Calculation of charge transfer rates

3.2 Computational details

3.2.6 Calculation of charge transfer rates

In Publications III and IV, the CT rates for the ED and CR processes were calculated with the Marcus theory (Equation 2.1) at a temperature of 293.15 K. The parameters used for calculating the rates were derived in the following manner. In organic systems, the intermolecular reorganization energy is typically divided into the inner (λ_i) and outer, i.e. external (λ_s) contributions. The inner contribution originates from the changes in the equilibrium geometries of the eD and eA compounds upon CT, whereas the external contribution is due to the changes in the electronic and nuclear polarizations and relaxation of the surrounding medium[91]. The inner reorganization energy can be determined either from a frequency analysis or by using the PESs of the molecular states involved in the CT processes considered[91]. The latter approach, whereλ_iis calculated as the difference between the energy of the reactants (products) in the geometry of the products (reactants) and that of their equilibrium geometry (see Figure 2.5), has been a standard approach in the studies of the PSC systems and has been thus employed in this work, as well. The external reorganization

energy is usually determined by the classical dielectric continuum model of Marcus [169]. However, as the accurate description ofλ_s is still rather challenging and is highly affected by the uncertainty of the calculated parameters used in this model, λ_swas kept as an adjusted parameter similarly to the previous studies[34, 170]. The CT rates were calculated withλ_sof 0.10–0.75 eV.

The Gibbs free energy (∆G^◦) is the energy difference between the complexes in their initial and final states (see Figure 2.5)[91]. Similarly to the previous studies of the PSC systems, the Weller’s equation[171]was employed to calculate∆G^◦from the energies of the individual eD and eA compounds, while taking the Coulombic attrac- tion (∆E_Coul) between their charged states into account[91, 167]. The full forms of the equations employed for calculatingλ_i[91, 167, 172],∆G^◦[91, 167], and∆E_Coul [91, 167]can be found from the original Publications III and IV.

In Publication III, electronic couplings for the ED and CR processes of the eD–eA complexes were calculated with the two and multi-state GMH[48, 49]and FCD [50]schemes. In Publication IV, only the FCD schemes were employed. In GMH, the diabatic states localized at different sites (e.g. local and CT states) are assumed to have a zero transition dipole moment (µ^diab_if ) between them, whereasH_ifbetween the diabatic states localized at the same sites is zero. In the limiting case of the 2- state GMH scheme, the coupling between the initial and final charge-localized, i.e.

diabatic states is defined as

H_if= µ₁₂∆E₁₂

∆µ^diab_if = µ₁₂∆E₁₂

p(∆µ₁₂)²+4µ₁₂² (3.5) where diabaticH_if and the difference between the diabatic state dipole moments (∆µ^diab_if ) can be defined by the adiabatic terms, i.e. adiabatic transition dipole moment (µ₁₂), vertical excitation energy difference (∆E₁₂), and the electric dipole moment difference (∆µ₁₂=µ₁−µ₂) between the adiabatic states |1>and |2>.

The dipole moment vectors are typically projected either on the direction defined by∆µ₁₂(or the average of the electric dipole moment differences in the multi-state

were projected along the CT vector, which is defined as the vector connecting the mass centers of the eD and eA, i.e.eR_eD−eA:

proj_R

eD−eAµ_ij= (µ_ij·R_eD−eA)

|︁

|︁R_eD−eA|︁

|︁²

R_eD−eA. (3.6)

Similarly to GMH, in FCD, the transition densities (∆q_if) between the diabatic states localized at different sites are zero.

The 2-state FCD has the similar form as the 2-state GMH:

H_if= |∆q₁₂|∆E₁₂

Æ(∆q₁−∆q₂)²+4∆q₁₂² . (3.7) In FCD, the studied system is partitioned into two fragments corresponding to the eD and eA. An adiabatic eD–eA charge difference matrix,∆q^ad, is defined by its elements

∆q_ij^ad=

∫︂

r∈eD

ρ_ij(r)dr−

∫︂

r∈eA

ρ_ij(r)dr (3.8)

whereρ_ij(r)is the one-particle density (if i=j) for the diagonal elements∆q_ii^adand

∆q_jj^ad defined as the eD–eA charge differences in the adiabatic states |i>and |j>, respectively, or the transition density for the off-diagonal elements∆q_ij^ad(if i̸=j).

For calculating the multi-state electronic couplings, the approach proposed by Yang and Hsu[51], which is based on the similar 3-state approaches[52, 53], was followed.

This method was selected, because it allows for calculating the multi-state GMH and FCD couplings without the need of manual selection and assignment for the states and has resulted in the couplings consistent with the experimental ones for the complexes of small and medium sized organic compounds and donor–bridge–

acceptor systems[51]. Here, the equations will be presented only for the multi-state FCD scheme, but those of the multi-state GMH scheme are similar and have been presented in Publication III. First,∆q^ad, (or similarly the adiabatic dipole moment

matrixµâd in GMH) was diagonalized with a unitary transformation matrixU₁, which is composed of the eigenvectors of∆qâd(or similarly toµâd):

U^T₁∆q^adU₁=U^T₁

⎛

⎜

⎝

∆q₁₁ ∆q₁₂ ∆q₁₃ . . .

∆q₂₁ ∆q₂₂ ∆q₂₃ . . .

∆q₃₁ ∆q₃₂ ∆q₃₃ . . . ... ... ... ...

⎞

⎟

⎠ U₁

⎛

⎜

⎝

∆q_l 0 0 . . . 0 ∆q_m 0 . . . 0 0 ∆q_n . . . ... ... ... ...

⎞

⎟

⎠

(3.9)

The same transformation was applied to the corresponding adiabatic Hamiltonian, i.e. the diagonal adiabatic energy matrixE, to obtain the Hamiltonian (H):

U^T₁EU₁=U^T₁

⎛

⎜

⎝

E₁ 0 0 . . . 0 E₂ 0 . . . 0 0 E₃ . . . ... ... ... ...

⎞

⎟

⎠ U₁=

⎛

⎜

⎝

H_ll H_lm H_lm . . . H_ml H_mm H_mn . . . H_nl H_nm H_nn . . . ... ... ... ...

⎞

⎟

⎠

(3.10)

In the limiting case of the 2-state schemes, the diabatic charge difference matrix,

∆q^diab, (or similarly the diabatic dipole moment matrix,µ^diab) and diabatic Hamil- tonian (H^diab) can be obtained already from Equations 3.9 and 3.10, respectively.

However, in the multi-state schemes, there may exist several states localized on the same site, i.e. with the same nature, and thusHshould be localized within the same- site states. The states obtained in this manner are adiabatic within one block, but diabatic with respect to the states localized at different sites.

According to Yang and Hsu[51], in the FCD scheme, the∆q values for the ideal CT states should be 2 or−2, while those of the local states should be zero. Similarly, in GMH, the local states should have smallµvalues, whereas those of the CT states should approach the ideal dipole moment defined as∆µ^id_if =eR_eD–eA, whereR_eD–eA is the distance between the mass centers of eD and eA. Similarly to the work of Yang and Hsu, the averages of the∆qvalues of the ideal local (zero) and CT states (±2), i.e. ±1, were used as the thresholds for∆q to assign the local and CT subspaces in Publications III and IV. However, in the case of the multi-state GHM schemes, instead of using the half of the ideal dipole moment for assigning the states like sug- gested by Yang and Hsu, the threshold of 10.0 D was used in Publication III, because otherwise there would not have been any CT states in some cases.

After assigning the states,H(obtained from Equation 3.10) is re-diagonalized within each block (i.e. local and CT) to define theH^diab:

U^T₂

⎛

⎝

H_LS H_LS,CT H_CT,LS H_CT

⎞

⎠U₂=

⎛

⎜

⎝

E_CT1 He_CT1,LS He_CT1,CT2

He_LS,CT1 E_LS He_LS,CT2 He_CT2,CT1 He_CT2,LS E_CT2

⎞

⎟

⎠

(3.11)

where each bold letter refers to a matrix in the local and CT subspaces defined by the subscript,Eis a diagonal matrix, and the final coupling values (H_if) are obtained as the corresponding off-diagonal matrix elements inHe_LS,CT. Furthermore,∆q^diab can be obtained by applying the same transformationU₂to the diagonalized∆q(or µ) matrix obtained from Equation 3.9:

U^T₂

⎛

⎜

⎝

∆q_l 0 0 . . .

0 ∆q_m 0 . . . 0 0 ∆q_n . . . ... ... ... ...

⎞

⎟

⎠ U₂=

⎛

⎜

⎝

∆q^diab_CT1 0 ∆q^diab_CT1,CT2

0 ∆q^diab_LS 0

∆q_CT2,CT1^diab 0 ∆q_CT2^diab

⎞

⎟

⎠

(3.12)

The adiabatic electronic and transition dipole moments and charge differences for the GS and selected number of the singlet excited states were calculated by using the GMH and FCD schemes as implemented in Q-Chem 4.2. The Q-Chem software yields the 2-state couplings between each state, but only the 2-state FCD couplings were used as such. In Publication IV, the coupling calculations were carried out with the grid of 99 Euler–Maclaurin radial grid points and 302 Lebedev angular grid points, SCF convergence criterion of 10⁻⁶, and the cutoff for neglect of two electron integrals of 10⁻¹²to keep the calculations carried out with Gaussian 16 and Q-Chem consistent. Moreover, the radii from the Universal Force Field with a scaling factor of 1.1 were employed in conjunction with CPCM. Due to the SCF convergence problems of larger SMA-based complexes, the iterative conjugate gradient solver was employed together with the Precond, NoMatrix, and UseMultipole keywords.

4 RESULTS AND DISCUSSION

This chapter will summarize the key findings of this work, which have been presented in more detail in Publications I–IV. First, the OTωvalues determined for all the studied compounds and their interfacial complexes are presented. After this, structural and optoelectronic properties of the individual PSC compounds will be examined to understand better their functionality and performance in the PSCs.

Then, a step towards understanding of the interactions between the eD and eA compounds at their local interfaces will be taken by examining the structures and CT characteristics of the selected eD–eA complexes. Finally, the results of the multi- state electronic coupling calculations together with the other CT rate parameters and rates for the ED and CR processes taking place in the eD–eA complexes are presented. In all cases, the effect of the functional and dispersion corrections on the features is closely examined. Furthermore, the effect of other calculation parameters and models used will be also considered.

No documento Molecular Level Understanding of Conjugated Organic Solar Cell Materials: a Computational Study (páginas 70-77)