupling. D2 does not have any geometries available at room temperature that result in a

significant change in coupling. Based on this analysis, we conclude that an ideal coupling for a

DPIBF type molecule is greater than or equal to that in D5 (including thermal motion). We also

predict a stronger temperature dependence will exist for singlet fission in D5 than in D2 because

of this temperature dependent transfer element.

Pentacene and tetracene crystals have been shown to be very efficient at producing

triplets via singlet fission. Recent work has examined the tetracene CCP T5/T6 as well as two

other tetracene CCPs coupled by phenyl rings. These reports singlet fission yields of at best

3%. 53, 54 Because the tetracene IC is more exothermic for singlet fission, we predict that a more strongly coupled polyacene system would be promising for singlet fission. T2, T4, or analogs with fewer or more steric groups are promising because they have large coupling but may allow room temperature singlet fission. Pentacene based analogs are even more promising, as uncoupled pentacene has a Gf almost half an electron volt more favorable than uncoupled

tetracene. Finally, carotenoids, or more generally, polyenes represent a class of molecules we

have not considered because a low second triplet level could lead to parasitic intersystem

crossing, 58 but which remain of interest to us because of very exothermic free energies for fission

for ICs.

48 2.4 Summary and Conclusion

In this chapter we have examined how coupling individual chromophores can affect the transfer integral, t, and have found that for several promising dyes it is easy to achieve an electronic transfer integral ranging from almost zero to nearly half an electron volt. It has been shown that a simple analysis of the IC HOMO and LUMO of a chromophore monomer can provide a very good preliminary guess as to how strong a given chemical coupling will be based on orbital overlap. We have also discussed how coupling dye molecules can lead to excimer formation, which makes singlet fission less thermodynamically favorable. The free energy of fission has been studied using TDDFT, and we have shown how a simple one electron HOMO to

LUMO model can produce reasonable estimates of this quantity. In every CCP, there is a trade- off between the electron transfer integral and the free energy of fission, which has to be balanced to optimize triplet yield. Tetracene and pentacene analogs of T2 and T4 are considered the most promising candidates for further experiments. Future work will investigate the mechanism for singlet fission, in an attempt to more exactly understand how to design the best systems for singlet fission.

Chapter 3

Development of a Mechanism for Singlet Fission and Analysis of

Fission Dynamics and Yield In The Regime of Coherent Electron

Transfer

50 3.1 Introduction

In the previous chapter we outlined a framework for evaluating the promise of a given

CCP based on the free energy of singlet fission and the electronic matrix element, using the assumptions that an endothermic free energy of fission was bad, and that a large electronic matrix element was good. These assumptions are generally sound for electronic processes in molecular systems, but the exact dependence of singlet fission on these parameters is not clear without knowing the mechanism.

In this chapter we propose a two step mechanism composed of electron transfer from one chromophore to its partner, forming a charge transfer intermediate, followed by transfer of an electron of opposite spin in the opposite direction, resulting in a pair of charge neutral triplets.

We analyze every possible electronic state in a four electron, four orbital model of a coupled chromophore pair (CCP), and reduce the complexity of the problem to the most important states.

The dynamics of the excitation are modeled with density matrix theory, assuming fast coherent electron transfer. Using our proposed mechanism, we scan the space of all reasonable CCP properties to understand in general how molecular properties affect fission yield. Finally, we use density functional theory to evaluate all of the kinetic and thermodynamic parameters used in our model for several promising molecules, and predict fission yields for each of them.

3.2 Methodology

To predict singlet fission yield in various molecular species we (i) created a ten state dynamic model by analyzing singlet fission in a CCP with a four electron, four orbital basis, (ii) 51 used electronic structure computations to evaluate the energy of every state and electronic coupling elements for a few promising chromophores, and (iii) applied the density matrix formalism to examine the evolution of the excitation over time for several test cases as well as for real molecules. Each of these three steps is discussed in further detail below.

3.2.1 Creation of a Model System

We analyze singlet fission in molecular CCPs in the limit of two uncoupled chromophores. We use this basis because chapter 2 has shown that ideal CCPs for singlet fission may be only very weakly coupled. Our treatment of singlet fission begins by focusing on the frontier molecular orbitals (HOMOs and LUMOs) of two chromophores. In order to consider all possible mechanisms for singlet fission, we examine every electron configuration for this model, working in the basis of two uncoupled chromophores. (Figure 13) 52

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30 69

31 32 33 34 35 36 70

37 38 39 40 53 54 55 56

41 42 43 44 57 58 59 60

45 46 47 48 61 62 63 64

49 50 51 52 65 66 67 68

Figure 13 There are 70 possible electron configurations in a 4 electron 4 orbital basis comprised of a pair of chromophore HOMOs and LUMOs. 36 states have two neutral chromophores, 32 are a single electron transfer charge transfer pair, and 2 states put all four electrons on one chromophore. Highlighted states are those that are most energetically accessible.

Ignoring charge-transfer events, each half of the system has two orbitals and two electrons, yielding six possible electron configurations. A pair of neutral chromophores therefore has 36 possible electron configurations. If we consider a single electron transfer from one chromophore to the other, we have to include 32 more configurations, and there are two configurations with double electron transfers. In order to evaluate which states are most likely to play a role in singlet fission, we assume (because of the energy gap law) 72 that only states close 53 in energy to the initial and final states are likely to be accessed by a significant population

during the lifetime of the excitation. We know that the energy of the first singlet excitation is

equal to roughly twice the energy of the first triplet excited state and also ideally approximately

2.2 eV for solar cell applications. 57 Most configurations with two neutral chromophores can be

easily eliminated from consideration, such as state 15, which has two singlets, or state 19, that

has a singlet and a triplet excitation. We would estimate these states as having energies of 4.4 or

3.3 eV from our primitive model.

Only eight of the thirty six neutral pair states are kept, four that are of S 1S0 (17,23,27,28) character, and four that have TT character (1,2,7,8). We include the eight single charge transfer states that are in their ground electronic state (37,38,45,46,53,54,61,62), but remove all other single charge transfer states as they contain multiple excited electrons and are expected to be significantly higher in energy than the other states considered in this model. The two double charge transfer states are ruled out based on the extreme unfavorability of putting two extra electrons on the types of molecules that will be considered for singlet fission. In all, fifty four of the seventy states can be removed from consideration because they are at least an order of magnitude more than k BT (thermal energy) away from the initial and final electronic state. Of

the sixteen states that remain, four have one excited singlet chromophore and one ground state

chromophore (S 1S0), eight are ground state singly charge transfer pairs (CT), and four states represent a pair of chromophore triplets (TT). 54

S1 States CT States TT States

K T L 1 H 5

K TD1 KTT R 2 6 9 TH

TD2

K T KTT R 3 H 7 10 TD1

KL 4 8 T H

Figure 14 Ten state system for singlet fission with all allowed electron transfers shown. States have been renumbered.

We use this set of sixteen states as our reduced basis for singlet fission, and consider all possible transitions between these states that involve only a single electron transfer. Each S 1

state can convert into two different CT states, one by transferring an electron from the LUMO of

one chromophore to the LUMO of the other, and the second by transferring an electron from the

HOMO of one chromophore to the HOMO of the other. (Figure 14) Two of the TT states can

convert to two different CT states by a single electron transfer, both by sending an electron from

the LUMO of one chromophore to the HOMO of the other chromophore. Four CT states can

each convert to two S 1 states or one TT. Two TT states (all electrons up or all down) and four

CT states (those with overall spin of ±1) can not be reached by any combination of one-electron 55 processes between energy-allowed states. To reach these two TT states would require two

concerted electron spin flips, while reaching these four CT states would require an electron to

hop from one molecule to the other while simultaneously switching its spin state. These six

states are dismissed from our reduced scheme, as we believe second (and higher) order processes

are unlikely to be important when first order processes provide a viable pathway. Our reduced

ten state scheme for singlet fission, with renumbered states, is shown in Figure 14.

Because we assume that orbital occupancy does not affect the orbital character (taking

Koopmans’ approximation) all four electron transfers from a HOMO (LUMO) orbital of one

chromophore to the HOMO (LUMO) of the other have the same electronic matrix element, TH

(TL). The four electronic transitions from CT to TT states that involve a diagonal electron

transfer can be divided into two types, either left to right HOMO to LUMO or LUMO to HOMO.

If the two halves of the CCP are not the same, these two transitions could have different matrix

elements, TD1 and TD2 . We consider two possible ways for an excitation to leave the system,

either by singlet decay or triplet-triplet decay. Singlet decay encompasses several processes,

such as fluorescence, internal conversion, intersystem crossing, and injection to an electrode,

while triplet-triplet decay is presumed to consist entirely of injection into an electrode. We use a

rate constant to represent triplet-triplet decay and two different rate constants to represent singlet

decay for the two different types (left and right localized) of singlet states. The two singlet decay

rates will usually be similar but could differ in the case of a hetero-CCP or if the CCP was

tethered to the electrode through only one of the two chromophores. Triplet-triplet decay most

likely begins with injection of a single triplet, but we assume the second triplet will live long

enough to also inject. A more complicated model could be constructed that considers other 56 pathways or transitions, but we argue this simple model captures the essential behavior of

singlet fission.

3.2.2 Electronic Structure Computations

We use electronic structure techniques to ascertain all of the parameters (energies and

matrix elements) in the ten state model for several promising CCPs. We chose three CCPs based

on three IC scaffolds from the experimental and theoretical literature and examined the effects of

adding electron donating and withdrawing groups to each IC.

In order to model the dynamics of TT formation, we computed the energy of our ICs in

their ground state, first excited singlet state, and first triplet state, as well as in their cation and

anion ground states. All computations were done in the optimized ground state geometry

because we assume the dynamics are fast and coherent, preventing geometry relaxation. We

continue to approximate the CCPs as a sum of two ICs. This approximation gives fairly accurate

energies for the localized singlet and triplet pair CCP states as long as the ICs are weakly

coupled, as all of the systems we study are. One advantage of this model is that the energy of

many CCPs in many different electronic states can be computed in a more rapid fashion. If the

energy of any given CCP is just the sum of two halves plus a correction term, one can compute

energies for N 2 CCPs in just N computations. The energy of the pair of triplets states and the localized excited singlet state are generated by adding the energies of two ICs in the appropriate electronic states.

57 E=2 E − 2 E (3.1) TT T S 0

E= E − E (3.2) SS10 s 1 S 0

where E and E are the energy of the triplet pair and localized singlet CCP states, and TT S1 S 0

E , E , E are the IC triplet, ground state and first excited singlet state energies. In order to T S0 s1 accurately model the energy of the CT states we had to account for the Coulombic stabilization from the two halves interacting.

E+−= EE + + − − 2 E (3.3) CA C A S0

* E+−= EE + + − −2 E − (3.4) CA C A S0

where E and E* are the uncorrected and corrected energy of a CCP CT state, E , C+ A − C+ A − C+

E − and E are IC cation, anion and ground state energies, and is the correction energy. We A S0 use Constrained DFT (CDFT) 57, 73-75 to find the correction energy because the two charged chromophores are so close together. CDFT restricts the charge on each half of the CCP (to +1 or

-1) and iteratively adjusts the electron density of each half based on the field from the other half.

=(EE+ +− − 2)( E − E − E ) (3.5) C A S0 CDFT CCPS 0

58 We run one CDFT computation for each scaffold CCP, and compare the CDFT energy to the neutral CCP energy, E to find the correction term, . The same correction is then CCPS 0 applied to all variants of a given CCP scaffold with a given geometry. This allows a good estimate for the Coulombic stabilization without having to run a costly CDFT computation on every pair of ICs.

The electronic matrix elements, TH and TL , for the three basic CCP scaffolds were computed using the half energy level splitting method described in the methods section of chapter 2. The half energy level splitting method can only be used to compute electronic matrix elements for orbitals that are close in energy, however. We thus can not use this model to solve for the diagonal (HOMO to LUMO) electronic matrix elements. We instead rely on the

76-78 Longuet-Higgins-Roberts approximation, that states that the electronic matrix element Hab is proportional to the orbital overlap, Sab .

Hab=<Ψ a| H | Ψ>= b kS ab = k <Ψ ab | Ψ> (3.6)

This is exactly true if Ψa and Ψb are eigenstates of the Hamiltonian, in which case k equals Eb. Because we are examining the overlap of orbitals that are eigenstates of the IC

Hamiltonians, but computing matrix elements for the CCP basis this equation is not exact, but

can be used as an approximation. In order to compute the overlap, Sab , we split the geometry optimized CCP into two ICs by cutting the connecting bond and hydrogen terminating each half.

We then compute the wave function of the HOMO and LUMO for each IC. The overlap 59 between the two HOMOs, and two LUMOs is found by integrating over space, and we use

these values and the known H ab values from the half-splitting method (valid only when the two orbitals are close in energy) to compute the proportionality constant k. The integrated overlap

Sab for the diagonal electron transfer is then used with this k, to find H ab for diagonal electron transfer. The same k computed for a scaffold CCP is used to compute matrix elements for the substituted versions of the scaffold from their overlap integrals.

All electronic structure computations were done with density functional theory (DFT) using the B3LYP functional with a 6-31G** basis set. Time dependent DFT (TDDFT) was used to evaluate the excited singlet states for all systems, and constrained DFT (CDFT) was used to compute the energy of charge transfer dimer states. All computations were done using either

QChem 3.0 62 or NWChem 5.0 63, 64 . We used the CDFT module written by Van Voorhis and coworkers for implementation in NWChem 5.0. 57, 73-75

3.2.3 Time Evolution of Excitation

In this chapter we examine coherent evolution of the excitation using density matrix formalism and a purely electronic Hamiltonian based on the ten state scheme discussed previously. In order to estimate the percent of the initial excitation that leaves the system through singlet fission as opposed to other pathways, we introduce absorbing boundary conditions. 79 We investigate how the excitation can evolve in a coherent system with the

Liouville-von-Neumann equation, adding in a term for dissipation 80

60 dρ ih =[][] H,ρ + H , ρ , (3.9) dt 0 D +

where H0 is the Hamiltonian for the system, H D is the dissipative matrix, and ρ is the density matrix. The Hamiltonian and dissipative correction matrix derived from our ten state model are:

 ESS1 0 000 TH T L 0000    0E 00 T T 0000  SS0 1 LH   00E 000 T T 00   SS0 1 H L   000ESS1 0 00 T LH T 00    TTH L00 E AC 0000 T D 2 H0 =   (3.10)  TTLH 000 ECA 00 T D 1 0   00TT 00 E 00 T   HL CA D 1   00TTLH 000 ETAC D 2 0   00000T 0 T E 0   D1 D 2 TT     0000TD2 0 T D 1 00 E TT 

 −iK L 0 0 000000 0    0−iK 0 000000 0  R   0 0−iK 000000 0   R   0 0 0−iK L 00000 0   0 0 0 000000 0  H D =   (3.11)  0 0 0 000000 0   0 0 0 000000 0     0 0 0 000000 0   0 0 0 00000−iK 0   TT     0 0 0 00000 0 −iK TT 

where H0 s the Hamiltonian governing evolution within the ten state manifold, and H D represents population leaving the ten state system by fluorescence, internal conversion, 61 intersystem crossing, or injection. The system is set up with some initial density matrix ρ0 , where all of the population is equally distributed among the four local S 1 states, and evolves in small finite timesteps until all the population has left the system. At the end of the simulation we can examine the overall fission yield, based on how much population left the system via triplet pair decay as well as looking at the population of each state as a function of time. We interpret the percent of the excitation that has left the system by the triplet pair decay path to represent the percent of excitations in a given type of CCP that would undergo singlet fission as opposed to other decay mechanisms.

We run several simulations using fictitious, but molecularly relevant, state energies E n, electronic matrix elements T n and decay rates K n to understand how variation within the parameter space affects fission yield in this model. We subsequently use parameters that were computed for real CCPs to predict fission yields. The combination of these sets of data are used to create design parameters for molecular singlet fission and to suggest future experiments.

3.3 Results and Discussion

3.3.1 Electronic Structure

In this chapter, we focus on one CCP each for three different ICs, as shown in Figure 15.

Pentacene was chosen because the best experimental singlet fission yield reported was in polyacene single crystals,29 so the molecular analogs are a natural choice. DPIBF was chosen because recent experimental 58 and theoretical 57 reports have illustrated some promise for weakly coupled DPIBF CCPs. The polyene molecule was chosen for two reasons; if a polyene could 62 undergo singlet fission, the process would be significantly exothermic based on its singlet and triplet energy levels, which is unusual. Also, singlet fission has been suggested to explain high triplet yield reported in some protein-bound carotenoids. 52

R R 5 5 R Polyene CCP O DPIBF CCP

O R R R Pentacene CCP

Figure 15 Structures of three promising CCPs. We examine these three CCPs, as well as some variants that have amino (a), or nitro (n) ‘R’ groups

Each of these three CCPs is coupled in a similar manner, through a bond that would lead to conjugation across the CCP if not for steric forces rotating the two ICs nearly ninety degrees out of plane from each other. All three of these molecules can rotate about the connecting bond at room temperature, and this rotation has a significant effect on the overlap and hence coupling between ICs. We have examined each CCP at its ground state geometry as well as the most coplanar geometry it can assume at room temperature. 63 Table 5 Electronic Matrix Elements for Several Bare and Functionalized CCPs. The prefixes aa, an, and nn are diamino, aminonitro, and dinitro functionalized variants.

TH (eV) TL (eV) TD1 (eV) TD2 (eV) DPBIF 0.068 0.068 0.066 0.066 PENT 0.068 0.082 0.056 0.056 POLY 0.065 0.059 0.065 0.065 aaDPIBF 0.061 0.070 0.064 0.064 anDPIBF 0.056 0.058 0.044 0.089 nnDPIBF 0.066 0.034 0.046 0.046 aaPENT 0.063 0.085 0.056 0.056 anPENT 0.056 0.058 0.044 0.089 nnPENT 0.066 0.056 0.051 0.051 aaPOLY 0.044 0.071 0.058 0.058 anPOLY 0.069 0.039 0.034 0.084 nnPOLY 0.096 0.022 0.047 0.047

Each of the three base CCPs has a similar overlap integral and similar electronic matrix element owing to the similarity of the connections (Table 5). As one would expect, the electron donating and withdrawing groups affect the overlap, and hence matrix elements of each CCP scaffold. Most notable is the strong electron withdrawing property of the nitro groups, especially in the LUMO. This leads to nn-CCPs with very low TL values. The diagonal matrix elements are highly altered in the an-CCPs because the amino and nitro groups withdraw electron density from the HOMO and LUMO, but push more electron density out into the system in the LUMO and HOMO. This means the two diagonal electron transfer matrix elements (as well as TH and

TL) are very different in the an-CCPs.

Table 6 shows the energies of every state in the ten state model for the twelve CCPs, three base CCPs, and several variants with electron donating and withdrawing groups added.

The bare DPIBF CCP has the highest CT and TT state energies. The POLY CCP has the lowest

TT state energy, with fission being significantly exothermic, while the PENT CCP has both the 64 CT and TT states lower in energy than the locally excited S 1 states. Homo-CCPs of DPIBF and PENT that incorporate amine and nitro functional groups tend to have lower charge transfer energies than the bare CCPs, while functionalized POLY CCPs have higher energy CT states.

The lower CT states could result from the heteroatoms providing a good location for excess charge to reside, while it is possible that the POLY CCP is more stabilized in its local excited states by the functional groups than in the charge transfer states. Almost every functionalized

CCP has singlet fission values more endothermic (or less exothermic) than the unfunctionalized version. This trend is expected and has been discussed previously. 57 Briefly, the electron donating and withdrawing groups tend to have different electron densities in the HOMO and

LUMO of the ICs, and thus the exchange stabilization of the local triplet is reduced, so the triplet goes up in energy. When the two chromophores in a CCP are different (for example with an amino group on one side, and a nitro group on the other side) the symmetry of the system is reduced. The two localized singlet states and the two charge transfer states are no longer degenerate. Each of these three hetero-CCPs has one CT state that is very low in energy relative to the localized singlet states, and a second CT state that is very high in energy, as expected.

Table 6 Energies of Electronic States for several CCPs (in eV)

+ - - + S1S0 S0S1 C A A C TT DPIBF 0 0 1.005 1.005 0.303 PENT 0 0 0.354 0.354 -0.240 POLY 0 0 0.595 0.595 -0.712 aaDPIBF 0 0 0.988 0.988 0.384 anDPIBF 0 -0.207 -0.144 1.823 0.286 nnDPIBF 0 0 0.898 0.898 0.397 aaPENT 0 0 0.345 0.345 -0.144 anPENT 0 -0.089 -0.506 1.116 -0.169 nnPENT 0 0 0.353 0.353 -0.105 aaPOLY 0 0 0.668 0.668 -0.222 anPOLY 0 -0.191 -0.484 1.674 -0.257 nnPOLY 0 0 0.712 0.712 -0.101 65

3.3.2 Time Evolution of Model Systems

3.3.2.1 Basic Model

In order to understand the type of dynamics that arise from this ten state coherent system,

we first examine a simple case in detail. In this example, simulation 1 (Sim1), all ten states are

isoenergetic, and every matrix element is equal (0.027 eV). The decay matrix elements for both

singlet exciton states and the triplet pair state are equal and set to 2.7x10 -4 eV, which corresponds to a mean lifetime of 690 fs if singlet decay were the only pathway. The matrix elements are

Figure 16 Fission yield is ~11% if all states are isoenergetic, all matrix elements are equal, and all decay pathways are equal. The population oscillates between different states very quickly, with fission yield based on average state populations and the ratio of the rate-limiting singlet and triplet decay processes. The inset shows the first 0.1 ps in detail. S1, CT, and TT labels refer to the sum of all population in a S1, CT or TT state. Fission and Fluorescence are the percent of the population that has left by either triplet pair or singlet decay routes, and Percent is the percent of population that has left the system through the fission route as opposed to the singlet decay route. 66 close to those of several promising CCPs, and the decay rates are chosen to be on the order of very fast internal conversion events, while the state energies are chosen to be isoenergetic for simplicity. For a homo-CCP we assume that states 1-4 begin equally occupied. As can be seen in Figure 16, population quickly leaves the S 1 states, enters the CT states, and then moves into the TT states. Under these conditions, the CT population reaches its peak in less than twenty femtoseconds, and the TT population peaks in just over thirty femtoseconds. Unlike an isoenergetic two state coherent system, where all the population would transfer to the second state before reversing direction, the TT population peaks at around 20% of the total population before all of the population flows back through the CT states to the S 1 states. Because of the many electron transfer pathways, the oscillatory nature of the state populations is not simple. As expected, however, at all points in time, the four identical S 1 states have the same population, as do the four identical CT states, and the two identical TT states. After one picosecond, roughly half the population of the system has exited through one of the decay channels, and after five picoseconds more than 90% of the population has left. Because the decay rates are equal, the ratio of triplet pair injection to singlet decay is proportional to the ratio of the integrated populations of the TT and S 1 states over time. The ratio begins low because initially all population is in the S 1 states, peaks as the TT population is first beginning to tail off (at 40 fs), and then settles on a very slowly rising curve, yielding ~11% fission at infinite time. Physically, this means that if a molecular system with these parameters (isoenergetic states, weak coupling and extremely slow decay rates) were to be made, one would expect 22% triplet yield, which exceeds any known experimental result for molecular singlet fission. To understand why the triplet yield is this promising in this example, as well as to see if we can get higher yields

(perhaps even up to 200%) we do several other simulations with hypothetical molecular 67 parameters below. These examples have been selected because they illustrate some of the complex behavior of this interconnected ten state coherent system. An illustrative selection of simulations is summarized in Table 7, as well as discussed in the text below.

Figure 17 (Left) Doubled triplet-triplet decay rates lead to increased fission yield, while (Right) doubled singlet decay rates decrease fission yield.

68 Table 7 Summary of several simulations with various parameters for coherent dynamics. Each row is a different simulation, and the figure in which more details can be found is listed. The electronic coupling matrix elements are listed first, then the decay rate constants, followed by the energies of the states (E14 is states 1 and 4), and finally the fission percent is listed. More detail on every simulation is in the text.

Figure TH TL TD1 TD2 KTT KL, K R E14 E23 E58 E67 E90 Fission % 16 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0 0 0 11.0 17 0.027 0.027 0.027 0.027 0.0003 0.0005 0 0 0 0 0 7.5 17 0.027 0.027 0.027 0.027 0.0005 0.0003 0 0 0 0 0 15.2 18 0.027 0.027 0.027 0.027 0.0027 0 0 0 0 0 0 25.0 18 0.027 0.027 0.027 0.027 0.0027 0 N/A 0 0 0 0 50.0 18 0.027 0.027 0.027 0.027 0.0027 0 N/A, 0 N/A N/A, 0 N/A N/A, 0 100.0 19 0.054 0.054 0.054 0.054 0.0003 0.0003 0 0 0 0 0 11.0 20 0.027 0.054 0.027 0.027 0.0003 0.0003 0 0 0 0 0 19.0 20 0.027 0.027 0.027 0.054 0.0003 0.0003 0 0 0 0 0 22.5 20 0.027 0.054 0.027 0.054 0.0003 0.0003 0 0 0 0 0 28.3 21 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0.081 0.081 0 11.0 21 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0.27 0.27 0 11.0 22 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0 0 0.027 11.0 22 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0.081 0.081 0.027 7.5 22 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0.27 0.27 0.027 2.4 23 0.027 0.027 0.027 0.027 0.0003 0.0003 0.027 0 0 0 0 15.3 23 0.027 0.027 0.027 0.027 0.0003 0.0003 0 0 0 0.027 0 16.0 23 0.027 0.027 0.027 0.027 0.0003 0.0003 0.027 0 0 0.027 0 20.8

3.3.2.2 Effect of Decay Rates on Singlet Fission

One of the easiest ways to increase or decrease fission yield within this model is to increase or decrease the ratio of the triplet pair and singlet decay constants, KTT , KL, and KR.

Using the parameters outlined above, if KTT is doubled, then the fission yield increases to 15%, while if KL and KR are instead doubled, the fission yield drops to 8% (Figure 17). In both simulations the relative populations of each state are the same, as they are controlled by the matrix elements and energy levels of the states. The fission yield changes because a greater or lesser percent of the population in a given state exits per time period when the decay constants are changed. If all three decay matrix elements are increased to 2.7x10 -3 eV, the fission yield drops to 8%, because the combination of this short lifetime and the fact that all population begins in an S 1 state leads to a larger percent of singlet decay. If the decay elements are all lowered to

2.7x10 -5 eV, however, the fission yield stays at 11%. As long as the decay rate is significantly 69 slower than the rate of electron transfer, changing all of the decay constants together does not affect fission yield. Physically, this means that limiting nonradiative decay of singlets, singlet injection into an electrode, and fluorescence will be important for achieving high fission yields in a dye sensitized solar cell. It is worth noting that it is possible to increase KTT and have fission yield decrease. The ideal triplet pair decay matrix element would match the rate at which population is being fed into the TT states. In the isoenergetic example with all matrix elements equal, the ideal decay rate constant equals the matrix elements. A higher KTT would actually decrease fission yield. This is not true for KL and KR, which always lead to increased singlet decay when increased.

70 This difference exists because the population begins in S 1 – if we artificially had all population begin in the TT states, the opposite would be true.

One of the most surprising and important results of this model is that if both KL and KR are set to zero, the fission yield is only 25% (Figure 18). The simulation begins as one would expect, with population oscillating between S 1, CT and TT states, but as population leaves the system via KTT , the coherence between the CT and TT states disappears, and population stops entering states 9 and 10. The remaining 75% of the initial excitation is trapped oscillating between the S 1 and CT states. This is an example of multiple pathways creating quantum interference in the system. To confirm that this is a result of interference between multiple pathways, and to examine how the myriad pathways create this parasitic (to fission) interference we ran several simulations. If we reduce the system to one S 1 state, one CT state, and one TT state, with KL = KR = 0 and KTT ≠ 0, we find that the entire population of the system eventually leaves by triplet-triplet decay. (Figure 18) If we remove only states 1 and 4 (S 1 localized on the left chromophore) and electronic matrix elements that lead to and from them, and start with all population in states 2 and 3 the ultimate fission yield is 50%. If we remove only states 5 and 8

(the charge transfer state with the anion on the left side) in a similar manner, the fission yield is

50%. Finally, if we remove only one of the two TT states (state 9) the fission yield is 25%.

From these simulations we learn that there is a destructive interference created by the left vs. right localized singlet states, and interference from the left vs. right charge transfer states. The overall fission yield can be increased by breaking the symmetry of this system in either of both levels. While instructive, none of these simulations provide a realistic solution to the problem of destructive interference in real molecules, as we can not simply ignore certain states. One can imagine, however, that a similar effect could be realized by making one set of localized S 1 or CT 71 states inaccessible, either by manipulating certain matrix elements or energy levels. As we explore the parameter space of system further we will search for methods to circumvent this problem in order to realize efficient singlet fission yield in molecular systems.

3.3.2.3 Effect of Matrix Elements on Singlet Fission

Previous research has examined the strength of the electronic coupling between two chromophores in a CCP, since this coupling partly determines the rate of electron and energy transfer between the two halves. We examine how fission yield changes in a simple system

(Sim1, five isoenergetic states, slow decay rates, matrix elements = 0.027 eV unless mentioned) when the matrix elements are altered. Increasing every matrix element to 0.054 eV doubles the rate of oscillations relative to Sim1, but the minimum and maximum populations for each state remain unchanged, and the overall fission yield is also unchanged. (Figure 19) This result is somewhat surprising, because a first guess would be that maximizing the electronic matrix element would be critical. In this example, the decay rates are far slower than the electron transfer rates, so the dynamic equilibrium between all states is established much quicker than decay regardless of the coupling in this regime. The fission yield is determined by the relative average populations of the S 1 and TT states and the ratio of the singlet and triplet pair decay rates. In both cases the average populations on S 1 and TT states are the same, and the only difference is the rate at which population goes back and forth. The absolute magnitude of the electronic matrix elements is only important if there is a competing decay pathway that occurs on the same timescale as electron transfer. If decay were very quick (< 100 fs), significant 72 population could decay from S 1 before any population reached TT. In this case large electronic matrix elements would be important.

Figure 19 Doubling all of the matrix elements in an isoenergetic system with slow decay rates does not affect fission yield, although it does increase the frequency of population oscillation 73

When only some of the electronic matrix elements are altered, however, the fission yield

-4 changes even when the decay rate is left the same (KL = KR = KTT = 2.7x10 eV). If either the TH or the TL matrix elements are doubled, but the other elements are unchanged from Sim1, the fission yield increases to ~19%. (Figure 20) The population of states over time becomes much more complicated than in Sim1, with longer oscillatory periods than when TH and TL were the same. This increase in fission yield occurs because the destructive interference has been reduced, and some of the pathways ( TH vs. TL) from S 1 states to CT states are oscillating on a different timescale. In a similar vein, if all matrix elements are the same, except for either TD1 or

TD2 , then the yield jumps to 23%. One sign that these enhanced yields are due to symmetry breaking is that if KL = KR = 0, we see a fission yield of 50% if just one matrix element is doubled, not 25% as when all are equal. Additionally, the fission yield increases as the ratio of

TH to TL increases up to 2, but if the ratio of the matrix elements increases further the fission yield drops again, down to 15% when TH = 3 TL. If both one of TH or TL and one of TD1 or TD2 are 74 doubled while the other two remain the same, the fission yield jumps to 27% (and if KL = KR in this scenario, 100% fission yield is realized). In contrast to these beneficial modifications, if TH and TL are doubled but TD1 or TD2 are left unchanged, the fission yield drops to 5%. This is easy to understand, as the oscillations from S 1 to CT states have sped up owing to the increased matrix elements, but the CT to TT transitions have not. These two transitions got out of phase resulting in more population in S 1 and CT states and less in TT states compared to the case when all matrix elements are equal.

In terms of physical meaning, these examples provide several possible strategies for constructing an ideal CCP. Fission yield is increased when TH and TL are not equal. This will be the case when the CCP has more coupling (overlap) between the two chromophores in either the

HOMO or LUMO. Previous molecules that displayed this tendency were BQD-type molecules, as well as CCPs of DPIBF that had heteroatom linkers. Fission yield is also increased when the two diagonal electron transfer elements are not equal. To realize this, the CCP should be designed such that one HOMO1-LUMO2 overlap is better than the other HOMO2-LUMO1 overlap. This could be done either by connecting two identical chromophores to each other in different locations, or by binding two different chromophores together to make a hetero-CCP.

3.3.2.4 Effect of Non-degenerate State Energies on Singlet Fission

Until now we have only discussed systems where every state is isoenergetic. This has let us study some of the behavior of our ten state coherent system, but it is not realistic to think this would be the case for a real molecular system. We now examine the effects of changing the energy levels of various states using similar parameters to previous simulations, with all matrix elements set to 0.027 eV, all decay rate constants equal 2.7x10 -4 eV. When the energies of both of the CT states are increased above that of the S 1 and TT states, there is no change in percent fission for small CT energies.(Figure 21) As the energy of the CT states moves further away from the S 1 and TT states there are two noticeable differences in the population of various states over time. The CT states gain less population as the energy gap increases, and the major period of oscillation for all three types of states increases. This is similar to coherent electron transfer 76 between two states of different energy – as the energy gap increases, the period of oscillation increases, but the amount of population that is transferred decreases. Importantly, however, the relative populations of the S 1 and TT states are unaffected by the barrier. Given isoenergetic S 1 and TT states, CT barrier height only affects fission yield if the barrier height and singlet decay rate are such that all population leaves the system before one full period of S 1 to TT oscillations.

At a barrier of 0.27 eV, the singlet would have to decay completely (by fluorescence, or injection, for example) on the order of 90 fs in order to have fission yield affected by the charge transfer energy barrier. If the barrier were 1.5 eV the fission yield would decrease if the singlet exciton decayed in less than 750 fs. If both CT states are an equal energy below the S 1 and TT states instead of above them, the exact same behavior in population over time and overall fission yield is seen. In the regime of fast coherent electron transfer the energy difference between states (and matrix elements) determine the rate and magnitude of electron transfer, and the direction of energy change (uphill vs. downhill) is irrelevant.

If the S 1 and CT states are isoenergetic but the TT state is shifted in energy, the fission yield decreases, regardless of whether the product TT state is higher or lower in energy. From

11% fission yield in the isoenergetic case, the fission yields drops to 5% with a 0.081 eV offset, and < 1% with a 0.27 eV offset. (Figure 22) If the CT and TT states are isoenergetic, and the S 1 states are shifted instead, there is an identical drop in fission yield. If we combine an elevated

CT state energy with a G between the S 1 and TT states, the fission yield is worse than if there were the same G but no CT barrier. In this case, the height of the CT barrier does affect fission yield. For a G of 0.027 eV, the fission yield is 11% with no barrier, 8% with a 0.081 eV barrier, and 2% with a 0.27 eV barrier. The main reason the barrier affects fission yield in this case is because the two electron transfer steps oscillate with different periods. While CT states at 77

Figure 22 If the S 1 and TT states are not isoenergetic, the fission yield is decreased. If there is a ∆G, then the presence of a moderate CT barrier does hurt fission yield. (Left) ∆G =0.027 eV. (Center) ∆G =0.027 eV and CT=0.081 eV (Right) ∆G =0.027 and CT=0.27 eV

.081 eV and TT states at 0.027 eV lead to only 8% fission yield, if the CT barrier is put at -0.081 eV, the yield is 11%. When the barrier is positive, the energy difference between S 1 and CT is twice the gap between CT and TT states, so the oscillations are very out of phase, whereas a negative barrier puts the oscillations of the S 1 to CT electron transfer step fairly close to the CT to TT electron transfer step, providing a higher fission yield.

In a hetero-CCP, it would be possible for the left and right localized S 1 and charge transfer states to be different in energy. If one of two sets of localized S 1 (states1,4 vs. 2,3) or

CT (states 5,8 vs. 6,7) states is not equal in energy to the other, the fission yield is typically increased. If either the left (or right) localized S 1 (or CT) states are increased in energy by 0.027 eV, and all other parameters are unchanged, the fission yield increases from 11% to 16%. (Figure

23) If both one set of localized singlet states and one set of localized CT states are raised 0.027 eV the fission yield increases to 21%. Changing the initial population of the system will change the fission yield in any simulation where the two localized singlet states are not isoenergetic. If all of the population begins in the higher energy set of S 1 states in these examples, the yields are 78 11% and 15%. If all of the population begins in the S 1 states with lower energy (and isoenergetic with the TT states) the fission yields are 20 and 27%. In all of these cases the enhanced fission yield comes from symmetry breaking. In the unphysical case where KL = KR

=0, the fission yield is 50% if either the S 1 or CT states have broken symmetry, and 100% if both sets of states have broken symmetry.

Figure 23 Hetero-CCPs show promise because the different energy levels of the left and right localized S 1 and CT states lead to higher fission yields. (Left) States 1,4 are raised 0.027 eV above all others. (Center) States 5,8 are raised 0.027 eV. (Right) States 1,4,5,8 are all raised 0.027 eV.

From a molecular design point of view, these simulations suggest that hetero-CCPs may be attractive targets because the two sets of localized S 1 and CT states will have different energies. It is worth noting that unlike the case of the electronic matrix elements, even a very small breaking of the symmetry leads to a large increase in fission yield. Because of this, small differences in chromophores, such as electron donating or withdrawing groups, might suffice. 79

3.3.2.5 Lessons for Simulations

One of the major limitations of fission yield is the destructive interference that occurs when the system has high symmetry. This symmetry (and interference) is broken when left and right localized S 1 states and CT states have different energies, as is expected to be the case for a hetero-CCP. The symmetry can also be broken by having the matrix elements TH and TL uneven, and TD1 and TD2 uneven. Additionally, this symmetry could be broken in a real system by dephasing and a fluctuating environment. A second major factor in the fission yield is the relative decay rates of the TT and S 1 states. Because either half of a TT state can inject into an electrode first, while only the excited S 1 half of a S 1S0 state can inject, it might be beneficial to link one half of a CCP to the electrode more strongly than the other. If it is a hetero-CCP, by linking the higher energy localized S 1 to the electrode one might be able to retard singlet exciton injection into the electrode without adversely affecting triplet-triplet injection. The third major conclusion we can draw from these simulations is that coherent dynamics with no dephasing and zero temperature suggest that having the G of singlet fission as close to zero as possible will help fission yield. We now examine several real CCPs using this model to predict which of several possible synthetic targets is expected to perform best.

3.3.3 Time Evolution of Real Molecules

Table 8 shows the percent fission yield for each of the twelve CCPs found using coherent dynamics within this ten state scheme. All of these simulations were run with KL = KR

-7 -4 = 2.7x10 eV (690 ps lifetime), while KTT = 2.7x10 eV (690 fs lifetime). These parameters are 80 chosen to represent fluorescent decay of the localized singlet states, and a quick triplet pair injection into an electrode. All three of the DPIBF homo-CCPs display very poor singlet fission yield. The three DPIBF homo-CCPs all have very high energy charge transfer states, and very little population enters the CT states because of this, while the an-DPIBF chromophore has one

CT state very close in energy to one of the S 1 states, and as a result has a dramatically higher singlet fission yield.

Table 8 Fission Yield of Real CCPs with State Energies Provided (in eV) + - - + S1S0 S0S1 C A A C TT Fission % DPIBF 0 0 1.005 1.005 0.303 4 PENT 0 0 0.354 0.354 -0.240 24 POLY 0 0 0.595 0.595 -0.712 1 aaDPIBF 0 0 0.988 0.988 0.384 2 anDPIBF 0 -0.207 -0.144 1.823 0.286 42 nnDPIBF 0 0 0.898 0.898 0.397 1 aaPENT 0 0 0.345 0.345 -0.144 42 anPENT 0 -0.089 -0.506 1.116 -0.169 46 nnPENT 0 0 0.353 0.353 -0.105 42 aaPOLY 0 0 0.668 0.668 -0.222 7 anPOLY 0 -0.191 -0.484 1.674 -0.257 69 nnPOLY 0 0 0.712 0.712 -0.101 21

The most distinct trend in this data is that the CCPs with TT energies closest to their S 1 energies had the highest singlet fission yield. In order of increasing absolute value of free energy of fission, the chromophores are ranked (including only the low energy S 1 for hetero-CCPs because it dominates fission yield): anPOLY, anPENT, nnPOLY, nnPENT, anPENT, aaPOLY,

PENT, DPIBF, aaDPIBF, nnDPIBF, anDPIBF, and POLY. Only three of the CCPs in this list are not in the place we would expect if we were to approximate that the only thing that led to 81 high fission yield was a small absolute value free energy of fission (energy of TT-S0S1). aaPOLY and nnPOLY would both be expected to have a greater singlet fission yield based solely on their free energies of fission, however their CT energy states are roughly twice as far away from their S 1 and TT states as any of the more efficient CCPs. anDPIBF would be expected to be less efficient at singlet fission based solely on free energy of fission, but we note that it has a

CT level far closer to its S 1 energy level than any other chromophore. This means more population transfers from the S 1 to the CT state, which limits singlet decay, and helps produce greater population in the TT state.

Overall, having a free energy of fission close to zero was the most important predictor of high fission yield, and having a CT state close in energy to the S 1 and TT states was the second most important factor. This is what one would expect based on the model simulations.

All three hetero-CCPs performed better than their homo-CCP analogs, owing to a combination of having a TT states in better resonance with the S 1 states, and by nature of having a CT state closer in energy to the localized S 1 states.

3.4 Summary and Conclusion

In order for singlet fission to be efficient within the framework of coherent, purely electronic dynamics it is critical for the S 1 and TT states to be as close to isoenergetic as possible. If the TT state is not close in energy to the initial S 1 state energy, the time average population in the TT state is very low, leading to low fission efficiency. In addition to this constraint on the free energy of the reaction, it is essential for the CT states to lie relatively close to the S 1 and TT states energies in order for the population to be able to oscillate back and forth 82 from S 1 to TT states on a timescale comparable to or faster than excitation decay. The magnitude of the various electron transfer matrix elements is not nearly as important as the state energies, and thus CCP design should be done with energy in mind first. In most cases this will mean a relatively weak coupling is preferable in order to prevent fission from being highly endothermic, but some chromophores, such as twisted polyenes, might perform better with stronger coupling because of low IC triplet energies. The energy of the CT intermediate states can be controlled both by the ICs, where using biased chromophores provides a reduction in CT energy, or by controlling the Coulombic stabilization of a CCP CT state by moderating the distance between charged chromophores. We find that CCPs that combine these strategies to place as many energy levels as close together as possible are the ones that illustrate the highest singlet fission yields in the regime of coherent electron dynamics. 83

Chapter 4

Singlet Fission Yield and Dynamics In the Regime of Incoherent

Electron Transfer

84 4.1 Introduction

In chapter 3 we developed a mechanism for singlet fission based on a ten state scheme

(Figure 14). The mechanism involves two sequential electron transfer events, and we investigated how to optimize singlet fission yield in the case where the electron transfer steps were fast and coherent, with no molecular reorganization or dephasing from the environment. In a complex system such as a dye sensitized solar cell (DSSC), it is likely that molecular motions and environmental fluctuations could play a very large role in electron transfer effects. In this chapter we examine the same ten state scheme, but through the context of incoherent electron transfer events. We use Marcus theory to compute semiclassical rate constants and use a set of classical rate laws to observe the time evolution of the excitation. We explore how various molecular and environmental parameters affect fission yield for both imaginary and real molecular systems. Finally, we propose design parameters for an ideal molecular singlet fission system within the regime of incoherent electron transfer.

4.2 Methodology

In this chapter we compute the energy and electronic coupling for several CCPs in multiple geometries. These parameters are used to compute electron transfer rates within the framework of Marcus theory. The time evolution of the excitation is studied using a previously derived ten state system. We examine how singlet fission yield depends on the energy and coupling of various states within this framework, predict singlet fission yields for molecular

CCPs, and provide design rules for optimal molecular singlet fission. 85

4.2.1 Electronic Structure Computations

As in the previous chapter, we use a pair of non-interacting chromophores as our basis set for electronic structure computations. This basis matches well with our model, which has been formulated in a basis of a localized pair of chromophores, and allows us to make computations for ICs and combinatorially combine them to form many CCPs without having to perform a large number of expensive computations on large molecules. We also focus on the same basic ICs and

CCP scaffolds in order to compare the results between the two dynamic regimes (fast, coherent, purely electronic dynamics vs. vibrationally mediated, incoherent electron transfer).

Each IC is first optimized in its ground state, first triplet state, cation, and anion geometries. We calculate the first excited singlet at the same geometry as the ground state singlet because of difficulties in accurately computing gradients for excited state geometry optimizations.62 While this is likely to slightly overestimate the energy of the relaxed singlet state, we do not expect a large reorganization energy for the molecules considered here. 81 The energy of each IC is computed for every electronic state in its optimized geometry, as well as in the optimized geometry of every state related by a one electron transfer. In total, we computed energies for S 1 at the S 0 optimized geometry (S 1@S 0), S 1@A, S 1@C, S 0@S 0, S 0@A, S 0@C,

A@S 0, A@A, A@T, C@S 0, C@C, C@T, T@T, T@A, and T@C for each IC. To find the energy for a given CCP in a given state and geometry, we simply add the energy of the two halves. As discussed in the previous chapter, summing the cation and anion energies in this manner will not result in an accurate charge transfer state energy, so we apply a correction factor obtained from CDFT. 86 All electronic coupling elements are computed exactly as in chapter 3. TH and TL are computed for CCPs by the energy splitting method, and used to find a proportionality constant, k, between the electronic matrix element and the orbital overlap. This proportionality constant is used to determine the diagonal electron transfer values TD1 and TD2 which represent electron transfer from the LUMO of one chromophore to the HOMO of the other and which can not be computed according to the energy splitting method. Based on a tight-binding argument, we approximate that the molecular orbitals do not change with occupancy, and thus the electronic matrix elements governing a given transition are independent of electronic state.

QChem 3.0 62 or NWChem 5.0. 63, 64 We used the CDFT module written by Wu and Van Voorhis for implementation in NWChem 5.0. 73-75

4.2.2 Time Evolution of the Excitation

Marcus theory provides a framework for computing nonadiabatic electron transfer rates between various states in our model while taking geometric relaxation into account. In the semiclassical limit, which is valid for high temperature electron transfer, the charge transfer rate is expressed as 60, 82, 83

87 o 2 2π 2 1   k= ( )| V |( )exp −( G + λ ) , (4.1) h 4λkB T  4πλ kB T  

where Go is the Gibbs free energy of the reaction, V is the electronic coupling between the initial and final states, and λ is the sum of the molecular relaxation and the solvent reorganization energy. The semiclassical approximation neglects tunneling, which can affect the electron transfer rate, most notably at low temperatures. To include the effects of tunneling on the electron transfer rate we use the Marcus-Levich-Jortner formalism, which includes a single quantum mechanical vibrational mode representing internal reorganization, and is expressed as 84,

ν ' o 2 2π 2 1 S −(G +λ + ν 'h ω )  k=( ) | V | ( )∑ exp( − S ) exp s (4.2) h 4λsk B T  4πλ kB T ν ' ν '!  

where S is the Huang-Rhys factor and is directly related to the internal reorganization

h energy ( S = λi / ω , where λi is computed to be typically 0.1-0.3 eV). We use a single quantum vibrational mode with an energy hω of 1500 cm -1 (0.19 eV), close to the skeletal stretching vibration energies of several aromatic systems. 86 This expression assumes no thermal vibrational excitation, which is reasonable for the temperature regimes of interest.

These electron transfer rates are then used to study the dynamics of singlet fission using the previously derived ten state system composed of four localized singlet states, four localized 88 charge transfer states, and two triplet pair states. The excitation starts evenly spread across the four localized singlet states, and we study the evolution of the excitation according to simple rate laws using a finite-difference of the time propagator, and finishing when 99.9+% of the population has left the system via either singlet or triplet pair decay. Electron transfer rates are used as computed from Marcus theory, while the decay parameters are varied to represent physically meaningful timescales for fluorescence, internal conversion, and charge injection.

Triplet pair decay is assumed to be injection into an electrode, and the percent of the initial excitation that leaves the system via triplet pair decay is taken to be equal to the singlet fission yield. The complete list of classical rate laws used for the evolution of our system is as follows:

dS D =KPP( ++ )( KP + P ) (4.3) dt L14 R 23

dT D =K( P + P ) (4.4) dt TT 9 0

dP 1 =KPKP + −++( K K KP ) (4.5) dt 515 616L 15 161

dP 2 =KPKP + −++( K K KP ) (4.6) dt 525 626R 25 262

dP 3 =KPKP + −++( K K KP ) (4.7) dt 737 838R 37 383

dP 4 =KPKP + −++( K K KP ) (4.8) dt 747 848L 47 484

dP 5 =KPKPKP + + −++( K K KP ) (4.9) dt 151 252 050 51 52 50 5 89 dP 6 =KPKPKP + + −++( K K KP ) (4.10) dt 161 262 969 61 62 69 6

dP 7 =KPKPKP + + −++( K K KP ) (4.11) dt 373 474 070 73 74 70 7

dP 8 =KPKPKP + + −++( K K KP ) (4.12) dt 383 489 989 83 84 89 8

dP 9 =KPKP + −( K ++ K KP ) (4.13) dt 696 898TT 96 989

dP 0 =KPKP + −( K ++ K KP ) (4.14) dt 505 707TT 05 070

th where PN is the population in the N state according to the ten state model ( P0 is state ten), KXY being the calculated rate constant from state X to Y, KL, KR, KTT are the left and right localized singlet decay rates and the triplet pair decay rate, and SD and TD are the amount of population that has left the system via either singlet or triplet pair decay.

4.3 Results and Discussion

4.3.1 Electronic Structure Computations

As in chapter 3, we examine three basic ICs, DPIBF, pentacene, and a polyene, and we focus on one twisted CCP for each. Table 9 shows the gas phase energies for 12 ICs based on these chromophores with amino, hydroxyl, and nitro functionalized variants. For each IC the energy is listed for each electronic state in its optimized geometry as well as in all connected electronic states, both vertically and adiabatically. Each IC is normalized such that the ground 90 state singlet at the optimized ground state geometry (S 0@S 0) energy is zero. The cation and anion energies are given as isolated energies without the computed CDFT stabilizations added, because the CDFT stabilizations are based on CCP geometries, and cannot be assigned to individual ICs. All three unfunctionalized chromophores have E≥ 2 E when including S1 T 1 relaxation, indicating that singlet fission should be exothermic. The DPIBF chromophore is least exothermic, followed by pentacene, with singlet fission being most exothermic for the polyene chromophore. Simply adding the cation and anion energies for all three chromophores yields a charge transfer state that is significantly higher in energy than the localized singlet and triplet pair states, but this energy will be lowered when we consider the mutual stabilization of the two charged ICs in CCP. (For reference, the Coulombic stabilization from CDFT for the polyene

CCP we consider is the smallest at ~1.3 eV, with the DPIBF correction close to 1.8 eV, and the pentacene CCP correction the largest at 2.3 eV.) Each of the three base chromophores is generally affected in the same way by the three functional groups we consider. The nitro group stabilizes the anion energies and destabilizes the cation energies, while the hydroxyl and amino groups destabilize the anions and stabilize the cation, as one would expect from their relative strengths as electron-withdrawing and -donating groups. None of these three groups has a large effect on the triplet state energy, and all three lower the first excited singlet state, with the hydroxyl having the smallest effect and the nitro group the largest. 91 Table 9 Energy of 16 ICs at every Relevant Electronic State and Geometry. Molecules are labeled by IC type (PO=polyene, DP=DPIBF, PE=pentacene) and functional group (a=amino, h=hydroxy, n=nitro). All values are in eV

State PO DP PE aPO hPO nPO aDP hDP nDP aPE hPE nPE S0@S00 0 0 0 0 0 0 0 0 0 0 0 S0@C 0.18 0.08 0.05 0.17 0.17 0.13 0.14 0.10 0.07 0.06 0.06 0.05 S0@A 0.19 0.16 0.07 0.19 0.21 0.22 0.17 0.17 0.17 0.09 0.08 0.11 A@A -1.21 -0.48 -1.13 -0.83 -1.06 -1.99 -0.27 -0.36 -1.37 -0.92 -1.10 -1.78 A@S1 -1.03 -0.33 -1.08 -0.67 -0.88 -1.82 -0.09 -0.21 -1.23 -0.85 -1.03 -1.70 A@S0 -1.03 -0.33 -1.08 -0.67 -0.88 -1.82 -0.09 -0.21 -1.23 -0.85 -1.03 -1.70 A@T -1.08 -0.41 -1.07 -0.69 -0.90 -1.86 -0.15 -0.28 -1.26 -0.84 -1.02 -1.68 C@C 5.87 6.14 5.90 5.24 5.59 6.29 5.65 5.94 6.58 5.52 5.76 6.30 C@S1 5.97 6.22 5.94 5.41 5.76 6.42 5.83 6.05 6.67 5.59 5.83 6.36 C@S0 5.97 6.22 5.94 5.41 5.76 6.42 5.83 6.05 6.67 5.59 5.83 6.36 C@T 5.99 6.26 5.98 5.40 5.75 6.43 5.83 6.07 6.72 5.63 5.86 6.39 T@T 0.85 1.44 0.78 0.84 0.86 0.81 1.40 1.42 1.36 0.77 0.76 0.76 T@A 0.99 1.52 0.85 1.00 1.02 0.97 1.51 1.53 1.52 0.86 0.85 0.89 T@C 1.03 1.55 0.85 0.99 1.01 0.96 1.55 1.54 1.49 0.87 0.85 0.83 S1@S1 2.88 3.00 2.09 2.68 2.77 2.54 2.76 2.89 2.75 1.97 2.03 1.91 S1@C 2.99 3.08 2.14 2.85 2.94 2.67 2.94 3.01 2.84 2.04 2.09 1.97 S1@A 3.01 3.06 2.13 2.83 2.94 2.70 2.96 3.00 2.89 2.03 2.09 1.94

The ten states in our dynamic model are all combinations of two IC states. The change in energy and the reorganization energy for each electron transfer in the ten state model is based on the sum of the two IC halves. For example, an electron transfer from state 1 to state 5 in our dynamic model would involve an electron transfer from (S 1@S 1 + S 0@S 0) to (A@A + C@C), with reorganization energy computed based on (S 1@A +S 0@C) and (A@S 1 + C@S 0). We switch to a more compact notation where each electron state and geometry are listed by the numbering scheme of the dynamic model, so the transition from state 1 to state 5 is written 1@1 to 5@5, with reorganization energy computed based on 1@5 and 5@1. Table 10 shows the energies of sixteen selected CCPs listed in terms of the energy of each total CCP electronic state in every relevant geometry. 92

Table 10 Energies for 16 CCPs for each state and geometry. The CDFT correction energy for CT states has been applied. All energies are given in eV relative to the lowest S 1S0 state. Each state is listed as a State@Geometry. The CCP scaffolds are abbreviated PO=polyene, DP=DPIBF, PE=pentacene, with variants aa=diamino, hh=dihydroxy, nn=dinitro, an=aminonitro. State PO DP PE aaPO hhPO nnPO anPO aaDP hhDP nnDP anDP aaPE hhPE nnPE anPE 1@1 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1@5 0.30 0.13 0.08 0.32 0.34 0.28 0.28 0.34 0.21 0.21 0.27 0.12 0.12 0.08 0.11 1@6 0.30 0.24 0.12 0.36 0.38 0.35 0.39 0.35 0.29 0.26 0.35 0.16 0.15 0.17 0.18 2@2 0.00 0.00 0.00 0.00 0.00 0.00 -0.14 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.05 2@5 0.30 0.24 0.12 0.36 0.38 0.35 0.17 0.35 0.29 0.26 0.25 0.16 0.15 0.17 0.09 2@6 0.30 0.13 0.08 0.32 0.34 0.28 0.18 0.34 0.21 0.21 0.27 0.12 0.12 0.08 0.04 5@5 0.47 0.93 0.36 0.42 0.46 0.45 1.47 0.90 0.95 0.74 1.83 0.33 0.33 0.30 1.11 5@1 0.75 1.17 0.47 0.75 0.80 0.75 1.76 1.25 1.22 0.96 2.09 0.46 0.47 0.44 1.23 5@2 0.75 1.17 0.47 0.75 0.80 0.75 1.76 1.25 1.22 0.96 2.09 0.46 0.47 0.44 1.23 5@10 0.72 1.12 0.52 0.72 0.77 0.71 1.61 1.19 1.17 0.98 1.97 0.51 0.51 0.49 1.20 6@6 0.47 0.93 0.36 0.42 0.46 0.45 -0.74 0.90 0.95 0.74 -0.20 0.33 0.33 0.30 -0.54 6@1 0.75 1.17 0.47 0.75 0.80 0.75 -0.40 1.25 1.22 0.96 0.12 0.46 0.47 0.44 -0.39 6@2 0.75 1.17 0.47 0.75 0.80 0.75 -0.40 1.25 1.22 0.96 0.12 0.46 0.47 0.44 -0.39 6@9 0.72 1.12 0.52 0.72 0.77 0.71 -0.46 1.19 1.17 0.98 0.08 0.51 0.51 0.49 -0.32 9@9 -1.17 -0.13 -0.53 -1.00 -1.05 -0.92 -1.03 0.04 -0.04 -0.02 0.01 -0.42 -0.50 -0.40 -0.44 9@6 -0.85 0.07 -0.38 -0.69 -0.74 -0.60 -0.72 0.30 0.18 0.26 0.31 -0.24 -0.32 -0.19 -0.20 9@8 -0.85 0.07 -0.38 -0.69 -0.74 -0.60 -0.72 0.30 0.18 0.26 0.25 -0.24 -0.32 -0.19 -0.27

The three base CCPs vary from roughly zero change in free energy (DPIBF) from the localized singlet (1@1) to the triplet pair state(9@9) to more than one eV exothermic for the polyene CCP. DPIBF has the highest energy charge transfer states (5@5 and 6@6), while pentacene has the lowest energy charge transfer intermediate. The homo-CCPs with amino, hydroxy and nitro groups are all slightly less exothermic than the corresponding bare CCPs, but all have slightly lower charge transfer state energies. The hydroxy groups have the smallest effect on lowering the charge transfer state, while the nitro groups have the most dramatic effect.

The aminonitro-CCPs (an-CCPs) have one very low energy CT state when the amino- chromophore is positively charged and the nitro-chromophore is negatively charged, and one very high energy charge transfer state when the amino-chromophore is negatively charged and 93 the nitro-chromophore is positively charged. The hetero-CCPs also have non-degenerate left and right localized S 1 states, but the energy difference is very small in all cases studied.

4.3.2 Time Evolution of Model Systems

4.3.2.1 Basic System

In order to understand how each molecular parameter affects the dynamics and fission yield in this ten state Marcus Theory model we run several simulations using fictitious but molecularly relevant parameters. These hypothetical simulations decouple some of the various degrees of freedom in this system and illustrate how each variable affects fission yield. After examining the effects of varying each parameter on its own, we investigate cooperative behavior among several parameters and use these simulations to predict “ideal” properties for singlet fission in a molecular system.

We first examine a simple set of parameters, with the energy of every state equal and independent of molecular geometry (no inner reorganization energy) and with every matrix element equal (set to 0.027 eV). Both singlet decay constants KL and KR as well as the triplet

-5 decay constant KTT are set equal to 2.7x10 eV (These K values correspond to a lifetime of 2.42 ps). We run the simulation at 300 K, assume the solvent reorganization energy is 0.1 eV for every electron transfer, and that the excitation begins evenly split between the four S 1 states. (If the initial state is a different linear combination of the singlet states, or all one state, theresults are the same as this case.) These default parameters are used throughout these simulations unless other parameters are explicitly mentioned. In this example, (Figure 24) the population very 94 quickly transfers from the S 1 states to the CT states and then the TT states. Because every state is isoenergetic, the system reaches a steady state where each of the ten states has an equal population. Because there are twice as many S 1 as TT states, the sum of the S 1 state populations is double the sum of the TT state populations. With KL = KR = KTT , once equilibrium is reached, twice as much population leaves the system by singlet decay as by triplet pair decay.

Before equilibrium is reached there is more singlet decay than triplet pair decay owing to the initial state populations, so the actual yield is less than this. Equilibrium is reached very quickly

(~ 0.1 ps), however, so the fission yield is very close to one third.

If the temperature is lowered to 200 K there is a very small decrease in fission yield because the time to reach equilibrium is increased due to decreased electron transfer rates. With all states isoenergetic, the temperature dependence is small, coming only from the prefactor and not the exponential term in the Marcus equation. If the solvent reorganization energy is increased to 0.2 eV under these conditions there is a miniscule decrease in fission yield, also because there is a slower approach to equilibrium. With the current decay constants, matrix elements and state energies, the equilibrium populations are reached before even one percent of the population has left the system. If the singlet decay rates are increased tenfold while the triplet pair decay rate is held constant, just over 95% of the excitation leaves via singlet decay, while if instead we increase the triplet pair decay rate tenfold and not the singlet decay rate, over

80% of the excitation leaves via triplet pair decay. Clearly, within this simple system the ratios of the decay rates are a large factor in determining fission yield. Several other simulations are compared to this basic case in Table 11 and in the text below.

Figure TH TL TD1 TD2 KTT KL, K R E14 E23 E58 E67 E90 Fission % 24 & 25 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0 0 32.8 25 0.054 0.054 0.027 0.027 0.00027 0.00027 0 0 0 0 0 32.9 25 0.054 0.054 0.054 0.054 0.00027 0.00027 0 0 0 0 0 33.1 26 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.054 0.054 0 31.7 26 0.027 0.027 0.027 0.027 0 0 0 0 0.054 0.054 0 N/A 26 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.11 0.11 0 24.6 27 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0 0.027 14.7 27 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0 0 -0.027 57.7 27 0.027 0.027 0.027 0.027 0.00027 0.00027 0 0 0.108 0.108 -0.027 35.0 28 0.027 0.027 0.027 0.027 0.00027 0.00027 0.081 0 0.027 0.135 -0.027 69.7 28 0.027 0.027 0.027 0.027 0.00027 0.00027, 0 0.081 0 0.027 0.135 -0.027 97.6

96 4.3.2.2 Effect of Changing Matrix Elements on Singlet Fission

One of the sets of parameters we discuss in our model is the electronic coupling matrix elements that couple the two ICs in a CCP. If we vary the matrix parameters in our model, keeping the parameters from above, we can see that this affects the rate at which the populations of given states equilibrate. (Figure 25) When all of the coupling matrix elements are doubled relative to the base case, and we can see that the ten states reach equilibrium four times as quickly. If only TH and TL are doubled, but not TD1 and TD2 , we see that the CT states equilibrate with S 1 states earlier than in simulation 1, but the TT states do not equilibrate sooner. The ratio of populations at equilibriums is not changed when the electronic coupling matrix elements are changed, whether they are all changed equally, or to differing degrees. Both of these observations are as expected from Marcus Theory. The electron transfer rate (forward and backward) is proportional to the matrix element squared, and since the equilibrium constant between two states is proportional to the forward rate over the backward rate, and the coupling affects forward and backward electron transfer rates equally, changing the values for coupling has no effect on the equilibrium constant.

4.3.2.3 Non-Isoenergetic State Schemes

We have briefly examined how fission yield depends on temperature, decay rates, solvent reorganization energy, and electronic coupling in the trivial case where all states are isoenergetic, and there is no internal geometry relaxation. We now examine how changing the energy of the various states affects fission yield. Based on Marcus Theory for a two-state system, we know that the energetic difference between two states will affect the forward and reverse electron transfer rates as:

2 −(G + (λo + λ i ))  k f ∝ exp   (4.15) 4(λo+ λ i ) k B T 

2 −−(G + (λo + λ i ))  kr ∝ exp   (4.16) 4(λo+ λ i ) k B T  98

where G is the free energy of the electron transfer, λi is the internal reorganization

energy, λo is the solvent reorganization energy, and kB T is Boltzmann’s constant times temperature. If we set the singlet and triplet pair decay rates KL = KR = KTT = 0 and simply observe the equilibrium populations in our system, we can see how the relative energies of states affect the populations. If we assume no internal reorganization, but 0.1 eV solvent reorganization energy and a CT state energy of 0.027 eV, each S 1 and TT state has roughly three times the population as each CT state, while raising the CT state energy to 0.054 eV yields a population ratio of roughly eight times as much population per S1 or TT state as per CT state.

(Figure 26) If we apply the previous decay rates, we see the same ratios, and the fission yield is almost unchanged from when there was no CT barrier. This is because the ratio of S 1 to TT populations at equilibrium is unchanged, and these small CT barriers do not delay reaching equilibrium for very long. If the CT state energy is 0.27 eV, however, the fission yield is roughly zero because the high CT energy prevents population from reaching the TT state before the lifetime decay of the S 1 state. With an intermediate CT energy of 0.11 eV the fission yield is

24%, down from 33% because before the CT and TT states reach equilibrium 50% of the excitation has already decayed, ~40% by singlet decay and ~10% by triplet-triplet decay.

If the CT state is lower in energy than the S 1 and TT states (which are isoenergetic) then each CT state will hold more population than each S1 and TT state. Regardless of how low in energy the CT states are, the fission yield is 33%. This is because there is no decay from the CT state in our model, and the S 1 and TT states do drain population out of the system. An improved model could consider decay from a CT state, but because the real molecules we consider tend to have high energy CT states, this is not likely to cause a significant error in our simulations. 99

If the TT states are higher in energy than the S 1 states, the equilibrium population of the

TT states is lower than that of the S 1 states, which depresses fission yield. If the TT states are

0.027 eV higher (with S 1 = CT = 0) the fission yield drops from 33% at isoenergetic, to 15%. If the TT states are instead lower in energy than the S 1 states by 0.027 eV, the fission yield is 58% because the equilibrium population of the TT states is higher than that of the S 1 states. (Figure

27) If we introduce a CT barrier of 0.11 eV to the scenario where the TT state is at -0.027 eV, the fission yield drops to 35%. The equilibrium population ratio is the same as before but much of the singlet decays before the equilibrium is reached due to the CT barrier.

100

It is also interesting to consider molecules where the left and right localized S 1 states and the left and right charge transfer states are not degenerate, as would be expected for hetero-

CCPs. If one S 1 state is at 0.027 eV and the other is at -0.027 eV, with all CT and TT states at 0, and with all states having 0.027 eV internal reorganization energy, the fission yield is 23%, compared to 33% when both S 1 states are at 0. As one would expect, at equilibrium the low energy S 1 state gains more population than the high energy S1 loses relative to the isoenergetic case. Because of this, more population rests in the S 1 states, and the fission yield is decreased.

Physically, this means that in the case of a hetero-CCP, more of the excitation will localize on the side with the lower energy singlet. If the two types of CT states are not degenerate, the overall behavior of the system is dominated by whichever one is lower in energy. For example, if all S 1 and TT states are set to zero energy, and the two CT states lie at 0.027 and 0.081 eV, the fission yield is 32%. When both CT states are 0.027 eV, the yield is 33%, only a modest change. 101 Even more impressive is that if one CT state stays at 0.027 eV and the other jumps up to 2.7 eV, the fission yield only drops to 31%. If both CT states are at 2.7 eV there is zero fission yield, because there are no good paths from S 1 to TT, but apparently one good path is almost as good as two. Physically, this means that hetero-CCPs should be promising systems if they lower one CT state at the cost of raising the other.

We also examine a scenario in which each of the five types of states has a different energy, a model hetero-CCP. When S 1S0 is at 0 eV, S 0S1 at 0.081, the two types of CT states at

0.027 and 0.135, and TT at -0.027 eV (with all states being .027 eV higher in their non- optimized geometry) the fission yield is 70%. (Figure 28) If we set the singlet decay rate from the lower energy singlet to be 2.7x10 -7 eV instead of 2.7x10 -5 eV, the fission yield jumps to 98%.

If we start all of the population in the low energy S 1 state, instead of evenly spread in both, the fission yield increases slightly. One could imagine that if charge injection were much quicker than other singlet decay rates, by tethering one side of the chromophore to the electrode and not the other, the singlet decay rates could be made unequal. If the high energy localized singlet were tethered, there would be little singlet injection because little population would reside in the high energy singlet. The triplet injection rate would not be affected because both sides of the molecule carry a local triplet, and once the first triplet injects, the second will most likely live long enough to also inject because of a lack of good decay routes.

Finally, we examine how temperature affects singlet fission in a complicated set of energy states as opposed to the initial isoenergetic system. If the temperature is decreased to 200

K or increased to 400 K, the fission yield changes from 70% to 76% or 64% respectively, if we assume that the electronic couplings are not temperature dependent. Because this simulation is overall exothermic, increased temperature actually hurts fission yield. If we look instead at an 102 endothermic set of energies (S 1 = CT = 0, and TT = 0.027 eV) fission yield varies from 9% to

15% to 18% at 200, 300, and 400 K respectively. In general, we see that fission yield is increased at higher temperatures if the process is endothermic, and decreased at higher temperatures if fission is exothermic. The exception to this rule would be if fission were exothermic, but a high energy CT state prevents the population from reaching its equilibrium.

For example if S 1 = 0, CT = 0.189 eV, and TT = -0.027 eV the fission yield is 0, 1, and 8% at

200, 300 and 400 K. Because the equilibrium population ratios for TT and S 1 states are never reached, it doesn’t matter that higher temperature would lead to more S 1 population relative to

TT. The main effect of temperature in this simulation is to help cross the CT energy barrier. To further complicate the temperature dependence of fission, we point out that for certain molecules

(for example the ones we focus on in this study) the electronic couplings will be temperature dependent because of vibrational motion about a key torsional degree of freedom, and the fission yield is expected to change more dramatically with temperature.

Figure 28 (Left) In a model hetero-CCP with the S 1 states at 0 and 0.081 eV, the CT states at -0.027 and 0.135 eV and the TT states at -0.027 eV the fission yield is very high. (Right) If the low energy localized singlet cannot inject into the electrode, the fission yield jumps to nearly 100%. 103 4.3.2.4 Conclusions from Model Simulations

In every simulation the population begins in the S1 states and moves towards equilibrium.

Once the equilibrium is reached, the percent fission is based on the ratio of the sum of all TT state populations times the triplet pair decay rate KTT , to the sum of all S 1 population times their decay parameters. (states 1 and 4 times KL plus states 2 and 3 times KR). The energy levels of the states determines the equilibrium populations, so low energy TT states will encourage singlet fission, while a high energy TT state will encourage singlet decay.

In order to maximize fission yield, the amount of population that leaves the system by singlet decay before equilibrium is reached must be minimized. The time taken to reach equilibrium depends on the coupling elements, and on the energy of the CT states. Larger electronic coupling matrix elements leads to quicker equilibration, while a high energy CT state, will slow equilibration.

4.3.3 Time Evolution of Real Molecules

-5 All calculations on real molecular systems were run first with KL = KR = KTT = 2.7x10 eV, corresponding to a lifetime of 2.42 ps, much faster than typical fluorescence, but on the timescale of electron injection, and the fastest of intersystem crossing and internal conversion

events. λo is set to 0.1 eV. When we consider this fast decay scenario, only the three an-CCPs have nonzero fission yields, as the other CCPs have CT states that are simply too high in energy to populate on this timescale. The an-CCPs all provide a CT state with lower energy than the localized singlet state, and all provide a lower energy TT state than the lowest localized singlet state energy. Because of this, population initially decays from the singlet state and 104 simultaneously transfers to the CT states. For anPOLY, the TT state is lower in energy than the CT state, and thus the population quickly funnels from the CT states to the TT states, where it undergoes triplet pair decay, resulting in a very high fission yield of 90%. (Figure 29) The anPENT and anDPIBF have their TT states lower in energy than the S 1 states, but higher in energy than the lowest CT states. This means that population pools in the CT states, and slowly drains out to the TT and S 1 states from where the excitation can leave the system. Both of these chromophores also have fission yields above 90% in our model. We point out that the fission yield for these two would drop precipitously, however, if there were any decay route from the charge transfer state, which we have not included in our model. If we add a simple decay route from the CT states in the same manner we allow population to decay from the S 1 and TT states, even with a CT state lifetime of 24.2 ps (ten times longer than the S 1 and TT lifetimes) the singlet fission yield for anPENT and anDPIBF drops to 10% and 0%, respectively, for these two chromophores, and the anPOLY chromophore is virtually unaffected, remaining at near 90% fission yield. The best of the homo-CCPs is nnPENT, which has a S 1 to CT electron transfer rate

6 -1 constant of 5.6x10 s , compared to the hetero-CCPS, which have S 1 to CT rate constants from

1.7x10 13 s -1 (anPOLY) to 3.7x10 12 s-1 (anPENT). Even if singlet injection, intersystem crossing and internal conversion could be entirely suppressed, a fluorescence lifetime as long as 60 ns would still prevent nonzero singlet fission yield. Similarly, increasing the coupling between two chromophores in a homo-CCP would accelerate electron transfer, but no reasonable amount of coupling would enhance the rate enough to yield observable singlet fission. Additionally, any increase in coupling is likely to decrease the free energy of fission. Because the electron transfer rate depends exponentially on the free energy of electron transfer, and only quadratically upon 105 the coupling, we anticipate that CCPs with weak coupling will display the best singlet fission yields.

4.4 Design Principles for Singlet Fission in DSSCs

Several conclusions can be drawn about ideal CCP design for molecular singlet fission in

DSSCs. The IC should be chosen first, primarily based on having the T1 energy lower than half of the S1 energy. While too strong a driving force could lead one into the inverted region of electron transfer, this is not likely to be a problem with realistic molecules. Additionally, as has been previously mentioned, if the T 2 energy level lies below or only slightly above the S 1 energy level, that could open up efficient intersystem crossing pathways which decrease singlet fission.

The polyene molecule we examined has a T 2 level below S 1, which could make it an unattractive 106 chromophore if the intersystem crossing from S 1 to T 2 is efficient. High experimental yield

(~30%) in biological carotenoids attributed to singlet fission hints that a low T 2 level might be acceptable. 50

Once an IC has been selected, a CCP should be designed in order to yield a downhill energetic stairway from S 1 to CT to TT states. Weak coupling will be desired to keep TT below

S1, while polar or polarizable functional groups (or a polar solvent) can bring the energy of the

CT states near or below S 1. If any decay route exists out of the CT state, for example charge injection to the electrode, one must be careful not to stabilize the CT states below the TT states in energy. We have illustrated this case with anPENT; if we had chosen to form hnPENT, based on Table 9, the CT states would lie below S 1 and above TT. The geometry of the CCP will also be important for the energy of the CT states, as a significant Coulombic stabilization is needed to bring the CT states down in energy relative to a pair of isolated ions. We have examined CCPs that are directly bonded, creating a large Coulombic stabilization. A longer bridge, such as the phenylene linker investigated by Bardeen and coworkers 53 would yield a weaker Coulombic stabilization because the ion pair would be further separated, and would raise the energy of the

CT state. Finally, the solvent/environment of the CCP should be taken into account, as more polar solvents will stabilize the states differently, with the CT state being most notably lowered in energy, and will also affect the total reorganization energy of the system. Based on our simulations and these design principles, we anticipate CCPs such as anPOLY or hnPENT will be among the most promising systems for molecular singlet fission.

In terms of overall cell design, it would be beneficial to limit the rate of singlet decay without slowing the rate of triplet injection. One way to selectively lower singlet decay would be by using an electrode that did not have available states in the conduction band near the S 1 107 energy level of the singlet fission dye, but which did provide open states for singlets from a lower energy dye, and for triplets that result from fission. An alternate approach would be to only couple one half of a CCP to the electrode to limit singlet injection. In a hetero-CCP, if only the side of the molecule with the higher energy localized singlet were tethered to the electrode, singlet injection could be dramatically lowered, as most of the singlet exciton population would reside on the far end of the chromophore. The triplet pair state would always have one triplet on the chromophore near the electrode, and once the first triplet injected, the fission process would be irreversible, and the second triplet would only have to inject quicker than its phosphorescence lifetime.

4.5 Limits of the Model

While we believe the model we have described, as well as the conclusions drawn from it, are correct in most ways, it is important to consider the possible weaknesses of this approach.

The entire model is based upon the assumption that we can treat the molecular system as a sum of two uncoupled chromophores. Because strong coupling is likely to hurt the exothermicity of singlet fission, and because we have only considered cases with weak coupling this approximation should not cause too much trouble, however, if one wanted to consider a CCP with strong coupling (for example the highly exothermic polyenes) our model would not be appropriate, and even for weak coupling this approximation leads to small errors.

The other simplification our model makes is our representation of electronic states as electronic configurations. To truly consider this problem correctly, one would have to consider the actual states of the full coupled chomophore Hamiltonian. Our approximation greatly 108 simplifies the problem, but introduces some error in our computation of electronic coupling elements, as the transition from a S 1S0 to CT state is not truly a single electron transfer from one uncoupled chomophore HOMO (LUMO) to the other uncoupled chromophore HOMO (LUMO).

From a computational point of view, we have made several simplications that could be improved upon. As in almost any case, more expensive computational methods, such as multi- reference electronic structure computations could be ued to yield more accurate state energies.

The addition of solvation would also improve the exact numerical predictive power of this model, as would more advanced methods of computing electronic coupling elements.

This model is not presented as a perfect model, but rather as a first step towards understanding the process of singlet fission. Our model allows one to evaluate singlet fission yield given any set of energies, electronic coupling matrix elements, initial state populations and decay rates imaginable. We believe the approximations made are appropriate, and that this model correctly predicts the trends and overall behavior of various classes of molecules for singlet fission.

4.6 Perspective for Experimentalists

The model we discuss here shows good agreement with current experimental results, and provides obvious guidelines for future experiments. Existing coupled chromophore pairs (CCPs) of tetracene and diphenylisobenzoburan have prohibitively high energy charge transfer (CT) states, and because of this have zero or near zero singlet fission yield. Pentacene and tetracene single crystals, which have near perfect singlet fission yield, are respectively exothermic and slightly endothermic for singlet fission. Additionally, they support low energy delocalized charge 109 transfer states, allowing the excitation to go from singlet excitation to pair of triplet states quicker than singlet fluorescence. The pair of triplets state can quickly decay into isolated triplets that spatially separate very quickly in the crystal, and this leads to almost total fluorescence quenching. If high triplet yields observed in carotenoids stem from singlet fission, this results is also compatible with or model. Polyenes, such as carotenoids, have very low pair of triplet energy levels, as well as low CT state energies, making them promising substrates for molecular singlet fission.

Choose monomers with low triplet energies. Alternant hydrocarbons and Exothermic Fission biradicaloids are good candidates. Weak electronic coupling keeps E(S1S0) > E(TT) delocalized S1 from dropping.

Add heteroatoms, especially asymmetric functionalization of the two halves of a Low energy charge coupled chromophore pair. Spatial arrangement of two chromophores can transfer intermediate affect Coulomb stabilization. Polar solvents also stabilize charge transfer states.

Limit singlet injection to the electrode by using an electrode without unoccupied Prevent unwanted states at the first singlet energy, but with empty states at the triplet energy. decay Use a hetero-CCP and tether the high energy localized S1 to the electrode.

Stronger electronic coupling between the two chromophores will increase Enhance electron electronic transfer rates if it does not hurt the energetic conditions transfer described above

Figure 30 Flowchart for designing promising molecules for singlet fission. The important molecular properties are displayed at right, with possible means of achieveing those properties at left.

Our model predicts which molecular properties are most important for singlet fission yield. We discuss the relative importance of various molecular parameters in this section, and 110 provide a complemtary flowchart for the design of promising molecules for singlet fission.

The most important molecular parameter is the relative energy of the states, with an ideal system having both localized first excited singlet state energies higher than at least one of the two charge transfer state energies, with the pair of triplets state energy being lowest. The localized singlet and pair of triplet state energies depend on the individual chromophores, while the charge transfer state energies depend on several factors. In order to have a low energy pair of triplet states, the two chromophores must have triplet energies less than half of their first excited singlet energies, and there must be only weak coupling (t) between the two chromophores to avoid dropping the delocalized first excited singlet state energy too much. The easiest way to lower the CT state energies is by asymmetrically functionalizing the two chromophores to yield a biased molecule with one low energy charge transfer state. Placing the two chromophores close together in the CCP will affect the CT energy because of Coulombic stabilization, and a polar solvent should also affect the energy of the CT states.

The second critical parameter is the ratio of the decay rates from the localized singlet, pair of triplets, and charge transfer states. The triplet injection rate would ideally be as fast as possible, and the singlet and charge transfer decay rates should be as low as possible. The electronic coupling between the molecule and the electrode will affect injection of all decay rates, but energy matching with the electrode could also affect this. In a hetero-CCP, tethering the half with the higher energy localized singlet to the electrode is likely to lower the singlet decay rate, while tethering the chromophore most likely to be a cation in the CT state will limit injection from the CT state. If an electrode material could be selected that did not allow injection of the high energy singlet exciton, but which had a high density of unoccupied states near the triplet energy level, singlet fission yield could be dramatically increased. 111 The electronic coupling between the two chromophores is much less important than the previous two parameters. If the electronic coupling matrix element could be increased with no other effects, the electron transfer rates would increase, helping singlet fission yield in cases where the singlet decay rate was comparable to the electron transfer rates. The largest increase in rate from increased electronic coupling is at most a factor of one hundred, as compared to increases of more than a factor of a million for changes in the energy levels discussed above.

Increasing the electronic coupling element between two chromophores has other effects on the system, however, as strongly coupled CCPs can not accurately be treated in terms of localized IC orbitals and localized excited states. This means that the model outlined here will become less accurate for predicting singlet fission yield with increasing electronic coupling. Additionally, as discussed in Chapter 2, increasing the electronic coupling between two chromophores is likely to decrease the energy of the delocalized singlet state in a proper, delocalized state basis, and this will hurt fission yield because the state energies are critical.

In summary, to maximize singlet fission, an individual chromophore should be chosen first, based primarily upon having its triplet energy as low as possible compared to its first excited singlet state. (In extreme cases a molecule could have its triplet energy too low if fission either entered the inverted Marcus region, or if the triplet energy was below the bottom of the electrode conduction band.) If it is possible to maintain a low triplet energy level while adding functional groups to make one chromophore more stable accepting extra charge and the other chromophore more stable donating charge this should be done to stabilize the CT intermediate.

Then the two chromophores should be connected in a manner that establishes a weak electronic coupling, and which positions the two chromophores close together to maximize the Coulombic stabilization of the CT state. 112

4.7 Summary and Conclusion

In this chapter we have analyzed how singlet fission yield depends on various molecular parameters using Marcus theory to predict electron transfer rates for a ten state dynamic scheme.

Within this theoretical framework, fission yield is most strongly dependent upon the relative energy levels of the localized singlets, charge transfer states, and triplet pair states. The optimal molecular system would have at least one charge transfer state lower in energy than all of the localized singlet exciton states, and the triplet pair states should be the lowest energy states. We have also discussed how the fission yield depends, to a lesser degree, on the electronic matrix elements between two coupled chromophores, the temperature, and the solvent/environment of the system. We conclude that the ideal molecular systems for singlet fission will be weakly coupled CCPs with asymmetric functional groups to stabilize the charge transfer intermediate.

We propose that CCPs of pentacene and a linear polyene will be among the best molecular candidates and that they could display fission yields above 90% in DSSCs.

113 References

1. Eisenberg, R.; Nocera, D. G., Preface: Overview of the forum on solar and renewable energy. Inorganic Chemistry 2005, 44, (20), 6799-6801.

2. Wigley, T. M. L.; Richels, R.; Edmonds, J. A., Economic and environmental choices in the stabilization of atmospheric CO2 concentrations. Nature 1996, 379, (6562), 240-243.

3. Lewis, N. S.; Nocera, D. G., Powering the planet: Chemical challenges in solar energy utilization. Proceedings of the National Academy of Sciences of the United States of America 2006, 103, (43), 15729-15735.

4. Lewis, N. S., Toward cost-effective solar energy use. Science 2007, 315, (5813), 798- 801.

5. Barnham, K. W. J.; Mazzer, M.; Clive, B., Resolving the energy crisis: nuclear or photovoltaics? Nature Materials 2006, 5, (3), 161-164.

6. Dresselhaus, M. S.; Thomas, I. L., Alternative energy technologies. Nature 2001, 414, (6861), 332-337.

7. Gratzel, M., Solar energy conversion by dye-sensitized photovoltaic cells. Inorganic Chemistry 2005, 44, (20), 6841-6851.

8. Gratzel, M., Perspectives for dye-sensitized nanocrystalline solar cells. Progress in Photovoltaics 2000, 8, (1), 171-185.

9. Gratzel, M., Photoelectrochemical cells. Nature 2001, 414, (6861), 338-344.

10. Oregan, B.; Gratzel, M., A Low-Cost, High-Efficiency Solar-Cell Based on Dye- Sensitized Colloidal Tio2 Films. Nature 1991, 353, (6346), 737-740.

11. Alstrum-Acevedo, J. H.; Brennaman, M. K.; Meyer, T. J., Chemical approaches to artificial photosynthesis. 2. Inorganic Chemistry 2005, 44, (20), 6802-6827.

12. Meyer, G. J., Molecular approaches to solar energy conversion with coordination compounds anchored to semiconductor surfaces. Inorganic Chemistry 2005, 44, (20), 6852- 6864.

13. Law, M.; Greene, L. E.; Johnson, J. C.; Saykally, R.; Yang, P. D., Nanowire dye- sensitized solar cells. Nature Materials 2005, 4, (6), 455-459.

114 14. Hanna, M. C.; Nozik, A. J., Solar conversion efficiency of photovoltaic and photoelectrolysis cells with carrier multiplication absorbers. Journal of Applied Physics 2006, 100, (7), -.

15. Klimov, V. I. M., A.A.; McBranch, D.W.; Leatherdale, C.A.; Bawendi, M.G., Quantization of Multiparticle Auger Rates in Semiconductor Quantum Dots. Science 2000, 287, 1011-1013.

16. Nozik, A. J., Spectroscopy and hot electron relaxation dynamics in semiconductor quantum wells and quantum dots. Annual Review of Physical Chemistry 2001, 52, 193-231.

17. Schaller, R. D.; Klimov, V. I., High efficiency carrier multiplication in PbSe nanocrystals: Implications for solar energy conversion. Physical Review Letters 2004, 92, (18), -.

18. Ellingson, R. J.; Beard, M. C.; Johnson, J. C.; Yu, P. R.; Micic, O. I.; Nozik, A. J.; Shabaev, A.; Efros, A. L., Highly efficient multiple exciton generation in colloidal PbSe and PbS quantum dots. Nano Letters 2005, 5, (5), 865-871.

19. Singh, S.; Jones, W. J.; Siebrand, W.; Stoichef.Bp; Schneide.Wg, Laser Generation of Excitons and Fluorescence in Anthracene Crystals. Journal of Chemical Physics 1965, 42, (1), 330-&.

20. Jundt, C.; Klein, G.; Sipp, B.; Lemoigne, J.; Joucla, M.; Villaeys, A. A., Exciton Dynamics in Pentacene Thin-Films Studied by Pump-Probe Spectroscopy. Chemical Physics Letters 1995, 241, (1-2), 84-88.

21. Groff, R. P.; Avakian, G. P.; Merrifie.Re, Coexistence of Exciton Fission and Fusion in Tetracene Crystals. Physical Review B 1970, 1, (2), 815-&.

22. Lopezdelgado, R.; Miehe, J. A.; Sipp, B., Fluorescence Decay Time Measurements in Tetracene Crystals Excited with Synchrotron Radiation. Optics Communications 1976, 19, (1), 79-82.

23. Arnold, S.; Whitten, W. B., Temperature-Dependence of the Triplet Exciton Yield in Fission and Fusion in Tetracene. Journal of Chemical Physics 1981, 75, (3), 1166-1169.

24. Arnold, S.; Alfano, R. R.; Pope, M.; Yu, W.; Ho, P.; Selsby, R.; Tharrats, J.; Swenberg, C. E., Triplet Exciton Caging in 2 Dimensions. Journal of Chemical Physics 1976, 64, (12), 5104-5114.

25. Vonburg, K.; Altwegg, L.; Zschokkegranacher, I., Influence of Surface Effects and Temperature on the Interaction of Triplet Excitations in Molecular-Crystals. Physical Review B 1980, 22, (4), 2037-2049.

115 26. Frankevich, E. L.; Lesin, V. I.; Pristupa, A. I., Rate Constants of Singlet Exciton Fission in a Tetracene Crystal Determined from Rydmr Spectral Linewidth. Chemical Physics Letters 1978, 58, (1), 127-131.

27. Fleming, G. R.; Millar, D. P.; Morris, G. C.; Morris, J. M.; Robinson, G. W., Exciton Fission and Annihilation in Crystalline Tetracene. Australian Journal of Chemistry 1977, 30, (11), 2353-2359.

28. Pope, M.; Geacintov, N.E.; Vogel, F., Singlet Exciton Fission and Triplet-Triplet Exciton Fusion in Crystalline Tetracene. Molecular Crystals and Liquid Crystals 1969, 6, (1), 83-&.

29. Swenberg, C. E.; Stacy, W. T., Bimolecular Radiationless Transitions in Crystalline Tetracene. Chem. Phys. Lett. 1968, 2, (5), 327-328.

30. Geacintov, N.; Pope, M.; Vogel, F., Effect of Magnetic Field on the Fluorescence of Tetracene Crystals: Exciton Fission. Physical Review Letters 1969, 22, (12), 593-596.

31. Geacintov, N. E.; Burgos, J.; Pope, M.; Strom, C., Heterofission of Pentacene Excited Singlets in Pentacene-Doped Tetracene Crystals. Chem. Phys. Lett. 1971, 11, (4), 504-508.

32. Katoh, R.; Kotani, M., Fission of a Higher Excited-State Generated by Singlet Exciton Fusion in an Anthracene Crystal. Chemical Physics Letters 1992, 196, (1-2), 108-112.

33. Yarmus, L.; Rosentha.J; Chopp, M., Epr of Triplet Excitons in Tetracene Crystals - Spin Polarization and Role of Singlet Exciton Fission. Chemical Physics Letters 1972, 16, (3), 477-&.

34. Klein, G.; Voltz, R.; Schott, M., Magnetic-Field Effect on Prompt Fluorescence in Anthracene - Evidence for Singlet Exciton Fission. Chemical Physics Letters 1972, 16, (2), 340.

35. Swenberg, C. E.; Van Metter, R.; Ratner, M., Comments on Exciton Fission and Electron Spin Resonance in Tetracene Crystals. Chem. Phys. Lett. 1972, 16, (3), 482-485.

36. Merrifield, R. E.; Avakian, P.; Groff, R. P., Fission of Singlet Excitons into Pairs of Triplet Excitons in Tetracene Crystals. Chem. Phys. Lett. 1969, 3, (3), 155-157 and 728.

37. Swenberg, C. E.; Ratner, M. A.; Geacintov, N. E., Energy Dependence of Optically Induced Exciton Fission. Journal of Chemical Physics 1974, 60, (5), 2152-2157.

38. Tomkiewi.Y; Groff, R. P.; Avakian, P., Spectroscopic Approach to Energetics of Exciton Fission and Fusion in Tetracene Crystals. Journal of Chemical Physics 1971, 54, (10), 4504-&.

39. Klein, G.; Voltz, R.; Schott, M., Singlet Exciton Fission in Anthracene and Tetracene at 77Degrees K. Chemical Physics Letters 1973, 19, (3), 391-394.

116 40. Katoh, R.; Kotani, M.; Hirata, Y.; Okada, T., Triplet exciton formation in a benzophenone single crystal studied by picosecond time-resolved absorption spectroscopy. Chemical Physics Letters 1997, 264, (6), 631-635.

41. Zenz, C.; Cerullo, G.; Lanzani, G.; Graupner, W.; Meghdadi, F.; Leising, G.; De Silvestri, S., Ultrafast photogeneration mechanisms of triplet states in para-hexaphenyl. Physical Review B 1999, 59, (22), 14336-14341.

42. Austin, R. H.; Baker, G. L.; Etemad, S.; Thompson, R., Magnetic-Field Effects on Triplet Exciton Fission and Fusion in a Polydiacetylene. Journal of Chemical Physics 1989, 90, (11), 6642-6646.

43. Lanzani, G. C., G.; Zavelani-Rossi, M.; De Silvestri, S.; Comoretto, D.; Musso, G.; Dellepiane, G., Triplet-Exciton Generation Mechanism in a New Soluble (Red-Phase) Polydiacetylene. Physical Review Letters 2001, 87, 187402.

44. Osterbacka, R.; Wohlgenannt, M.; Chinn, D.; Vardeny, Z. V., Optical studies of triplet excitations in poly(p-phenylene vinylene). Physical Review B 1999, 60, (16), R11253-R11256.

45. Osterbacka, R.; Wohlgenannt, M.; Shkunov, M.; Chinn, D.; Vardeny, Z. V., Excitons, polarons, and laser action in poly(p-phenylene vinylene) films. Journal of Chemical Physics 2003, 118, (19), 8905-8916.

46. Wohlgenannt, M.; Graupner, W.; Osterbacka, R.; Leising, G.; Comoretto, D.; Vardeny, Z. V., Singlet fission in luminescent and nonluminescent Pi-conjugated polymers. Synthetic Metals 1999, 101, (1-3), 267-268.

47. Kraabel, B.; Hulin, D.; Aslangul, C.; Lapersonne-Meyer, C.; Schott, M., Triplet exciton generation, transport and relaxation in isolated polydiacetylene chains: subpicosecond pump- probe experiments. Chemical Physics 1998, 227, (1-2), 83-98.

48. Dellepiane, G.; Comoretto, D.; Cuniberti, C., Chemical modulation of the electronic properties of polydiacetylenes. Journal of Molecular Structure 2000, 521, 157-166.

49. Wohlgenannt, M.; Graupner, W.; Leising, G.; Vardeny, Z. V., Photogeneration action spectroscopy of neutral and charged excitations in films of a ladder-type poly(para-phenylene). Physical Review Letters 1999, 82, (16), 3344-3347.

50. Gradinaru, C. C.; Kennis, J. T. M.; Papagiannakis, E.; van Stokkum, I. H. M.; Cogdell, R. J.; Fleming, G. R.; Niederman, R. A.; van Grondelle, R., An unusual pathway of excitation energy deactivation in carotenoids: Singlet-to-triplet conversion on an ultrafast timescale in a photosynthetic antenna. Proceedings of the National Academy of Sciences of the United States of America 2001, 98, (5), 2364-2369.

117 51. Papagiannakis, E.; Kennis, J. T. M.; van Stokkum, I. H. M.; Cogdell, R. J.; van Grondelle, R., An alternative carotenoid-to-bacteriochlorophyll energy transfer pathway in photosynthetic light harvesting. Proceedings of the National Academy of Sciences of the United States of America 2002, 99, (9), 6017-6022.

52. Papagiannakis, E.; Das, S. K.; Gall, A.; van Stokkum, I. H. M.; Robert, B.; van Grondelle, R.; Frank, H. A.; Kennis, J. T. M., Light harvesting by carotenoids incorporated into the B850 light-harvesting complex from Rhodobacter sphaeroides R-26.1: Excited-state relaxation, ultrafast triplet formation, and energy transfer to bacteriochlorophyll. Journal of Physical Chemistry B 2003, 107, (23), 5642-5649.

53. Muller, A. M.; Avlasevich, Y. S.; Mullen, K.; Bardeen, C. J., Evidence for exciton fission and fusion in a covalently linked tetracene dimer. Chemical Physics Letters 2006, 421, (4-6), 518-522.

54. Muller, A. M.; Avlasevich, Y. S.; Schoeller, W. W.; Mullen, K.; Bardeen, C. J., ASAP. Journal of the American Chemical Society ASAP .

55. Shaakov, S.; Galili, T.; Stavitski, E.; Levanon, H.; Lukas, A.; Wasielewski, M. R., Using spin dynamics of covalently linked radical ion pairs to probe the impact of structural and energetic changes on charge recombination. Journal of the American Chemical Society 2003, 125, (21), 6563-6572.

56. Dance, Z. E. X.; Mi, Q. X.; McCamant, D. W.; Ahrens, M. J.; Ratner, M. A.; Wasielewski, M. R., Time-resolved EPR studies of photogenerated radical ion pairs separated by p-phenylene oligomers and of triplet states resulting from charge recombination. Journal of Physical Chemistry B 2006, 110, (50), 25163-25173.

57. Paci, I.; Johnson, J. C.; Chen, X. D.; Rana, G.; Popovic, D.; David, D. E.; Nozik, A. J.; Ratner, M. A.; Michl, J., Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found? Journal of the American Chemical Society 2006, 128, (51), 16546-16553.

58. Johnson, J. C.; Chen, X.; Rana, G.; Paci, I.; Ratner, M. A.; Michl, J.; Nozik, A. J., Photophysical properties of covalently linked molecular dimers based on biradicaloids: evidence for singlet fission. IN PREP IN PREP .

59. Dexter, D. L., A Theory of Sensitized Luminscence in Solids. The Journal of Chemical Physics 1953, 21, (5), 836-850.

60. Marcus, R. A., Theory of Oxidation-Reduction Reactions Involving Electron Transfer.1. Journal of Chemical Physics 1956, 24, (5), 966-978.

61. Scholes, G. D., Long-range resonance energy transfer in molecular systems. Annual Review of Physical Chemistry 2003, 54, 57-87.

118 62. Shao, Y.; Molnar, L. F.; Jung, Y.; Kussmann, J.; Ochsenfeld, C.; Brown, S. T.; Gilbert, A. T. B.; Slipchenko, L. V.; Levchenko, S. V.; O'Neill, D. P.; DiStasio, R. A.; Lochan, R. C.; Wang, T.; Beran, G. J. O.; Besley, N. A.; Herbert, J. M.; Lin, C. Y.; Van Voorhis, T.; Chien, S. H.; Sodt, A.; Steele, R. P.; Rassolov, V. A.; Maslen, P. E.; Korambath, P. P.; Adamson, R. D.; Austin, B.; Baker, J.; Byrd, E. F. C.; Dachsel, H.; Doerksen, R. J.; Dreuw, A.; Dunietz, B. D.; Dutoi, A. D.; Furlani, T. R.; Gwaltney, S. R.; Heyden, A.; Hirata, S.; Hsu, C. P.; Kedziora, G.; Khalliulin, R. Z.; Klunzinger, P.; Lee, A. M.; Lee, M. S.; Liang, W.; Lotan, I.; Nair, N.; Peters, B.; Proynov, E. I.; Pieniazek, P. A.; Rhee, Y. M.; Ritchie, J.; Rosta, E.; Sherrill, C. D.; Simmonett, A. C.; Subotnik, J. E.; Woodcock, H. L.; Zhang, W.; Bell, A. T.; Chakraborty, A. K.; Chipman, D. M.; Keil, F. J.; Warshel, A.; Hehre, W. J.; Schaefer, H. F.; Kong, J.; Krylov, A. I.; Gill, P. M. W.; Head-Gordon, M., Advances in methods and algorithms in a modern quantum chemistry program package. Physical Chemistry Chemical Physics 2006, 8, (27), 3172-3191.

63. High_Performance_Computational_Chemistry_Group A computational chemistry package for parallel computers. Version 5.0, Pacific Northwestern National Laboratory.

64. Kendall, R. A.; Apra, E.; Bernholdt, D. E.; Bylaska, E. J.; Dupuis, M.; Fann, G. I.; Harrison, R. J.; Ju, J. L.; Nichols, J. A.; Nieplocha, J.; Straatsma, T. P.; Windus, T. L.; Wong, A. T., High performance computational chemistry: An overview of NWChem a distributed parallel application. Computer Physics Communications 2000, 128, (1-2), 260-283.

65. Bredas, J. L.; Beljonne, D.; Coropceanu, V.; Cornil, J., Charge-transfer and energy- transfer processes in pi-conjugated oligomers and polymers: A molecular picture. Chemical Reviews 2004, 104, (11), 4971-5003.

66. Bally, T.; Borden, W. T., In Reviews in Computational Chemistry , Lipkowitz, K. B.; Boyd, D. B., Eds. Wiley-VCH: 1999; Vol. 13, pp 1-99.

67. Coquet, R.; Willock, D. J., Physical Chemistry Chemical Physics 2005, 7, 3819-3828.

68. Kobko, N.; Masunov, A.; Tretiak, S., Chemical Physics Letters 2004, 392, 444-451.

69. Lambert, C.; Risko, C.; Coropceanu, V.; Schelter, J.; Amthor, S.; Gruhn, N.; Durivage, J. C.; Brédas, J. L., Journal of the American Chemical Society 2005, 127, 8508-8516.

70. Lundberg, M.; Siegbahn, P. E. M., Journal of Chemical Physics 2005, 122, 224103.

71. Pacchioni, G.; Frigoli, F.; Ricci, D., Physical Review B: Condensed Matter and Materials Physics 2000, 63, 054102.

72. Englman, R.; Jortner, J., Energy Gap Law for Radiationless Transitions in Large Molecules. Molecular Physics 1970, 18, (2), 145-&.

73. Wu, Q.; Van Voorhis, T., Extracting electron transfer coupling elements from constrained density functional theory. Journal of Chemical Physics 2006, 125, (16), -. 119 74. Wu, Q.; Van Voorhis, T., Direct calculation of electron transfer parameters through constrained density functional theory. Journal of Physical Chemistry A 2006, 110, (29), 9212- 9218.

75. Wu, Q.; Van Voorhis, T., Direct optimization method to study constrained systems within density-functional theory. Physical Review A 2005, 72, (2), -.

76. Longuet-Higgins, H. C.; Roberts, M. d. V., The Electronic Structure of the Borides MB6. Proceeedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 1954, 224, (1158), 336-347.

77. Venkataraman, L.; Klare, J. E.; Nuckolls, C.; Hybertsen, M. S.; Steigerwald, M. L., Dependence of single-molecule junction conductance on molecular conformation. Nature 2006, 442, (7105), 904-907.

78. Woitellier, S.; Launay, J. P.; Joachim, C., The Possibility of Molecular Switching - Theoretical-Study of [(Nh3)5Ru-4,4'-Bipy-Ru(Nh3)5]5+. Chemical Physics 1989, 131, (2-3), 481-488.

79. Bixon, M.; Jortner, J., Electron transfer - From isolated molecules to biomolecules. Electron Transfer-from Isolated Molecules to Biomolecules, Pt 1 1999, 106, 35-202.

80. Ter Haar, D., Theory and Applications of the Density Matrix. Reports on the Progress of Physics 1961, 24, 304-362.

81. Van Vooren, A.; Lemaur, V.; Ye, A. J.; Beljonne, D.; Cornil, J., Impact of bridging units on the dynamics of photoinduced charge generation and charge recombination in donor-acceptor dyads. Chemphyschem 2007, 8, (8), 1240-1249.

82. Barbara, P. F.; Meyer, T. J.; Ratner, M. A., Contemporary issues in electron transfer research. Journal of Physical Chemistry 1996, 100, (31), 13148-13168.

83. Marcus, R. A., On Theory of Electron-Transfer Reactions.6. Unified Treatment for Homogeneous and Electrode Reactions. Journal of Chemical Physics 1965, 43, (2), 679-&.

84. Jortner, J.; Bixon, M., Intramolecular Vibrational Excitations Accompanying Solvent- Controlled Electron-Transfer Reactions. Journal of Chemical Physics 1988, 88, (1), 167-170.

85. Jortner, J., Temperature-Dependent Activation-Energy for Electron-Transfer between Biological Molecules. Journal of Chemical Physics 1976, 64, (12), 4860-4867.

86. Markel, F.; Ferris, N. S.; Gould, I. R.; Myers, A. B., Mode-Specific Vibrational Reorganization Energies Accompanying Photoinduced Electron-Transfer in the Hexamethylbenzene Tetracyanoethylene Charge-Transfer Complex. Journal of the American Chemical Society 1992, 114, (15), 6208-6219. 120

Appendix A

Fortran Codes

121 A-1 Fortran (f90) Code To Compute Orbital Overlap

! This program is used to compute orbital overlap Sab.

! This program must be fed two orbital densities, orb1, orb2 as spit out by

! the QChem command "IANLTY" as well as an input file that specifies

! the x,y,z min and max, and step size that were used to generate the orbital

! plots.

! Note: Overlap of an orbital with itself goes to ~0.145 (1 /2pi) as the grid

! becomes infinitely finer and extends farther and farhter away from the

! molecules. I normalize to correct. 3A border in each direction is usually

! more than adequate for convergance, as is 0.1A grid spacing.

program Overlap

implicit none

integer nx,ny,nz,j

logical scanok,scan1

character*80 line 122 real bot,top real lengthx,lengthy,lengthz,dx,dy,dz real orb1,orb2,overlap

open(80,file="qchem.in") scanok=scan1(80,1,6,'$plots',line) if(.not.scanok) then

print*,'Error. You are missing a qchem.out file or a $plots section'

goto 300 endif read(80,*) read(80,*)nx,bot,top lengthx=abs(top-bot) read(80,*)ny,bot,top lengthy=abs(top-bot) read(80,*)nz,bot,top lengthz=abs(top-bot) close(80)

123 dx=(lengthx)/nx

dy=(lengthy)/ny

dz=(lengthz)/nz

! Read in files

overlap=0E0

open(90,file="orb1")

open(91,file="orb2")

do j=1,nx*ny*nz

read(90,*)orb1

read(91,*)orb2

overlap=overlap + orb1*orb2*dx*dy*dz

enddo

close(90)

close(91)

print*,overlap 124 print*,'Normalization (dx*dy*dz): ',dx*dy*dz

300 print*,' '

end program Overlap

!***************************************************

function scan1 (nu,i,j,str,line)

! **** function to scan1 input till a pattern is matched ****

! ****

! **** nu= file to read from

! **** i,j= columns of file in which to find the string

! **** str= string to search for

! **** line= the input line on which string is found (output)

! **** scan1= true if found, false otherwise

! **** 125

character*80 line

character*(*) str

integer i,j,nu

logical scan1

scan1= .true.

100 read (nu,'(a80)',end=200) line

if (line(i:j).eq.str) return

goto 100

200 scan1= .false.

return

end

!*****************************************

A-2 Fotran (f90) Code for Coherent Dynamics

! This code uses density matrix formalism and finite steps of hte time propagator

! to examine excitation evolution over time for coherent singlet fission. 126 !

! Parameters are customized for each molecule.

program CoherentDynamics

implicit none

character namer*3,namei*3,name1r*8,name1i*8

integer k,timeslice,savestep, topk,m,n

parameter(timeslice=40000000,topk=2000000001, savestep=12000)

double precision dt,FinalPercentFission,FinalPercentFluorescence,FinalPercentLeftSystem

double precision fluorpop,fisspop

parameter (dt=.0005) !Timestep in au

double complex E13, E24, E57, E68, E90

double complex TH, TL, UPR, UPL

double complex s1_6, s11_6, s16_6, s2_7, s12_7, s17_7, s3_13, s8_13, s18_13, s4_14, s9_14, s19_14, s5_20, s10_20, s15_20

double complex K9_12, K1_11, K2_11

double complex rho0(12,12),rho1(12,12),ham(12,12),sink(12,12),csink(12,12) 127 double complex i

double complex trace

double complex source(12,12)

parameter (i=(0E0,1E0))

!10 states energies. each is by definition isoenergetic with another state. Energies are in hartree!

parameter(E13= (0,0))

parameter(E24= (0,0))

parameter(E57= (.0369,0))

parameter(E68= (.0369,0))

parameter(E90= (.0111,0))

!Matrix Elements in hartree

parameter(TH= (.00125,0)) !HH

parameter(TL= (.00125,0)) !LL

parameter(UPR= (.00122,0)) !diag up and right

parameter(UPL= (.00122,0)) !diag up and left

128 !Decay Constants

parameter(K1_11= (.0000001,0)) !fluorescence from Left S1

parameter(K2_11= (.0000001,0)) !other fluorescence (Right S1)

parameter(K9_12= (.0001,0)) !fission from TT

!States in the density matrix are as follows:

!1,3 left local S1

!2,4 right local S1

!5,6 are cts from 1,2

!7,8, are cts from 3,4 (5,7 left anion, 6,8 right anion)

!9,10 tt

!11 fluoresced

!12 tt sink

!NOTE THAT UNFORTUNATELY THE 4 SINGLET AND CT STATES ARE

NUMBERED DIFFERENTLY THAN IN PAPERS

!Where 1,4 are left localized and 2,3 are right localized.

!And 5,8 and 6,7 go together. 129 !

open(50,file="pop1.dat")

open(51,file="pop2.dat")

open(52,file="pop3.dat")

open(53,file="pop4.dat")

open(54,file="pop5.dat")

open(55,file="pop6.dat")

open(56,file="pop7.dat")

open(57,file="pop8.dat")

open(58,file="pop9.dat")

open(59,file="pop10.dat")

open(70,file="s1pop.dat")

open(71,file="ctpop.dat")

open(72,file="ttpop.dat")

open(73,file="fluorpop.dat")

open(74,file="fisspop.dat")

open(75,file="totalpop.dat")

open(76,file="ratio.dat") 130 open(77,file="time.dat")

open(78,file="percent.dat")

!INITIAL DENSITY MATRIX rho0=(0E0,0E0) rho0(1,1)=(.25,0) rho0(2,2)=(.25,0) rho0(3,3)=(.25,0) rho0(4,4)=(.25,0)

!CREATING A HAMILTONIAN ham=(0E0,0E0)

!S1 --> CT by lumo electron movement, all geometries ham(1,5)=TH ham(2,6)=TH ham(3,7)=TH 131 ham(4,8)=TH

!S1 --> CT by homo electron movement, all geometries

ham(1,6)=TL

ham(2,5)=TL

ham(3,8)=TL

ham(4,7)=TL

!CT --> TT, both td routes, all geometries

ham(5,10)=UPR

ham(6,9)=UPL

ham(7,9)=UPR

ham(8,10)=UPL

!Make matrix symmetric (note this doubles diagonal values, so its done before define diagonals)

ham=ham+transpose(ham)

!Define diagonal elements, state energies 132 ham(1,1)=E13

ham(2,2)=E24

ham(5,5)=E57

ham(6,6)=E68

ham(9,9)=E90

ham(3,3)=E13

ham(4,4)=E24

ham(7,7)=E57

ham(8,8)=E68

ham(10,10)=E90

!CREATING SINK TERMS (to drain population out of the ten states into fission and fluor collection bins)

sink=(0E0,0E0)

sink(1,1)= -i*K1_11 !fluorescence from S1

sink(2,2)= -i*K2_11 133 sink(3,3)= -i*K1_11 !fluorescence from S1

sink(4,4)= -i*K2_11

sink(9,9)= -i*K9_12 !fission from TT

sink(10,10)= -i*K9_12 !fission from TT

csink=conjg(sink) !complex conjugate of sink

source=(0E0,0E0)

rho1=rho0 ! rho1 is the "in-progress" form of the density matrix

do k=1,topk ! This loop defines the steps in the evolution

!Save whole matrix in comma delmited text every 10,000 states - Sometimes I graph to see coherences

if (mod(k,timeslice).eq.1) then

write(namei,1000)k/timeslice+100

write(namer,1000)k/timeslice+100

1000 format(I3)

name1r = namer // 'r.dat' 134 name1i = namei // 'i.dat'

open(80,file=name1r)

open(81,file=name1i)

do m=1,12

do n=1,12

write(80,'(E14.6)',ADVANCE='NO')real(rho1(m,n))

write(81,'(E14.6)',ADVANCE='NO')imag(rho1(m,n))

enddo

write(80,*)

write(81,*)

enddo

close(80)

close(81)

endif

!source is the totally diagonal (noncoherent) term used to allocate population to the fissioned and fluored states

source=(0E0,0E0) 135 source(11,11)=2*(K1_11*(rho1(1,1)+rho1(3,3))+K2_11*(rho1(2,2)+rho1(4,4

)))

source(12,12)=2*(K9_12*(rho1(9,9)+rho1(10,10)))

!Evolution. density matrix master equation with a leakage term added.

rho1 = rho1 - i*(matmul(ham,rho1)-matmul(rho1,ham))*dt - i*(matmul(sink,rho1)- matmul(rho1,csink))*dt + source*dt

!Write down Populations of Interest

if (mod(k,savestep).eq.0) then

write(50,*)real(rho1(1,1))

write(51,*)real(rho1(2,2))

write(52,*)real(rho1(3,3))

write(53,*)real(rho1(4,4))

write(54,*)real(rho1(5,5))

write(55,*)real(rho1(6,6)) 136 write(56,*)real(rho1(7,7))

write(57,*)real(rho1(8,8))

write(58,*)real(rho1(9,9))

write(59,*)real(rho1(10,10))

write(70,*)real(rho1(1,1)+rho1(2,2)+rho1(3,3)+rho1(4,4)) !S1@S1

write(71,*)real(rho1(5,5)+rho1(6,6)+rho1(7,7)+rho1(8,8)) !CT@CT

write(72,*)real(rho1(9,9)+rho1(10,10)) !TT@TT

fisspop=real(rho1(12,12)) !Fissioned

fluorpop=real(rho1(11,11)) !Fluored

write(73,*)fluorpop

write(74,*)fisspop

write(75,*)real(trace(rho1)) !All population should equal 1

write(76,*)real(fisspop/(fluorpop+00000001))

write(78,*)real(fisspop/(fluorpop+fisspop+.00000001)) !Could add an "if" term so no divide by 0 error instead.

write(77,*)(k-1)*dt

endif

137

end do

close(50) close(51) close(52) close(53) close(54) close(55) close(56) close(57) close(58) close(59) close(70) close(71) close(72) close(73) 138 close(74)

close(75)

close(76)

close(77)

close(78)

!Spit out some numbers about final state after all time steps

FinalPercentFission = real(rho1(12,12))

FinalPercentFluorescence = real(rho1(11,11))

FinalPercentLeftSystem =FinalPercentFission + FinalPercentFluorescence open(90,file="finalpercent.dat") write(90,*)'Final Percent Fission: ',FinalPercentFission write(90,*)'Final Percent Fluorescence: ',FinalPercentFluorescence write(90,*)'Final Percent Left System: ',FinalPercentLeftSystem close(90)

end program Coherent Dynamics

!end of program

!****************************** 139 ! This program finds the trace of a 12x12 matrix

function trace(matrix)

implicit none

double complex trace,matrix(12,12)

integer j

trace=(0E0,0E0)

do j=1,12

trace=trace+matrix(j,j)

enddo

end function trace

!******************************

A-3 Fortran (f90) Code for Marcus Theory Dynamics

! Marcus Kinetics Scheme for Singlet Fission Dynamics

! Conventions for states are as published in Coherent and Marcus Paper on Fission.

! E11 is state 1 at geom 1, E24 is state 2 at geom 4 140 !

! Solves semiclassical and one quantum vibration mode tunneling Marcus rates from energy inputs

! Does time evolution by small finite timesteps.

! Tracks how population leaves the system.

program MarcusKinetics

!VARIABLES ARE PROBABLY OVERFLY DEFINED, DOUBLE COMPS INSTEAD

OF REAL - might help with numercial instability

implicit none

integer k,savestep, topk,m,n

parameter(topk=10000001, savestep=500)

double precision dt,FinalPercentFission,FinalPercentFluorescence,FinalPercentLeftSystem

double precision vFinalPercentFission,vFinalPercentFluorescence,vFinalPercentLeftSystem

double precision placeholder(4) 141 double precision E11,E22,E33,E44,E55,E66,E77,E88,E99,E00

double precision E15,E16,E25,E26,E37,E38,E47,E48

double precision E51,E52,E61,E62,E73,E74,E83,E84

double precision E50,E69,E70,E89,E05,E07,E96,E98

double precision v15,v16,v26,v25,v37,v38,v47,v48,v50,v69,v89,v70

double precision k15,k16,k26,k25,k37,k38,k47,k48,k50,k69,k89,k70

double precision k51,k61,k62,k52,k73,k83,k74,k84,k05,k96,k98,k07

double precision kv15,kv16,kv26,kv25,kv37,kv38,kv47,kv48,kv50,kv69,kv89,kv70

double precision kv51,kv61,kv62,kv52,kv73,kv83,kv74,kv84,kv05,kv96,kv98,kv07

double precision kf1, kf2, kf3,kf4,kf9,kf0 !fluor and fiss rates states i

double precision pop1, pop2, pop3, pop4, pop5, pop6,pop7,pop8,pop9,pop0

double precision popfluor, popfiss !populations

double precision tpop1, tpop2, tpop3, tpop4, tpop5, tpop6,tpop7,tpop8,tpop9,tpop0

double precision tpopfluor, tpopfiss !temp pops

double precision popv1, popv2, popv3, popv4, popv5, popv6,popv7,popv8,popv9,popv0

double precision popfluorv, popfissv !populations with vibes

double precision tpopv1, tpopv2, tpopv3, tpopv4, tpopv5, tpopv6,tpopv7,tpopv8,tpopv9,tpopv0

double precision tpopvfluor, tpopvfiss !temp pops with vibes 142 parameter (dt=.10D0)

!State Energies

parameter(E11= 0D0) !S1S0

parameter(E22= 0D0) !S0S1

parameter(E55= .017D0) !CT1

parameter(E66= .017D0) !CT2

parameter(E99= -.043D0) !TT

parameter(E15= .011D0)

parameter(E16= .011D0)

parameter(E25= .011D0)

parameter(E26= .011D0)

parameter(E51= .028D0)

parameter(E52= .028D0)

parameter(E61= .028D0)

parameter(E62= .028D0)

parameter(E50= .027D0)

parameter(E69= .027D0) 143 parameter(E05= -.031D0)

parameter(E07= -.031D0)

!Habs

parameter(v15= .00119D0) !HH

parameter(v16= .00109D0) !LL

parameter(v50= .00118D0) !Td1

parameter(v69= .00118D0) !Td2

!Decay Rates

parameter (kf1= .00001D0) !fluorescence from Left S1

parameter (kf2= .00001D0) !other fluorescence (Right S1)

parameter (kf9= .00001D0) !fission from TT

!Initial Populations

pop1=.25D0

pop2=.25D0

pop3=.25D0

pop4=.25D0 144 pop5=0D0

pop6=0D0

pop7=0D0

pop8=0D0

pop9=0D0

pop0=0D0

!popv are population for simulation using one quantum mode

popv1=pop1

popv2=pop2

popv3=pop3

popv4=pop4

popv5=pop5

popv6=pop6

popv7=pop7

popv8=pop8

popv9=pop9

popv0=pop0

145 ! Sets energy of redundant states equal to their partners, defined above

E00=E99

E44=E11

E33=E22

E88=E55

E77=E66

E48=E15

E47=E16

E38=E25

E37=E26

E84=E51

E83=E52

E74=E61

E73=E62

E89=E50

E70=E69

E98=E05

E96=E07

v26=v15 146 v37=v15

v48=v15

v25=v16

v38=v16

v47=v16

v70=v69

v89=v50

kf4=kf1

kf3=kf2

kf0=kf9

!Rates - computed by RATES function (below)

!k15 is semiclassical rate from 1@1-->5@5

!kv15 is the one quantum mode rate

call RATES(E11,E15,E55,E51,v15,placeholder)

k15= placeholder(1)

k51= placeholder(2) 147 kv15= placeholder(3) kv51= placeholder(4)

call RATES(E11,E16,E66,E61,v16,placeholder) k16= placeholder(1) k61= placeholder(2) kv16= placeholder(3) kv61= placeholder(4)

call RATES(E22,E26,E66,E62,v26,placeholder) k26= placeholder(1) k62= placeholder(2) kv26= placeholder(3) kv62= placeholder(4)

call RATES(E22,E25,E55,E52,v25,placeholder) k25= placeholder(1) k52= placeholder(2) kv25= placeholder(3) 148 kv52= placeholder(4)

call RATES(E33,E37,E77,E73,v37,placeholder) k37= placeholder(1) k73= placeholder(2) kv37= placeholder(3) kv73= placeholder(4)

call RATES(E33,E38,E88,E83,v38,placeholder) k38= placeholder(1) k83= placeholder(2) kv38= placeholder(3) kv83= placeholder(4)

call RATES(E44,E47,E77,E74,v47,placeholder) k47= placeholder(1) k74= placeholder(2) kv47= placeholder(3) kv74= placeholder(4) 149

call RATES(E44,E48,E88,E84,v48,placeholder) k48= placeholder(1) k84= placeholder(2) kv48= placeholder(3) kv84= placeholder(4)

call RATES(E55,E50,E00,E05,v50,placeholder) k50= placeholder(1) k05= placeholder(2) kv50= placeholder(3) kv05= placeholder(4)

call RATES(E66,E69,E99,E96,v69,placeholder) k69= placeholder(1) k96= placeholder(2) kv69= placeholder(3) kv96= placeholder(4)

150 call RATES(E88,E89,E99,E98,v89,placeholder)

k89= placeholder(1)

k98= placeholder(2)

kv89= placeholder(3)

kv98= placeholder(4)

call RATES(E77,E70,E00,E07,v70,placeholder)

k70= placeholder(1)

k07= placeholder(2)

kv70= placeholder(3)

kv07= placeholder(4)

! Write down a list of the electron transfer rates

open(49,file="rates.dat")

write(49,*)'k15: ',k15

write(49,*)'k51: ',k51

write(49,*)'kv15: ',kv15

write(49,*)'kv51: ',kv51

write(49,*)'******' 151 write(49,*)'k16: ',k16 write(49,*)'k61: ',k61 write(49,*)'kv16: ',kv16 write(49,*)'kv61: ',kv61 write(49,*)'******' write(49,*)'k26: ',k26 write(49,*)'k62: ',k62 write(49,*)'kv26: ',kv26 write(49,*)'kv62: ',kv62 write(49,*)'******' write(49,*)'k25: ',k25 write(49,*)'k52: ',k52 write(49,*)'kv25: ',kv25 write(49,*)'kv52: ',kv52 write(49,*)'******' write(49,*)'k50: ',k50 write(49,*)'k05: ',k05 write(49,*)'kv50: ',kv50 write(49,*)'kv05: ',kv05 152 write(49,*)'******'

write(49,*)'k96: ',k96

write(49,*)'k69: ',k69

write(49,*)'kv96: ',kv96

write(49,*)'kv69: ',kv69

close(49)

!open files to write populations

open(51,file="pop1.dat")

open(52,file="pop2.dat")

open(53,file="pop3.dat")

open(54,file="pop4.dat")

open(55,file="pop5.dat")

open(56,file="pop6.dat")

open(57,file="pop7.dat")

open(58,file="pop8.dat")

open(59,file="pop9.dat")

open(60,file="pop0.dat") 153 open(71,file="s1pop.dat") open(72,file="ctpop.dat") open(73,file="ttpop.dat") open(74,file="fluorpop.dat") open(75,file="fisspop.dat") open(76,file="totalpop.dat") open(77,file="ratio.dat") open(78,file="percent.dat") open(79,file="time.dat") open(61,file="popv1.dat") open(62,file="popv2.dat") open(63,file="popv3.dat") open(64,file="popv4.dat") open(65,file="popv5.dat") open(66,file="popv6.dat") open(67,file="popv7.dat") open(68,file="popv8.dat") open(69,file="popv9.dat") open(70,file="popv0.dat") 154 open(81,file="s1popv.dat")

open(82,file="ctpopv.dat")

open(83,file="ttpopv.dat")

open(84,file="fluorvpop.dat")

open(85,file="fissvpop.dat")

open(86,file="totalvpop.dat")

open(87,file="ratiov.dat")

open(88,file="percentv.dat")

open(89,file="timev.dat")

! This loop is where dynamics happen do k=1,topk

!STORE TEMPORARY VALUES

tpop1=pop1

tpop2=pop2

tpop3=pop3

tpop4=pop4

tpop5=pop5 155 tpop6=pop6 tpop7=pop7 tpop8=pop8 tpop9=pop9 tpop0=pop0

tpopfluor=popfluor tpopfiss=popfiss

tpopv1=popv1 tpopv2=popv2 tpopv3=popv3 tpopv4=popv4 tpopv5=popv5 tpopv6=popv6 tpopv7=popv7 tpopv8=popv8 tpopv9=popv9 tpopv0=popv0 156 tpopvfluor=popfluorv

tpopvfiss=popfissv

!EVOLUTION (KINETIC EQNS, classical first, quantum vibe second)

popfluor= tpopfluor + kf1*tpop1*dt + kf2*tpop2*dt + kf3*tpop3*dt + kf4*tpop4*dt

popfiss= tpopfiss + kf9*tpop9*dt + kf0*tpop0*dt

pop1= tpop1 + (k51*tpop5 + k61*tpop6)*dt - (kf1*tpop1 + k15*tpop1 + k16*tpop1)*dt

pop2= tpop2 + (k52*tpop5 + k62*tpop6)*dt - (kf2*tpop2 + k25*tpop2 + k26*tpop2)*dt

pop3= tpop3 + (k73*tpop7 + k83*tpop8)*dt - (kf3+ k37 + k38)*tpop3*dt

pop4= tpop4 + (k74*tpop7 + k84*tpop8)*dt - (kf4+ k47 + k48)*tpop4*dt

pop5= tpop5 + (k15*tpop1 + k25*tpop2 + k05*tpop0)*dt - (k51 + k52 + k50)*tpop5*dt

pop6= tpop6 + (k16*tpop1 + k26*tpop2 + k96*tpop9)*dt - (k61 + k62 + k69)*tpop6*dt

pop7= tpop7 + (k37*tpop3 + k47*tpop4 + k07*tpop0)*dt - (k73 + k74 + k70)*tpop7*dt 157 pop8= tpop8 + (k38*tpop3 + k48*tpop4 + k98*tpop9)*dt - (k83 + k84 + k89)*tpop8*dt

pop9= tpop9 + (k69*tpop6+k89*tpop8)*dt - (kf9 + k96 + k98)*tpop9*dt

pop0= tpop0 + (k70*tpop7+k50*tpop5)*dt - (kf0 + k07 + k05)*tpop0*dt

popfluorv= tpopvfluor + kf1*tpopv1*dt + kf2*tpopv2*dt + kf3*tpopv3*dt + kf4*tpopv4*dt

popfissv= tpopvfiss + kf9*tpopv9*dt + kf0*tpopv0*dt

popv1= tpopv1 + (kv51*tpopv5 + kv61*tpopv6)*dt - (kf1*tpopv1 + kv15*tpopv1 + kv16*tpopv1)*dt

popv2= tpopv2 + (kv52*tpopv5 + kv62*tpopv6)*dt - (kf2*tpopv2 + kv25*tpopv2 + kv26*tpopv2)*dt

popv3= tpopv3 + (kv73*tpopv7 + kv83*tpopv8)*dt - (kf3+ kv37 + kv38)*tpopv3*dt

popv4= tpopv4 + (kv74*tpopv7 + kv84*tpopv8)*dt - (kf4+ kv47 + kv48)*tpopv4*dt

popv5= tpopv5 + (kv15*tpopv1 + kv25*tpopv2 + kv05*tpopv0)*dt - (kv51 + kv52 + kv50)*tpopv5*dt

popv6= tpopv6 + (kv16*tpopv1 + kv26*tpopv2 + kv96*tpopv9)*dt - (kv61 + kv62 + kv69)*tpopv6*dt

popv7= tpopv7 + (kv37*tpopv3 + kv47*tpopv4 + kv07*tpopv0)*dt - (kv73 + kv74 + kv70)*tpopv7*dt 158 popv8= tpopv8 + (kv38*tpopv3 + kv48*tpopv4 + kv98*tpopv9)*dt - (kv83 + kv84

+ kv89)*tpopv8*dt

popv9= tpopv9 + (kv69*tpopv6+kv89*tpopv8)*dt - (kf9 + kv96 + kv98)*tpopv9*dt

popv0= tpopv0 + (kv70*tpopv7+kv50*tpopv5)*dt - (kf0 + kv07 + kv05)*tpopv0*dt

!WRITE TO .dat FILES

if (mod(k,savestep).eq.0) then

write(51,*)pop1

write(52,*)pop2

write(53,*)pop3

write(54,*)pop4

write(55,*)pop5

write(56,*)pop6

write(57,*)pop7

write(58,*)pop8

write(59,*)pop9

write(60,*)pop0

write(71,*)pop1+pop2+pop3+pop4

write(72,*)pop5+pop6+pop7+pop8 159 write(73,*)pop9+pop0

write(74,*)popfluor

write(75,*)popfiss

write(76,*)pop1+pop2+pop3+pop4+pop5+pop6+pop7+pop8+pop9+pop0+popfiss+popfluor

!Should equal 1

write(77,*)popfiss/(popfluor+00000001)

write(78,*)popfiss/(popfluor+popfiss+.00000001)

write(79,*)(k-1)*dt

write(61,*)popv1

write(62,*)popv2

write(63,*)popv3

write(64,*)popv4

write(65,*)popv5

write(66,*)popv6

write(67,*)popv7

write(68,*)popv8

write(69,*)popv9

write(70,*)popv0 160 write(81,*)popv1+popv2+popv3+popv4

write(82,*)popv5+popv6+popv7+popv8

write(83,*)popv9+popv0

write(84,*)popfluorv

write(85,*)popfissv

write(86,*)popv1+popv2+popv3+popv4+popv5+popv6+popv7+popv8+popv9+popv0+popfissv

+popfluorv !Should equal 1

write(87,*)popfissv/(popfluorv+00000001)

write(88,*)popfissv/(popfluorv+popfissv+.00000001)

write(89,*)(k-1)*dt

endif

end do

close(51)

close(52)

close(53) 161 close(54) close(55) close(56) close(57) close(58) close(59) close(60) close(71) close(72) close(73) close(74) close(75) close(76) close(77) close(78) close(79) close(61) close(62) close(63) 162 close(64)

close(65)

close(66)

close(67)

close(68)

close(69)

close(70)

close(81)

close(82)

close(83)

close(84)

close(85)

close(86)

close(87)

close(88)

close(89)

!Spit out some numbers about final state after all time steps 163 FinalPercentFission = popfiss

FinalPercentFluorescence = popfluor

vFinalPercentFission = popfissv vFinalPercentFluorescence = popfluorv vFinalPercentLeftSystem =vFinalPercentFission + vFinalPercentFluorescence open(91,file="vfinalpercent.dat") write(91,*)'Final Percent Fission: ',vFinalPercentFission write(91,*)'Final Percent Fluorescence: ',vFinalPercentFluorescence write(91,*)'Final Percent Left System: ',vFinalPercentLeftSystem close(91)

164 end program MarcusKinetics

!end of program

!***************************

! Find Marcus rates, forward and backward, semiclassical and one quantum vibe subroutine RATES(E1,E2,E3,E4,Hab,placeholder)

! NOTE - NUMBERS MUST BE INPUT IN HARTREES

implicit none

double precision hbar,pi,kB

double precision deltaG,lambdaO,lambdaIf,lambdaIr,temp,omega,E1,E2,E3,E4,Hab

double precision Sf,Sr

double precision placeholder(4),factorial,ratetemp(4) 165 integer m hbar=1D0 pi=3.1415926535897932D0 kB=3.169790518D-6

!E1 is state 1, opt geom

!E2 is state 1, other geom

!E3 is state 2, opt geom

!E4 is state 2, other geom

!RATES(1) is kf, (2)kr, (3)kfvibe, (4)krvibe

!Solve for deltaG and lambda inners from input numbers deltaG=(E3-E1) lambdaIf=(E4-E3) if (lambdaIf < 0) then

lambdaIf = 0 endif 166 lambdaIr=(E2-E1) if (lambdaIr < 0) then

lambdaIr = 0 endif

! Parameters to change lambdaO= .1/27.21D0 ! .1 eV in hartrees omega = .186/27.21D0 ! 1500cm-1 in hartrees temp= 300D0 ! 300 K

! End stuff to change

Sf = lambdaIf/(hbar*omega) ! Huang-Rhys

Sr = lambdaIr/(hbar*omega) placeholder(1)=0D0 placeholder(2)=0D0 placeholder(3)=0D0 placeholder(4)=0D0 167

! Solve for rates based on inputs and conistants

placeholder(1)=2D0*pi/hbar * Hab**2 * 1D0/sqrt(4D0 * pi * (lambdaO+lambdaIf) * kB * temp) * exp(- (deltaG + (lambdaO+lambdaIf))**2/(4D0*(lambdaO+lambdaIf) * kB * temp) )

placeholder(2)=2D0*pi/hbar * Hab**2 * 1D0/sqrt(4D0 * pi * (lambdaO+lambdaIr) * kB * temp) * exp(- (-1*deltaG + (lambdaO+lambdaIr))**2/(4D0*(lambdaO+lambdaIr) * kB * temp) )

do m=0,20

ratetemp(3)= 2D0*pi/hbar * Hab**2 * exp(- (deltaG + lambdaO + m*hbar*omega)**2/(4D0*lambdaO * kB * temp) )

ratetemp(3) = ratetemp(3)* (exp(-Sf) *Sf**m / factorial(m)) * 1D0/sqrt(4D0 * pi * lambdaO * kB * temp)

placeholder(3) = placeholder(3) + ratetemp(3)

ratetemp(4)= 2D0*pi/hbar * Hab**2 * exp(- (-1*deltaG + lambdaO + m*hbar*omega)**2/(4D0*lambdaO * kB * temp) ) 168 ratetemp(4) = ratetemp(4)* (exp(-Sr) *Sr**m / factorial(m)) * 1D0/sqrt(4D0 * pi * lambdaO * kB * temp)

placeholder(4) = placeholder(4) + ratetemp(4)

enddo

end subroutine RATES

!**********************

!***********************

! gamma function - needed for definition of factorial function

function gamm(xx)

implicit none

double precision gammln,xx,gamm

double precision ser,stp,tmp,x,y,cof(6)

integer j 169 save cof,stp

data cof,stp/76.18009172947146d0,-86.50532032941677d0, 24.01409824083091d0,-

1.231739572450155d0,.1208650973866179d-2, -.5395239384953d-5,2.5066282746310005d0/

x=xx

y=x

tmp=x+5.5d0

tmp=(x+0.5d0)*log(tmp)-tmp

ser=1.000000000190015d0

do j=1,6

y=y+1.d0

ser=ser+cof(j)/y

enddo

gammln=tmp+log(stp*ser/x)

gamm=dexp(gammln)

return

end function gamm

!************************

!****************************** 170 ! factorial: x! function factorial(xx)

implicit none

integer xx

double precision factorial,gamm

if(xx>=0) then

factorial = gamm((xx+1)*1.D0)

else

factorial = gamm(0D0)

endif

end function factorial

!******************************

171