analytical derivation with a model Hamiltonian, we propose proxies for estimating the coupling strength in both diabatic and adiabatic pictures for general SEIC reactions. In the diabatic picture, we demonstrated the existence of a bilinear relationship between the coupling strength and molecular orbital overlap, which provides a practical way to predict diabatic couplings. In the adiabatic picture, we showed that nonadiabatic couplings can be approximated by simple functions of the wave function CT coefficients. These approaches were verified through the investigation of singlet oxygen photosensitization, where both [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] Δg and 1 g oxygen states can be competitively generated by a triplet fusion reaction. The interplay between the CT-mediated mechanism, the spatial factors of the bimolecular complex, and the electronic structure of the oxygen molecule during the reaction explains the curiously small coupling to the 1 g state along specific incidence directions. The results from both the diabatic and adiabatic pictures provide a comprehensive understanding of the reaction mechanism, which applies to general SEIC problems.

INTRODUCTION

Spin-exchange internal conversion (SEIC) forms a class of reactions where the global spin of a molecular assembly is conserved, while the spin of the monomers is changed. The most famous example of such a reaction is the singlet fission (SF), [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF][START_REF] Zimmerman | A Correlated Electron View of Singlet Fission[END_REF][START_REF] Zimmerman | Mechanism for Singlet Fission in Pentacene and Tetracene: From Single Exciton to Two Triplets[END_REF][START_REF] Smith | Singlet Fission[END_REF] following the general formula

1 1 1 1 3 3 A B A B     + → +     (1)
which has been under scrutiny recently due to its potential for enhancing the efficiency of organic photovoltaics (usually A and B are the same molecule). [START_REF] Busby | A design strategy for intramolecular singlet fission mediated by charge-transfer states in donor-acceptor organic materials[END_REF][START_REF] Paci | Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found?[END_REF] Another example of SEIC is the internal-conversion triplet fusion reaction (TF)

1 3 3 1 1 1 A B A B .     + → +     (2)
This reaction yielding a singlet complex is one among nine possible outcomes, the other being five quintet complexes and three triplet complexes. Reaction ( 2) is expected to dominate the fusion rate because of internal conversion tends to be faster than intersystem crossing. [START_REF] Abdel-Shafi | Electronic to vibrational energy conversion and charge transfer contributions during quenching by molecular oxygen of electronically excited triplet states[END_REF][START_REF] Schmidt | Quantitative determination of (1)Sigma(+)(g) and (1)Delta(g) singlet oxygen in solvents very different polarity. General energy gap law for rate constants of electronic energy transfer to and from O-2 in the absence of charge-transfer interactions[END_REF] This reaction was recently applied on functional molecular devices as the formally inverse reaction of singlet fission. [START_REF] Kondakov | Triplet annihilation exceeding spin statistical limit in highly efficient fluorescent organic light-emitting diodes[END_REF][START_REF] Jia | Molecular Spacing Modulated Conversion of Singlet Fission to Triplet Fusion in Rubrene-Based Organic Light-Emitting Diodes at Ambient Temperature[END_REF] It is also at the core of singlet oxygen photosensitization, [START_REF] Bai | Divide to Conquer: A Kinetic Model for Singlet Oxygen Photosensitization[END_REF][START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF][START_REF] Schweitzer | Mechanism of photosensitized generation of singlet oxygen during oxygen quenching of triplet states and the general dependence of the rate constants and efficiencies of O-2((1)Sigma(+)(g)), O-2((1)Delta(g)), and O-2((3)Sigma(-)(g)) formation on sensitizer triplet state energy and oxidation potential[END_REF][START_REF] Schmidt | The Effect of Solvent Polarity on the Balance between Charge Transfer and Non-Charge Transfer Pathways in the Sensitization of Singlet Oxygen by ππ* Triplet States[END_REF] which happens through

1 3 3 1 1 1 1 1 2 2 PS O PS O or     + → +       ( ) (3)
where PS is a photosensitizer.

Theoretical and computational chemistry have contributed to the study of SEIC reactions since the early 1960s [START_REF] Merrifield | Theory of Magnetic Field Effects on the Mutual Annihilation of Triplet Excitons[END_REF][START_REF] Jortner | Triplet Excitons in Crystals of Aromatic Molecules[END_REF] and many recent advances in both fronts, SF [START_REF] Busby | A design strategy for intramolecular singlet fission mediated by charge-transfer states in donor-acceptor organic materials[END_REF][START_REF] Yost | A transferable model for singlet-fission kinetics[END_REF][START_REF] Feng | Fission of Entangled Spins: An Electronic Structure Perspective[END_REF][START_REF] Yang | First-Principle Characterization for Singlet Fission Couplings[END_REF] and TF, [START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF][START_REF] Serrano-Perez | Singlet oxygen generation in PUVA therapy studied using electronic structure calculations[END_REF] have brought new insights on the molecular mechanism and the reaction rate calculation. As usual in any unimolecular reaction, both the energy of the involved states and the coupling between them should be considered to estimate reaction rates. [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] State energies can be routinely calculated with quantum chemical methods, either to the entire complex or considering each monomer individually. [START_REF] Paci | Singlet fission for dye-sensitized solar cells: Can a suitable sensitizer be found?[END_REF][START_REF] Greyson | Singlet Exciton Fission for Solar Cell Applications Energy Aspects of Interchromophore Coupling[END_REF][START_REF] Smith | Recent Advances in Singlet Fission[END_REF] However, the calculation of the coupling strength for SEIC reactions-usually an intermolecular energy transfer processes in weak-coupling regime-is still a challenging task, especially considering that systems of interest may correspond to large molecules in floppy organic crystals. For instance, in the search for the optimal way to treat these couplings, topics like the role of charge transfer (CT) configurations and the choice between diabatic and adiabatic pictures have been intensely discussed. [START_REF] Zimmerman | A Correlated Electron View of Singlet Fission[END_REF][START_REF] Smith | Singlet Fission[END_REF][START_REF] Feng | Fission of Entangled Spins: An Electronic Structure Perspective[END_REF][START_REF] Beljonne | Charge-Transfer Excitations Steer the Davydov Splitting and Mediate Singlet Exciton Fission in Pentacene[END_REF][START_REF] Monahan | Charge Transfer-Mediated Singlet Fission[END_REF] The diabatic picture has been a popular option because the diabatic coupling can be directly applied in the rate models such as the Fermi's golden rule. [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF][START_REF] Yost | A transferable model for singlet-fission kinetics[END_REF] Many different approaches to estimate the diabatic coupling have been proposed mainly based on two strategies. The first strategy is the direct calculation from predefined diabatic wave functions, [START_REF] Grofe | Diabatic-At-Construction Method for Diabatic and Adiabatic Ground and Excited States Based on Multistate Density Functional Theory[END_REF] such as in the constrained density functional theory with configuration interaction (CDFT-CI) method, using the constraint of charge/spin density. [START_REF] Yost | A transferable model for singlet-fission kinetics[END_REF][START_REF] Wu | Configuration interaction based on constrained density functional theory: A multireference method[END_REF] Direct calculation holds a clear physical meaning, but it may miss intermolecular electronic correlations. [START_REF] Zimmerman | A Correlated Electron View of Singlet Fission[END_REF] The second strategy is diabatization based on the calculation of adiabatic states, e.g., with the fragment spin difference method, [START_REF] Yang | First-Principle Characterization for Singlet Fission Couplings[END_REF] and the N/D method based on nonadiabatic couplings (NAC) and differences of energy gradients. [START_REF] Varga | Direct diabatization based on nonadiabatic couplings: the N/D method[END_REF] Although globally applicable diabatization is still a challenging subject in theoretical chemistry, [START_REF] Van Voorhis | The Diabatic Picture of Electron Transfer, Reaction Barriers, and Molecular Dynamics[END_REF][START_REF] Karman | Diabatic states, nonadiabatic coupling, and the counterpoise procedure for weakly interacting open-shell molecules[END_REF] various effective diabatization approaches have been developed, [START_REF] Li | Model space diabatization for quantum photochemistry[END_REF][START_REF] Wittenbrink | A new approach for the development of diabatic potential energy surfaces: Hybrid block-diagonalization and diabatization by ansatz[END_REF] especially with recent breakthroughs for relative large molecular systems. [START_REF] Subotnik | The requisite electronic structure theory to describe photoexcited nonadiabatic dynamics: nonadiabatic derivative couplings and diabatic electronic couplings[END_REF][START_REF] Xie | Construction of Vibronic Diabatic Hamiltonian for Excited-State Electron and Energy Transfer Processes[END_REF] The adiabatic picture has also been widely applied to study SEIC. [START_REF] Zimmerman | A Correlated Electron View of Singlet Fission[END_REF][START_REF] Feng | Fission of Entangled Spins: An Electronic Structure Perspective[END_REF] Nonadiabatic couplings can be directly computed based on the wave function of the whole molecular complex, with the intermolecular correlation automatically considered. Nevertheless, the adiabatic picture also has two obvious disadvantages. First, NAC is not connected to the reaction rate as conveniently as the diabatic coupling [START_REF] Kolomeisky | A Simple Kinetic Model for Singlet Fission: A Role of Electronic and Entropic Contributions to Macroscopic Rates[END_REF] (although it is directly used in nonadiabatic mixed quantum-classical dynamics simulation [START_REF] Crespo-Otero | Recent Advances and Perspectives on Mixed Quantum-Classical Dynamics[END_REF] ). Second, NAC is not a quantity readily available for most of quantum chemical levels. [START_REF] Lischka | Multireference Approaches for Excited States of Molecules[END_REF] Such practical difficulty to calculate NACs has led to a search for suitable proxies to replace them. For instance, researching an SF reaction, Morrison and Herbert took the direct product of monomeric states as the wave function basis within the ab initio Frenkel-Davydov exciton model, [START_REF] Morrison | Analytic derivative couplings and first-principles exciton/phonon coupling constants for an ab initio Frenkel-Davydov exciton model: Theory, implementation, and application to compute triplet exciton mobility parameters for crystalline tetracene[END_REF] to calculate the adiabatic state and NAC, instead of the direct calculation of the entire dimer system. [START_REF] Morrison | Evidence for Singlet Fission Driven by Vibronic Coherence in Crystalline Tetracene[END_REF] In another study, Feng, Luzanov, and Krylov proposed a proxy for NAC based on the reduced one-particle transition between initial and final adiabatic states and used it to evaluate the NAC strength variation with different intermolecular arrangements. [START_REF] Feng | Fission of Entangled Spins: An Electronic Structure Perspective[END_REF] In a recent series of research works, we have focused on the TF reaction given in eq (3), the photosensitization reaction yielding singlet oxygen. [START_REF] Bai | Divide to Conquer: A Kinetic Model for Singlet Oxygen Photosensitization[END_REF][START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] To deal with weakly coupled floppy complexes, we proposed the Divide-to-Conquer (DtC) model for reaction rates in ref [START_REF] Bai | Divide to Conquer: A Kinetic Model for Singlet Oxygen Photosensitization[END_REF] (for convenience, it is also briefly discussed in the Supporting Information, SI). In the DtC model both diabatic and nonadiabatic couplings are available for the same reaction coordinates, and they are directly correlated to each other by a simple function. This feature provides us with an excellent opportunity to clarify the relationship between adiabatic and diabatic pictures in SEIC reactions. Moreover, to work with the heterodimer PS-O2 (rather than with homodimers usually adopted in this type of investigation) also enriches the prototypical range of SEIC research.

Our goal in this work is to derive proxies for diabatic and nonadiabatic couplings for SEIC reactions.

These proxies should play a double role of 1) allowing a straightforward estimate of couplings for large systems and 2) providing a direct interpretation of the reaction mechanisms. These proxies will be discussed and validated against quantum-chemical results from our previous investigations on the PS-O2 complex.

THEORETICAL FRAMEWORK

Diabatic Picture

In the diabatic picture, we consider the reaction with several different diabatic states: the initial state 𝜓 𝑖𝑛𝑖 , the final state 𝜓 𝑓𝑖𝑛 , and the intermediate CT state 𝜓 𝐶𝑇 . [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF][START_REF] Monahan | Charge Transfer-Mediated Singlet Fission[END_REF] To build a general picture of the SEIC reaction, we start with an AB complex consisting of molecules A and B, where A and B can be identical or not.

For SEIC reaction, the Förster interaction is spin forbidden, [START_REF] Skourtis | Dexter energy transfer pathways[END_REF] while Dexter interaction is necessary, [START_REF] Dexter | A Theory of Sensitized Luminescence in Solids[END_REF] and the diabatic coupling between the states arises from two types of interaction paths: a direct interaction and a CT-mediated superexchange interaction. [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] As an example, the interaction pathways of one special case belonging to the TF reaction are visualized in Figure 1. With these two CT states as the mediate, the total diabatic coupling is given by 1, 39, 41

     = - (4)
where CT1 and CT2 correspond to the two CT states (A -B + and A + B -, as shown in Figure 1), which can connect the initial and final states through one-electron transfer paths. Ĥ is the electronic Hamiltonian operator, and i CT E is the energy of the CT state i taking the initial state energy as the reference. From the perspective of A, these two interaction pathways are i) first-donate-then-accept-electron (top) and ii) firstaccept-then-donate-electron (bottom). In eq (4), the first term in the right side corresponds to coupling through the CT-mediated mechanisms, and the second term is the coupling through the direct mechanism.

Assuming that the quantum interstate electronic coupling is based on the orbital overlap, a linear relationship between the electronic coupling and the overlap of the corresponding diabatic state wave functions was proposed to evaluate electron transfer coupling approximately, [START_REF] Troisi | Hole Migration in DNA: a Theoretical Analysis of the Role of Structural Fluctuations[END_REF] which has been confirmed in various works. [START_REF] Gajdos | Ultrafast Estimation of Electronic Couplings for Electron Transfer between pi-Conjugated Organic Molecules[END_REF][START_REF] Hoffmann | An Extended Hückel Theory. I. Hydrocarbons[END_REF][START_REF] Cheung | Theoretical Study of the Organic Photovoltaic Electron Acceptor PCBM: Morphology, Electronic Structure, and Charge Localization[END_REF] In Ref. [START_REF] Gajdos | Ultrafast Estimation of Electronic Couplings for Electron Transfer between pi-Conjugated Organic Molecules[END_REF] they have tested the simple expression

âCT a CT a CT a CT H C C        = =  (5)
where aCT  is the electronic coupling between the initial state ( The linear relationship works very well for a very diverse set of donor-acceptor pairs in different random orientations and distances, sharing the same constant C. [START_REF] Gajdos | Ultrafast Estimation of Electronic Couplings for Electron Transfer between pi-Conjugated Organic Molecules[END_REF] For simplicity, we assume the term C as the same constant. (Although even using different C for different coupling terms, the conclusion will not change.) C was reported to lie between 3 and 15 eV in different studies, [START_REF] Troisi | Hole Migration in DNA: a Theoretical Analysis of the Role of Structural Fluctuations[END_REF][START_REF] Gajdos | Ultrafast Estimation of Electronic Couplings for Electron Transfer between pi-Conjugated Organic Molecules[END_REF][START_REF] Cheung | Theoretical Study of the Organic Photovoltaic Electron Acceptor PCBM: Morphology, Electronic Structure, and Charge Localization[END_REF] and the variation may be due to the accuracy of the overlap calculation.

The coupling terms for the CT-mediated mechanism in eq (4) are exactly the charge-transfer coupling appearing in eq (5). Therefore, we can rewrite the CT-mediated diabatic coupling in terms of the overlap of molecular orbitals

2 2 ' ' 1 '' ˆˆ11 ii i ini CT CT fin A B B A i CT A B A B A B B A HH C E E E              + - -+ =   =+   =  (6)
which is simpler to be analyzed. In this equation, ( )

2 1 1 A B A B C E E  + - -+ -- =+ 
As shown in eq (6), the two CT-mediated pathways give the same orbital overlap terms, because the frontier molecular orbitals involved in the two paths are exactly the opposite. For example, the overlap terms for the two CT-mediated pathways in Figure 1 are

* A B B A     and * B A A B     , which
are the same after a simple reordering. Therefore, with the determined initial and final states, only one total orbital overlap term will regulate the diabatic coupling through CT-mediated mechanism. To predict the magnitude of this term, up to only four MOs ( The last term in eq (4), corresponding to the direct coupling, is usually smaller than the CT-mediated terms, if the CT state energy is not too high. [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF][START_REF] Smith | Recent Advances in Singlet Fission[END_REF] It comes from the two-electron exchange integral, and we can simplify it with a two-electron/two-orbital picture as

µ '' 1 2 1 1 2 2 12 1 H ( ) ( ) ( ) ( ) || ini fin A B B A dr dr r r r r rr       = - 
. As we can see from this expression, this term is also related to the orbital overlap between the same four frontier MOs as before. If we assume that the orbital overlap of (A, B) and ('A, 'B) are distributed differently in the space (for most cases it is true), then we can find a fixed or averaged

µ '' 1 2 1 1 2 2 12 '' 1 1 1 2 2 2 '' 1 H ( ) ( ) ( ) ( ) || 1 ( ) ( ) ( ) ( ) 1 ini fin A B B A A B B A x A B B A x dr dr r r r r rr dr r r dr r r d d               = -  =   ( 7
)
Although this is a rough approximation, considering the tiny contribution of direct coupling compared to the CT-mediate coupling, this treatment still provides an effective way to include any direct coupling effect. The term dx should be mainly determined by the orbital shape and the distance between A and B.

Combing eqs ( 4), ( 6) and ( 7), now we have the expression of diabatic coupling from the orbital overlap:

' ' ' ' '' 1 1 SEIC A B B A A B B A x A B B A x V d d               =+  =+    (8)
which can be finally written as

'' SEIC total A B B A V      = (9) 1 total x d  - =+
includes the effects from both CT-mediate mechanism and direct mechanism ( )

1 x d  - ?
. In this work, we do not compute total  and exclusively focus on the essential role of the total overlap term to qualitatively explain certain features of the PS-O2 SEIC reaction.

As a conclusion of this part, we utilized the bilinear relationship between electron transfer coupling and frontier MO overlaps, [START_REF] Gajdos | Ultrafast Estimation of Electronic Couplings for Electron Transfer between pi-Conjugated Organic Molecules[END_REF] to derive a clear relationship between the final diabatic coupling and the total orbital overlap as a product of two MO overlap terms. This relationship can be applied to predict the diabatic coupling strength, especially its variation in intermolecular space.

We note here that Feng and Krylov also tested the effect of orbital overlap on the coupling strength, and found only a weak correlation between them. [START_REF] Feng | On couplings and excimers: lessons from studies of singlet fission in covalently linked tetracene dimers[END_REF] However, they considered the NAC strength, while our derivation in this sections is for the diabatic coupling. Moreover, with our derivation, we found a direct relation between the diabatic coupling an the product of orbital overlaps (eq (8)), while in their work they took the sum of the orbital overlaps instead.

Adiabatic Picture

In the adiabatic picture, the transition between adiabatic states is mediated by the nonadiabatic coupling vector

J IJ I    F R ( 10
)
where R is the nuclear coordinate.

For SEIC reaction, usually the CT state energy is higher than the other two states, [START_REF] Morrison | Evidence for Singlet Fission Driven by Vibronic Coherence in Crystalline Tetracene[END_REF][START_REF] Chan | The Quantum Coherent Mechanism for Singlet Fission: Experiment and Theory[END_REF][START_REF] Mirjani | Theoretical Investigation of Singlet Fission in Molecular Dimers: The Role of Charge Transfer States and Quantum Interference[END_REF] and the diabatic coupling strength for intermolecular CT is limited, so we have the following relationship:

CT ini CT fin iCT fCT E E E E or -  -   ?
. Then, the adiabatic wave functions are

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1
c E E E c c c E c E E E E E c c c                = + - = + +      = - - -- = + + ( ) ( ) ( ) ( ) 2 3 3 3 3 3 3 ( ) ; ( ) fin fCT iCT CT CT ini fin CT CT fin CT CT ini ini fin fin CT EE c E E E c c c E E           = + + - = + +  (12)
In these expressions, ( )

1 ini c , ( )
2 fin c , and
( ) 3 CT c
are always near to the unity, which reflects that each adiabatic state is dominated by one of the three diabatic states in the weak coupling region, only with some minor components of other two states.

( ) We can derive the NAC between state 1 and state 2 according to eqs (10) and ( 12)

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 12 1 2 2 2 1 1 1 fin fin ini ini CT CT ini ini fin fin CT CT c c c c c c        =   + + = + +  F R R ( 13
)
Because 𝜓 𝑖𝑛𝑖 , 𝜓 𝑓𝑖𝑛 , and 𝜓 𝐶𝑇 are all diabatic states, all the terms like

y x    R
are zero and the gradient of wave functions in eq (13) are negligible. Only the terms with the gradient of the coefficients remain

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2
c c c c c c c c c c c c c c c c c c c                      +       +       ++     = F R R R R R R R R R ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 1 1 2 1 1 1 CT ini ini fin fin CT CT fin ini ini fin fin ini CT CT ini ini ini CT fin fin CT CT c c c c c c c c c c c             + + +   + + +   ++  R R R (14)
As we mentioned before, the coefficients CT ini  = . Therefore, the first term in the sum in eq (14) is negligible, and we can simplify the second part with the following equation

( ) ( ) ( ) ( ) 1
 ( ) ( ) 1 1
.

iCT CT CT CT ini CT CT iCT ini CT c E E c C C     =   →=   =   (15)
As mentioned before, C is a constant ranging from 3.5 eV to 18 eV depending on the reference. It should be related to the average of the two orbital energies. [START_REF] Hoffmann | An Extended Hückel Theory. I. Hydrocarbons[END_REF] Therefore, a value above 10 eV seems to be reasonable, especially when one of the orbital comes from an oxygen or nitrogen atom. Then, for a typical CT E value, this would make ini CT

 negligible compared to

(1) CT c . Although it is not always guaranteed that this term can be neglected, we have assumed that it is null to simplify the picture, which is proved to work well during our calculations later.

After all these approximations, the NAC in eq ( 14) is simplified to a much concise expression

( ) ( ) ( ) ( ) 22 11 CT 12 ini ini CT cc cc  +  F RR ( 16
)
This result tells that only four quantities determine the NAC strength: the coefficients of 𝜓 𝑖𝑛𝑖 and 𝜓 𝐶𝑇 in the adiabatic state 1, and the gradient of the coefficient of 𝜓 𝑖𝑛𝑖 and 𝜓 𝐶𝑇 in the adiabatic state 2. (We yet recall that ( )

1 ini
c is close to the unity.) Quantum chemical calculation at single point can provide all the coefficient, and just few points more is enough to generate the gradient.

Eq ( 16) is derived for an ideal three-state model, while in the quantum chemical calculation of real molecules, instead of the coefficient of diabatic states, we may have configuration interaction (CI) coefficients of configuration state functions (CSF), [START_REF] Lischka | Multireference Approaches for Excited States of Molecules[END_REF] and more than one CT state is involved. Nevertheless, for the SEIC problems, it is easy to assign the orbitals in the active space to molecule A or B and to determine the CSFs that hold the CT character. For the SEIC reaction in the weak-coupling regime, usually the MOs in the active space change very little and their order does not change, if the intermolecular distance does not become very short. Therefore, the character of the CSF is steady and accessible. As a result, we can take a set of CSFs as the approximation of initial, final, and CT states, and generalize eq (16) in the following practical way

( ) ( ) ( ) ( ) 22 11
12

pCoef ii ii i ini CSF i CT CSF cc cc      +   F RR ( 17
)
12 pCoef F is one of the proxies for NAC that we test later (pCoef stands for "proxy based on CI cefficients"). It provides us with a way to calculate the NAC from simple quantum chemical calculations.

However, the need of gradient of the coefficients is somewhat inconvenient. Can we derive an expression without the gradient? The answer is yes if we take advantage of the expression of the coefficient in eq

E E E EE cc E c c c E E E E E E E E E E  +   -   -       - -     =+     -  - -     = + + -- -  -  + F RR RR R R R R ( ) ( ) ( ) 11 2 CT fCT CT CT fin CT fin E cc E EE  -  + - - R (18)
The third and fifth term containing the gradient of CT state energy, which is proportional to the distance, will be negligible compared to other terms. For intermolecular CT coupling, a previous work has concluded that it decreases exponentially along intermolecular distance R. [START_REF] Reed | Intermolecular interactions from a natural bond orbital, donor-acceptor viewpoint[END_REF] Then, we can take

xx xx d A dR      R
, where x A is a constant to simplify eq (18) as follows

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1
CT ini f fin A A A F c c c EE E E E E E E E A A c A c E E E E E E E AA E -  -    + - - --                = + - - - + -         --        = - + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 12
A c c EE E A A c c EE cc    +    -  = - -   = (19)
where ( )

1 CT fin CT ini CT fin fin E E A E E A  -  - - - 
. Note that in eq (19), NAC is a scalar quantity projected along the intermolecular distance. Note also that if we calculate the 21 F in the same way, we just obtain the same result with opposite sign, 12 21 FF =-, which means that our derivation preserves an important property of the NAC.

With eq (19), we find a correlation between the NAC strength and the coefficients of CT states. Conveniently, no coefficient of initial and final states appears. This result bears some similar structure as that in eqs. ( 7) and ( 9), where the diabatic coupling is correlated to the orbital overlap between initial/final state and the CT states. The similarity should come from the same superexchange mechanism through CT states, and implies similar performances when the superexchange mechanism is dominant. The term CT  is a combination of state energy, ini A , and fin A . Therefore, it is determined by the molecular complex and the intermolecular direction. For a fixed intermolecular direction, we can take CT  as a constant.

In practical terms, to work with CI coefficients of CSFs from a quantum chemical calculation, we should generalize eq (19) to consider all CSF coefficients holding CT character:

,2 12 12

pCoef CT CSF F  = (20)
with the definition

(1) (2) 12 CT ii i CT CSF cc     (21) CSF 
groups all other terms dependent on energies and energy derivatives and will be assumed to be a constant for a single incidence direction throughout this paper. For this reason, we will exclusively discuss CT IJ  , which just requires the coefficient of few CSFs in the adiabatic states I and J, which are provided by even just one single point calculation. If eq (21) works well, we can use it to describe the NAC strength for different intermolecular distance within a determined direction.

To summarize, after the derivations based on the adiabatic wave functions of SEIC reaction, we proposed two proxies to express the NAC just adopting coefficients of the wave functions directly from quantum chemical calculations. The first NAC proxy, pCoef IJ

F

, is given by eq (17) and should deliver the absolute NAC values, based on CI coefficients and CI coefficient gradients. The second NAC proxy, CT IJ  , is given by eq (21) and should give relative NAC values, based on CI coefficients of CT configurations.

As far as we know, these two proxies for NAC are proposed by the first time, and we will test their performance in the next section by comparing them to the first-principle calculated NAC.

RESULTS AND DISCUSSION

The TF reaction in singlet oxygen photosensitization

In this work, we take the singlet oxygen photosensitization by thiothymine, an example of triplet fusion reaction (see reaction (3)), as a prototype to study several features in SEIC. The choice of thiothymine as PS holds a couple of advantages: first, the PS-O2 complex is relatively small, so that different quantum chemical levels, including multireference methods, are affordable. Second, O2 has two possible final states, 1 g and 1 Δg, which delivers valuable additional information on the electronic interactions. This reaction is still far from being satisfactorily treated by theoretical computational chemistry, mainly due to the intrinsic open-shell nature, while new investigations with direct quantum chemical calculation are always welcome. [START_REF] Schmidt | Quantitative determination of 1 g + and 1 g singlet oxygen in solvents very different polarity. General energy gap law for rate constants of electronic energy transfer to and from O 2 in the absence of charge-transfer interactions[END_REF][START_REF] Paterson | Invited review -Overview of theoretical and computational methods applied to the oxygen-organic molecule photosystem[END_REF] We explore the developments and conclusions from our previous research on this reaction, [START_REF] Bai | Divide to Conquer: A Kinetic Model for Singlet Oxygen Photosensitization[END_REF][START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] to help us focus on the present goal, to test and validate the coupling proxies proposed in the previous sections. Nevertheless, despite the focus on reaction (3), we anticipate that the current findings contribute to a general understanding of SEIC, including SF.

Reaction (3) can generate either [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] Δg or 1 g singlet oxygen if the triplet energy ET of PS exceeds ~1.6

eV (the excitation energy for 1 g oxygen). For this TF reaction, the initial state is 3 PS-3 O2 (TT), the final state is 1 PS-1 O2 (SS), and the CT state to enable the CT-mediated mechanism is either + PS -O2 or -PS + O2.

In a conventional four-electron view, these states can be described with four frontier molecular orbitals (detailed shown in Section 3.3): the π 𝑝 and 𝜋 𝑝 * (the two singly occupied orbital of an organic PS in the T1 state) and the 𝜋 𝑥 * and 𝜋 𝑦 * (the two singly occupied degenerated orbitals of O2 in the ground state).

Although O2( 1 g) is quickly deactivated into O2( 1 g) within solvents, [START_REF] Scurlock | Chemical reactivity of singlet sigma oxygen (b 1 g + ) in solution[END_REF][START_REF] Bodesheim | Chemical reactivity of sigma singlet oxygen O2( 1 g + )[END_REF] advanced spectroscopic techniques have been developed to distinguish between the two states [START_REF] Bodesheim | Triplet-state Energy-dependence of the Competitive Formation of O2( 1 g + ), O2( 1 g) and O2( 3 g -) In the Sensitization of O2 by Triplet-states[END_REF][START_REF] Chou | Direct spectroscopic evidence of photosensitized O2 765 nm ( 1 g + → 3 g -) and O2 dimol 634 and 703 nm (( 1 g)2→( 3 g -)2) vibronic emission in solution[END_REF] and provide experimental data for 1 g / 1 Δg ratio during singlet oxygen generation. From the computational standpoint, using the DtC model, [START_REF] Bai | Divide to Conquer: A Kinetic Model for Singlet Oxygen Photosensitization[END_REF] we showed that the 1 g / 1 Δg rate ratio might range from near to zero up to about 20, depending on the activation energies ‡ E  and diabatic coupling strength from different intermolecular incidence 16 direction, orientation, and distance between O2 and PS. [START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] We found that in some directions the 1 g coupling was always near to zero, while the 1 Δg coupling is normally large, leading to the relatively small 1 g / 1 Δg rate ratio (as plotted in Figure 2 with blue circles). Calculated diabatic couplings for 1 Δg and 1 g oxygen generation at the maximum rate distance obtained for 15 incidence directions. The color code indicates the 1 g/ 1 Δg rate ratio. Results from ref. [START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] Considering the fact that the 1 Δg and 1 g oxygens are constructed from the same orbitals with a different spin, [START_REF] Paterson | Invited review -Overview of theoretical and computational methods applied to the oxygen-organic molecule photosystem[END_REF][START_REF] Ogilby | Singlet oxygen: there is indeed something new under the sun[END_REF] the phenomenon of near zero 1 g coupling but much larger [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] Δg coupling is intriguing. Nevertheless, in that previous computational study, we did not directly discuss the underlying mechanism responsible for it, while in this work it will be revealed in detail, thanks to the theoretical tools proposed in Section 2. In what follows, we picked out two points among the fifteen incidence directions-"perp-S-i" (for which the 1 g coupling is little) and "para-61-o" (for which both couplings are large)-and restrict the analysis to them. These two directions are indicated in Figure 2 by the green filled circles (their geometries are shown in the SI).

Analysis in the adiabatic picture

In this section, we test the performance of the NAC proxies given by eqs ( 17) and ( 21) in comparison to the full NAC directly computed at the CASSCF level for the PS-O2 complex. During the whole calculation, the orbitals from PS and O2 do not mix much, so it is easy to designate the four MOs in the active space to 𝜋 𝑝 * , π 𝑝 , 𝜋 𝑥 * and 𝜋 𝑦 * respectively. These MOs for the two directions are plotted in Figure 3. With these MOs, we determined the four CSFs that carry the CT characters according to the electron occupation of the CSFs. Their CI coefficients in different adiabatic states were obtained after the CASSCF calculation. The initial state is the adiabatic state S4, and the final state may be any of the adiabatic states S1 to S3. The final states S1 and S2 correspond to the generation of [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] Δg oxygen, while the final state S3

corresponds to the generation of 1 g oxygen. We took eqs ( 17) and ( 21) to calculate pCoef IJ F and CT IJ  , and investigated their performance as the NAC proxy quantitatively along the intermolecular distance.

According to eq (17), the coefficients of the CT and TT CSFs in the initial state as well as their gradient are needed to calculate pCoef IJ

F

. Ten CSFs with the largest weights in the initial state S4 at the first point (with longest intermolecular distance) were taken as the TT CSFs. We are interested in the projection pCoef IJ F of the NAC vector on the intermolecular distance. Thus, the gradient is simply the derivative along the intermolecular coordinate, and it was calculated by finite difference along the linearly interpolated pathway:

( ) ( ) ( )

11 2 I I I i n i n i n dc R c R c R dR R +- -   (22)
where R  is step length used for the calculation of this direction. Following eq (21), the calculation of

CT IJ
 is simpler and only requires the CI coefficients of the four CT CSFs. F (DC stands for "directly computed"), which is the pro- jection of the NAC vector on the intermolecular direction R.

We start the analysis with the directions para-61-o, whose results are plotted in F , especially when the intermolecular distance is around or larger than the distance of maximum rate point, which is always larger than 2.9 Å. [START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] When the molecules get closer, the deviation might increase, although the approximation still works well for some of the couplings. The deviation may come from taking the CSFs as the approximation of diabatic states and supposing they are constant. However, the small distance will increase the activation energy a lot and make it unimportant for the final reaction rate.

So, pCoef IJ

F

works very well as a proxy for NAC. gives nearly the same value as the directly calculated NAC, if the intermolecular distance is not too small. 21) along the perp-S-i direction. The distance of the maximum rate is 3.14 Å. [START_REF] Bai | Spatial Factors for Triplet Fusion Reaction of Singlet Oxygen Photosensitization[END_REF] The use of For some CT configurations,

0 i c  (3)
but [START_REF] Smith | Singlet Fission[END_REF] 0 i c  , while for others it is just the opposite,

0 i c  but (3) 0 i c  . (4)
As a result, each term in the sum

(3) (4) ii i CR CSF cc  
is near zero. In other words, some CT CSFs are involved in S4, but they do not appear in S3, while for others, it is just the opposite. Therefore, although the CT configurations are not absent in both adiabatic states, there are no CT configurations that can connect the two states successfully.

Why is the nonadiabatic coupling zero only for S3, not for S1 or S2 state? As shown in eq (12), the coefficients of the CT CSFs in these two adiabatic states are determined by the coupling strength terms iCT  and fCT  , which are the electronic couplings between the intermediate CT state and the initial TT and the final SS states, respectively. Then, the distribution of the four CT CSFs on S3 and S4 is transformed to the diabatic coupling problem: no CT configuration has a large coupling with both the initial TT and final 1 g SS state. As we discussed in Subsection 2.1, these coupling strengths can be evaluated by the orbital overlaps, and we already build a correlation between the total SEIC diabatic coupling and the product of two orbital overlap term in eq (8). Therefore, it is more convenient to study this problem in the diabatic picture, as we do in next subsection.

Analysis in the diabatic picture

In the diabatic picture, we need to consider the initial TT state and final SS diabatic states and the CTmediated interaction paths between them. To keep the analysis in a clear and practical way, we investigate the interaction path based on the configurations directly constructed with frontier orbitals. For the initial [START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] TT state from the product of triplet state of PS (T1) and triplet state of O2 ( 3 g  ), the wave function of the complex is a linear combination of configurations whose the spin projection of PS and O2 sum to zero: 1

  2 - 1 0 0 3 1 3 (T ) T T T T T T PS O g  + - -+ +- - = (23)
In the (4,4) space, T+T-and T-T+ configurations are represented by single determinants (Figure 6), while T0T0 is written in terms of four determinants. For the final 1 SS states, the wave function of the complex is written in terms of the product of the closeshell ground state of PS (S0) and the singlet state of O2 ( 1 g or 1 g). In the (2,2) space of O2, these wafve functions are: [START_REF] Paterson | Invited review -Overview of theoretical and computational methods applied to the oxygen-organic molecule photosystem[END_REF][START_REF] Kearns | Physical and chemical properties of singlet molecular oxygen[END_REF]      

                - - - - - = -  = -  = + + (24)
where SS1 for instance is

* * y 0 S 1 (1) (2) 2 x  
and SS1 to SS4 are illustrated at the right side of Figure 7.

For the CT-mediated mechanism, we also need to define the CT configurations (CT1 to CT4), which are illustrated in the middle of Figure 7. In a way analogous to Figure 1, we generate interaction paths between all TT and all SS configurations through CT configurations. Taking the T+T-configuration, for example, all interaction paths connecting it to SS1 to SS4 are shown in Figure 7. The diabatic couplings between configurations can now be written. For instance, based on eq (9), the coupling term of T+T-with SS1 through CT1 (the first pathway in Figure 7) is As we mentioned before, the zero 1 g coupling along perp-S-i direction is intriguing. As shown in Fig-

CT1 * * * * T T SS1 SEIC total p x x p V      +-  →=   (25)
ure 8(a), the 1 g coupling along this direction is always near to zero, while the other two diabatic coupling strengths increase when the intermolecular distance decreases. We can now use the diabatic coupling given in eq (9) to explain the reason underlying effect.

From eq (24), we know that the only the SS2 and SS4 configurations are needed for 1 g singlet oxygen generation. As shown in Figure 7, the diabatic coupling from T+T-to SS2 through either CT1 or CT2 is With the four MOs from the CASSCF calculations plotted in Figure 3(a), we can estimate the magnitude of these overlap terms for the perp-S-i direction. The overlap occurs along the sulfur-O2     become very small, which means the interaction through CT-mediated mechanism is not effective for both SS2 and SS4 configurations, explaining the zero 1 g coupling along this direction.

According to the wave function for PS-O2( 1 0 1 S g - ) state in eq [START_REF] Monahan | Charge Transfer-Mediated Singlet Fission[END_REF], the 1 Δg1 coupling also depends on the SS2 and SS4 configurations, and it should be near to zero too. On the other hand, the     should have a large overlap value, implying that the 1 Δg0 coupling should be much larger than that for 1 g.

Nevertheless, the NACs for adiabatic states S1 and S2 states of perp-S-i are both not null, because both S1 and S2 states are not pure 1 Δg1 or 1 Δg0, but a mixture of them. To prove this point, we analyzed the components of these two states. If the coupling for 1 Δ g1 is near zero, the coupling strength for the mixed state should only depend on the 1 Δg0 contribution. We took the 1 Δg0 coefficients of the S1 and S2 states and calculated the ratio between them. This coefficient ratio was compared to the ratio of the NAC, as plotted in Figure 9. These two ratios are nearly the same, strongly supporting our interpretation. For the para-61-o direction, the orbitals were taken from Figure 3

CONCLUSION

Spin-exchange internal conversion (SEIC) is a class of reactions including singlet fission (SF) and triplet fusion (TF) as particular cases. Based on the understanding of SEIC reaction, we proposed proxies to evaluate the coupling strength in reliable but simple ways, and tested them on the PS-O2 singlet oxygen generation problem, an example of the TF reaction. In the diabatic picture, we built a bilinear relationship between the diabatic coupling and a product of two orbital overlap terms, and applied it together with related MOs to interpret the curious behavior of diabatic coupling along some intermolecular directions; in the adiabatic picture, two proxies of the NAC related to the CI coefficients of specific CSFs were proposed, and their results were compared with the directly calculated full NAC. Only the singlet oxygen photosensitization from the T1 state of PS was investigated because of the fast and high quantum-yield relaxation to T1 state after photoexcitation of PS. In the case of long-lived higher excited states, a weakcoupling reaction could still be treated with a similar approach, while other reactions may need additional methodological developments.

In the adiabatic picture, the full NAC and the two NAC proxies were computed along the PS-O2 intermolecular distance. The excellent performance of both proxies on this reaction system is confirmed from the direct comparison with the full NAC. The proxies are derived for a general SEIC reaction, which implies that we can employ them to make inexpensive semiquantitative analysis for SEIC reactions in large molecular ensembles and aid the qualitative analysis of the interstate interaction, under the conditions that the coupling from CT-mediated mechanism is dominant compared to that from the direct mechanism.

In the diabatic picture, we have employed the diabatic coupling proxy based on the orbital overlap term to qualitatively explain why the 1 g coupling is always zero in some incidence directions of the PS-O2

reaction. There are still some challenges, including the exact computation of the CT state energy and orbital overlap, to quantitatively calculate the diabatic coupling value, which remains for the future research. However, the evaluation from orbital overlap could be a simple way to predict the coupling strength along the various intermolecular directions, providing practical guideline for related molecular design.

The new proxies in adiabatic and diabatic pictures have their own advantages. For NAC proxies, they are calculated based on quantum chemical calculations and provide robust results comparable to the full NAC strength. On the other hand, the orbital overlaps in the diabatic picture are easier to analyze even without resorting to explicit calculations. In the example studied here, just the shapes of four key MOs were enough to deliver a qualitative analysis. Although the MOs used here are from CASSCF calculations, the analysis should be valid with Kohn-Sham orbitals even from isolated molecules as well. It is straightforward to extend these analyses to other SEIC problems with larger systems, if the dominant frontier MOs for the related excited states during the reaction are already known.

The approaches are derived for general SEIC reactions, although we take one special TF reaction case to prove the validity. Since photosensitization of singlet oxygen generation is an important research topic for experiments and theorists, [START_REF] Schmidt | Quantitative determination of 1 g + and 1 g singlet oxygen in solvents very different polarity. General energy gap law for rate constants of electronic energy transfer to and from O 2 in the absence of charge-transfer interactions[END_REF] the current findings during the verification process are also timely and appealing to researchers in this field.

ASSOCIATED CONTENT

Supporting Information

Details on quantum chemical calculation, molecular orbitals in the active spaces. This material is available free of charge via the Internet at http://pubs.acs.org.

1

  Mechanism of Spin-Exchange Internal Conversion: Practical Proxies for Diabatic and Nonadiabatic Couplings Shuming Bai* and Mario Barbatti* Aix Marseille Univ, CNRS, ICR, Marseille, France ABSTRACT: Spin-exchange internal conversion (SEIC) is a general class of reactions having singlet fission and triplet fusion as particular cases. Based on a charge-transfer (CT)-mediated mechanism and

Figure 1 .

 1 Figure 1. Interaction paths for one an example of TF reaction in the AB complex. Top and bottom pathways indicate CT-mediated mechanisms. The pathway in the middle is the direct mechanism.

a

  ) and charge transfer ( CT  ) states, and C is a constant. a CT  is the overlap between the two wave functions, and it is approximated by a CT  , the overlap between the two frontier molecular orbitals (MOs), and a CT  . Because of the electron transfer nature, a  and CT  are mainly located at the donor and acceptor molecules, respectively.

  are necessary to calculate it. (Four MOs are needed if ' AA   and ' BB   , as shown in Figure S3 of SI.)

3 CTc

 3 are always near the unity, and their gradient is very small. The mixture terms ( ) 2 ini c and ( ) 1 fin c are close to zero. From the diabatic picture, we also know that 0 fin ini  → and 1

Figure 2 .

 2 Figure 2.Calculated diabatic couplings for[START_REF] Casanova | Theoretical Modeling of Singlet Fission[END_REF] Δg and 1 g oxygen generation at the maximum rate distance

Figure 3 .

 3 Figure 3. The four frontier molecular orbitals from the CASSCF calculation of PS-O2 along (a) perp-S-i and (b) para-61-o directions.

Figure 4 F

 4 of three reaction paths are very close to DC IJ

Figure 4 .Figure 5 

 45 Figure 4. (a) First-principle NAC DC SS TT F -, the proxy pCoef IJ F

Figure 5 .

 5 Figure 5. (a) First-principle NAC DC SS TT F -, the proxy pCoef IJ F

(

 34 CT as well as 34 DCF

 3434 as NAC proxies goes beyond to have an inexpensive estimate of the NAC.Unlike the full NAC, which is difficult to analyze directly, these two proxies are determined by simple equations, being much easier to interpret, especially IJ CT  . For instance, the proxies tell us why perp-S-i directions have tiny 1 g couplings. As shown in Figure5, are always near zero along intermolecular distance. With its definition in eq(21), we know that 34 CT  is determined by the coef- ficient of CT CSFs in S3 and S4.We already have all of these coefficients in from the CASSCF calculation, so it is forthright to determine why to zero: it occurs due to the complementary distribution of the four involved CT CSFs (configurations shown in next subsection) on S3 and S4.

Figure 6 .

 6 Figure 6.The electronic configurations for the 1 TT state of PS-O2 with the four frontier orbitals.

Figure 7 .

 7 Figure 7.Interaction paths based on CT-mediated mechanism for triplet fusion of PS-O2 system from the

   . (CT3 and CT4 do not contribute to this coupling.) the diabatic coupling from T+T-to SS4 through either CT3 or CT4 is proportional to * from CT1 or CT2). If we further consider the diabatic couplings from T-T+ or T0T0 configurations using the same procedure (FigureS1and FigureS2in the SI), we see that only the same overlap terms are essential for the 1 g singlet oxygen generation.

Figure 8 .

 8 Figure 8. Calculated diabatic couplings for 1 Δg and 1 g oxygen generation along the (a) perp-S-i and (b) para-61-o direction.

Figure 9 .

 9 Figure 9.The ratios of NAC and 1 Δg0 coefficient of adiabatic S1 and S2 states for the perp-S-i direction.

  (b). Here, both 𝜋 𝑥 * and 𝜋 𝑦 * are not perpendicular or parallel to the ring plane but are sloped. Consequently, all overlaps are not zero, and 1 g coupling is larger than zero, which agrees with the fully calculated coupling as shown in Figure 8(b).

  To derive a proxy for NAC specially adapted for SEIC reactions, we start from the diabatic initial state 𝜓 𝑖𝑛𝑖 , final state 𝜓 𝑓𝑖𝑛 , and the intermediate CT state 𝜓 𝐶𝑇 as mentioned in subsection 2.1. The adiabatic wave functions are obtained by diagonalizing the following system Ham-

	iltonian
			ini E  = 00 iCT 
	H	SEIC	0 iCT  fin fCT fCT CT E E     =

  , the mixture of CT to state1 or state 2. These relations hold because the CT state is directly connected to the initial and final states via diabatic couplings, while 𝜓 𝑖𝑛𝑖 and 𝜓 𝑓𝑖𝑛 are connected by the superexchange effect of the CT state.

	2, are much smaller than	( ) 1 CT c and ( ) 2 CT c
		c	2	, the mixture of 𝜓 𝑖𝑛𝑖 to state
		ini

1 fin c

, the mixture of 𝜓 𝑓𝑖𝑛 to state 1, and

( )

  )

	22
	cc
		11
	ini		CT
		cc
	12	ini		CT
		iCT	fCT
							fCT
		fin	CT	fin	11 CT fin
					ini		CT
							fCT
		iCT					CT
			T			iCT	iCT	fCT
				1			1	1
							2
				ini			ini	ini
	fin	CT		fin	fin	CT	fin
							fin	CT	fin
		fCT
	CT

  direction, and the orbital 𝜋 𝑝 * and π 𝑝 around the sulfur atom are perfectly oriented to maximize the overlap with 𝜋 𝑥 * ,

	but minimize the overlap with 𝜋 𝑦 * . Thus, both	** | p y   	and	 	* | π y	p	 tend to be tiny. With these two
	terms near to zero, both overlap terms * *     and * * *		*
	p	x	y	p		p	y		x	p

1

  Δg0 coupling is determined by the overlap terms with the SS1 and SS3 configurations,

									*     * *	and
									p	x	x	p
	*     . Although * *	* p     * * || yy  p 	should be small, * *	*
	p	y	y	p		p	x	x	p

ACKNOWLEDGMENT

We are grateful to Prof. Hans Lischka and Dr. Felix Plasser for discussions and valuable suggestions. This work was supported by Excellence Initiative of Aix-Marseille University (A*MIDEX) and the project Equip@Meso (ANR-10-EQPX-29-01), both funded by the French Government "Investissements d'Avenir" program. The authors also acknowledge funding from HPC resources from GENCI-CINES (Grant 2017-A0010810012). MB also acknowledges funding from and of the WSPLIT project (ANR-17-CE05-0005-01).

AUTHOR INFORMATION

Corresponding Author E-mail: shuming.bai@univ-amu.fr E-mail: mario.barbatti@univ-amu.fr; website:www.barbatti.org

Notes

The authors declare no competing financial interests.


Ran Liu

Li Yang

Tongtong Yang

Yan Huang

Mario Barbatti

Jun Jiang

Guozhen Zhang

Modulating Electron Transfer in an Organic Reaction via Chemical Group Modification of the Photocatalyst

Tuning electron transfer (ET) rates from catalysts to substrates is important for modulating photocatalytic organic reactions. In this work, we have taken pyrene-based photocatalysts (Py) for photocatalytic hydrodefluorination of polyfluoroarenes (FA) as model systems, and conducted a first-principle study on modulating ET rates from Py to FA via chemical modification of Py with different electron donating/withdrawing groups (EDGs/EWGs). The computed spatial distributions of frontier Kohn-Sham orbitals suggest that ET is energetically more favorable for Py-EDGs than for Py-EWGs. The estimated ET rates by a simplified Marcus model show that they are appreciably enhanced by EDGs substitution and weakened by EWGs substitution. Noticeably, the associated Gibbs free energy change plays a dominant role. Our findings of tuning ET rates for Py-FA complexes via chemical group modifications cast new insight into the rational design of metal-free photocatalysts for organic transformations.

Light-driven organic transformations have been a vibrant area in photocatalysis towards quite a few challenging synthetic goals. [START_REF] Arceo | Photochemical Activity of a Key Donor-Acceptor Complex Can Drive Stereoselective Catalytic Alpha-Alkylation of Aldehydes[END_REF][START_REF] Schultz | Solar Synthesis: Prospects in Visible Light Photocatalysis[END_REF][START_REF] Jin | Alcohols as Alkylating Agents in Heteroarene C-H Functionalization[END_REF][START_REF] Romero | Organic Photoredox Catalysis[END_REF][START_REF] Liu | Recent Advances in Visible-Light-Driven Organic Reactions[END_REF] Generally, they involve energy and matter conversion accompanied by a series of electron processes, including photo-induced electronic excitation, excitation energy transfer, and electron transfer (ET). [START_REF] Ravelli | A Multi-Faceted Concept for Green Chemistry[END_REF][START_REF] Arias-Rotondo | The Photophysics of Photoredox Catalysis: A Roadmap for Catalyst Design[END_REF] Noticeably, in photocatalytic organic reactions, ET from photocatalysts to substrates is a central step, which activates reactants and generates reactive species. [START_REF] Ravelli | Carbon-Carbon Bond Forming Reactions Via Photogenerated Intermediates[END_REF] Therefore, the mechanistic understanding of the ET and subsequent modulation of ET rates have drawn intensive attention in recent years. [START_REF] Ischay | Efficient Visible Light Photocatalysis of [2+2] Enone Cycloadditions[END_REF][START_REF] Du | Photocatalytic Reductive Cyclizations of Enones: Divergent Reactivity of Photogenerated Radical and Radical Anion Intermediates[END_REF][START_REF] Neumann | A Cooperative Hydrogen-Bond-Promoted Organophotoredox Catalysis Strategy for Highly Diastereoselective, Reductive Enone Cyclization[END_REF][START_REF] Lima | Organic Synthesis Enabled by Light-Irradiation of Eda Complexes: Theoretical Background and Synthetic Applications[END_REF][START_REF] Silvi | Enantioselective Organocatalytic Alkylation of Aldehydes and Enals Driven by the Direct Photoexcitation of Enamines[END_REF][START_REF] Lu | Pi-Hole-Pi" Interaction Promoted Photocatalytic Hydrodefluorination Via Inner-Sphere Electron Transfer[END_REF][START_REF] Pirnot | Photoredox Activation for the Direct Beta-Arylation of Ketones and Aldehydes[END_REF] In present work we will demonstrate how different types of functionalization of a promising class of metal-free organic chromophores can control the ET process.

Metal-free photosensitizers and photocatalysts are promising alternatives to conventional ones for photocatalytic organic synthesis, which usually comprise inorganic semiconductors [START_REF] Lang | Heterogeneous Visible Light Photocatalysis for Selective Organic Transformations[END_REF] or organometallic compounds [START_REF] Twilton | The Merger of Transition Metal and Photocatalysis[END_REF] , because they may be inexpensive for scaling-up production, easily tunable in light adsorption, structurally flexible, and most importantly, environmentally friendly. These features have made metal-free organic chromophores appealing in recent years, drawing increasing attention in an active research field. [START_REF] Romero | Organic Photoredox Catalysis[END_REF][START_REF] Arias-Rotondo | The Photophysics of Photoredox Catalysis: A Roadmap for Catalyst Design[END_REF] For example, Miyake and co-workers successfully developed a couple of tunable organic photosensitizer based on the dihydrophenazine and phenoxazine. [START_REF] Theriot | Organocatalyzed Atom Transfer Radical Polymerization Driven by Visible Light[END_REF][START_REF] Pearson | Organocatalyzed Atom Transfer Radical Polymerization Using N-Aryl Phenoxazines as Photoredox Catalysts[END_REF][START_REF] Mccarthy | Structure-Property Relationships for Tailoring Phenoxazines as Reducing Photoredox Catalysts[END_REF] Further, Lu et al. performed the challenging hydrodefluorination of polyfluoroarenes (FA) using pyrene-based photocatalysts (Py), which are not only purely organic but also plays the dual role of photosensitizer and catalyst in a single compound. [START_REF] Lu | Pi-Hole-Pi" Interaction Promoted Photocatalytic Hydrodefluorination Via Inner-Sphere Electron Transfer[END_REF] Importantly, they proposed that the "π-hole-π" interaction between Py and FA plays a crucial role in overcoming otherwise unfavorable energetic ET reaction that activates the C-F bond of FA. Despite of these successful examples, metal-free organic photosensitizers and photocatalysts are still scarce and not well understood yet. Therefore, it is urgent to conduct comprehensive study of the basic physical chemistry of such compounds, as we have delivered in our work. Specifically, Lu et al.'s discovery motivated us to conduct a theoretical study towards a comprehensive understanding of the mechanism of ET from Py to FA and the subsequent modulation of ET through molecular design.

ET reactions have been subjected to many theoretical studies. [START_REF] Kumpulainen | Ultrafast Elementary Photochemical Processes of Organic Molecules in Liquid Solution[END_REF] Here we are interested in a qualitative description of the tuning effect on the ET rate due to the addition of different chemical groups to the catalyst, which can be gauged through Marcus theory. [START_REF] Marcus | Electron Transfer Reactions in Chemistry: Theory and Experiment (Nobel Lecture)[END_REF] Nevertheless, in the framework of conventional density functional theory (DFT), the tendency of delocalization of the electron density makes it difficult to clearly define the initial and final electronic diabatic states for an ET reaction. To address this type of challenge, Van Voorhis and co-workers developed constrained density functional theory (CDFT), [START_REF] Kaduk | Constrained Density Functional Theory[END_REF] which enables building approximate diabatic states to model the initial and final states of ET and calculate the ET rate. Therefore, we used the CDFT method in conjunction with a simplified Marcus model to describe the thermodynamics and kinetics of the ET reaction of non-bonded Py-FA complexes.

Chemical modifications with different functional groups have been a common strategy for tuning electronic properties of a molecular catalyst. Herein we selected a couple of representative electron donating groups (EDGs: -OH and -NH2) and electron withdrawing groups (EWGs: -CN, -NO2, -CHO), all of which possess smaller steric hindrance compared to -C≡C-C(CH3)3 used in the previous experimental study. [START_REF] Lu | Pi-Hole-Pi" Interaction Promoted Photocatalytic Hydrodefluorination Via Inner-Sphere Electron Transfer[END_REF] To have a more comprehensive understanding of the impact of functional groups on ET rates of Py-FA complexes, we also studied Py with substitution of -C≡C-C(CH3)3, -C≡C-CH3, and -CH=CH2. The schematic model of Py derivatives with various substitutions (X) is depicted in Figure 1. We built a simplified model system consisting of one Py and two FA, i.e., Py-(FA)2, which resembles the crystal structure reported in the Lu et al.'s work. To model the ET using CDFT framework, we specifically assign one of FA molecules as the electron acceptor and the other serves as an environmental factor that keeps the activated FA from moving (Figure S1). We first studied the light absorption of Py(X) and Py(X)-FA2, and the formation and electron affinities of [Py(X)-FA2] -anions. Then, we studied the spatial distribution of frontier Kohn-Sham orbitals (FMOs) of these anions. Finally, we computed Gibbs free energy change (∆G 0 ), reorganization energy (λ), and electron coupling (Vif) using the CDFT method, which were used to estimate the corresponding ET rate. We have found that the substitution of selected EDGs appreciably strengthens the ET from Py to FA, while the substitution of EWGs (including alkynyl and alkenyl groups) remarkably restrains the ET. In both cases, the underlying reason for the rate variation is linked with the relative values of the free energy change.

Effective light absorption by Py(X) is the prerequisite for the photocatalytic HDF of FA. We found that both EDGs and EWGs can cause a measurable red shift of the maximum absorption of Py(X) towards the visible light region and enhanced absorbance, compared to unmodified Py (Table S1). Particularly, Py with alkynyl groups exhibits an appreciable All Py-(FA)2 complexes of interest are found to have a negative binding energy ranging from -0.61 to -0.88 eV at the B97XD/6-311+G(d,p) level, suggesting that they are non-covalent van der Waal's (vDW) complexes and the interaction is not significantly affected by the identity of the added group (Table S3). Further, the light absorption of each Py species is almost unchanged as it forms Py-(FA)2, indicating that the substrate has little impact on the light absorption of the photocatalyst (Table S1 andTable S2). Therefore, both pathways are equivalent for preparing the complex, which is ready for subsequent ionization.

As Lu et al. proposed, the excited Py-FA complex is then reductively quenched by a sacrificial electron donor and becomes an anion. [START_REF] Lu | Pi-Hole-Pi" Interaction Promoted Photocatalytic Hydrodefluorination Via Inner-Sphere Electron Transfer[END_REF] All excited Py-(FA)2 complexes undergo a substantial exothermic relaxation as they receive an electron (Table S4). Specifically, EWGs increases the electron affinity of the excited complex, while EDGs reduces it, compared to unmodified Py. This result suggests that EWGs tend to stabilize the excess electron attached to the complex, while EDGs have the opposite effect. Further, we compared the energy change for losing the excess electron on these relaxed Py(X) -anions. Since the ET can be viewed as transition from the state of [Py] --(FA)2 to the state of Py-[FA] -FA], we calculated the energy ( 1 E  ) needed to detach the excess electron from the anions, to compare the tendency of ET of different Py(X): To further understand the impact of different functional groups on the electronic structure of the catalyst-substrate complex, we analyzed the spatial distributions of frontier Kohn-Sham orbitals of all [Py-(FA)2] -(Figure 3). The singly occupied Kohn-Sham orbitals (SOMO) of all anions is distributed at the Py moiety, while the lowest unoccupied Kohn-Sham orbital (LUMO) can be located on different parts of the complex that depends on which specific functional group is added. For the pristine Py, Py-OH, and Py-NH2, the LUMO is distributed over the FA moiety. For other Py(X), the LUMO is distributed over the Py moiety, and only higher virtual Kohn-Sham orbitals have been found in the FA part, e.g., LUMO+1 for Py-CH=CH 2 , Py-C≡C-CH 3 , and Py-C≡C-C(CH 3 ) 3 ; LUMO+3 for Py-CN and Py-CHO; and LUMO+4 for Py-NO2 (Figure S2). Given these spatial distributions of frontier Kohn-Sham orbitals, it is more difficult for an electron in the SOMO of a Py(EWGs) to jump to FA than it is for an electron in the SOMO of a Py(EDGs). Therefore, we can conclude from this analysis that the ET process from Py(X) -to FA will be energetically more favorable for Py-EDGs than for Py-EWGs.

    1 catalyst catalyst E E E -  = - . The values of 1 E  in
As mentioned, to simulate the ET process from Py(X) to FA, the initial and final states are Weller's equation, [START_REF] Lu | Pi-Hole-Pi" Interaction Promoted Photocatalytic Hydrodefluorination Via Inner-Sphere Electron Transfer[END_REF] which is appreciably lower than 0.67 eV by direct calculation using CDFT method. This difference may be due to our simplified model system, which ignores the explicit solvent environment and adopts an incomplete basis set to describe the anion.

We analyzed the ET reaction using a simplified Marcus model. To do so, we noted that a specific C-F bond is longer and deviated more from the FA plane at the final state (Figure S4). This is indicative of the C-F bond activation that may initiate the subsequent HDF reaction. Using the vertical distance change of the activated C-F bond of FA as the onedimensional reaction coordinate driving the ET, we obtained the potential energy profiles for the initial and final states of [Py-(FA)2] -, as shown in Figure 4 and Figure S3. The potential energy profiles of both states are near parabolic, which is a requirement to describe the ET with a Marcus model. Obviously, the ET reactions in all these vdW belong to the normal region, [START_REF] Kumpulainen | Ultrafast Elementary Photochemical Processes of Organic Molecules in Liquid Solution[END_REF] where the ET rate decreases with the increase of the Gibbs free energy change.

Figure 4: The one-dimensional potential energy profiles of the initial state (black dots) and the final state (red dots) of the catalyst-reactant complex as a function of the reaction coordinate for the ET reaction. E denotes potential energies of the catalyst-reactant complex calculated by CDFT method at B97XD/6-311+G** level. R denotes the reaction coordinate for the ET reaction defined in Figure S4.

The initial and final states refer to the states with the excess electron constrained in the catalyst and the reactant, respectively. The solid curves are parabolic fitting of the data included only to guide the eyes.

( ) ( ) ( ) ( )   + + + + 1 D A |D A D A |D A D A|D A D A|D A 2 E E E E  - - - -     = - + -     (1)
where ( )

X|Y E stands for the potential energy of electronic state X at optimized geometry of electronic state Y, D refers to the electron donor, and A refers to the electron acceptor. The electron coupling coefficients were computed with the direct coupling method implemented in Q-Chem. [START_REF] Shao | Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package[END_REF] Based on the calculated 0 G  ,  , and if V , we computed the ET rate (

ET k ) using Eqn. (2). 2 ET ‡ 21 exp 4 if B B G kV kT kT    - =   h (2)
where h is the reduced Planck constant, kB is the Boltzmann constant, and T is the temperature. ‡ G  is the activation energy given as ( )

2 0 ‡ 1 4 GG    =  + .
These results have been collected in Table 1. Note that this simplified formulation of the Marcus model is intended only to provide a qualitative estimate of the ET rates, indicating the general effect of adding an EDG or EWG to the catalyst (See note in the "Computational Method" section). Therefore, the relative ET rates is more important for understanding the impact of added chemical groups than their absolute rate values. Table 1 shows that the ET time ( 1 ET k -) for pristine Py falls in the microsecond regime. The rate is significantly altered as substitution groups are added to Py. While Py-EDGs substantially boost the ET rate, by up to a factor 1000 in -NH2, Py-EWGs dramatically restrain it. The trend is line with what we found in the electron binding energy of Py(X) -anions and their frontier Kohn-Sham orbital distributions as presented above. Importantly, this comparison suggests that chemical group modification can be an effective knob to tune the ET property of organic photocatalysts.

Among all quantities contributing to the rate, the free energy change of ET plays a dominant role in determining the relative value of ET k , indicating that it can be used as a descriptor that facilitates us to study the modulation of a specific type of ET reaction.

Table 1. Comparison of reorganization energies, free energy changes, activation energies, electron coupling coefficients, the absolute and relative ET rates for various of Py(X)-(FA)2 anionic complexes.

Py(X)-(FA)2 1.71×10 -25 In the framework of the search for metal-free organic photosensitizers and photocatalysts, the For Py(X)-(FA)2 anions, the LUMO is distributed over the FA moiety for Py and Py-EDGs (EDGs: -OH and -NH2), but it stays at the Py moiety for Py-EWGs (EWGs: -CN, -NO2, -the microsecond regime. An EDG substitution may increase the ET rate by hundreds or even thousands of times. An EWG substitution, on the other hand, inhibits the ET. We note that 0 G  plays a decisive role in determining the trend of ET among different complexes, while the inner-sphere  is insensitive to chemical modification and if V 's variation with substitution is negligible. We propose that the electron binding energy of catalyst and the free energy change of ET can be used as descriptors for the estimation of relative ET rates for a family of similar species. Finally, we also note that the qualitative differences in the rates between the two classes of chromophores is so striking that it implies that adoption of more involved computational models than the bare Py-FA2 used in our simulations should not significantly change the finds reported here.

 /eV 0 G  /eV ‡ G  /eV if V /eV ET k /s -1 rel k -H 1.
In short, our findings provide a comprehensive understanding on the tuning of ET rates for Py-(FA)2 complexes using chemical group modifications. The information derived from our work can set heuristic rules to be applied to other classes of metal-free organic systems. By exploring more organic chromophores using the same strategy, we are on the way of rational design of new organic photocatalysts.

Computational Method

Density functional theory (DFT) and linear-response time-dependent density functional theory (TDDFT), as implemented the Gaussian 09 program [START_REF] Frisch | Revision D.01[END_REF] , were used to study the ground and excited states of Py, FA, and their complexes. To study the ET from Py to FA, the CDFT method was employed using Q-Chem 5.0 program [START_REF] Shao | Advances in Molecular Quantum Chemistry Contained in the Q-Chem 4 Program Package[END_REF] , to model the approximate diabatic states in which the excess electron is explicitly constrained in either Py or FA. All geometry optimizations and single-point energy calculations employed the range-separated and dispersion-corrected ωB97XD functional [START_REF] Chai | Long-Range Corrected Hybrid Density Functionals with Damped Atom-Atom Dispersion Corrections[END_REF] with the 6-31G(d) [START_REF] Francl | Self-Consistent Molecular Orbital Methods. Xxiii. A Polarization-Type Basis Set for Second-Row Elements[END_REF] (for optimization) and 6-311+G(d,p) [START_REF] Clark | Efficient Diffuse Function-Augmented Basis Sets for Anion Calculations. Iii. The 3-21+G Basis Set for First-Row Elements[END_REF] (for single points) basis sets. Normal mode analysis was also carried out at the same level of geometry optimization to obtain Gibbs free energy for all converged structures.

The electronic coupling was calculated with the direct coupling method. [START_REF] Farazdel | Electric-Field Induced Intramolecular Electron Transfer in Spiro .Pi.-Electron Systems and Their Suitability as Molecular Electronic Devices. A Theoretical Study[END_REF] The solvent effect of dieethylacetamide, which was used as the solvent in Lu et al.'s experimental work, was implicitly included by using the corresponding polarizable continuum model (PCM). [START_REF] Tomasi | Quantum Mechanical Continuum Solvation Models[END_REF] For the use of simplified formulation of the semiclassical Marcus model, we keep in mind that it intends only to provide a big picture of the impact of adding an EDG or EWG to the catalyst on the ET rates. This approach will serve as the starting point of future work of accurate calculation of quantitative ET rates involved in organic photocatalytic systems. In addition to the uncertainties in the free energy changes already mentioned, our estimates also do not account for entropic and enthalpic effects in the reorganization energies and completely neglects the outer-sphere variations. The model is also only valid for parabolic potentials sharing comparable harmonic frequency in both states (Table S5), and with this frequency being smaller than the thermal energy. Albeit these latter conditions are all approximately satisfied (see discussion of Table S5), we think the relative ET rates is more important for understanding the impact of added chemical groups than their absolute rate values. For systems with larger frequencies, the rates can be estimated with the help of the Bixon-Jortner model. [START_REF] Brédas | Charge-Transfer and Energy-Transfer Processes in π-Conjugated Oligomers and Polymers: A Molecular Picture[END_REF]

Figure 1 .

 1 Figure 1. The model structure of the photocatalyst molecules, Py(X). (X = (-H, -OH, -NH2; -CH=CH2, -C≡C-CH3, -C≡C-C(CH3)3; -NO2, -CN, -CHO).)

  Scheme 1. The chat flow of the photoinduced hydrodefluorination of FA catalyzed by Py

  visible light absorption around 395 nm, in good agreement with experimental data in Lu et al.'s work. They suggest two parallel routes of the formation of excited Py-FA complexes: I. Py → Py * + (2 FA) → [Py-(FA)2] * ; II. Py + (2 FA) → Py-(FA) 2 → [Py-(FA) 2 ] * .

Figure 2

 2 Figure 2 are indicative of the electronegativity of Py(X), which is appreciably strengthened as EWGs are added and weakened as EDGs are added. This distinction causes a drastic change in the subsequent ET reaction, as we shall see.

Figure 2 .

 2 Figure 2. The energy needed for removing an electron from various [Py(X)] -anions.

Figure 3 .

 3 Figure 3. The spatial distribution of frontier Kohn-Sham orbitals of all [Py-(FA)2] -complexes. For the complexes structure, the lower part is the Py molecule and the upper part is the FA molecule.



  assigned as [Py(X)] --(FA)2 and Py(X)-[FA] -[FA], respectively. The computed Gibbs free energy change between the minimum of each state, 0 G  , have been collected in Table1. This dataset clearly shows that the variation of 0 to more positive values, with strong EWGs (-NO2, -CN) being more positive than weak EWGs (alkynyl and alkenyl groups). We note that Lu et al. reported a ) of 0.36 eV for Py-C≡C-C(CH3)3 using experimental data and

  photocatalytic hydrodefluorination of polyfluoroarenes (FA) by pyrene-based photocatalysts as a representative model system and conducted a first-principle study on the modulation of ET rates in the framework of Marcus theory. We demonstrate how the chemical modification of Py can affect the ET from Py to FA, a key step in the photochemical process.

Acknowledgments

This work was financially supported by MOST (No. 2016YFA0400904), NSFC (21703221, 21633006), and the Fundamental Research Funds for the Central Universities (WK2060030027). RL and GZ are grateful for the computing resources from the Supercomputing Center of University of Science and Technology of China. MB thanks the support of the Excellence Initiative of Aix-Marseille University (A*MIDEX) and the project Equip@Meso (ANR-10-EQPX-29-01), both funded by the French Government "Investissements d'Avenir" program. MB also acknowledges funding from the WSPLIT project (ANR-17-CE05-0005-01).

Supporting Information

The Supporting Information is available free of charge on the ACS Publication website at DOI: 10.1021/xxxx. Chemical structure of the Py-(FA)2 complexes (Figure S1); light absorption properties of all Pyrene derivatives (Table S1) and all complex composed of Pyrene derivatives and FA molecule (Table S2); the computed binding energy of each Py-(FA)2 complex (Table S3); the energy change for the excited complex receiving an excess electron (Table S4); the spatial distribution of higher virtual molecular orbitals of [Py-(FA)2] -complexes (Figure S2); the one-dimensional potential energy profiles of the initial and final state of the catalyst-reactant complex as a function of the reaction coordinate for the ET reaction (Figure S3); the reaction coordinate for the ET reaction, R.(Figure S4); the harmonic frequency of the normal mode of the initial state and final state associated to the ET coordinate depicted in Figure 4 (Table S5); The Becke populations of the final state of ET of each [Py(X)-FA2] -complex given by CDFT calculations (Table S6); and

AUTHOR INFORMATION

Corresponding Author

*(G.Z.) E-mail: guozhen@ustc.edu.cn. *(M.B.) E-mail: mario.barbatti@univ-amu.fr.

ORCID

Guozhen Zhang: 0000-0003-0125-9666 Mario Barbatti: 0000-0001-9336-6607

Note

The authors declare no competing financial support.