Estimation of Electronic Coupling for Singlet Excitation Energy

Article pubs.acs.org/JPCC

Estimation of Electronic Coupling for Singlet Excitation Energy Transfer Alexander A. Voityuk* Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain Institute of Computational Chemistry, Universitat de Girona, 17071 Girona, Spain S Supporting Information *

ABSTRACT: Electronic coupling is a key parameter that controls the efficiency of excitation energy transfer (EET) and exciton delocalization. A new approach to estimate electronic coupling is introduced. Within a two-state model, the EET coupling V of two chromophores is expressed via the vertical excitation energies (Ei and Ej), transition dipole moments (Mi and Mj) of the system and transition moments (μA and μB) of the individual chromophores: V = (Ei − Ej){[(MiMj)(μA2 − μB2) − (μAμB)(Mi2 − Mj2)]/[(Mi2 − Mj2)2 + 4(MiMj)2]}. These quantities are directly available from quantum mechanical calculations. As the estimated coupling accounts for both short-range and long-range interactions, this approach allows for the treatment of systems with short intermolecular distances, in particular, π-stacked chromophores. For a system of two identical chromophores, the coupling is given by V = (Ei − Ej)[(EiFj − EjFi)/(EiFj + EjFi)][1/(2 cos θ)] where Fi and Fj are the corresponding oscillator strengths and cos θ is determined by the relative position of the chromophores in the dimer. Thus, the coupling can be derived from purely experimental data. The developed approach is used to calculate the EET coupling and exciton delocalization in two π-stacks of pyrimidine nucleobases 5′-TT-3′ and 5′-CT-3′ showing quite different EET properties.

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INTRODUCTION Theoretical and computational methods have been successfully applied to study excitation energy transfer (EET) in various molecular systems. Recent developments in this field were considered in detail in several reviews.1−4 The efficiency of EET between two chromophores A and B, A*−B → A−B*, is controlled by electronic coupling of two diabatic states φA and φB. In the reference states, the exciton is localized on the corresponding molecules. Commonly, φA and φB are defined as excited states of the separated chromophores. The electronic coupling of the reference excited states is a key parameter that controls delocalization of excited states and the probability of energy transfer between the chromophores.1−4 The coupling of singlet excited states V includes both a shortrange term, which depends on the orbital overlap between the excited states, and a long-range Coulomb contribution. The Coulomb coupling of two molecules A and B can be estimated as the interaction of their transition dipole moments μA and μB Vdd =

μA⃗ μB⃗ 3 RAB

−

distance RAB. More accurate values can be computed using transition atomic charges6 Vtc =

i∈A j∈B

(1)

where RAB is the intermolecular distance. This widely used dipole−dipole scheme suggested by Förster 60 year ago5 provides good estimates when the spatial extension of the transition densities in the molecules is much smaller than the © 2014 American Chemical Society

qiqj R ij (2)

In eq 2, i runs over all atoms of the molecule A, and j runs over all atoms of B; qi and qj are transition charges derived from quantum mechanical (QM) calculations of excited states φA and φB of the individual molecules. At large distances between the chromophores, Vdd ≈ Vtc. Both eq 1 and eq 2 account only for the long-range interaction of the excited states, whereas the orbital and exchange terms are neglected. The performance of the transition charge (TC) model was discussed in several papers.7,8 In particular, it was shown that the TC scheme works well if RAB ≥ 4.5 Å. For π-stacked chromophores, however, RAB ∼ 3.5−4.0 Å and the short-range interaction may be significant.1−4,7,8 To obtain the total coupling, the complex containing both chromophores should be treated. For symmetric systems, the coupling V is equal to half of the excited-state gap Δ1,4

⃗ )(μB⃗ RAB ⃗ ) 3(μA⃗ RAB 5 RAB

∑

Received: November 2, 2013 Revised: January 2, 2014 Published: January 2, 2014 1478

dx.doi.org/10.1021/jp410802d | J. Phys. Chem. C 2014, 118, 1478−1483

The Journal of Physical Chemistry C |Vsplit| =

|Ei − Ej| Δ = 2 2

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chromophores A and B, respectively. Such states can be approximately represented by excited states of individual molecules. The algorithm differs substantially from schemes where the diabatic states are represented by the wave functions of isolated molecules and the direct calculation of the coupling is performed V = HDA − SDA((HDD + HAA)/(1 − S2DA)). In this case the effects of nonorthogonality (if the overlap SDA ≠ 0) are significant and should be accounted for.19,20 In contrast, we use orthogonal diabatic states generated by rotation of adiabatic states, SDA = 0. However, such diabatic states are not completely localized. Subotnik et al. have performed a detailed analysis of the diabatic states for singlet electronic excitations.13 It was shown that small excitation tails stemming from highenergy excited states do not lead to substantial changes in the diabatic coupling. The results13 provide some justification for the approximation used in our scheme. As shown below, the nonorthogonality effects within the proposed scheme are rather small and can be neglected in most cases. A two-state model implies that φA and φB can be expressed via two adiabatic states ψi and ψj. A unitary transformation connects the adiabatic and diabatic states

(3)

In eq 3, the excitation energies Ei and Ej correspond to the (φA + φB) and (φA − φB) combinations of the excited states. When the molecules A and B are different or the position of identical molecules is nonsymmetric, the half-splitting scheme provides only an upper limit of the coupling and may significantly overestimate the matrix element. Hsu et al.9 introduced a fragment excitation difference (FED) scheme to estimate coupling, including both the long- and short-range interactions of the excited states. The FED approach is an extension of the fragment charge difference method10 used to calculate the electronic coupling for hole and excess electron transfer. Alternative approaches based on the transformation of adiabatic states have also been recently introduced.11−13 These schemes are applied in combination with quantum chemical methods based on single-electron excitation.4,9 Additionally, the constrained DFT method can be employed to derive electronic coupling.14 Recently, Lischka and co-workers demonstrated that the EET coupling of two excited states ψi and ψj can be estimated using transition densities corresponding to the ψ0 →ψi and ψ0 →ψj excitations (where ψ0 is the ground state).15,16 The treatment is exact for single-electron excitations and provides reasonable estimates for the coupling when the contributions of two- and multielectron excitations are relatively small. In the following section, we introduce a method that expresses the electronic coupling for singlet excitation energy transfer A*B → AB* in a system consisting of two chromophores, A and B, via excitation energies Ei and Ej, transition dipole moments Mi and Mj of the system, and transition dipole moment μA and μB of the individual chromophores V = (Ei − Ej)

φA = ψi cos ω + ψj sin ω φB = −ψi sin ω + ψj cos ω ψi = φA cos ω − φBsin ω ψj = φA sin ω + φBcos ω

V=

Fj − Fi 1 1 ≈ (Ei − Ej) Fi + Fj 2 cos θ EjFi + EiFj 2cos θ

Ei − Ej 2

sin 2ω

(7)

In line with eq 7, the coupling is determined by the difference of the adiabatic energies and by the angle ω. Using eq 6b, one can express the adiabatic transition moments Mi and Mj via the diabatic transition moments μA and μB.

(4)

The suggested scheme can be directly used with any quantum mechanical approach that provides excitation energies and transition dipole moments. In addition to singly excited configurations, the states of interest may include two-electron and higher excitations. If the chromophores A and B are identical V = (Ei − Ej)

(6b)

By definition (eqs 6a and 6b), the states φA and φB are orthonormalized. The matrix element V = ⟨φA|H|φB⟩ is related to the difference of adiabatic energies Ei − Ej:

(MiMj)(μA2 − μB2 ) − (μA μB )(Mi2 − M2j ) (Mi2 − M2j )2 + 4(MiMj)2

(6a)

Mi = μA cos ω − μB sin ω Mj = μA sin ω + μB cos ω

(8)

Then

EiFj − EjFi

(5)

Mi2 − M2j = (μA2 − μB2 ) cos 2ω − 2(μA μB ) sin 2ω

In eq 5, Fi and Fj are the oscillator strength and cos θ controls the relative position of the chromophores in the dimer (in πstacks, θ is the twist angle). Thus, the excitonic coupling of two chromophores in the dimer can be derived from purely experimental data. Note that if the chromophores are symmetrically arranged one of the oscillator strengths, Fi or Fj becomes equal to zero (see below) and eq 5 reduces to eq 3. To illustrate the use of the method, we estimate the excitonic coupling in two π-stacks of nucleobases using the MS-CASPT2 method for the excited-state calculation.

2(MiMj) = 2(μA μB ) cos 2ω + (μA2 − μB2 ) sin 2ω Mi2 + M2j = μA2 + μB2

(9)

where (MiMj) and (μAμB) are the dot product of the transition dipoles. On arrangement, one obtains sin 2ω = 2

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(MiMj)(μA2 − μB2 ) − (μA μB )(Mi2 − M2j ) (Mi2 − M2j )2 + 4(MiMj)2

(10)

It is easily followed that

METHOD A general approach to calculate electronic coupling is based on the orthogonal transformation of the adiabatic states {ψ} to diabatic states {φ}.9−12,17,18 We define the diabatic states φA and φB to be the states where the exciton is localized on

(Mi2 − M2j )2 + 4(MiMj)2 = (μA2 − μB2 )2 + 4(μA μB )2

(11)

Thus, the denominator in eq 10 can also be expressed via the transition moments of separated chromophores. Finally, we have 1479


The Journal of Physical Chemistry C V = (Ei − Ej) = (Ei −

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(MiMj)(μA2 − μB2 ) − (μA μB )(Mi2 − M2j )

(MiMj)(μA2 − μB2 ) − (μA μB )(Mi2 Ej) (μA2 − μB2 )2 + 4(μA μB )2

−

M2j )

|V | = |Ei −

(12)

−

M2j )| (12a)

μB̃ =

1 ⎧⎛ 1 ⎨⎜ − 2 ⎩⎝ 1 + S

⎛ ⎞ 1 1 + ⎟μ + ⎜ 1−S⎠A ⎝ 1+S

⎞ ⎫ 1 ⎟μ ⎬ 1 − S ⎠ B⎭

(μA μB ) 2

1−S

−

2 2 μ2 + μB2 1 (μA + μB )S S ≈ (μA μB ) − A 2 2 1−S 2

μA2 − μB2 2

1−S

≈ μA2 − μB2

Note that in most cases |S| < 0.1 and the corresponding correction to electronic coupling, eq 12, is not significant. However, when (μAμB) = 0 the nonorthogonal effect may be important. If a molecular system cannot be separated into individual chromophores and thus QM calculations of the subunits are prohibited, μA and μB must be derived with an alternative approach. In particular, applying some constraints to localize the excitation on fragment A and then on fragment B (such as in the constrained DFT14) one can derive the quantities μA and μB from the calculation of the whole system. The formalism developed here cannot be applied to systems beyond the two-state approximation. For an n-state model, the unitary transformation matrix of size n × n is determined by n(n − 1)/2 variables. On the other hand, there are only (n − 1) independent relations between n adiabatic and diabatic transition dipole moments (the Σi n= 1M2i = Σan= 1μ2a constraint should be accounted for). Thus, the transformation matrix can be unambiguously defined only for n = 2. Nevertheless, the twostate model can be used to treat multichromophore systems. In such cases, subsystems consisting of only two chromophores are treated independently. Exciton Delocalization. When one knows the coupling, the extent of exciton delocalization in the system can be estimated. Equation 6b suggests that in the adiabatic state ψi the exciton density on chromophores A and B, Xi(A) and Xi(B), is equal to cos2 ω and sin2 ω, respectively. Using eq 7 and the relation Xi(A) + Xi(B) = 1 and Xi(A) − Xi(B) = cos 2ω we obtain

(13)

(14)

(15)

Fj − Fi

1 Fi + Fj 2 cos θ

⎞ ⎫ 1 ⎟μ ⎬ 1 − S ⎠ B⎭

(20)

Here, Fi and Fj are the oscillator strength of the transitions ψ0 → ψi and ψ0 → ψj in the complex. Equation 15 suggests that the coupling in the dimer can be estimated from the spectroscopic data and the structural parameter cos θ determined by the relative orientation of the monomers. Assuming that in eq 15 Ei/Ej ≈ 1, one obtains V ≈ (Ei − Ej)

⎛ ⎞ 1 1 − ⎟μ + ⎜ 1−S⎠A ⎝ 1+S

μA2̃ − μB̃ 2 =

EiFj − EjFi

1 EjFi + EiFj 2 cos θ

1 ⎧⎛ 1 ⎨⎜ + 2 ⎩⎝ 1 + S

(μÃ μB̃ ) =

Taking into account eq 9 and the relation between the oscillator strength and transition moment F = 2/3·E·M2 V = (Ei − Ej)

μÃ =

where S is the overlap integral of the diabatic states. Then

M2j − Mi2 4μ2 cos θ

(18)

(19)

where θ is the angle between the directions of μA and μB. This angle depends on the relative position of the molecules in the dimer. In particular, θ is the twist angle between π-stacked monomers. From eqs 11 and 13 one obtains V = (Ei − Ej)

2(μA μB )

When, however, a system consists of two identical chromophores μ2A = μ2B, which are perpendicular to each other, (μAμB) = 0, the scheme cannot be applied to estimate the coupling. In this special case, eq 3 can be used to estimate the coupling. The proposed method, eq 12, employs the transition moment μA and μB of individual chromophors. When the diabatic states are nonorthogonal, the moments μA and μB should be corrected. Using the Löwdin orthogonalization procedure one can define the diabatic quantities in the orthogonal basis

M2j − Mi2

μ2 cos θ (Mi2 − M2j )2 + 4(MiMj)2 A

(17)

M2j − Mi2

V = (Ei − Ej)

In any symmetric system, the adiabatic states are expressed as (φA ± φB) combinations of the diabatic state; thus, Mi = 1/√2 (μA − μB) and Mj = 1/√2 (μA + μB). Substituting these expressions for Mi and Mj into eq 12 and taking into account that μ2A = μ2B, one directly obtains eq 3. This means that eq 12 reduces to eq 3 for any symmetric dimer independent of the relative orientation of the chromophores. In a system where |μA| = |μB|, eq 12 can be written as V = (Ei − Ej)

μA2 − μB2

If μ2A = μ2B but (μAμB) ≠ 0

Equation 12 expresses the coupling through the adiabatic energies Ei and Ej, transition dipole moments Mi and Mj of the complex, and transition moments μA and μB of individual chromophores. The vectors μA and μB are computed for molecules A and B using atomic coordinates that are the same as those in the complex AB. As the sign of the transition dipole moments is not determined, some care should be taken by estimation of the coupling. If in eq 6b ω = 0, the adiabatic states ψi and ψj and diabatic states φA and φB are ordered by energy Ei < Ej and εA < εB, then ψi = φA, ψj = φB Mi = μA, and Mj = μB. In line with eq 7, V = 0; therefore, the terms (μAμB)(M2i − M2j ) and (MiMj)(μ2A − μ2B) in eq 12 must have the same sign. Alternatively, one may estimate the absolute value of the coupling as |(MiMj)(μA2 − μB2 ) − (μA μB )(Mi2 Ej| (Mi2 − M2j )2 + 4(MiMj)2

2(MiMj)

V = (Ei − Ej)

(Mi2 − M2j )2 + 4(MiMj)2

(16)

Some Limitations. For a system where two chromophores are perpendicular, (μAμB) = 0 but μ2A ≠ μ2B, eq 12 reduces to 1480


The Journal of Physical Chemistry C Xi(A) =

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1 + cos 2ω 1 1 4V 2 = + 1− 2 2 2 (Ei − Ej)2 X i(B) = 1 − X i(A)

(21)

As seen from eq 6b, Xj(A) = Xi(B) = sin2 ω and Xj(B) = Xi(A) = cos2 ω, and the reverse exciton distribution should be observed in the state ψj. Does the Two-State Model Hold for a System of Interest? Equations 12−16 were derived for the “ideal” twostate model which implies that two adiabatic states of interest are represented by linear combination of only two diabatic states φA and φB and therefore the transformation described by eqs 6a and 6b is exact. However, when there is another interacting state φX (e.g., a bridge state in D−B−A systems) which is energetically close to φA and φB, the adiabatic states have contributions of this state. Although the two-state model is not exact anymore, in many cases it provides reasonable values of electronic couplings.13,21,22 Several relationships, μ2A + μ2B = M2i + M2j , fA + f B = Fi + Fj, or eq 11, may be used to check whether the contributions of other diabatic states to the adiabatic states of interest is small enough and whether the twostate approximation is still applicable. The model should work well if the ratio R, e.g., R = (Fi + Fj)/( fA + f B) or R = (M2i + M2j )/(μ2A + μ2B), does not significantly deviate from the unit, |R − 1| ≤ 0.2. We note that because eqs 9 and 11 for “real” systems become approximate, the results obtained with eqs 13−15 might differ.

Figure 1. Structure of the stacks 5′-T-T-3′ and 5′-C-T-3′.

Table 1. MS-CASPT2 Energy E (eV), oscillator strength f, and transition dipole moment μ (au) for S1 (π,π*) Excited State of the C and T Bases nucleobase

E

f

|μ|

μx

μy

μz

5′-C 5′-T T-3′

4.749 5.232 5.232

0.231 0.453 0.454

1.410 1.880 1.882

0.215 1.416 1.874

1.394 −1.237 −0.168

0.000 0.000 0.000

with previous CASPT225 data. Negligibly small differences in f and |μ| values of 5′-T and T-3′ are caused by rounding off the atomic coordinates to 3 decimal places. 5′-TT-3′ Stack. Although the thymine bases in the stack have identical internal geometries (bond lengths and bond angles), they are not equivalent. The excitation energies, oscillator strengths, and transition dipole moments for the stack are listed in Table 2. The S1 and S2 energies of the dimer are similar to the S1 energy of thymine. The S2 − S1 energy difference is 0.109 eV. The oscillator strengths for S0 → S1 and S0 → S2 are significantly different and amount to 0.095 and 0.837, respectively. The sum F1 + F2 (0.932) is close to that of the monomers (0.907). As the ratio ((F1 + F2)/2f T) is close to 1, the two-state model can be applied. In the ideal case, eqs 13−16 are equivalent and provide the same coupling value. In practice, however, the derived values may differ. To estimate the coupling, the structural parameter cos θ must be defined. In the ideal B− DNA stacks, θ = 36°; thus, cos θ = 0.809. For any dimer consisting of the same molecules, cos θ can be easily derived from the relative position of the molecules. Alternatively, it can be expressed via the transition moment calculated for the individual monomers, cos θ = (μA1μA2)/(|μA1||μA2|). Using the transition moments for 5′-T and 3′-T (Table 1), one gets cos θ = 0.809. The estimated excitonic coupling values are listed in Table 3. As expected, the coupling values derived with eqs 13−16 are close to each other, within the range of 0.053−0.055 eV. The Coulomb coupling calculated with eq 2 is 0.057 eV, which is very close to the total coupling value. The transition charges required for this estimation are provided in the Supporting Information. The dipole−dipole approximation, eq 1, gives a reasonable value of 0.042 eV. We note that the situation found for the stack 5′-TT-3′ (the short-range interactions are weak and the total and Coulomb couplings are similar) appears to be rather unusual. It changes significantly for the stack 5′-CT-3′ considered in the next section.

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RESULTS OF QUANTUM MECHANICAL CALCULATIONS Below, we apply the proposed scheme to derive the electronic coupling for singlet excitation energy transfer and to estimate the extent of the exciton delocalization in two nucleobase πstacks. To this end, one computes the excited state of interest (usually S1 and S2 of the complex A−B and S1 of each chromophore). The QM calculation of the complex provides the excitation energies Ei and Ej and the transition moments Mi and Mj, whereas the calculation of molecules A and B provides the data (εA, εB, μA, and μB) for individual chromophores. Note that the orientation of A and B should remain unchanged as compared to that in the complex. Alternatively, the transition moments μA and μB should be properly rotated. We apply the MS-CASPT2 (multistate formulation of CASPT2) method23 to compute the π → π* excited states of two stacks 5′-T-T-3′ and 5′-CT-3′ consisting of pyrimidine nucleobases thymine (T) and cytosine (C) (see Figure 1). The relative position of the nucleobases in the stacks corresponds to that in the ideal B-DNA. The atomic coordinates of the complexes are listed in the Supporting Information. The calculations were performed with MOLCAS 7.624 using the ANO-S basis set. For both complexes, the active space (8,8) comprising 8 electrons distributed among 8 π orbitals was used. The active space (4,4) was employed for individual nucleobases C and T. The CASSCF wave functions of the monomers were averaged over 4 states and those of the dimers over 8 states. Monomers. The results of the MS-CASPT2 calculations for the (π,π*) excited states of the nucleobases (vertical excitation energies, oscillator strengths, transition dipole moments) are shown in Table 1. For thymine, the data are provided for two different orientations of the molecule corresponding to the 5′ and 3′ position. The computed values are in good agreement 1481


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Table 2. MS-CASPT2 Energy E (eV), Oscillator Strength F, and Transition Dipole Moment M (au) for the S1 and S2 Excited State of the 5′-TT-3′ and 5′-CT-3′ Stacks stack

state

E

F

|M|

Mx

My

Mz

5′-TT-3′

S1 S2 S1 S2

5.237 5.346 4.790 5.100

0.095 0.837 0.267 0.384

0.858 2.527 1.507 1.752

0.097 2.375 0.819 1.700

0.852 −0.864 1.265 −0.424

0.014 −0.018 0.008 −0.012

5′-CT-3′

coupling and exciton delocalization have been estimated for two π-stacks of pyrimidine bases 5′-TT-3′ and 5′-CT-3′ using MS-CASPT2 calculations. It has been found that even in these related systems, the long-range and short-range contributions may be quite different. In particular, the Coulomb interaction provides the main contribution to the coupling in 5′-TT-3′, whereas it is much smaller than the short-range term in the 5′CT-3′ stack.

Table 3. Excitonic Couplings (meV) in the 5′-TT-3′ and 5′CT-3′ Stacks Vdd

Vtc

Vsplit

stack

eq 1

eq 2

eq 3

eq 12

eq 15

V eq 16

5′-TT-3′ 5′-CT-3′

42.4 7.6

56.6 0.8

55.0 155.0

53.5 103.6

53.3

53.6

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The character of exciton delocalization in 5′-TT-3′ can be found within the scheme described by eq 21. In particular, using V = 0.054 and Δ = 0.109 eV, one obtains that in the excited state S1, X5′T = 0.64 and X3′T = 0.36 whereas X5′T = 0.36 and X3′T = 0.64 in S2. 5′-CT-3′ Stack. The excitation energies, oscillator strengths, and transition dipole moments for the stack are given in Table 2. The vertical gap S2 − S1, calculated to be 0.31 eV, is smaller than the difference in excitation energies of monomers, 0.48 eV (Table 1). The sum of oscillator strengths for the S0 → S1 and S0 → S2 transitions in the complex, F1 + F2 = 0.651, is similar to that for the monomers, f C + f T = 0.685. The ratio (F1 + F2)/ (f C + f T) ≈ 0.95 suggests that the two-state model may be applied. Using eq 12 and the data listed in Tables 1 and 2, one gets the total coupling value to be 0.104 eV. The Coulomb coupling values estimated with eqs 1 and 2, Vdd = 0.765 × 10−2 and Vtc = 0.838 × 10−3 eV, are significantly smaller than the total coupling. Despite a large difference in the excitation energies of cytosine and thymine (4.75 and 5.23 eV, Table 1) the relatively strong coupling of the states, 0.104 eV, leads to a partial delocalization of the exciton between the nucleobases. Substituting Δ = 0.310 eV and V = 0.104 eV in eq 21 predicts that in the excited state S1 of the stack, XC = 0.87 and XT = 0.13. For comparison, insertion of the Coulomb coupling value, Vtc ∼10−3 eV, in eq 21 suggests that the excited states S1 and S2 are completely localized on cytosine and thymine, respectively. Overall, the Coulomb coupling does not provide a reasonable estimate for the excited-state interaction in the 5′-CT-3′ stack. Note that the DNA π-stacks considered here are only illustrative examples. For discussions of excited states in DNA, see refs 26−33 and references therein.

ASSOCIATED CONTENT

S Supporting Information *

Atomic coordinates of the π-stacks 5′-TT-3′ and 5′-CT-3′ and the transition charges in the pyrimidine bases derived from MSCASPT2 calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

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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) was provided by grant CTQ 2011-26573. REFERENCES

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CONCLUSIONS Electronic coupling is a key parameter that controls the efficiency of energy transfer and the extent of exciton delocalization in molecular systems. It has been shown that the coupling can be expressed (eq 12) via vertical excitation energies as well as transition moments of the system and individual chromophores directly available from quantum mechanical calculations. The suggested approach accounts for both short-range and long-range interactions and allows the treatment of systems with short interchromophore distances. When the chromophores in the dimer are identical, the general formula is reduced to eq 15 and the coupling in such cases can be derived from purely experimental data. The electronic 1482


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Estimation of Electronic Coupling for Singlet Excitation Energy

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