Estimation of Electronic Coupling for Photoinduced Charge

Article pubs.acs.org/JPCC

Estimation of Electronic Coupling for Photoinduced Charge Separation and Charge Recombination Using the Fragment Charge Difference Method Alexander A. Voityuk* Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain and Institute of Computational Chemistry, Universitat de Girona, 17071 Girona, Spain S Supporting Information *

ABSTRACT: Photoinduced electron transfer reactions play an important role in chemistry, biochemistry, and material sciences. Electronic coupling of donor and acceptor is a key parameter that controls the rate of charge separation and charge recombination processes. The fragment charge difference (FCD) method is extended to calculate the electronic couplings and diabatic energies for the photoinduced reactions. It is shown that FCD provides consistent values of the ET parameters for any 3-state model system. We compare the matrix elements obtained within the 2- and 3-state treatment for different situations and suggest how to check adiabatic states included in the diabatization procedure. Two examples demonstrate the use of the FCD method in combination with MS-CASPT2 calculations to derive the ET parameters.

■

formation ε = U+EU allows one to construct the diabatic matrix ε from a diagonal matrix E of adiabatic energies. The diagonal elements of ε represent the energy of diabatic states, whereas the off-diagonal elements are electronic coupling of the diabatic states. Different transformations {ψ} → {φ} have been developed.5,7,15,17 In the most known Generalized Mulliken− Hush (GMH) scheme introduced by Cave and Newton, the transition dipole moment between diabatic states is assumed to be zero.5 Several years ago, Cave and co-workers provided a detailed analysis of multistate effects arising in estimating electronic couplings using the GMH approach.20 They considered different cases depending on the interaction and relative energies of three diabatic states (the ground state φGS, locally excited state φEX, and charge separated state φCS) and developed a scheme to assess when these multistate effects will be important. Also, they suggested an effective 2-state approximation to the n-state GMH model, which allows to obtain quite accurate estimates of the ET parameters.20 The FCD method is based on the condition that in diabatic states the excess charges are localized as much as possible.7 To apply this approach, one should define molecular fragments corresponding to the donor and acceptor sites. The algorithm is computationally feasible for large molecules and has been used in combination with ab initio,7 DFT,23,24 and semiempirical

INTRODUCTION Photoinduced electron transfer (PET) is a fundamental process in chemistry and biology. Light absorption by a donor− acceptor complex [D−A] leads to population of locally excited states [D*−A] and [D−A*], which can decay by electron transfer from D to A resulting in a charge separated state [D+− A−]. The photoinduced charge separation (CS) process is of great interest in different areas ranging from photosynthesis to materials manufacture. Decay pathways of [D+−A−] include fluorescence, irradiative charge recombination (CR) leading to the ground state, [D+−A−] → [D−A], and electron and hole migrations. Thus at least three electronic states of the system, the ground state [D−A], excited state [D*−A] (or [D−A*]), and CS state [D+−A−], should be considered to describe PET. Theoretical and computational methods are widely employed to explore the electron transfer (ET) dynamics in molecular systems.1−4 A key parameter that controls the rate of the ET reactions is electronic coupling V. Its accurate estimation is quite challenging. Several approaches can be applied to calculate electronic couplings.5−19 In most cases, one employs a two state model. In this article, we consider an extension of the fragment charge difference method (FCD)7 to PET. This scheme is based on linear transformation (rotation) of adiabatic states {ψ} to diabatic states {φ}, φ = ψU. The diabatic states can be defined as the initial and final states in ET and correspond to the states with excess charges localized on the donors and acceptor.8 In some cases, more than two adiabatic states have to be included in the transformation.5,20−22 The unitary trans© 2013 American Chemical Society

Received: December 4, 2012 Revised: January 21, 2013 Published: February 4, 2013 2670

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calculations25 of radical cation and radical anion systems to estimate electronic couplings for thermal ET. The FED scheme,26 an extension of FCD, has been applied to treat singlet and triplet excitation energy transfer in different systems.15,27,28 The goal of the present work is to extend the FCD scheme to photoinduced CS and CR reactions. The method is shown to provide consistent values of CS and CR couplings for three state systems. We compare the matrix elements obtained within the 2- and 3-state treatment for different situations and suggest how to check adiabatic states included in the diabatization procedure. The use of the FCD method in combination with MS-CASPT2 calculations is demonstrated by two examples.

the transition one-electron density.7 By definition, the charge on D and A is zero in the diabatic states φGS and φEX, whereas the charges on D and A in the state φCS are +1 and −1, respectively. Then in the adiabatic state ψi, the charges Qii(D) and Qii(A) are determined by a contribution of φCS to ψi, Qii(D) = +CCSi2 and Qii(A) = −CCSi2, which leads to ΔQii = 2CCSi2. Similarly ΔQ ij = CCSiCCSj − ( −CCSiCCSj) = 2CCSiCCSj

where CCSi and CCSj are coefficients given by eq 2. Expression 4 can be written as ⎛0

■

+

(5)

C ΔQC + = C(C +dC)C + = (CC +)d(CC +) = d

In other words, for any 3-state system, the FCD transformation allows one to restore the diabatic Hamiltonian h0. Since two eigenvalues of ΔQ are equal, d11 = d33 = 0, the transformation U is not unique. Any transformation U′, U′ = UR with ⎛ cos ω 0 −sin ω ⎞ ⎜ ⎟ R=⎜ 0 1 0 ⎟ ⎝ sin ω 0 cos ω ⎠

(6)

will also diagonalize the matrix ΔQ. Therefore, the diabatic Hamiltonian h = U+EU will depends on the rotation angle ω in eq 6. This angle may be easily derived from the condition VGS,EX  ⟨φGSHφEX⟩ = 0.20 Thus, the following procedure can be applied to obtain the ET parameters: (1) calculate the ground and excited states of a system by a suitable quantum mechanical method; (2) calculate matrix ΔQ, with elements ΔQij = Qij(D) − Qij(A) = ∫ r∈Dρij(r)dr − ∫ r∈Aρij(r)dr, where i and j label the adiabatic states; (3) compute the rotation matrix U by diagonalizing ΔQ; (4) construct h′ by h′ = U+EU; (5) compute the diabatic matrix by h = R+h′R, where R is defined by eq 6 with ω = 1/2 tan−1(2h12 ′ /(h11 ′ − h22 ′ )). The last step is required to ensure that VGS,EX = 0. For any 3-state system, the diabatic matrix h constructed by the FCD scheme is equal to the matrix h0 specified by eq 1. Note that in practice a number of diabatic electronic states (N > 3) can contribute to the adiabatic states of interest. In this case, a 3-state model provides an approximation, and the derived ET parameters will deviate from the reference values. Another important class of PET processes is photoinduced charge shift, where an extra charge localized on one site (donor or acceptor) is transferred to the other. For instance, by electronic excitation of a system DA+ (both D and A+ are closed-shell moieties) charge shift D(A+)* → D+(A) occurs leading to radicals D+ and A. Lappe at al. considered recently computational treatment of such ET reactions within GMH.29 The FCD scheme described above for CS and CR reactions, (DA)* → D+A and D+A → DA, may also be applied to the charge shift process. Even though the charge difference on D

(1)

(2)

leads to adiabatic states ψ1, ψ2, and ψ3, ψ = φC with energies E1, E2, and E3. When couplings VGS,CS and VEX,CS in eq 1 are significantly smaller than the corresponding gaps εCS − εGS and εEX − εCS, the energies E1, E2, and E3 are close to εGS, εCS, and εEX, respectively, and ψ1, ψ2, and ψ3 resemble diabatic states φGS, φCS, and φEX. The transposed matrix C+ defines the reverse transformation of the adiabatic to diabatic states, h0 = CEC+. Thus, if one uses U = C+, the couplings and diabatic energies estimated from E1, E2, and E3 will be exactly equal to the corresponding values of the diabatic Hamiltonian h0. In the FCD method, the transformation U is obtained by diagonalization of the charge difference matrix ΔQ with elements ΔQ ij = Q ij(D) − Q ij(A)

⎞ ⎟ 2 ⎟C 0⎠

The diagonal matrix d with elements 0, 2, and 0 represents the charge difference on D and A in the diabatic states φGS  [DA], φCS  [D+A−] and φEX  [D*A], respectively. Equation 5 suggests that diagonalization of ΔQ, d = U+ΔQU may be performed by U = C+

with the diabatic energies εGS, εCS, and εEX and electronic couplings VGS,CS and VEX,CS. Note that, in eq 1, VGS,EX  ⟨φGSHφEX⟩ = 0 because the states φGS and φEX are locally adiabatic.20 Diagonalization of h0, E = C+h0C, with ⎛CGS1 CGS2 CGS3 ⎞ ⎜ ⎟ C = ⎜CGS1 CGS2 CGS3 ⎟ ⎜ ⎟ ⎝C EX1 C EX2 C EX3 ⎠

+⎜

ΔQ = C dC = C ⎜ ⎝

MODEL Think of a system for which the diabatic Hamiltonian matrix h0 is completely defined. Diagonalization of h0, E = C+h0C, provides adiabatic states of the system. Using some adiabatic properties of the system, one may construct a transformation matrix U to defined a new diabatic Hamiltonian h, h = U+EU. Comparing h and h0, one may estimate the performance of the applied diabatization procedure. Let us apply this approach to test the FCD treatment of CS and CR reactions in a simple model that has 3 diabatic states: the ground state φGS  [DA], excited state φEX  [D*A], and charge separated state φCS  [D+A−]. The states are assumed to be orthonormalized. Transitions φEX → φCS and φCS → φGS describe the CS and CR reactions. The diabatic Hamiltonian h0 is defined by the symmetric matrix of order 3 × 3 ⎛ εGS VGS,CS 0 ⎞ ⎜ ⎟ h0 = ⎜ VGS,CS εCS VEX,CS ⎟ ⎜⎜ ⎟ VEX,CS εEX ⎟⎠ ⎝ 0

(4)

(3)

The diagonal element ΔQii is equal to the difference of the charges on donor and acceptor, Qii(D) and Qii(A), in the adiabatic state ψi. Off-diagonal elements are expressed through 2671

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only two adiabatic states ψ2 and ψ3 can be used to describe CS φEX → φCS. Similarly, adiabatic states ψ1 and ψ2 represent a suitable basis to treat CR φCS → φGS when the admixture of φEX is relatively small. In other words, the 2-state model may be applied when adiabatic states ψi and ψj are well represented by linear combinations of two diabatic states of interest and the contribution of other states is small. If the 2-state model is applied to adiabatic states ψi and ψj, the electronic coupling V of the initial and final ET is given by7

and A in the diabatic states φGS  [DA+], φCS  [D+A], and φEX  [(DA+)*] (it equals −1, 1, and −1, respectively) differs from the corresponding quantities (0, 2, and 0) for CS and CR, the just explained FCD algorithm (steps 1−5) remains valid for the charge shift processes. Effective Two-State Hamiltonian. ET processes in molecular systems are usually described with the effective 2state Hamiltonian.1,2 Any N-state ET system can be treated (at least formally) by the effective two state model.30 Such an effective Hamiltonian can be derived by applying the Löwdin partitioning method31 to the quasi-diabatic 3-state Hamiltonian. For instance, in a system with states R, P, and X, the ET reaction R → P may be described by the effective Hamiltonian

|V | =

(Ej − Ei)|ΔQ ij| [(ΔQ ii − ΔQ jj)2 + 4ΔQ ij 2]1/2

The free energy ΔF is determined by

⎛ ε eff V eff ⎞ R RP ⎟ h (E ) = ⎜ ⎜ eff eff ⎟ ⎝ V RP εP ⎠ eff

−ΔF =

⎛ εR VRP ⎞ ⎛ VRX ⎞ 1 ⎟⎟ + ⎜ ⎟ (VRXVXP) = ⎜⎜ ⎝ VRP εP ⎠ ⎝ VXP ⎠ E − εX

(7)

eff εEX = εEX

VGSCS2 E − εGS

eff V EX,CS = VEX,CS

(8) + −

Similarly, for charge recombination [D A ] → [DA], using φCS → φGS and eq 7 with R = φCS, P = φGS, and X = φEX, we have

α=

VCSEX 2 E − εEX

eff = VGSCS V GSCS

(11)

1 |ΔQ ii + ΔQ jj| 2

(12)

In the numerical tests below, we find a range of α where the electronic couplings derived with the 2-state model agree well with the reference 3-state data. Numerical Tests. Let us consider a 3-state system with the diabatic parameters εGS = 0, εEX = 1 eV, VGSCS = VCSEX = 0.10 eV, and εCS ranging from 0 to 1 eV. As shown above, the FCD scheme restores properly the diabatic Hamiltonian from adiabatic states of any 3-state system. The 2-state approach, eq 10, applied to the same adiabatic states provides, however, row estimates of the couplings. How close are the values of couplings V2st computed with eq 10 to the reference data V3st. Figure 1 shows the ratio R = (V2st/V3st) and the parameter α, eq 12, calculated for CS and CR as functions of εCS. Obviously, the 2-state treatment provides accurate values of the couplings when R is close to 1. For CR (φCS → φGS) when εCS < 0.8 eV (the gap εEX−εCS > 0.2 eV), 1 ≤ (V2st/V3st) < 1.05 and αCR > 0.85. If εCS becomes larger (the gap between φCS and the third state φX = φEX decreases), the parameter α and the ratio R = (V2st/V3st) deviates significantly from 1. In particular, α= 0.78, 0.70, and 0.59 and R = 1.07, 1.12, and 1.20 at εCS = 0.85, 0.90, and 0.95 eV (the gap εEX−εCS is equal to 0.15, 0.10, and 0.05 eV, respectively). Thus, the two-state model provides good estimates of the CR coupling when α > 0.85. Similar results are found also for CS (φEX → φCS); see Figure 1. In this case, the initial and final states are φEX and φCS, and the gap between φCS and the third state φX = φGS affects the accuracy of the 2-state model. Accurate values of V2st, 1 ≤ (V2st/ V3st) < 1.05, are obtained for εCS > 0.2 eV (the gap εCS−εGS >

eff εGS = εGS eff εCS = εCS +

(Ej − Ei)2 − 4V 2

This raises the question how to recognize situations where this simple 2-state scheme might be applied and where it will fail to provide satisfactory results. A detail analysis of this point has been given within the GMH approach by Cave et al.20 Here, we consider this question in the context of the FCD model. A sum of contributions from φCS to adiabatic states ψi and ψj, α = CCSi2 + CCSj2, is a convenient parameter to characterize the applicability of a 2-state model. The model may be employed if the parameter α is close to 1. In this case, a contribution of diabatic state φX to the adiabatic states ψi and ψj should be small. Obviously, α decreases with increasing contribution of φX to the adiabatic states. One can express α through the difference of the donor and acceptor charges in the adiabatic states ψi and ψj

The parameter E is determined by iterative solution of eq 7 and is equal to one of the adiabatic energies. Often, E is approximated by 1/2(εR + εP).30,32 For charge separation [D*A] → [D+A−], R = φEX, P = φCS, and X = φGS, εRVRX = VRX,EX = 0, and VXP = VEX,CS and eq 7 can be written as

eff εCS ≃ εCS +

(10)

(9)

As seen from eqs 8 and 9, the effective coupling for both CS and CR is equal to the value found within the 3-state model, the same is true also for εEX and εGS. Only the diabatic energy εCS should be changed. The correction of εCS depends on the energy E. In practice, replacing E in eqs 8 and 9 by εCS, one obtains the appropriate values of εeff CS. The free energy for both CS and CR estimated as the difference of the effective energies of final and initial states is less negative than that obtained within the 3-state model. In most cases, however, the correction of εCS is relatively small and can be neglected. FCD Treatment of Two Adiabatic States. It was shown that the two-state model does not provide accurate description of excited-state ET reactions, and a third state should be taken into account to properly explain the photophysical behavior of D−A systems.33,34 For instance, even relatively small mixing of CT and locally excited states can considerably affect fluorescence intensity of the complexes.33,34 However, in many cases, electronic couplings for the ET reactions can be computed properly within a two-state approach, which implies that only two adiabatic states are treated simultaneously. If the ground state φGS does not strongly interact (mix) with φCS, 2672

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electrostatic potential, cannot be used within FCD because of the difficulty in estimating off-digonal matrix elements of ΔQ. Computational details. We consider two π stacks: a neutral dimer (A1,A2) consisting of two adenine bases and a radical cation (Ind,G)+ comprising indole and guanine (see Figure 2). Atomic coordinates of the systems are provided in the Supporting Information.

Figure 1. Performance of the 2-state FCD scheme, eq 10, by calculating couplings for the CS and CR reactions. The ratio R = V2st/ V3st characterizes the deviation of the estimated couplings from the reference value. The parameter α is calculated with eq 12. The 2-state model provides good estimates of the coupling when α > 0.85.

Figure 2. Structure of π stacks (A1,A2) and (Ind,G)+.

MS-CASPT2 (multistate formulation of CASPT2, which accounts for the nonorthogonality of the CASPT2 wave function35) calculations were performed with MOLCAS 7.636 using the basis set of atomic natural orbital (ANO-S) type with the primitive set C,N,O(10s6p3d)/H(7s3p),37 contracted to C,N,O[3s2p1d]/ H[2s1p]. For (A1,A2), the active space (12,12) comprising 12 electrons distributed among 12 π orbitals, averaging over eight states and imaginary level shift parameter38 of 0.1 were used. For (Ind,G)+, we employed the active space with 11 electrons on 12 π orbitals (six orbitals on each molecule), averaging over five states, and real level shift parameter of 0.2. Adenine Dimer. In the dimer, apart from neutral states with localized and delocalized excitations, CS states [A1+A2−] and [A1−A2+] exist. The probability of the nonadiabatic process [A1A2*] → [A1+A2−] depends on the coupling squared VCS2. Charge recombination leads back to the ground state, [A1+A2−] → [A1A2]. The rate of this process is determined by VCR2. The MS-CASPT2 results are listed in Table 1. The data are in

0.2 eV); note that αCS > 0.85 at this condition. Thus, parameter α indicates clearly whether a simple 2-state scheme is accurate enough or a more complicated 3-state treatment should be used. Paramter α α=

1 2

N

∑ ΔQ ii i=1

(13)

can also be applied in a more general context to estimate whether N selected adiabatic states (usually N = 2 or 3) of a neutral system form an appropriate basis to calculate the ET parameters. The N states can be treated if α is close to 1, e.g., |α − 1| ≤ 0.10.

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RESULTS OF MS-CASPT2 CALCULATIONS Below are two examples that illustrate how to apply the FCD method in combination with the MS-CASPT2 calculations. We note that electronic couplings obtained within FCD correspond to orthogonal diabatic states and no further correction of their values is required. The following approach is used. First, one computes the ground and excited states of a system. Then one defines the donor and acceptor sites (one just specifies which atoms belong to the D and A) and analyzes the charges on D and A in the adiabatic states. On the basis of these data, one selects adiabatic states corresponding to the ground, chargetransfer, and locally excited states (ψ1, ψ2, and ψ3, respectively). In many cases, states ψ1 and ψ3 have only insignificant admixture of the charge transfer giving ΔQ11 and ΔQ33 to be close to zero, but ΔQ22 ≈ 2, which means that ψ2 is an almost pure CT state. Then, the 2-state model can be applied, and CS and CR couplings are derived with eq 10 for {ψ2,ψ3} and {ψ1,ψ2} pairs of states. When, however, the diabatic state φCS contributes to all three adiabatic states, the 3-state model should be used. The parameter α, eq 13, may be helpful to distinguish these situations. In the FCD approach, we employ the Mullikene population analysis to estimate the ΔQ matrix. The Mullikene charges describe properly the changes in the electron density on D and A by passing from one adiabatic state to another, and they are typically listed in the standard output of commonly used quantum mechanical programs. In contrast, the ESP charges, which are derived by fitting the molecular

Table 1. Adiabatic State Energies E, Oscillator Strengths f, Charges on A1 Q(A1), and Fragment Charge Differences ΔQii Calculated in the Dimer (A1,A2) with MS-CASPT2 state b

1 2 3 4 5 6 7 8

E (eV)

f

Q(A1)

ΔQiia

0.0 5.447 5.489 5.755 5.913 6.532 7.197 7.341

0.152 0.264 0.148 0.916 × 10−1 0.970 × 10−2 0.176 0.133

0.005 0.005 0.006 −0.017 −0.020 −0.898 −0.010 0.009

−0.010 −0.010 −0.012 0.034 0.040 1.796 0.020 −0.018

The charge on the second base Q(A2) = −Q(A1); ΔQii = Qii(A2) − Qii(A1) = −2Qii(A1). bThe ground state. a

agreement with previous calculations.39 Among the calculated states, there is only one, state 6, with significant charge transfer [A1−A2+], where bases A2 and A1 are electron donor and acceptor, respectively. The charge difference for this state, ΔQ66 = Q(D) − Q(A) ≈ 1.80. Let us consider two CS reactions, CS1 and CS2, involving states 7 and 8, respectively. The oscillator strength of electronic transitions from the ground state to states 7 and 8 is found to be 0.18 and 0.13 (Table 1), and therefore, 2673

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The N-state model is applicable if α is close to 1, e.g., |1 − α| ≤ 0.10. It should be noted that eq 14 for charge shift differs from eq 13 for CS by a factor of 1/2. This is because for charge shift the difference of charges on D and A in diabatic states φGS, φCS, and φEX is 1, −1, and 1 giving the sum of 1, whereas for charge separation, the corresponding differences are 0, 2, and 0, and their sum equals 2. Equation 15 can be used to determine α for both types of systems

the states should be populated by UV absorption. By contrast, the charge transfer state (state 6) cannot be directly populated (the oscillator strength for this transition is less than 0.01). Because the energy of states 7 and 8 is higher than that of state 6, the free energy of CS1 and CS2 is negative. Parameter α, eq 13 calculated for the 2- and 3-state models of CS1 and CS2, is found to be about 0.9. This suggests that both 2- and 3-state FCD schemes should provide similar values of the coupling. For CS1, V2st is found to be 0.068 eV (adiabatic states 6 and 7 were treated); the simultaneous transformation of adiabatic states 1, 6, and 7 gives the same value of the coupling, V3st = 0.068 eV. Since the electronic coupling is much smaller than the corresponding adiabatic splitting, the free energy ΔF estimated as the difference of the adiabatic energies, E6 − E7 = −0.665 eV is very similar to that provided by eq 11, ΔF = −0.651 eV. For CS2, the 2- and 3-state models give almost identical values for the electronic coupling, V2st ≈ V3st ≈ 0.044 eV. The free energy values calculated as the difference (E6 − E8) and within eq 11 are −0.809 and −0.804 eV. The ET parameters for CR were derived in a similar way. The 2-state model (adiabatic states 1 and 6) predicts CR coupling to be 0.177 eV. The same value is also obtained by the 3-state approach applied to adiabatic states and 1, 6, and 8. A large negative value of the free energy, ca. −6.5 eV, clearly suggests that the ET reaction in the gas phase occurs in the inverted Marcus regime. π-Stacked Radical Cation (Ind,G)+. Although the hole transfer between Ind and G appears to be thermally activated process, formally it should be treated with the 3-state model because the locally excited state Ind* affects the ET rate.40 The parameters of the ground and three low-lying excited states are listed in Table 2. The ground and two first excited states in the

α=

a

E

1a 2 3 4

0.0 0.434 0.753 1.996

f

Q(Ind)

Q(G)

ΔQii

0.123 × 10−1 0.969 × 10−1 0.775 × 10−1

0.845 0.838 0.310 0.853

0.155 0.162 0.690 0.147

0.690 0.676 −0.380 0.706

reaction 2st

V V3st

Ind+·G → Ind·G+

Ind+*·G → Ind·G+

0.291 0.234

0.125 0.144

■

CONCLUSIONS The fragment charge difference method has been applied to calculate the ET parameters (site energies and electronic couplings) for photoinduced charge separation, charge recombination, and charge shift reactions. It has been shown that the FCD scheme provides consistent values of the ET parameters for any 3-state system. The derived couplings can be used to construct the 2-state effective Hamiltonian for CS and CR (no correction of these matrix elements is required). We have suggested how to check whether N selected adiabatic states form an appropriate basis to calculate the ET parameters. Two examples have been provided to illustrate the use of the FCD method in combinations with MS-CASPT2 calculations to derive the ET parameters.

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ASSOCIATED CONTENT

S Supporting Information *

N i=1

(15)

i=1

seen, the 2-state model overestimates the coupling for Ind·G+ → Ind+·G and underestimates it by 13% for Ind+*·G → Ind·G+ as compared to the coupling obtained by the 3-state treatment. The calculated couplings are comparable in magnitude with the adiabatic energy gap, and therefore, the free energy cannot be properly estimated as the energy difference of the adiabatic states. The diabatic energies derived with the 3-state model are ε[Ind+·G] = 0.10, ε[Ind+*·G] = 0.50, and ε[Ind·G+] = 0.588 eV. Then, ΔF = −0.49 and −0.09 eV for hole transfer Ind·G+ → Ind+·G and Ind·G+ → Ind+*·G, respectively. It should be noted that the diabatic parameters for the (Ind,G)+ stack are very sensitive to mutual arrangement of the subunits.40

radical cation lie within 1 eV. In the adiabatic states, a hole is delocalized over both moieties. Transition 1 → 2 corresponds to a locally excited state of Ind and the charge distribution in the stack remains almost unchanged. Transitions 1 → 3 and 2 → 3 are associated with hole transfer from Ind to G. Three diabatic states may be defined φGS  [Ind+,G], φCS  [Ind,G+], and φEX  [Ind+*,G]. In these states, the hole (positive charge) is confined to Ind or G. The sum of charges on these moieties over the diabatic states is 2 and 1, respectively. The N-state model (N = 2 or 3) should work if the adiabatic states are represented by a linear combination of N diabatic states. It means that the sum of the charges on each fragment over N adiabatic states should be close to that for diabatic states (to 2 and 1 for Ind and G). This condition may be expressed using parameter α

∑ ΔQ ii

∑ ΔQ ii

Table 3. Electronic couplings for thermal and photoinduced HT in radical cation (Ind, G)+ calculated within 2- and 3state FCD scheme

The ground state.

α=

N

where QS is the total charge on the D−A system. Equation 15 reduces to eq 13 for neutral systems; QS = 0. Using ΔQii from Table 2, one can directly obtain α = 1.007 for the 3-state model, but the parameter deviates significantly from 1, when only two adiabatic states are considered (α = 0.845 for states 1 and 3, and α = 0.852 for states 2 and 3). This test suggests that the 3state scheme should work well. Although the 2-state model appears to be still applicable, the derived estimates may not be very accurate. The calculated couplings are listed in Table 3. As

Table 2. MS-CASPT2 Calculation of Adiabatic State Energies E, Oscillator Strengths f, and the Fragment Charges in the π-Stack (Ind,G)+ state

1 NQ S − 2

Atomic coordinates of the studied systems (Tables SI1 and SI2), MS-CASPT2 energies and ΔQ matrix for (A1,A2) (Tables SI3 and SI4), and MS-CASPT2 energies and ΔQ matrix for

(14) 2674

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(Ind,G)+ (Tables SI5 and SI6). This material is available free of charge via the Internet at http://pubs.acs.org.

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(36) Aquilante, F.; De Vico, L.; Ferre, N.; Ghigo, G.; Malmqvist, P.A.; Neogrady, P.; Pedersen, T. B.; Pitonak, M.; Reiher, M.; Roos, B. O.; Serrano-Andrés, L.; Urban, M.; Veryazov, V.; Lindh, R. J. Comput. Chem. 2010, 31, 224. (37) Pierloot, K.; Dumez, B.; Widmark, P.-O.; Roos, B. O. Theor. Chim. Acta 1995, 90, 87. (38) Forsberg, N.; Malmqvist, P.-Å. Chem. Phys. Lett. 1997, 274, 196. (39) Olaso-Gonzalez, G.; Merchan, M.; Serrano-Andres, L. J. Am. Chem. Soc. 2009, 131, 4368. (40) Butchosa, C.; Simon, S.; Blancafort, L.; Voityuk, A. A. J. Phys. Chem. B 2012, 116, 7815.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support from MICINN (Ministry of Science and Innovation, Spain) and the FEDER fund (European Fund for Regional Development) was provided by grants CTQ 201126573, UNGI08-4E-003, and UNGI10-4E-801.

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