FcMe 9 PTM in solution: the need for a three state model . 112

1.7 Conclusion

2.1.6 FcMe 9 PTM in solution: the need for a three state model . 112

112 FcPTM and FcMe₉PTM: solution and crystals

absorptions in both compounds. In the absence of additional data (like, e.g.

the absorption from the first excited state) in order to avoid the proliferation of free parameters, the Hamiltonian matrix elements that describe the electron hopping from the bridge to the acceptor 〈DπA|h⁽³⁾|Dπ⁺A⁻〉 = −˜τ_πand from the donor to the bridge 〈D⁺πA|h⁽³⁾|Dπ⁺A⁻〉 = −˜τ_D, are set equal: ˜τ_D=τ˜_π=τ.˜

In close analogy with the two state model, the only relevant element of the dipole moment operator are the diagonal element for the CT states:

ˆ µ₍₃₎=







0 0 0

0 αµ₀ 0

0 0 µ₀







(2.14)

where ˜µ₀is the dipole moment of the |D⁺A⁻〉 state. The dipole moment of the

|Dπ⁺A⁻〉 state, αµ0is defined as a fraction of the first, and, again, to avoid too many parameters, is set to half µ₀, so α =¹₂. This choice locates the centroid of positive charge of the bridge state just halfway between the D and A centers.

As in the two-state model, the coupling to an effective molecular vibration is introduced assigning harmonic PESs with same frequency but different equi-librium geometries to the basis states, as shown in the left panels of figure 2.12.

The description is made with a single vibrational coordinate q, corresponding to a single effective mode, that couples to the two electronic basis states. The Hamiltonian including the molecular vibration then reads:

ˆh⁽³⁾_el+ph=2˜z ˆρ_D+2˜x ˆρ_π− ˜τ ˆσ₍₃₎− ωvp2˜ǫ_v ρˆ_D+a_πρˆ_π q +1

2ω²_vq² (2.15) where the aπfactor accounts for the possible different coupling of the |Dπ⁺A⁻〉 compared to the |D⁺πA⁻〉 state. Different geometries can occurs for each of the three basis states, leading to two independent vibrational relaxation ener-gies for Dπ⁺A⁻and D⁺πA⁻states. However, getting reliable information on the vibrational coupling of the third state is difficult, and we impose the same ge-ometry on the two charge-separated states (see figure 2.12, left panel), so that the same relaxation energy, ˜ǫv, applies to both states (i.e. a_π =1). Different choices are of course possible but do not alter main results.

Polar solvation is treated again in the framework of the reaction field model, with the reaction field proportional to the molecular dipole moment. Since the dipole moment of the third state is set to a fixed fraction of ˜µ₀, polar solvation is described by the single parameter ˜ǫ_or. The total Hamiltonian, accounting for

114 FcPTM and FcMe₉PTM: solution and crystals

-10 0 10

q (τ unit) 0

2 4 6

E (eV)

-10 0 10

q (τ unit)

0 2 4 6

E (eV)

ε_v ε_v

∼ ∼

∼

λ_v1 λ_v2

D π A D⁺π A

-D π⁺ A

-g e₁ e₂

Figure 2.12: Basis states (left panels) and eigenstate (right panel) PES for the three-state model in equation (2.16) with ˜z= 1 eV , ˜x= 2 eV , ˜τ= 0.5 eV ˜ω_v= 0.2 eV, ˜ǫ_v=0.4 eV and ˜ǫ_or=0 eV.

both vibrational coupling and polar solvation, finally reads:

ˆh⁽³⁾(q, f ) =

2˜z −p2˜ǫ_vω_vq − f ˆ ρ_D +

2˜x −p2˜ǫ_vω_vq −1 2f

ˆ ρ_π

−˜τ ˆσ₍₃₎+1

2ω˜_vq²+ 1

4˜ǫ_orf² (2.16) where again the tilde marks symbols relevant to the three-state model. For each f , the diagonalization of the dependent Hamiltonian leads to three q-dependent eigenstates, that describe the PES relevant to the ground state, g, and to the first and second excited states, e₁ and e₂, respectively. The geome-try of equilibrium for each adiabatic state is: q_eq,i =

p_2˜_ǫ

ω_v 〈ρD〉i+ 〈ρπ〉i with i = g, e₁, e₂ The right panel of figure 2.12 shows the PES obtained for a spe-cific set of parameters. The vibrational eigenstates are constructed on each PES in the harmonic approximation [84, 85] and optical spectra are finally calcu-lated. The calculation is repeated for different f , and the spectra are obtained as Boltzmann averages with the same approach described for other compounds.

The proposed procedure to calculate spectra exploits the adiabatic approx-imation, as mentioned in sect. 1.1 for the two state model. The vibrational relaxation energy λ_viis shown in the left panel of figure 2.12, and can be calcu-lated at each f coordinate value. More details about spectra calculation can be

4 ^C⁶^H⁶

CCl₄ CH₂Cl₂

DMSO 2

C₆H₁₂ 4

CCl₄ CH₂Cl₂ ArNO₂

5000 10000 15000 ω (cm^-1) 0

2 4 ε (103 M-1 cm-1 )

0.00 0.20 0.45 0.80

10000 15000 20000 ω (cm^-1)

0 2 4

ε (103 M-1 cm-1 )

0.00 0.25 0.38 0.65

ε

_or

ε

_or

∼ ∼

Figure 2.13: Optical spectra of FcPTM (left panel) and Me₉FcPTM (right panel) in different solvents. Top panels: experimental spectra. Bottom panels: spectra calculated in the three state model with parameters of table 2.2, and setting the intrinsic bandwidth Γ = 0.07 eV.

found in the appendix sect. B.1.2.

Results

Bottom panels of figure 2.13 show spectra calculated for the parameters in ta-ble 2.2. The vibrational frequencies and the intrinsic line width are set to the same values as in the two-state model. To facilitate the comparison, experimen-tal spectra are reported in the upper panels of the same figure. As expected, the three-state model results in two absorption bands, which reproduce the ex-perimental observation of two solvatochromic CT absorption bands. A detailed comparison of calculated and experimental spectra is hindered, particularly for FcPTM, by the overlap of the secondary CT band with nearby localized absorp-tion bands. It is certainly possible to improve the quality of calculated spectra by increasing the number of parameters, like adopting two different τ for the two matrix elements (allowing for different τ_D and τ_π values.) However the basic role of bridge states is captured already in the simplest model.

From table 2.2, it turns out that ˜x > ˜z for both compounds, as required on

116 FcPTM and FcMe₉PTM: solution and crystals

Table 2.2: Molecular parameters for FcPTM and FcMe₉PTM described in the three-state model.

molecular parameter FcPTM FcMe₉PTM

z/ eV 0.78 0.50

x / eV 0.87 0.69

τ / eV 0.47 0.47

µ₀/ D 15.0 15.5

ǫ_v/ eV 0.06 0.07

ω_v/ eV 0.18 0.18

solvent parameter FcPTM FcMe₉PTM

ǫ_or/ eV cyclohexane 0.0 0.0

CCl4 0.10 0.08

dichloromethane 0.20 0.20

ArNo₂ 0.42

DMSO 0.65

physical basis. Moreover ˜τ, which measures the direct charge hopping from ei-ther the D or the A site to the bridge, is larger than the bridge-mediated hopping, τ, in the two-state model (see table 2.1). The effective strength of the vibra-tional coupling is roughly halved in the three-state model, suggesting that the effective ǫ_vestimated in the two-state model is roughly the sum of the contribu-tions from the two excited states. The values of the solvent relaxation energy in the three-state model are larger than the corresponding two-state model results (see table 2.1). This increase compensates for the reduction of the mesomeric dipole moment in the three-state model. The sizable weight of the Dπ⁺A⁻in the first excited state in fact leads to a decrease of the relevant dipole moment with respect to the two-state model, so that larger solvent relaxation energies are required in the three-state model with respect to the two-state model to repro-duce the same solvatochromism. While microscopic models relate the solvation relaxation energy to the solvent dielectric constant, refractive index and to the size and shape of the cavity occupied by the solute [27, 19, 33], this analysis demonstrates that ǫor is best treated as an adjustable parameter, whose spe-cific value also depends on the model adopted to describe the solute electronic structure.

Finally the effective dipole lengths extracted in the three-state model, ˜µ₀in table 2.2, are about twice the corresponding estimate in the two-state model

basis states energy dipole

|N〉 D b

₁

b

₂

b

₃

A 0 0

2.1.7 The role of low energy bridge states

An electronic model to account for active bridge states

The increase of the estimated µ₀by a factor ∼ 2 when going from a two-state to a three-state model for either FcPTM and FcMe₉PTM suggests a way to solve a long-standing paradox in model for DA chromophores. It is well known in fact that dipole lengths estimated from the analysis of DA spectra always severely underestimate the DA distance [87, 88, 89, 90, 91]. To generalize the result we consider here a purely electronic model for a n-site molecule, where the electron is transferred from D to A via n − 1 hops involving only adjacent n − 2 bridge states [83]. The resulting n states are schematically shown in figure 2.14 for n = 5. The same energy 2˜x is assigned to all bridge states, while the D⁺A⁻ (|Z〉) state has energy 2˜z < 2˜x. The same hopping integral ˜τ describes the CT between all adjacent sites along the chain. The relevant Hamiltonian is a trivial

118 FcPTM and FcMe₉PTM: solution and crystals extension of the three-state electronic Hamiltonian in equation (2.11):

ˆh_n=







0 −˜τ 0 0 · · · 0 0

−˜τ 2˜x −˜τ 0 · · · 0 0 0 −˜τ 2˜x −˜τ · · · 0 0 0 0 −˜τ 2˜x ... ... ... ... ... ... ... ... −˜τ 0 0 0 0 · · · −˜τ 2˜x −˜τ 0 0 0 · · · 0 −˜τ 2˜z







(2.17)

Consistent with the three-state model described in the previous section, the dipole moment of the |Z〉 state is set to ˜µ0, while the dipole moments of bridge states are fractions of ˜µ₀, as relevant to a system with equally spaced sites (see figure 2.14), leading, for the general case in equation (2.17), to the following values:

ˆ µ_n=







0 0 0 0 · · · 0 0

0 ^µ^˜⁰

(n−1) 0 0 · · · 0 0

0 0 ^2˜^µ⁰

(n−1) 0 ... 0 0

0 0 0 ^3˜^µ⁰

(n−1) ... ... ... ... ... ... ... ... 0 0 0 0 0 · · · 0 ⁽^n−2)˜µ⁰

(n−1) 0

0 0 0 · · · 0 0 µ˜₀







(2.18)

We limit our attention to a systems with an almost neutral ground state, in which bridge sites are weaker donors than D, so that 0 < ˜z < ˜x. For ˜x − ˜z ≫

τ bridge states become very high in energy and a perturbative treatment on

τ/(2˜x−2˜z) reduces the n-state model to an effective two-state model with z0=˜z, µ₀=˜µ₀and

τ = τ˜ⁿ⁻¹

(2˜x − 2˜z)ⁿ⁻² (2.19)

where, as before, the tilde applies to symbols relevant to the n-state model, while bare symbols refer to the two-state model. n-site (or n-state) molecules have n−1 optical excitations whose energies and transition dipole moments are shown in figure 2.15 as a function of ˜x−˜z for systems with n = 3, 4, and 5 sites.

In all cases, model parameters have been chosen as to converge, in the large

x limit, to the two-state model with z₀=τ = 1, and µ₀=1(in this section, we work fixing τ as the energy unit and µ₀as the unit dipole moment). With this

0 10 20

ω fi (eV)

n=3 n=4 n=5

0 5 10

x 0

0.05 0.1

|µfi| (µ0 unit)

5 10

x 5 10 15

~ ~ x~

Figure 2.15: Transition energies (top panels) and squared transition dipole mo-ments (bottom panels) calculated for n = 3, 4 and 5 state models (panels from left to right) with ˜z= 1, ˜µ₀=1, variable ˜x and ˜τ=

τ(2˜x)ⁿ⁻² ¹

n−1. Red lines refer to the lowest energy (main CT) transition. Dotted lines show the (˜x-independent) results relevant to the limiting two-state model (see text).

choice, we expect convergence, in the large ˜x limit, to a two-state model with ρ ≈ 0.15 (see equation 1.5). The corresponding limiting values of the transition frequency and squared transition dipole moments (ωC T ≈ 2.8 and µC T ≈ 0.13) are shown as dotted lines in figure 2.15.

The lowest energy transition, corresponding to the main CT band (marked by red lines in figure 2.15) is well separated from higher energy transitions in-volving bridge states and has by far the largest intensity: the main CT transition dominates the low energy portion of the spectrum. The corresponding energies and transition dipole moments (red lines in figure 2.15) properly converge to-ward the two-state limit (dotted lines) for ˜x − ˜z→ ∞ (or equivalently ˜x → ∞).

The convergence becomes slower with increasing n.

To investigate the effect of bridge states in the definition of an effective two-state model and in particular on the estimate of the relevant dipole length, we focus attention on the lowest energy, main CT absorption, disregarding higher energy transitions involving the bridge. Indeed, this is consistent with the treat-ment of experitreat-mental data, where we concentrate on the lowest energy tran-sitions modeled in term of a two state model. In other terms, we analyze the data relevant to the main CT transition, obtained from the diagonalization of

120 FcPTM and FcMe₉PTM: solution and crystals

0 5 10

x - z

0.5 0.75 1

µ

₀

n=3 n=4 n=5

5 10 15

x - z

0.5 0.75 1

µ

₀

~ ~

Figure 2.16: Dipole length extracted from the two-state analysis of the main CT band of the n-state models described in figure 2.15. Left panel shows results ob-tained by extracting the two-state model parameters (x, τ and µ₀) from the fre-quency, the squared transition dipole moment and the mesomeric dipole moment;

right panel shows similar results obtained using the ground state dipole moment instead of the mesomeric dipole moment.

the n-state model, to extract an effective two-state model in an analogous way as usually done by analyzing experimental absorption spectra. In particular, the parameters of the effective two-state model, τ, z (or equivalently ρ), and µ₀, can be estimated from three spectral properties. Equations (1.8), (1.7) and (1.10) are therefore used to extract ρ, τ, and µ₀out of the transition frequency, transition and mesomeric dipole moment (difference between ground and ex-cited state dipoles, a quantity directly related to solvatochromism) calculated in the n-state model. If d = ^∆µ_µ

ge it results:

ρ_± = d²+4 ± |d|p d²− 4

2d²+8 µ₀= µ_ge pρ(1 − ρ) τ = ω_geµ_ge

µ₀ (2.20)

The right panel of figure 2.16 shows the (˜x − ˜z) dependence of the effective µ₀estimated along these lines for the n =3, 4, and 5 state models in figure 2.15.

In all cases the effective µ₀converges toward the exact limit, µ₀=1, for ˜x → ∞, but it is always underestimated for any finite ˜x, i.e. when bridge states are closer in energy to the D⁺· · · A⁻(|Z〉) state. It is worth noting that µ0deviates more from the limiting value with increasing the number of bridge state.

Different estimates of the effective two-state model parameters can be ob-tained if a different choice is made about the reference spectral properties, i.e.

the quantities that represent the set of experimentally accessible information.

In fact, using equations (1.7), (1.8) and (1.9), the parameters of the effective two-state model can be extracted by reproducing the transition frequency, tran-sition dipole moment and ground state dipole moment (instead of mesomeric dipole moment). In this case, the following equation for ρ stands:

ρ = 1

µ_ge µ_g

(2.21)

The effective µ₀obtained according to this alternative procedure (right panel of figure 2.16) shows a qualitatively similar behavior to the previous one and even larger deviations from the limiting value than before.

This discussion clearly demonstrates that the unphysically small D–A dis-tances, extracted from the two-state model analysis of optical spectra of DA molecules [87, 88, 89, 90, 91], results from the presence of low-lying bridge states, playing an active role in the D to A CT. The larger discrepancy be-tween the geometrical and spectroscopic values of µ₀obtained for FcPTM and Me₉FcPTM with respect to common organic DA chromophores, is then natu-rally explained by the presence of (at least) a secondary low-lying CT transition in the absorption spectra. Moreover this picture explains why this discrepancy is not observed in the µ0values obtained from the optical spectra of CT com-plexes and crystals. In CT complex and crystals, the intermolecular CT occurs trough space because of the direct overlap of frontier orbitals. Optical spectra of these systems can be described by the same two state model of DA molecules.

However in these systems, where the D to A CT is not mediated by any virtual state, the dipole lengths obtained from the intensity of the CT bands are well comparable with geometrical distances. [104, 105, 106, 107]

Nel documento CHARGEANDENERGYTRANSFERINFUNCTIONALMOLECULARMATERIALS:SPECTROSCOPYANDMODELS UNIVERSITÀDEGLISTUDIDIPARMA (pagine 120-129)