Triplet-Triplet Coupling in Chromophore Dimers: Theory and

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A: Spectroscopy, Molecular Structure, and Quantum Chemistry

Triplet-Triplet Coupling in Chromophore Dimers: Theory and Experiment Daniel A Hartzler, Lyudmila V. Slipchenko, and Sergei Savikhin J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b04294 • Publication Date (Web): 24 Jul 2018 Downloaded from http://pubs.acs.org on July 27, 2018

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The Journal of Physical Chemistry

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Triplet-Triplet Coupling in Chromophore Dimers: Theory and Experiment Daniel A. Hartzler1, Lyudmila V. Slipchenko2*, Sergei Savikhin1* 1

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Department of Physics, Purdue University, West Lafayette, IN 47907, United States

Department of Chemistry, Purdue University, West Lafayette, IN 47907, United States

*Corresponding Authors: Lyudmila V. Slipchenko, email: [email protected]; Sergei Savikhin, email: [email protected]

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ABSTRACT

Knowledge of triplet state energies and triplet-triplet (T-T) interactions in aggregated organic molecules is essential for understanding photochemistry and dynamics of many natural and artificial systems. In this work, we combine direct phosphorescence measurements of triplet state energies, which are challenging due to the spin-forbidden nature of respective transitions and applicable to only a limited number of systems, with quantum chemical computational tools that can provide valuable qualitative and quantitative information about triplet states of interacting molecules. Using hexatriene, protoporphyrin, pheophorbide and chlorophyll dimers as model systems, we demonstrate a complicated dependence of TT coupling on a relative orientation of chromophores, governed by a nodal structure of overlapping electronic wave functions, that modulates inter-pigment interactions by orders of magnitude. It is also shown that geometrical relaxation of the triplet state is one of the critical factors for predictive modeling of T-T interactions in molecular aggregates.

Introduction

Knowledge of triplet state energies and triplet-triplet interactions in aggregated organic molecules is essential for understanding photochemistry and dynamics in many natural and artificial systems. In some systems, e.g. photosynthetic proteins, triplet states can lead to generation of highly toxic singlet oxygen. Triplet-triplet (T-T) interactions in these systems are crucial for triplet energy transfer (TET) to molecular quenchers providing efficient photoprotection.1–8 In fact, some herbicides function by promoting accumulation of non-photoprotected tetrapyrroles (e.g. Chls, porphyrins, etc.)9, allowing generation of toxic levels of singlet oxygen. In other systems, such as organic photovoltaics or organic LEDs (OLEDs), triplet states are essential for function.10–15 It has been shown, for example, that under suitable energetic arrangement singlet-triplet exciton fission in aggregates of closely spaced pigments can occur resulting in


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photovoltaic quantum efficiency approaching 200%,10,11,16–19 and intricate singlet-mediated transport mechanisms of the produced triplets were observed.20 While the triplet state energies of a number of isolated monomeric pigments and few aggregates have been measured directly by phosphorescence spectroscopy10,21–24, the spin forbidden nature and resulting low quantum yield of the radiative transition makes this a challenging task. On the other hand, modern quantum chemical computational methods could provide this information from the available structural information.25–33 In this work, general dependencies of T-T interactions on the structure of representative dimeric systems are analyzed computationally and compared with experimental data. It is known that the magnitude of the T-T coupling, and thus the TET rate, depend exponentially on the distance between triplet donor and acceptor.34 However, the exponential distance dependence of T-T coupling is only a part of the story. For monomers in close contact, the oscillatory nature of the molecular wave functions can lead to dramatic decrease in the coupling depending on relative monomer orientation even for closely spaced molecules.34–38 This implication has not been explored extensively due to the difficulty of calculating the couplings at close range34,36 unlike the singlet-singlet coupling responsible for the Fӧrster Resonance Energy Transfer (FRET) mechanism and singlet exciton formation that can be computed with a reasonable precision based just on the transition dipole moments and center-to-center distance between the pigments.36,39 However, it is speculated that the morphology of the molecular aggregates and, as a consequence, the molecular orbital overlap, affect electronic processes such as singlet-triplet exciton fission. 32,40,41 In this work we report electronic structure calculations of triplet state energies of several dimeric model systems with known structures and compare the results with the direct experimental assessment of these energies using phosphorescence measurements. As expected, quantum mechanical calculations reveal that while the inter-planar distances between aromatic molecules in dimers or aggregates are important, the horizontal displacements between the interacting molecules are crucial and can modulate inter-pigment



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interactions by orders of magnitude. In addition, it is shown that accurate prediction of the triplet state energies in dimeric systems requires explicit treatment of structural reorganization in the triplet state that may lead to significant additional splitting of the excitonic triplet states and cause dominant localization of the triplet state on one of the pigments.

Theoretical methods The triplet excited state energy upon aggregation of two or more pigments is affected by both environmental changes and T-T coupling, VT, between closely spaced pigments. The latter leads to excitonic delocalization of excited triplet states among nearby pigments and to a formation of a manifold of triplet states that spans over a broader spectral range than triplets in noninteracting molecules. It is shown in You et al.42, that the leading contribution to VT is given by the following integral:

VT ≈ − ∫ dx1dx2 χ D* ( x1 ) χ A ( x1 )

1 * χ D ( x2 ) χ A ( x2 ) r12

(1)

where x1 and x2 are electron position-spin coordinates, r12 is a spatial distance between these coordinates, χ are β spin-orbitals corresponding to lowest unoccupied molecular orbitals (LUMO), χ are α spin-

orbitals corresponding to the highest occupied molecular orbitals (HOMO), and subscripts D and A denote donor and acceptor molecules, respectively. Using eq. 1 You et al.

42

predicted that coupling

depends exponentially on the separation between the interacting molecules. However, no detailed investigation of the coupling dependence on the overlap between the molecules was performed. This topic is of a special interest as aromatic complexes exhibit conformations where both inter-planar separation and orientation of chromophores are varied.


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(a) LUMO

HOMO

(b)

slip

(c)

(d)

Figure 1. Hexatriene dimer separated by 3.5Å. (a) LUMO and HOMO orbitals of the interacting molecules at three slip positions (0Å, 1Å, 2.6Å). Red and blue ellipses indicate areas of positive and negative signs of the overlapping orbital integrals (Еq. 1), respectively. (b) The T-T coupling VT (black) and displacement energy D (blue) for hexatriene dimer calculated as a function of slip distance. Dashed purple line is an asymptote demonstrating quadratic dependence of the coupling on the overlap area. (c) E1 and E2 are the energies of triplet excitonic states; ε denotes the site energies of noninteracting pigments. (d) ES and ET are the binding energies of the dimer in the singlet ground state and in the lowest excited triplet state, respectively. The internal geometries of the two molecules monomers were kept rigid.

Figure 1 presents the results of calculations of T-T coupling in a model system of two hexatriene molecules facing each other while the molecules “slip” horizontally with respect to each other. Vertical



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separation between the molecules was fixed at 3.5 Å. Dimer T-T coupling was computed using the Fragment Spin Difference (FSD) method43 at the CIS/6-31G* level of theory44,45, as described in more detail in the Experimental and Computational Details section. The maximum coupling of ~0.1 eV is observed when the molecules fully overlap (slip=0Å, Fig. 1b). As the slip increases, the T-T coupling rapidly drops to zero at ~1Å and maximizes again at 2.6 Å. The oscillatory behavior of VT can be * understood in terms of the integral in Eq. 1 that contains intermolecular overlap products χ D ( x1 )χ A (x1 )

* and χ D ( x 2 )χ A ( x 2 ) of the LUMO and HOMO orbitals of monomers. As shown in Fig. 1a, in the case of * * full overlap (Fig. 1a, slip=0), χ D (x1 )χ A (x1 ) is positive, and χ D ( x 2 )χ A ( x 2 ) is negative, the integrand in

Eq. 1 is negative across the integration range, and VT is maximized. For slip=1Å, both overlap products alternate sign across the integration range, different parts of the integral cancel and VT approaches zero. The next maximum occurs at 2.6Å when the integrand in Eq. 1 is again negative across the integration range. The maxima in VT are roughly proportional to a square of the overlap area (dashed purple curve in Fig. 1b), as expected from Eq. 1. In a strong coupling regime, i.e., when T-T coupling is larger than the system-bath fluctuations, two delocalized excitonic states are formed in a dimer with the energies given by:5,21,46

E1 = ε + D − V12 E2 = ε + D + V12

(2)

where ε is the excited state energy of one pigment in the absence of the other pigment (assuming identical pigments), V12 is the intermolecular coupling, and D is the displacement energy that accounts for shift in individual pigment transition energy due to the presence of the second pigment. Here we adhere to the definition of D given by van Amerongen et al.46, which is the shift of the average excitation energy of the excitonic levels with respect to the excitation energy of noninteracting molecules.


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Fig. 1b shows the displacement energy, D, calculated for triplet states of the hexatriene dimer as a function of slip between the two molecules. In analogy with the coupling VT, D also oscillates as a function of slip distance, though these two energies oscillate with opposite phases. As a result, in the case of hexatriene dimer, one of the triplet exciton energies is almost independent of the slip distance and is roughly equal to the noninteracting molecular site energy ε, while the other state oscillates around it (Fig. 1c, energies E2 and E1, respectively). In case of the hexatriene dimer, the displacement energy has comparable contributions from inter-monomer electrostatic interactions and from the exchange interactions due to antisymmetrization of the overlapping dimer wave function. In general, the instantaneous change in electronic configuration upon promotion of a system from the ground to excited state is followed by vibrational relaxation of the molecular geometry. In the case of a dimer this means that the interplanar distance, the slip between the two molecules, as well as the internal geometry of each monomer can be different in the ground and excited states. Figure 1d depicts the calculated binding energies of the ground state (singlet, ES) and the lowest excited triplet state (ET) of the hexatriene dimer separated by 3.5 Å as a function of a slip distance. Indeed, while the equilibrium configuration in the ground state is observed at slip = 1.45 Å, in the lowest triplet state the molecules will be driven to adopt a new configuration with slip = 0 Å to minimize the total energy of the dimer. The model also predicts that the dimer is significantly stronger bound in the excited triplet state than in the ground state. Additionally, the vibrational relaxation in the lowest excitonic triplet state may introduce significant asymmetry to structures of the two chromophores leading to larger splitting between two excitonic states and preferable localization of excitation on one chromophore, a process similar to self-trapping or autolocalization of excitons in crystals.47,48 Since the vibrational relaxation usually occurs within a few picoseconds49,50 while a triplet excited state lifetime is typically in the micro- to milli-second range,7,21,51,52 to properly calculate the energy of the triplet-to-singlet phosphorescence, these geometrical changes must



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be accounted for. Note that hexatriene dimer here is used for illustrative purposes, and to simplify this example both the interplanar distance and internal geometries of monomers were kept fixed. These parameters, however, are also subject to change upon excitation as demonstrated below for more complex dimeric systems. Experimental and Computational Details Samples. Three samples containing dimers of known structure were made. The procedures for making these samples and evidence of their structures will be discussed in the following sections. Note that for all three samples a mixture of monomers and dimers resulted from the sample preparation. The use of the mixture of species ensured that there was minimum solvent difference experienced by monomers and dimers. Monomers of protoporphyrin IX (PPIX) disodium salt, pheophorbide a (Pheide a), and chlorophyll a (Chl a) along with all solvents and NaOH were purchased from Sigma-Aldrich and used without further purification. Deionized water was obtained from a Millipore Milli-Q system. The structures of these monomers are shown in Figure 2, along with the mutual orientation of the molecules in their respective dimers reported earlier. Since monomers and dimers are characterized by different absorption spectra, phosphorescence spectra of dimers could be separated from that of monomers by appropriate choice of excitation wavelengths. See Supporting Information for details.


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Figure 2. Structures of (a) - Protoporphyrin IX (PPIX), (b) - Pheophorbide a (Pheide a), (c) - Chlorophyll a (Chl a). Lower plane shows respective dimer structures determined by other groups (see text for details).

PPIX has been shown to form dimers in basic conditions.53,54 Following the method of Scolaro et al.53 a mixture of PPIX monomers and dimers was prepared by diluting a concentrated solution of PPIX in DMSO in a solution of 40% aqueous NaOH at pH 11.5 with 60% glycerin by volume, the final solution contained less than 5% DMSO by volume. Absorption spectra of the resulting monomer and dimer mixture matches the spectrum of Scolaro et al.53 with a broad, blue shifted (relative to the monomer) Soret peak plus a long wavelength shoulder coincident with the Soret band of the monomer. Varying the PPIX concentration changed the relative intensities of the broadened Soret peak and shoulder as would be expected for an equilibrium mixture of dimers and monomers (see Supporting Information). The structure of the PPIX dimer (Fig. 2a) is available from the 1H-NMR data of Janson and Katz.55 While this dimer structure was determined in chloroform, the absorption spectrum of PPIX in chloroform is similar to the spectrum in aqueous basic solution including growth of a broad, blue shifted Soret band at higher concentrations with a long wavelength shoulder.56 A plane to plane separation distance of approximately



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3.4 Å can be inferred from the dataset of Scheidt and Lee57, by comparison to aggregates and dimers of similar molecules (i.e. metal free porphyrins). The structure of a dimer in Fig. 2a also matches well the minimized structure obtained in our computational model as described later. All spectral measurements were performed at 77 K. Pheide a forms dimers of known structure in chloroform and were made for this study using the method of Closs et al. 58 Samples were then either plunge frozen at 77 K for phosphorescence measurements or, due to the poor optical properties of frozen chloroform (opaque snow), lowered to a temperature just above the freezing point of chloroform (213 K) for absorbance measurements. The basic structure of the Pheide a dimer (Fig. 2B) is obtained from the 1H-NMR (proton-Nuclear Magnetic Resonance) data of Closs et al.58 and fluorescence studies of Kooyman et al.59 A similar structure was predicted by our computational model (see Results section). The presence of two absorption bands centered at 667 nm and 695 nm shows that this sample was an equilibrium mixture of monomers and dimers (see Supporting Information) similar to the Pheide a monomer / dimer mixtures in aqueous solution produced by Eichwurzel et al.60 Chl a dimers of known structure were prepared following the method of Shipman61 and Cotton62 by dissolving Chl a at room temperature in non-polar solvent with a small amount of a nucleophile of the form R-OH (water saturated toluene) and plunging into liquid nitrogen (LN2). The toluene was saturated by adding an excess of deionized water (~0.5 mL water in 2-3 mL toluene) to a vial of toluene, which was then mixed with a vortex mixer and allowed to sit overnight. Upon freezing an additional absorption band centered at 708 nm was observed to the red of the monomeric Qy absorption band (centered at 673nm), closely matching the mixed monomer plus dimer absorption spectrum observed by Shipman61 and Cotton62, indicating that this sample was a mixture of monomers and dimers (see Supporting Information).


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The Chl a dimer structure proposed by Shipman et al. 61 is supported by data collected by a variety of methods. NMR and x-ray crystallography work with aggregates of Chl a and Chl a like molecules63–66, show that the monomers primarily overlap at the E and C rings while x-ray crystallography and Raman data show that water molecules are bound between the Chl a in dimer as shown in Fig. 2c (the oxygen is coordinated to the Mg2+ of one monomer and hydrogen bonds to R131 ketone of the neighboring monomer).61,66 Although the crystalline aggregates are unlikely to have the same relative monomer orientation as the free dimer due to crystal packing forces,67 the involvement of water or other OH containing nucleophilic solvent in dimer formation is known from the solvation studies of Livingston et al. and Cotton et al.62,68 Note, that as in the case of PPIX and Pheide a dimers, the Chl a dimer structure in Fig. 2c is in good agreement with the minimized dimer structure predicted by our computational model. Optical measurements. Absorption spectra were recorded with a Cary Bio 300 UV-Vis absorption spectrometer and phosphorescence spectra were recorded using the home-built time-gated phosphorimeter described earlier.21 Samples for low temperature absorption measurements were contained in custom 1 mm pathlength cells made by flame sealing one end of borosilicate glass rectangular tubing (Friedrich & Dimmock, LRT-1-10-100), while samples for phosphorescence measurements were contained in 3 mm ID quartz EPR tubes (Wilmad-LabGlass, 707-SQ-100M) and loaded into an optical cryostat at 77K (Oxford Instruments, Optistat DN). Phosphorescence band maxima were measured with an absolute ±10 nm precision, but since spectra for dimers and monomers for each sample were measured in the same experiment by changing only excitation wavelength, the error in relative peak positions of respective phosphorescence bands was only ~1 meV. Computational methods. Hexatriene dimer was constructed from hexatriene monomers optimized using density functional theory (DFT) at the B3LYP/6-31G* level of theory69,70 and vertical separation was fixed at 3.5 Å.



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A porphyrin dimer was used as a model system for the PPIX dimer. For calculations with fixed monomer geometries, porphyrin dimers were constructed from monomers optimized with B3LYP/6311G**. Dimer T-T couplings were computed using the Fragment Spin Difference (FSD) method developed by You et al.43 in which T-T couplings are obtained from analysis of the spin-density of triplet excited states in the dimer:

( Em − En ) ∆smn 2 2 ( ∆snn − ∆smm ) + 4∆smn

VFSD =

,

(3)

where Em/n are excitation energies of triplet states m and n. ∆ is a spin difference operator:

∆smn = ∫

r∈D

σ mn ( r ) − ∫ σ mn ( r ) = smn ( D ) − smn ( A) r∈ A

(4)

D and A define the donor and acceptor spaces; = − is a spin density function

defined as the difference between one-electron α and β electron density functions . In case of m=n a spin density function of the nth electronic state is used, while for m≠n a transition spin density between the mth and nth electronic states is utilized. Thus, FSD is similar in spirit to the generalized MullikenHush (GMH)71 and fragment excitation density (FED)72 methods and describes the electronic coupling as the off-diagonal element of the Hamiltonian matrix in which diagonal elements are spin-localized states. Configuration Interaction Singles (CIS) method in the 6-31G* basis set was used for evaluating T-T couplings and displacement energies along a slip coordinate in porphyrin dimer. The ωB97x-d/6-31+G* level of theory73 was used to optimize geometries of the ground singlet state in the porphyrin monomer and dimer. Monomer and dimer triplet states were optimized with time-dependent DFT (TD-DFT) with the same functional and basis set ωB97x-d/6-31+G*. Employing TD-DFT instead of DFT for the triplet states was necessary for converging to the lowest-energy triplet and for avoiding instabilities due to mixing of several almost degenerate triplet states. T-T couplings were also computed


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at the optimized dimer geometries, using the FSD with TD-DFT ωB97x-d/6-31+G*. Adiabatic energy differences between singlet and triplet states in monomer and dimer were obtained as total electronic energy differences between singlet and triplet states computed using TD-DFT for energies of the triplet states. The second way of computing S-T adiabatic energy differences, using ground state DFT (unrestricted DFT in case of the triplet states), which is referred to as ∆DFT, was also used for the other dimers but not porphyrin (due to instabilities of the triplet state DFT solution). The Pheide a system was described in two ways. A base-free chlorin was used to model triplet couplings and displacement energies in the dimer with fixed monomer geometries, optimized at B3LYP/6-31G*. Similarly to porphyrin, couplings were computed with the FSD method at the CIS/631G* level. Full Pheide a molecules were used for explicit calculations of singlet-triplet energy gaps at fully optimized monomer and dimer geometries. In these calculations, DFT with ωB97x-d/6-31G* was used for both singlet and triplet states (unrestricted wave functions were employed for triplets). S-T adiabatic energy differences and T-T couplings were computed using these optimized geometries of Pheide a with ωB97x-d/6-31G*. The Chl a dimer was modeled as a Mg-chlorin dimer, i.e. as a stripped down Chl a molecules, where all substituent groups (e.g. R3 vinyl, R17 tail, etc.) are replaced by hydrogens except for the R131 ketone and where the monomers are bound by coordination and hydrogen bonding interactions with two water molecules. Coordinating water molecules were excluded in constructing slices along varying slip separations, but included in full dimer geometry optimizations. Two (fixed) geometries of monomers were used for building slices with varying slip distances. One was a ground state singlet monomer geometry optimized in gas phase with B3LYP/6-31G**; another one was monomer geometry extracted from the optimized structure of the chlorin dimer with coordinating water molecules (B3LYP/6-31G**). Couplings along the slices were computed with the FSD at the CIS/6-31G* level of theory. Full geometry optimizations of the chlorin monomer with one coordinating water and of the chlorin dimer with two



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coordinating waters were performed with ωB97x-V/6-311G*.74,75 Triplet states were optimized using unrestricted triplet state DFT. Using these geometries, ∆DFT S-T adiabatic energy differences were computed with ωB97x-V/6-311G*, and T-T couplings, displacement energies and TD-DFT S-T energy differences were computed with TD-DFT ωB97x-d/6-31+G*. All calculations were performed in the Q-Chem electronic structure package.76 Geometries of all relevant structures are provided in the Supporting Information. Results

(a)

(b)

(c)

Figure 3. The phosphorescence spectra (monomer - black, dimer – blue) for (a) PPIX, (b) Pheide a and (c) Chl a. Grey lines represent unsmoothed spectra.


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TABLE 1. Measured and computed triplet state energies of monomeric and dimeric molecules. ET – phosphorescence transition energy, ∆ET – difference between monomeric and dimeric phosphorescence transition energies. The energies were computed using time dependent density functional theory (TDDFT) or as a difference between DFT calculations for the singlet and triplet states (Δ DFT). Measured Molecule

PPIX

Computed

ET

∆ET

ET (TDDFT)

∆ET (TDDFT)

∆ET (∆ DFT)

(eV)

(eV)

(eV)

(eV)

(eV)

1.533

Monomer

1.570

Dimer

1.505

1.482

Monomer

1.312

1.317

Dimer

1.272

1.278

Monomer

1.285

1.320

Dimer

1.240

0.052

0.065

Pheide a

0.040

Chl a

0.045

1.244


0.039

0.041

0.077

0.069


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TABLE 2. T-T coupling VT, displacement energy D and the triplet state energy shift in dimer with respect to monomer, ∆ET,vert = VT-D, computed at the singlet (ground) state optimized geometry, i.e., vertical excitonic energy lowering. Reorganization energies in monomer (λ λmon) and dimer (λ λdim) due to vibrational relaxation of the triplet state and their difference ∆λ λ = λdim - λmon. The total (adiabatic) energy lowering of the triplet state in the dimer with respect to the monomer, ∆ET,adiab = VT – D + ∆λ, that can be directly compared with the phosphorescence measurements. In addition, slip distance (slip) between the centers of pigments and plane-to-plane separation (h) are given for singlet and triplet optimized dimer geometries. ∆ET,vert VT, meV

D meV

λmon / λdim

∆λ λ

meV

meV

(VT - D)

∆ET,adiab (VT – D + ∆λ λ)

meV

meV

slip / h

slip / h

singlet geom

triplet geom

Å

Å

PPIX

5

-12

17

158 / 195

37

54

1.697 / 3.280

1.754 / 3.247

Pheide a

2

-12

14

248 / 275

27

41

2.787 / 3.229

2.730 / 3.173

Chl a coordinated by water

6

-47

53

269 / 253

16

69

6.911 / 3.453

6.930 / 3.426

9

-32

41

238 / 227

-11

30

6.911 / 3.453

6.930 / 3.426

Chl a no water

PPIX

PPIX is closely related to (bacterio)chlorophyll molecules widely represented in natural photosynthesis, and its flat geometry also makes the results relevant to a wide class of organic molecules forming stacked aggregates that are actively studied for possible photovoltaic applications10,11,16. Phosphorescence spectra of the PPIX monomer and dimer are shown in Fig. 3a. The major blue-most peaks in these spectra correspond to the adiabatic transition between the vibrationally relaxed excited triplet state and singlet


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ground state, i.e. 0-0 transition (the longest red arrow in Fig. 4). This peak shifts by 65 meV upon dimerization of PPIX (Fig. 3a and Table 1). The long wavelength tail/peaks in phosphorescence spectra correspond to transitions to vibrationally hot ground state (shorter red arrows in Fig. 4). The higher intensity of the vibrational sideband in the dimer in comparison to the monomer indicates a larger extent of the change in the geometry of the excited triplet state in respect to the ground singlet state. This is consistent with our computational predictions as discussed below. Note that while the triplet state is excited via intersystem crossing from a significantly higher energy singlet excited state, phosphorescence will occur almost exclusively from the lowest triplet state, since vibrational relaxation in such molecules usually occurs within a picosecond or less 77–79, while triplet state lifetimes are typically in the order of 100 µs or longer 21,80.

Figure 4. A simplified energy diagram of the ground singlet and excited triplet states showing phosphorescence transitions and computed energy differences presented in Tables 1 and 2.



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A porphyrin molecule (inset Fig. 5b) was chosen to model PPIX. Porphyrin possesses the same πelectron ring structure, but lacks the sidechains of PPIX that are not critical for T-T coupling. As in the case of the hexatriene dimer (Fig. 1), the nodal structure of the porphyrin wave function leads to oscillations in both VT and D when the two molecules with fixed geometries are shifted relative to each other (see Fig. 5a and 5b).

Figure 5. (a) T-T couplings, VT, and (b) displacement energies, D, at three interplanar separations 3.4 Å (black), 3.7 Å (blue) and 4.0 Å (red) as a function of slip coordinate (diagonal displacement) in the freebase porphyrin dimer. (c) Binding energies of the ground singlet state (ES, blue) and the lowest triplet state (ET, red) at a fixed 3.4 Å interplanar separation and fixed internal monomeric geometries. Slip position 1.7 Å corresponding to minimum of the ground singlet state is indicated with a vertical dashed line.


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Similarly to the case of the hexatriene dimer, oscillations in VT and D are out of phase, but decay quicker with increasing slip, as expected for flat molecules in a planar contact vs linear hexatriene. Likewise, coupling and displacement decrease exponentially with the increasing interplanar distance between the molecules (Fig. 5a and 5b show energies at interplanar distances 3.4, 3.7 and 4.0 Å). Figure 5c also predicts that the geometry of the porphyrin dimer should change upon excitation from the ground singlet state into the triplet state – the binding energy in the ground state (ES) minimizes at a slip of ~1.7 Å, while in the excited triplet state two minima in the binding energy ET are observed at ~1 Å and ~2.2 Å. However, a full geometry optimization of the triplet state at the TD-DFT (ωB97x-d/6-31+G*) level of theory, where the monomers are free to rotate in addition and slip in any direction, predicts that the most stable structure corresponds to a slip of ~1.75 Å and an interplanar separation of 3.25 Å, which is quite similar to the singlet geometry with a diagonal slip of 1.70 Å and an interplanar distance of 3.28 Å, which are similar to that in experimentally determined geometry of the PPIX dimer

55

(see Fig. 2a for the

structure of PPIX dimer inferred from Janson et al. 55). Using these optimized slip and interplanar values, but fixed geometries of monomers as in Fig. 5, displacement and coupling energies at the CIS/6-31G* level of theory are 1 meV and 14 meV for the singlet geometry and -2 meV and 12 meV for the triplet geometry, respectively (See Table S1 in Supporting Information). Thus, if only changes in inter-monomer degrees of freedom are taken into account, the total lowering of the S-T energy due to excitonic interactions in the dimer is estimated to be 13 and 14 meV in the singlet and triplet geometries of the dimer, respectively. Interestingly, changes in the internal geometries of the monomers upon dimerization do not affect significantly the values of couplings and displacement energies. For example, couplings at the optimized singlet and triplet geometries are 11 meV, comparable to the discussed above 14 and 12 meV couplings at the fixed monomer’s geometries. At the ωB97x-d/6-31+G* level of theory, displacement and coupling at both singlet and triplet geometries are -12 meV and 5 meV (see Table 2 and



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Table S1). To summarize, if only electronic coupling and displacement energy are taken into account, expected S-T energy lowering upon dimerization would be 10-20 meV, which is significantly smaller than the measured value of 65 meV. As we show below, the major contribution to the observed spectral shift in dimer triplet energy originates from the internal vibrational relaxation of molecules. Our calculations show that the internal geometries of the two molecules constituting the dimer in the ground singlet state are identical (i.e., it is a homodimer). However, this symmetry is broken in the triplet excited state where the two molecules adopt significantly different geometries. While one molecule remains in the singlet state geometry, the other one relaxes to the geometry somewhat similar to that of a monomeric molecule in triplet excited state. As a consequence, the lowest triplet excitation in the porphyrin dimer becomes fully localized on the latter molecule, the upper triplet state is localized on the molecule that remains in the singlet state geometry, and the splitting between the two states (354 mV) is dominated by geometric asymmetry of the two molecules. The amount of localization of excitonic states can be deduced from natural transition orbitals (NTO) and corresponding participation ratio

81–83

– see

Supporting Information for detail. In both the monomer and dimer, the triplet state undergoes vibrational relaxation characterized by reorganization energy λ, which is defined here as the energy lowering of the triplet state due to conformational transition from the optimal singlet to optimal triplet geometries (Fig. 4). The amount of such vibrational relaxation in the excited state can be generally deduced from the shape of the Frank-Condon spectrum, with the larger intensity of the vibrational band reflected in larger values of λ. Indeed, the computed values λdim>λmon for porphyrin (Table 2) are in agreement with the higher intensity of the vibrational band in the phosphorescence spectrum of the PPIX dimer in comparison with that of the monomer (Fig. 3a). Adding the effect of the vibrational relaxation in the monomer and in the dimer to coupling and displacement energies brings the calculated energy lowering of the T-S gap in the dimer to ∆ET,adiab = 54 meV (direct TD-DFT calculations result in 52 meV, see Tables 1 and 2). This number is in reasonable


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21

agreement with the experimentally observed difference of 65 meV between phosphorescence maxima of the PPIX dimer and monomer. In the case of the PPIX dimer, the vibrational relaxation is the major contribution to the observed shift of T-S phosphorescence in respect to the monomer. Thus, triplet-triplet interaction in the PPIX dimer corresponds to the weak exciton coupling limit. Pheide a The phosphorescence spectra of the Pheide a monomer and dimer are shown in Fig. 3b. Upon dimerization the phosphorescence maximum shifts by 40 meV (Table 1).

Figure 6. T-T coupling VT and displacement energies D for the Pheide a dimer as a function of slip distance calculated for an interplanar separation of 3.4 Å. The internal geometries of the two monomers were kept rigid.

A base-free chlorin was used to model triplet couplings and displacement energies as a function of slip distance in the dimer with a fixed intramonomer geometry (Fig. 6), since the presence of bulky fixedgeometry tails would affect these dependencies and introduce artifacts. However, the full Pheide a molecule was used for explicit calculations of singlet-triplet energy gaps at optimized monomer and dimer geometries, as tails, to a large extend, determine the relative positions of monomers in the dimer.



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Figure 6 shows T-T coupling VT and displacement energy D as a function of the slip, while interplanar separation between two base-free chlorins is kept at 3.4 Å. The latter distance is only slightly larger than the interplanar distances of 3.23 Å in the optimized singlet geometry and 3.17 Å in the optimized triplet geometry of the Pheide a dimer. Similar to porphyrin, the free-base chlorin dimer experiences oscillatory behavior of coupling and displacement energies. Moreover, oscillations of the coupling in the chlorin dimer closely follow those in the porphyrin dimer (Fig. 5), with maxima and minima found at the same slip distances, at least up to a slip of ~4 Å. However, unlike in the porphyrin dimer, the displacement energy in the free-base chlorin is always negative (attractive), tentatively due to favorable dipole-dipole interactions between the polar carbonyl groups of the chlorins. At the slip distances ~2.7-2.8 Å (as in the optimized singlet and triplet geometries of Pheide a dimer), coupling and displacement energies in the chlorin dimer are 8 meV and -33 meV, respectively, resulting in 41 meV shift of the lowest triplet state energy in the chlorin dimer in respect to that of a monomer (at the CIS/6-31G* level of theory). The corresponding values with TDDFT ωB97x-d/6-31G* for the optimized Pheide a dimer are smaller (2 meV and -12 meV at the singlet geometry), but as in chlorin the large negative displacement energy supersedes the considerably weaker coupling. Using the latter values, the phosphorescence shift upon dimerization of Pheide a is expected to be 14 meV, which is significantly smaller than the experimentally measured value of 40 meV. Similarly to porphyrin, the remaining energy shift originates from the difference in vibrational relaxation of the triplet states in the monomer and dimer. In case of Pheide a, ∆λ = 27 meV (Table 2) and ∆ET,adiab becomes 41 eV, which is in excellent agreement with the experiment.


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Chlorophyll a The Chl a monomer and dimer phosphorescence spectra are shown in Fig. 3c. The monomeric emission maximum (964±10 nm) is in good agreement with the previously measured 6-coordinated monomer (973±4 nm21), while the measured dimeric peak at 1000±10 nm agrees well with the 990 nm peak reported by Krasnovsky et al.84 for a possible Chl a dimer in pentane.

Figure 7. Coupling VT and displacement energy D in the Chl a dimer as a function of slip distance with (a) flat monomer geometries and (b) with monomer geometries distorted due to coordination of Mg center by a water molecule. Inter-planar separation was fixed at 3.4 Å.

The Chl a dimer was modeled as a chlorin dimer with Mg atoms as described in Experimental and Computational Details. Coordinating water molecules were excluded in constructing slices along varying slip separations but included in the full dimer geometry optimizations. Note that even though in the case



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of both the Chl a dimer and Pheide a dimer the monomers are displaced along the line connecting the center of the ring and two N atoms, the relative orientation of the monomers is different in Pheide a and Chl a (Fig. 2). As with the other dimers, triplet state lowering in this dimer can be attributed to the T-T coupling, displacement and vibrational relaxation terms. The total computed triplet state energy lowering in the dimer is 77 meV (from TDDFT calculations) or 69 meV (from direct ∆DFT calculations and from analysis of various components of the interaction energies), which are in a reasonable agreement with the experimental observation of 45 meV. A unique feature of Chl a dimer is the presence of two water molecules coordinating the Mg atoms. To analyze the effect of this microhydration on the dimer triplet state energies, we computed the dependencies of coupling VT and displacement energy D in two dimers, one constructed of two flat unperturbed Chl a molecules (Fig. 7a) and the second constructed of two Chl a monomers whose structures are perturbed due to coordination of Mg center by water as in the optimized geometry of the water-coordinated chlorin dimer (Fig. 7b). As the result of water coordination, the Mg atom is pulled out of the macrocycle plane toward the coordinating oxygen, i.e. in the dimer, the Mg atoms of the two Chl a molecules shift toward each other. Note that these coordinating waters were not included in the computations shown in Fig. 7b, as they would preclude smooth change in the slip distance. While in general these two sets of curves (with flat and distorted monomers) look similar, there are important differences. At small slip distances, flat monomers interact stronger and the magnitudes of both coupling and displacement energies are larger. This is because in the perturbed Chl a geometries, the Mg atoms in the dimer are closer to each other resulting in a repulsive steric effect. However, at the slip separations corresponding to the optimal dimer structure (~6.9 Å), interaction between the perturbed monomers is stronger leading to a larger displacement energy (-13 meV for flat monomers vs -24 meV for perturbed monomers). Thus, one effect of microhydration is a favorable geometry change in the interacting monomers that lowers the energy of the triplet state in the dimer. However, as in case with


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other complexes, accounting for coupling and displacement energies only is not sufficient for explaining the experimentally observed shift of 45 meV. An alternative way to analyze the effect of microsolvation is to compare the values of coupling, displacement and vibrational relaxation energies at the fully optimized geometry (including intramolecular optimization) of the Chl a dimer coordinated by two water molecules and at the same geometry with coordinating waters being removed (Table 2). When coordinating waters are removed, the dimer experiences larger coupling (9 meV vs 6 meV) but smaller displacement energy (-32 meV vs -47 meV). The increase in the magnitude of the displacement energy upon microsolvation makes sense as the complex becomes more polar. Thus, the second effect of microsolvation is a further increase (by magnitude) in the displacement energy due to enhancement in electrostatic interactions between monomers. Additionally, coordinating waters contribute to vibrational relaxation. When these waters are excluded, vibrational relaxation destabilizes the triplet state in the dimer with respect to the triplet in the monomer. To summarize, microsolvation plays a significant role in lowering of the triplet state energy in the dimer, with the strongest effect originating from a significant increase in the displacement energy. Conclusions In conclusion, the QM modeling supported by experimental data reveals that the horizontal slip distance between the interacting molecules is crucial for computing triplet state properties as it can modulate both inter-pigment interactions and displacement energies by orders of magnitude. It is also shown that vibrational relaxation following the formation of a triplet state can cause a substantial additional shift of the triplet state energy in a coupled system, as well as lead to a significant asymmetry in the geometry of the two molecules constituting the dimer and consequent localization of the excitation on one of the monomers. In the case of water assisted dimerization, as in Chl a, the effect of microhydration on triplet energy shift is not negligible and has to be taken into account to properly



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compute the relevant transition energies. Based on the presented analysis, it is expected that these conclusions are rather general and should apply to a large class of molecular dimers and aggregates. Supporting Information Supporting Information contains information regarding the UV-Vis absorbance of PPIX under acidic and basic conditions and the separation of monomeric and dimeric phosphorescence emission spectra. Optimized molecular structures of considered molecules are also provided. This material is available free of charge via the Internet at http://pubs.acs.org. Acknowledgements DAH and SS acknowledge support by the Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences of the U.S. Department of Energy through grant DESC0001341. LVS acknowledges support of the National Science Foundation (grant CHE-1465154). This research was supported in part through computational resources provided by Information Technology at Purdue, West Lafayette, Indiana.


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