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Evaluating electronic couplings for excited state charge transfer based on maximum occupation method (mom)-#SCF quasi-adiabatic states Junzi Liu, Yong Zhang, Peng Bao, and Yuanping Yi J. Chem. Theory Comput., Just Accepted Manuscript • Publication Date (Web): 10 Jan 2017 Downloaded from http://pubs.acs.org on January 10, 2017

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Journal of Chemical Theory and Computation

Evaluating Electronic Couplings for Excited State Charge Transfer Based on Maximum Occupation Method(mom)-∆SCF Quasi-Adiabatic States Junzi Liu,† Yong Zhang,‡ Peng Bao,† and Yuanping Yi∗,† †Institute of Chemistry Chinese Academy of Sciences, Beijing 100190, P. R. China ‡Synfuels China, Beijing, 101407, P. R. China E-mail: [email protected]

Abstract Electronic couplings of charge-transfer states with the ground state and localized excited states at the donor/acceptor interface are crucial parameters for controlling the dynamics of exciton dissociation and charge recombination processes in organic solar cells. Here we propose a quasi-adiabatic state approach to evaluate electronic couplings through combining mom-∆SCF and state diabatization schemes. Compared with TDDFT using global hybrid functional, mom-∆SCF is superior to estimate the excitation energies of charge-transfer states; moreover it can also provide good excited electronic state for property calculation. Our approach is hence reliable to evaluate electronic couplings for excited state electron transfer processes, which is demonstrated by calculations on a typical organic photovoltaic system, oligothiophene/perylenediimide complex.

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INTRODUCTION

Comprehensive investigation of electron transfer processes is an essential issue to understand the basic mechanism of organic optoelectronic devices. 1–3 For instance, exciton dissociation and charge recombination at donor/acceptor interfaces, which involve excited state charge transfer, have significant influence on the power conversion efficiencies (PCE) for organic solar cells (OSCs). 3,4 In order to obtain optimal PCE, we need to choose proper materials and attentively tune device formation conditions to maximize the rate of advantageous mechanism (such as exciton dissociation) while to minimize the pernicious mechanism (such as charge recombination). In general Marcus-Hush theory, 5–8 the electronic transfer rate between initial state (before electron transfer) and final state (after electron transfer) can be written by kIF

1 −(λ + ∆G0 )2 2π 2 , |VIF | exp = ~ 4πλkB T 4λkB T

(1)

where ∆G0 is the Gibbs free energy, λ is the reorganization energy and VIF is the effective electronic coupling. Obviously, kIF is proportional to the square modulus of effective electronic coupling, which could governs the relative magnitude of transition rate with same Gibbs free energy and reorganization energy. So the electronic couplings of charge-transfer (CT) state with exciton state and the ground state in electron transfer processes are crucial factors that can affect the dynamics of the exciton dissociation and charge recombination at D/A interface. 9–11 Specifically, if the electronic coupling between exciton state and CT state increases (in exciton dissociation), the photocurrent (required to be maximized) would be enhanced. At the same time, however, the increasing of coupling between the CT state and the ground state (in charge recombination) can lead to serious energy loss because a large reverse saturation current in the dark and it should be minimized. Therefore accurate calculation of electronic coupling involves CT states becomes an essential topic of theoretical description of electron transfer. The theory of electron transfer is usually based on diabatic states which have particular

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spatial electron locality. 12 However, exact diabatic states do not always exist. 13 Therefore one has lots of approaches to construct “diabatic” states. 14–27 According to different ways of constructing diabatic states, there are two strategies for calculating electronic couplings. 28,29 One is relies on introducing a physical observation (usually an operator), and transform the adiabatic states (eigenstates) into corresponding diabatic states with particular electron locality. With invoking of dipole operator, generalized Mulliken-Hush (GMH) 30,31 method can define the diabatic states by diagonalizing the dipole moment matrix of corresponding adiabatic states. Since GMH needs to assume a central charge-transfer direction, it can not usually deal with multiple centers electron transfer problems. By employing charge difference operator, Voityuk et al. 32 proposed fragment charge difference (FCD) method which has very similar formula to GMH method. Because of flexibility of choosing fragments and evaluating their charge difference, FCD can be used beyond the two charge centers problem. It is noted that Subotnik et al. presented a diabatization approach using Boys localization algorithm. 33,34 This approach not only makes the GMH can extract electronic coupling elements for electron transfer with many charge centers but also gives the exact GMH formula. Another strategy of evaluating electronic coupling is to directly construct diabatic state based on certain intuitive charge-localized electronic states through specific definitions and procedures. The electron-localized broken symmetry wave functions stemming from unrestricted Hartree-Fock(UHF) theory have been used to denote the initial and final diabatic states and applied to the study of charge transfer process. 35–37 This approach is usually used in a particular electron transfer reaction or self-exchange reaction, so the electronic coupling involved excited states is difficult to be considered. It is remarkable that constrained DFT 38–41 can be applied to construct diabatic states with particular charge locality. In constrained DFT, a new Lagrange multiplier is obtained by adding a supplementary potential to conventional Kohn-Sham potential. Thus the potential in certain space region is optimized until the desired number of electrons are fulfilled in this area. Further, Van Voorhis et al. proposed an approach to estimate the coupling matrix element in long-range

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electron transfer reaction. 39,42 Recently, they evaluated the electronic couplings between exciton state and charge-transfer state based on time-dependent density functional theory (TDDFT) and CDFT. 43 It provided an enlightening physical pictures for illustrating the role of CT states in exciton-exciton transitions within organic semiconductors. Based on Mo’s block-localized wavefunction (BLW), 44,45 Gao et al. developed multistate density functional theory (MSDFT) 46–49 which can provide diabatic states to determine electronic coupling. In BLW, the molecular orbitals are block-localized, and the total energy is variational minimized with nonorthogonal orbitals between blocks. Therefore, the relevant electron density is constrained by direct construction. In MSDFT, the diabatic coupling is directly obtained from the block-localized wavefunctions. Recently, the electronic couplings of electron and hole transfer reaction were studied by two-state approach based on MSDFT 48 and BLW. 50 Of these two kinds of different approaches for computing electronic couplings, transformation scheme methods such as GMH and its variants are widely used for various systems because they are very simple to implement in quantum chemistry software and have fast computational speed. 33 For the electron transfer among excited states, the electronic couplings of GMH-like methods depend on the dipole moment matrix elements and excitation energies. In other words, the quality of electronic coupling calculation is determined by the properties of excited states. Time-dependent density functional theory (TDDFT) becomes a workhorse of studying excited states for relative large molecular systems because it can reach reasonable balance of accuracy and computational cost. 51,52 However, TDDFT using standard hybrid functionals usually underestimates excitation energies of charge-transfer states for donor-acceptor molecular systems. 42,53–56 Range separated hybrid (RSH) functionals have been proposed to ameliorate this situation by mixing in the exact Hartree-Fock exchange with particular tuning procedures, but it is inevitable to give virtual orbitals with undesirable property that they are too diffuse and probably inappropriate for describing excited states. 57 While this field is actively being pursued, no “standard” solution is available yet. Therefore, alternative approaches for investigating CT excited state are valuable and ∆SCF-

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based methods have emerged as a promising one. 39,58–68 In our previous work, 68 we proposed a self-consistent field theory with maximum occupation method (mom-∆SCF) to handle the CT states. Except for excitation energies, the excited electronic state can be supplied by mom-∆SCF. Thus it provides a practical foundation for calculating the properties related to excited states. Inspired by this point, we present an electronic coupling approach which integrates eminent features of mom-∆SCF and GMH-like electronic coupling approach. The basic idea is to utilize mom-∆SCF excited electronic state to evaluate electronic coupling based on Boys diabatization approach. The rest of this paper is organized as follows: maximum occupation method and electronic coupling approach are introduced in section 2. In section 3, we illustrate our computational results and discussions. The conclusions and prospect are given in section 4.

2

METHODS

Our approach can be divided into two steps: (i) calculating desired excited states by mom∆SCF; (ii) evaluating electronic couplings using dipole moment matrix elements and excitation energies of obtained excited states.

2.1

Maximum Occupation Method-∆SCF

The standard self-consistent field theory can be achieved by solving the eigenvalue equation iteratively, F(C)C = SCǫ.

(2)

Here F(C) is the Fock/Kohn-Sham matrix which depends on molecular orbital coefficients. And S and C are overlap matrix on atomic orbitals and molecular orbital coefficients matrix. The maximum overlap method(MOM) developed by Gilbert and his coworkers 66,67 is a simple protocol for controlling electron occupation for finding excited state solutions in eq 2. Instead of Aufbau principle, MOM guarantees the occupied orbitals are chosen to be those that 5



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overlap most with the span of the old desired occupied space {ψ˜i }N i=1 . Specifically, the j-th new occupied orbital is identified and selected to be the ones with the largest projection Pj . The projection Pj is defined based on overlap matrix between new and old orbitals

Pj =

N X

Oij = hψ˜i |ψj i.

Oij ,

i=1

(3)

However, the original MOM algorithm may failed in some cases because of the degenerate orbitals and wrong phases. 68 To remedy this defects, the maximum occupation method (mom) was given by nj =

N X i

2

|Cij | = hψj |P0 |ψj i,

P0 =

N X i

|ψ˜i ihψ˜i |,

(4)

where P0 is the projection operator that is constructed by prechosen orbitals {ψ˜i }N i=1 . By introducing this projection operator P0 , nj can be an quantity from 0 to 1 just as occupation number. Compared with original MOM, maximum occupation method obtained same results for small molecules and it is more robust for even large molecules with dense and degenerate orbitals. 68 Applying mom criteria to unrestricted Hatree-Fock/Kohn-Sham theory, charge-transfer state can be well described by single electron transfer determinant. 56 In that way, we not only obtain more reasonable CT excitation energy, but also the relaxed CT electronic state. However, the mom criteria is only able to maintain maximum overlap of preset occupation space, and it cannot guarantee higher energy solution of SCF equation orthogonal to reference state. So the mom-∆SCF electronic states are generally nonorthogonal and so-called quasiadiabatic state with non-zero overlap with reference states. As a matter of fact, mom-∆SCF wave functions are the eigenstates of the whole system’s Hamiltonian, so they should be in principle adiabatic states. Practically, the overlap of mom-∆SCF states is usually very small and can be ignored. On another hand, we can indeed construct new Lagrangian of SCF equation that guarantees both maximum occupation and orthogonal requirements to reference state. 59,61,64,69 Therefore it is reasonable to treat mom-∆SCF wave functions as 6


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adiabatic states in our method. Additionally, non-Aufbau electronic state obtained by mom∆SCF is not a spin eigenfunction, the energy of the singlet excited state can be obtained by the common spin purification formula 70

ES = 2E↑↓ − E↑↑ .

(5)

The corresponding singlet excited state is written by combination of spin-mixed states in the light of conventional spin-adapted formula 1 |1 Φai i = √ (|Φai i + |Φ¯ai¯ i) , 2

(6)

where |Φai i and |Φ¯ai¯ i are spin-mixed states independently calculated by unrestricted KohnSham theory with mom criteria. And i, a (¯i, a ¯) stand for the electron transfer from i-th occupied orbital with spin up (spin down) to a-th virtual orbital with spin up (spin down). The electronic characteristic of these states are not local but still diffuse all over the system. Hence it should carry out diabatization (localization) of these states to identify proper diabatic states for describing the electron transfer processes.

2.2

Effective Electronic Couplings

Within two-state model, 12 effective electronic coupling is straightforwardly related to the diabatic state with charge-localized characteristic. And it is given by 1 H (Q) + H (Q) II FF HIF (Q) − SIF (Q) , VIF (Q) = 2 1 − SIF (Q) 2

(7)

ˆ F i is the direct electronic coupling between initial diabatic state ΨI where HIF = hΨI |H|Ψ ˆ is the (before electron transfer) and final diabatic state ΨF (after electron transfer). H ˆ I i and HFF = hΨF |H|Ψ ˆ Fi total electronic Hamiltonian of the whole system. HII = hΨI |H|Ψ are total energies of two diabatic states. SIF = hΨI |ΨF i is the overlap of two diabatic 7



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states. According to Condon approximation, 71 the dependence of VIF on the nuclear degree of freedom which can be neglected at all nuclear configuration Q of the system. Effective electronic coupling comes from the second term of the right-hand side of eq 8. The vertical excitation energy between adiabatic potential energy curves is the difference of adiabatic energy E+ and E− ∆Ev = E+ − E− =

s

2 ∆EIF + 4VIF2 , 2 1 − SIF

(8)

where ∆EIF = EI − EF = HII − HFF is the energy difference of two diabatic states. In principle, ∆EIF also depends on nuclear coordinates of system. Particularly, at the crossing point of two diabatic potential energy curves, i.e., Q = Qt , ∆Ev = 2VIF where EI (Qt ) = EF (Qt ). The splitting adiabatic states energy E+ and E− can be obtained by solving generalized eigenvalue equation based on initial and final diabatic states 













 HII HIF   a   E+ 0   1 SIF   a   .    =   b SFI 1 0 E− b HFI HFF

(9)

According to eq 9, the electron transfer adiabatic state can be depicted by |Φi = a|ΨI i + b|ΨF i. Here we focus on the simple and most widely used GMH-like method which transforms adiabatic state into diabatic state by means of dipole operator. 30,31 In our approach, arbitrary (Φ1 , Φ2 )T calculated by mom-∆SCF are treated as adiabatic states. This means S12 = ˆ 2 i = 0. It is note that (Φ1 , Φ2 )T (quasi-adiabatic states) are hΦ1 |Φ2 i = 0, H12 = hΦ1 |H|Φ singlet electronic states combined by α and β spin-mixed states (with eq 6) which are directly calculated by UKS with mom criteria. Under this approximation, the diabatic states can be obtained through a rotation matrix U 













 Φ1   cosθ sinθ   Φ1   ΨI  .  =  = U  Φ2 −sinθ cosθ Φ2 ΨF 8


(10)

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By the unitary transformation, the new diabatic states behave with charge-localized characteristic on different centers. Thus the direct electronic coupling between two new diabatic states are given by ˆ F i = cos2θH12 − 1 sin2θ (H11 − H22 ) . HIF = hΨI |H|Ψ 2

(11)

Insert eq 11 and H12 into eq 7, effective electronic coupling can be rewritten as √ 2√ 1 | 1 − cos4θ||H11 − H22 |. |VIF | = |HIF | = |sin2θ| |(H11 − H22 )| = 2 4

(12)

The cosine of rotation angle θ can be obtained by state diabatization procedure with Boys localization algorithm. 72 Subotnik et al. 33,34 extended Boys algorithm beyond orbital localization to state diabatization (localization) which transform adiabatic states into diabatic states. Moreover, Boys localization can be utilized over arbitrarily many charge centers problems. So Ns diabatic states representation {ΨI } can be constructed by rotating same number of adiabatic states {ΦJ } with matrix U ΨI =

Ns X

ΦJ UJI ,

I = 1, . . . , Ns .

(13)

J=1

By maximizing the function of rotation matrix U in eq 14, the optimal matrix U is given by transforming adiabatic states into diabatic states which satisfy the condition of transition dipole moment among charge centers tends to be zero.

fBoys (U) =

Ns X

I,J=1

|hΨI |~µ|ΨI i − hΨJ |~µ|ΨJ i|2 .

(14)

Two-by-two “Jacobi sweep” algorithm is usually doing over pairs of states for optimization procedure of eq 14. 72–74 Take example of two state system, the solution of rotation matrix or rotation angle θ in eq 10 can be obtained by Jacobi sweep with very fast speed and they

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are written most compactly as follows: 1 1 1 |~µ11 − µ ~ 22 |2 A12 = |~µ12 |2 − |~µ11 |2 − |~µ22 |2 + µ ~ 11 · µ ~ 22 = |~µ12 |2 − , 4 4 2 4

(15)

B12 = µ ~ 12 · (~µ11 − µ ~ 22 ),

(16)

A12 , (cos4θ)12 = − p 2 2 A12 + B12 B12 (sin4θ)12 = p 2 , 2 A12 + B12

(17) (18)

where µ ~ 11 = hΦ1 |~µ|Φ1 i and µ ~ 22 = hΦ2 |~µ|Φ2 i are dipole moments for corresponding states. µ ~ 12 = hΦ1 |~µ|Φ2 i is the transition dipole moment between Φ1 and Φ2 . Here Φ1 and Φ2 are spin-adapted states that are composed of spin-mixed states. Jacobi sweep can give exact rotation matrix of diabatization for two state system. It means that eq 12 is equivalent to exact GMH formula 33 |µv ||H11 − H22 | , |HIF | = p 12 |~µ11 − µ ~ 22 |2 + 4(µv12 )2

(19)

where µv12 is the projection of µ ~ onto the direction v = (~µ11 − µ ~ 22 )/|~µ11 − µ ~ 22 |. Comparing with original GMH formula eq 20, the approximation is attributed to use µ ~ 12 instead of µv12 , |~µ12 ||H11 − H22 | GMH |HIF |= p = αµ |H11 − H22 |. |~µ11 − µ ~ 22 |2 + 4(~µ12 )2

(20)

Moreover it can observe that the effective electronic coupling are related to excitation energy and a prefactor calculated by dipole moments matrix elements. Here we appoint αBoys = 21 sin2θ and αµ = √

|~ µ12 | |~ µ11 −~ µ22 |2 +4(~ µ12 )2

to be called diabatization factors corresponding

to different formulas. In this way, these factors that influence effective electronic coupling can be quantitatively analyzed and discussed.

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3 3.1

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RESULTS AND DISCUSSION Electronic Structures and Excited States

The isosurface plots of frontier orbitals for 4T/PDI complex are shown in Figure 2. The spatial distributions of these orbitals have palpable localized characteristic, e.g. HOMO and LUMO are localized on 4T and PDI, respectively. Other frontier orbitals also retain their spatial distributions on 4T and PDI monomers. It could be expected that the excited state which originates from transition between molecular orbitals (MOs) on 4T and PDI (e.g. HOMO → LUMO) has typical charge-transfer feature. On the other hand, localized excited states on 4T or PDI would be generated by MO transitions on same monomers (e.g. HOMO → LUMO+1 and HOMO-1 → LUMO). The 100% HOMO → LUMO TDDFT transition fea-

LUMO(B)

LUMO+1(A)

LUMO+2(A)

HOMO(B)

HOMO-1(A)

HOMO-2(A)

Figure 2: The spatial distribution of frontier molecular orbitals from HOMO-2 to LUMO+2(the irreducible representations of MOs in parentheses). ture indicates that the lowest excited state is with characteristic of absolute charge transfer from donor to acceptor. Therefore, in mom-∆SCF calculation, the preset occupation pattern of the lowest CT state is determined as transition from HOMO to LUMO. For MSDFT, since the detailed informations are illustrated in ref 47 and 48, here we briefly demonstrate the calculation procedure with MSDFT in this work. For 4T/PDI complex system, it can 12


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be easily divided into two blocks of PDI (donor) and 4T (acceptor). The localized occupied orbitals of each block are constrained only in the space of its block basis set. There is no charge in each block for the localized ground state; and one electron is transferred from donor block to acceptor block for CT state. The CT state is the spin adapted state of the α and β electron transfer localized state with the form of eq 6. Then the energies of the localized ground state and broken symmetry electron transfer localized states are variationally minimized within block restriction. The excitation energies are the difference between the energies of the localized ground state and the CT state. The subsequent electronic coupling calculations are based on optimized block-localized wavefunctions (diabatic state) with an approximated formula of eq 7. Generally speaking, CT excitation energy can be approximated as IP + EA − 1/R (IP: ionization potential, EA: electron affinity, R: distance between charge centers), and the CT excitation energies should be increased with 1/R asymptotic as the distance of the separated charges is getting longer. 55 We first tested various functionals including global hybrid functionals (B3LYP, PBE0 83 and M062X 84 ), range separated hybrid functionals (CAM-B3LYP 85 and LC-ωPBE, 86 ω = 0.4000 Bohr−1 ) and gap-tuned range separated hybrid functional (LCωPBE, 87 ω=0.1982 Bohr−1 ) for CT state calculation, shown in Figure S1 in the Supporting Information. Here we emphasized that mom-∆SCF singlet excitation energies are calculated according to eq 5 based on two independent mom-∆SCF calculations with parallel spin and entiparallel spin occupation pattern. The TDDFT excitation energies with global hybrid functionals (such as B3LYP and PBE0) show a little decrease with the 4T-PDI separation distance and are tremendously underestimated by 0.5 ∼ 1.2 eV relative to the mom-∆SCF results. This is in accordance with our known about TDDFT failure when CT states are calculated using global hybrid functionals. 43 In contrast, the CT excitation energies calculated by mom-∆SCF increase the 4T-PDI distance and have a qualitative correct 1/R asymptote for all the density functionals. Moreover, all CT excitation energy profiles using individual functionals are almost parallel with each other and the differences among the excitation en-

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calculated by TDDFT are mostly contributed by more than one particle-hole(p-h) pairs, we still can track them by the main p − h pairs since their transition components surpass 70%. Figure 3 displays the mom-∆SCF and corresponding TDDFT excitation energies. In principle, localized excited states on monomers should not change much in energy as the donor-acceptor distance is increased. As expected, the excitation energies of these localized excited states remains nearly constant with respect to the 4T-PDI separation whatever using TDDFT or mom-∆SCF. Van Voorhis et al. assessed the accuracy of ∆SCF method for calculating HOMO→LUMO transition excited states for a set of organic dye molecules and similar accuracy was found for TDDFT and ∆SCF. 58 The energies of localized excited states on PDI are about 0.22 eV smaller for ∆SCF relative to TDDFT. Similarly, the excitation energies of the first localized excited states on 4T are about 0.26 eV smaller, whereas those of the second localized excited states on 4T are only 0.10 eV larger than corresponding TDDFT excitation energies. Unexpectedly, the mom-∆SCF excitation energy at 3.5 Å is 0.3 eV higher than ones at the other distances for the second localized excited state (HOMO-2→LUMO+1) on 4T. At such short 4T-PDI distance, the obritals involved in the excitation are no longer localized on 4T but diffuse on both 4T and PDI, which are different from the ones at the other distances (see Figure S2 of the Supporting Information). This is expected to influence drastically the electronic couplings. To conclude, in comparison with TDDFT, mom-∆SCF can provide almost similar excitation energies of low-lying localized excited states and much better description for the charge-transfer states.

3.2

Electronic Couplings

Charge recombination between the CT and ground states governs the magnitude of reverse saturation current that would lead to significant Voc loss. The electronic coupling between the CT and ground states is an essential parameter for describing charge recombination. The effective electronic couplings VIF between the lowest CT state and ground state are shown in Fig. 4. We recall that the effective electronic coupling will exponentially decay with the 15




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Table 1: Excitation energies(in eV) of lowest CT excited states, αq a , αµ b , αBoys c and electronic couplings(in meV) between lowest CT state and ground state D(Å) ETDDFT E∆SCF EMSDFT

αq

αµ

αBoys

VIFFCD VIFGMH VIFBoys VIFMSDFT

3.5

1.2020

1.7730

1.8494

0.0248 0.0220 0.0318 29.84

26.44 56.33

37.77

4.0

1.2150

1.9713

1.9839

0.0122 0.0110 0.0142 14.78

13.37 28.07

19.33

4.5

1.2020

2.1022

2.0940

0.0060 0.0055 0.0065

7.21

6.67 13.67

8.96

5.0

1.1943

2.2039

2.1916

0.0027 0.0025 0.0027

3.19

2.97

5.98

3.79

5.5

1.1897

2.3009

2.2840

0.0010 0.0010 0.0011

1.21

1.14

2.45

1.39

6.0

1.1900

2.3531

2.3614

0.0003 0.0003 0.0003

0.38

0.36

0.74

0.36

a b c

|VIFFCD | = αq |H11 − H22 |, |VIFGMH | = αµ |H11 − H22 |,

αq = √

|dq12 | |dq11 −dq22 |2 +4(dq12 )2 |~ µ12 |

αµ = √

|~ µ11 −~ µ22 |2 +4(~ µ12 )2

αBoys = sin2θ/2 Here we also observe some differences between αµ and αboys at short D-A distances.

Actually, diabatization factor is a comprehensive reflection of dipole moments and transition dipole moment for the electronic states that take part in electron transfer(see Table S1 in the Supporting Information). Table S1 tabulates dipole moments of ground and CT excited states, and transition dipole moments between them. Except for the dipole moments of the ground state, the transition dipole moments calculated by mom-∆SCF and TDDFT are very close and small too. However, the TDDFT excited state dipole moments are much larger than the mom-∆SCF ones. According to the formulas of diabatization factors as shown in tablenotes b and c in Table 1, the smaller excited state dipole moments directly lead to larger diabatization factors calculated by mom-∆SCF. It should be pointed out that the actual excited wave function is not available in most time-dependent response theories. So excited state properties are defined via the response of the ground state to time-dependent perturbation. For mom-∆SCF, all dipole moments including the ground and excited states 17




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couplings. More importantly, our method can calculate electronic couplings of CT state with both the ground state and localized excited states, and it is easy to extend to compute electronic couplings involved triplet states. This would be very helpful to comprehensively investigate charge recombination and exciton dissociation processes in organic solar cells. It is noted that ∆SCF is not a “black-box” method, so it needs to firstly find the transition feature of excited state that we concern. It does not work when the excited state can not be described by a single excited determinant. In future work, we are going to employ mom criteria to restricted open-shell Kohn-Sham (ROKS) approach 63 which can give more rigorous spin-adapted wave functions to compute electronic coupling. And we will further investigate the reliability and feasibility of two typical electronic coupling scheme.

ASOOCIATED CONTENT Supporting Informaiton The charge transfer excitation energy profile as 4T-PDI distance calculated by mom-∆SCF, TDDFT and MSDFT using various functionals (Figure S1); Dipole moments of the ground state and charge-transfer excited states, and transition dipole moments between them based on TDDFT and mom-∆SCF (Table S1); The spatial distributions, orbital energies and occupation numbers of frontier molecular orbitals for localized excited state on 4T (Figure S2).

AUTHOR INFORMATION Corresponding Author ∗

Y. Yi. E-mail: [email protected]

OCRID Yuanping Yi: 0000-0002-0052-9364

Funding This work has been supported by the National Natural Science Foundation of China (Grant No. 91333117), National Basic Research (973) program of the Ministry of Science and Tech20


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nology of China (Grant NO. 2014CB643506), and the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB12020200).

Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS J. L. thanks Prof. Zhigang Shuai, Prof. Wenjian Liu, Prof. Lan Cheng and Prof. Yunlong Xiao for enlightening discussions and very helpful suggestions for paper writing. J. L. also thanks Prof. BingBing Suo and Dr. Qiming Sun for assistance in coding on the BDF and Pyscf, respectively.

References (1) Clarke, T. M.; Durrant, J. R. Chem. Rev. 2009, 110, 6736–6767. (2) Brédas, J.-L.; Beljonne, D.; Coropceanu, V.; Cornil, J. Chem. Rev. 2004, 104, 4971– 5003. (3) Brédas, J.-L.; Norton, J. E.; Cornil, J.; Coropceanu, V. Acc. Chem. Res. 2009, 42, 1691–1699. (4) Zhu, X.-Y.; Yang, Q.; Muntwiler, M. Acc. Chem. Res. 2009, 42, 1779–1787. (5) Marcus, R. A. J. Chem. Phys. 1956, 24, 966–978. (6) Marcus, R. A. Annu. Rev. Phys. Chem. 1964, 15, 155–196. (7) Hush, N. S. Trans. Faraday Soc. 1961, 57, 557–580. (8) Hush, N. S. Electrochim. Acta 1968, 13, 1005–1023. (9) Lemaur, V.; Steel, M.; Beljonne, D.; Brédas, J.-L.; Cornil, J. J. Am. Chem. Soc. 2005, 127, 6077–6086. 21



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(10) Burquel, A.; Lemaur, V.; Beljonne, D.; Cornil, J. J. Phys. Chem. A 2006, 110, 3447– 3453. (11) Coropceanu, V.; Sánchez-Carrera, R. S.; Paramonov, P.; Day, G. M.; Brédas, J.-L. J. Phys. Chem. C 2009, 113, 4679–4686. (12) May, V.; Kühn, O. Charge and Energy Transfer Dynamics in Molecular Systems; WILEY-VCH Verlag GmbH & Co. KGaA: Weinheim, Germany, 2011. (13) Mead, C. A.; Truhlar, D. G. J. Chem. Phys. 1982, 77, 6090–6098. (14) Pacher, T.; amd H. Köppel, L. S. C. J. Chem. Phys. 1988, 89, 7367–7381. (15) Köppel, H. In Conical Intersections: Electronic structure, Dynamics and Spectroscopy; Domcke, W., Yarkony, D. R., Koppel, H., Eds.; World Scientific: New Jersey, 2004; p 175. (16) Cederbaum, L. S. In Conical Intersections: Electronic structure, Dynamics and Spectroscopy; Domcke, W., Yarkony, D. R., Koppel, H., Eds.; World Scientific: New Jersey, 2004; p 3. (17) Baer, M. Chem. Phys. Lett. 1975, 35, 112–118. (18) Top, Z. H.; Baer, M. J. Chem. Phys. 1977, 66, 1363–1371. (19) Baer, M. Mol. Phys. 1980, 40, 1011–1013. (20) Cederbaum, M. Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections; Wiley: New York, 2006. (21) Nakamura, H.; Truhlar, D. G. J. Chem. Phys. 2001, 115, 10353–10372. (22) Nakamura, H.; Truhlar, D. G. J. Chem. Phys. 2002, 117, 5576–5593. (23) Nakamura, H.; Truhlar, D. G. J. Chem. Phys. 2003, 118, 6816–6829. 22


Page 22 of 28

Page 23 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(24) Ruedenberg, K.; Atchity, G. J. J. Chem. Phys. 1993, 99, 3799–3803. (25) Atchity, G. J.; Ruedenberg, K. Theor. Chem. Acc. 1997, 97, 47–58. (26) Domcke, W.; Woywod, C. Chem. Phys. Lett. 1993, 216, 362–368. (27) Domcke, W.; Woywod, C.; Stengle, M. Chem. Phys. Lett. 1994, 226, 257–262. (28) Hsu, C.-P. Acc. Chem. Res. 2009, 42, 509–518. (29) You, Z.-Q.; Hsu, C.-P. Int. J. Quantum Chem. 2014, 114, 102–115. (30) Cave, R. J.; Newton, M. D. Chem. Phys. Lett. 1996, 249, 15–19. (31) Cave, R. J.; Newton, M. D. J. Chem. Phys. 1997, 106, 9213–9226. (32) Voityuk, A. A.; Rösch, N. J. Chem. Phys. 2002, 117, 5607–5616. (33) Subotnik, J. E.; Yeganeh, S.; Cave, R. J.; Ratner, M. A. J. Chem. Phys. 2008, 129, 244101. (34) Subotnik, J. E.; Cave, R. J.; Steele, R. P.; Shenvi, N. J. Chem. Phys. 2009, 130, 234102. (35) Newton, M. D. Int. J. Quantum Chem. Symp. 1980, 18, 363–391. (36) Newton, M. D. Chem. Rev. 1991, 91, 767–792. (37) Zhang, L. Y.; Friesner, R. A.; Murphy, R. B. J. Chem. Phys. 1997, 107, 450–459. (38) Wu, Q.; Van Voorhis, T. Phys. Rev. A 2005, 72, 024502. (39) Wu, Q.; Van Voorhis, T. J. Chem. Theory Comput. 2006, 2, 765–774. (40) Wu, Q.; Cheng, C.-L.; Van Voorhis, T. J. Chem. Phys. 2007, 127, 164105. (41) Kaduk, B.; Kowalczyk, T.; Van Voorhis, T. Chem. Rev. 2012, 112, 321–370. (42) Wu, Q.; Van Voorhis, T. J. Chem. Phys. 2006, 125, 164105. 23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(43) Difley, S.; Van Voorhis, T. J. Chem. Theory Comput. 2011, 7, 594–601. (44) Mo, Y.; Peyerimhoff, S. D. J. Chem. Phys. 1998, 109, 1687–1697. (45) Mo, Y.; Gao, J. J. Phys. Chem. A 2000, 104, 3012–3020. (46) Mo, Y.; Song, L.; Lin, Y. J. Phys. Chem. A 2007, 111, 8291–8201. (47) Cembran, A.; Song, L.; Mo, Y.; Gao, J. J. Chem. Theory Comput. 2009, 5, 2702–2716. (48) Ren, H.; Provorse, M. R.; Bao, P.; Qu, Z.; Gao, J. J. Phys. Chem. Lett. 2016, 7, 2286–2293. (49) Mo, Y.; Bao, P.; Gao, J. Phys. Chem. Chem. Phys. 2011, 13, 6760–6775. (50) Mo, Y.; Song, L.; Lin, Y.; Liu, M.; Cao, Z.; Wu, W. J. Chem. Theory Comput. 2012, 8, 800–805. (51) Dreuw, A.; Head-Gordon, M. Chem. Rev. 2005, 105, 4009–4037. (52) Laurent, A. D.; Jacquemin, D. Int. J. Quantum. Chem. 2013, 113, 2019–2039. (53) Tozer, D. J.; Handy, N. C. Phys. Chem. Chem. Phys. 2000, 2, 2117–2121. (54) Tozer, D. J. J. Chem. Phys. 2003, 119, 12697–12699. (55) Dreuw, A.; Weisman, J. L.; Head-Gordon, M. J. Chem. Phys. 2003, 119, 2943–2946. (56) Dreuw, A.; Head-Gordon, M. J. Am. Chem. Soc. 2004, 126, 4007–4016. (57) Baerends, E. J.; Gritsenko, O. V.; Meer, R. V. 2013, 15, 16408–16425. (58) Kowalczyk, T.; Yost, S. R.; Van Voorhis, T. J. Chem. Phys. 2011, 134, 054128. (59) Glushkov, V. N.; Levy, M. J. Chem. Phys. 2007, 126, 174106. (60) Glushkov, V. M.; Assfeld, X. J. Chem. Phy. 2010, 132, 204106. 24


Page 24 of 28

Page 25 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(61) Ziegler, T.; Seth, M.; Mykhaylo Krykunov, J. A.; Wang, F. J. Chem. Phys. 2009, 130 . (62) Park, Y. C.; Krykunov, M.; Ziegler, T. Mol. Phys. 2015, 113, 1636–1647. (63) Kowalczyk, T.; Tsuchimochi, T.; Chen, P. T.; Top, L.; Van Voorhis, T. J. Chem. Phys. 2013, 138, 164101. (64) Evangelista, F. A.; Shushkov, P.; Tully, J. C. J. Phys. Chem. A 2013, 117, 7378–7392. (65) Peng, B.; Van Kuiken, B. E.; Ding, F.; Li, X. J. Chem. Theory Comput. 2013, 9, 3933–3938. (66) Gilbert, A. T. B.; Besley, N. A.; Gill, P. M. W. J. Phys. Chem. A 2008, 112, 13164– 13171. (67) Besley, N. A.; Gilbert, A. T. B.; Gill, P. M. W. J. Chem. Phys. 2009, 130, 124308. (68) Liu, J.; Zhang, Y.; Liu, W. J. Chem. Theory Comput. 2014, 10, 2436–2448. (69) Xiao, Y. Private communication. (70) Ziegler, T.; Rauk, A.; Baerends, E. J. Theor. Chim. Acta 1977, 43, 261–271. (71) Kuznetsov, A. M.; Ulstrup, J. Electron Transfer in Chemistry and Biology; John Wiley & Sons: New York, 1999. (72) Boys, S. F. In Quantum Theory of Atoms, molecules and the Solid State; Lowdin, P., Ed.; Academic: New York, 1966; p 253. (73) Edmiston, C.; Ruedenberg, K. Adv. Mod. Phys. 1963, 35, 457–467. (74) Kleiler, D. A.; Halgren, T. A.; Jr, J. H. H.; Lipscomb, W. N. J. Chem. Phys. 1974, 61, 3905–3919. (75) Nielsen, C. B.; Holliday, S.; yang Chen, H.; Cryer, S. J.; Mcculloch, I. Acc. Chem. Res. 2015, 48, 2803–2812. 25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 28

(76) Liu, W.; Hong, G.; Dai, D.; Li, L.; Dolg, M. Theor. Chem. Acc. 1997, 96, 75–83. (77) Liu, W.; Wang, F.; Li, L. J. Theor. Comput. Chem. 2003, 2, 257–272. (78) Liu, W.; Wang, F.; Li, L. In Relativistic Density Functional Theory: The BDF Program Package. Recent Advances in Relativistic Molecular Theory; Hirao, Y., Ishikawa, Y., Eds.; Recent Advances in Computational Chemistry; World Scientific: Singapore, 2004; Vol. 5; pp 257–282. (79) Sun,

Q.

PySCF:

Python

module

for

quantum

chemistry.

https://github.com/sunqm/pyscf. (80) Shao, Y.; Gan, Z.; Epifanovsky, E.; Gilbert, A. T. B.; Wormit, M.; Kussmann, J.; Lange, A. W.; Behn, A.; Deng, J.; Feng, X.; Ghosh, D.; Goldey, M.; Horn, P. R.; Jacobson, L. D.; Kaliman, I.; Khaliullin, R. Z.; Kús, T.; Landau, A.; Liu, J.; Proynov, E. I.; Rhee, Y. M.; Richard, R. M.; Rohrdanz, M. A.; Steele, R. P.; Sundstrom, E. J.; Woodcock III, H. L.; Zimmerman, P. M.; Zuev, D.; Albrecht, B.; Alguire, E.; Austin, B.; Beran, G. J. O.; Bernard, Y. A.; Berquist, E.; Brandhorst, K.; Bravaya, K. B.; Brown, S. T.; Casanova, D.; Chang, C.-M.; Chen, Y.; Chien, S. H.; Closser, K. D.; Crittenden, D. L.; Diedenhofen, M.; DiStasio Jr., R. A.; Dop, H.; Dutoi, A. D.; Edgar, R. G.; Fatehi, S.; Fusti-Molnar, L.; Ghysels, A.; Golubeva-Zadorozhnaya, A.; Gomes, J.; Hanson-Heine, M. W. D.; Harbach, P. H. P.; Hauser, A. W.; Hohenstein, E. G.; Holden, Z. C.; Jagau, T.-C.; Ji, H.; Kaduk, B.; Khistyaev, K.; Kim, J.; Kim, J.; King, R. A.; Klunzinger, P.; Kosenkov, D.; Kowalczyk, T.; Krauter, C. M.; Lao, K. U.; Laurent, A.; Lawler, K. V.; Levchenko, S. V.; Lin, C. Y.; Liu, F.; Livshits, E.; Lochan, R. C.; Luenser, A.; Manohar, P.; Manzer, S. F.; Mao, S.-P.; Mardirossian, N.; Marenich, A. V.; Maurer, S. A.; Mayhall, N. J.; Oana, C. M.; Olivares-Amaya, R.; O’Neill, D. P.; Parkhill, J. A.; Perrine, T. M.; Peverati, R.; Pieniazek, P. A.; Prociuk, A.; Rehn, D. R.; Rosta, E.; Russ, N. J.; Sergueev, N.; Sharada, S. M.; Sharmaa, S.; Small, D. W.; Sodt, A.; Stein, T.; Stück, D.; Su, Y.-C.; Thom, A. J. W.; Tsuchi26


Page 27 of 28

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


mochi, T.; Vogt, L.; Vydrov, O.; Wang, T.; Watson, M. A.; Wenzel, J.; White, A.; Williams, C. F.; Vanovschi, V.; Yeganeh, S.; Yost, S. R.; You, Z.-Q.; Zhang, I. Y.; Zhang, X.; Zhou, Y.; Brooks, B. R.; Chan, G. K. L.; Chipman, D. M.; Cramer, C. J.; Goddard III, W. A.; Gordon, M. S.; Hehre, W. J.; Klamt, A.; Schaefer III, H. F.; Schmidt, M. W.; Sherrill, C. D.; Truhlar, D. G.; Warshel, A.; Xua, X.; AspuruGuzik, A.; Baer, R.; Bell, A. T.; Besley, N. A.; Chai, J.-D.; Dreuw, A.; Dunietz, B. D.; Furlani, T. R.; Gwaltney, S. R.; Hsu, C.-P.; Jung, Y.; Kong, J.; Lambrecht, D. S.; Liang, W.; Ochsenfeld, C.; Rassolov, V. A.; Slipchenko, L. V.; Subotnik, J. E.; Van Voorhis, T.; Herbert, J. M.; Krylov, A. I.; Gill, P. M. W.; Head-Gordon, M. Mol. Phys. 2015, 113, 184–215. (81) Schmidt, M. W.; Baldridge, K. K.; Baotz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J. Comput. Chem. 1993, 14, 1347–1363. (82) Stephens, P. J.; Devlin, F. J.; Chabalowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623–11627. (83) Adamo, C.; Barone, V. J. Chem. Phys. 1999, 110, 6158–6170. (84) Zhao, Y.; Truhlar, D. G. Theor. Chem. Acc. 2008, 120, 215–241. (85) Yanai, T.; Tew, D. P.; Handy, N. C. Chem. Phys. Lett. 2004, 393, 51–57. (86) Vydrov, O. A.; Scuseria, G. E. J. Chem. Phys. 2006, 125, 234109. (87) Körzdörfer, T.; Brédas, J.-L. Acc. Chem. Res. 2014, 47, 3284–3291. (88) Yi, Y.; Coropceanu, V.; Brédas, J.-L. J. Mater. Chem. 2011, 21, 1479–1486. (89) Yi, Y.; Coropceanu, V.; Brédas, J.-L. J. Am. Chem. Soc. 2009, 131, 15777–15783.

27




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