**1.7 Conclusion**

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

**112** **FcPTM and FcMe**_{9}**PTM: 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** _{π}*and 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 αµ*_{0} 0

0 0 *µ*_{0}

(2.14)

*where ˜µ*_{0}*is the dipole moment of the |D*^{+}*A*^{−}〉 state. The dipole moment of the

*|Dπ*^{+}*A*^{−}*〉 state, αµ*0is defined as a fraction of the first, and, again, to avoid too
*many parameters, is set to half µ*_{0}*, so α =*^{1}_{2}. 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)}

*− ω*

*v*

*p2˜ǫ*

_{v}*ρ*ˆ

*+*

_{D}*a*

_{π}*ρ*ˆ

_{π}*q +*1

2*ω*^{2}_{v}*q*^{2} (2.15)
*where the a**π**factor 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, ˜ǫ**v**, 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 ˜µ*_{0}, polar solvation
*is described by the single parameter ˜ǫ** _{or}*. The total Hamiltonian, accounting for

**114** **FcPTM and FcMe**_{9}**PTM: 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)

*2x*

*2z*

ε* _{v}*
ε

_{v}∼ ∼

∼

∼

λ* _{v1}*
λ

_{v2}D π A
D^{+}π A^{}

-D π^{+} A^{}

*-g*
*e*_{1}*e*_{2}

*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*_{v}*ω*_{v}*q − f*
ˆ
*ρ** _{D}*
+

*2˜x −p2˜ǫ*_{v}*ω*_{v}*q −*1
2*f*

ˆ
*ρ*_{π}

−˜*τ ˆσ*_{(3)}+1

2*ω*˜_{v}*q*^{2}+ 1

4˜*ǫ*_{or}*f*^{2} (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, e*_{1} *and e*_{2}, respectively. The
*geome-try of equilibrium for each adiabatic state is: q** _{eq,i}* =

p_{2˜}_{ǫ}

*v*

*ω*_{v}*〈ρ**D*〉*i**+ 〈ρ**π*〉*i*
with
*i = g, e*_{1}*, e*_{2} 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** _{vi}*is 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*

2

4 ^{C}^{6}^{H}^{6}

CCl_{4}
CH_{2}Cl_{2}

DMSO 2

C_{6}H_{12} 4

CCl_{4}
CH_{2}Cl_{2}
ArNO_{2}

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 Me*_{9}*FcPTM (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 FcMe**_{9}**PTM: solution and crystals**

*Table 2.2: Molecular parameters for FcPTM and FcMe*_{9}*PTM described in the *
*three-state model.*

*molecular parameter* FcPTM FcMe_{9}PTM

˜

*z*/ eV 0.78 0.50

˜

*x* / eV 0.87 0.69

˜

*τ / eV* 0.47 0.47

˜

*µ*_{0}/ D 15.0 15.5

˜

*ǫ** _{v}*/ eV 0.06 0.07

˜

*ω** _{v}*/ eV 0.18 0.18

*solvent parameter* FcPTM FcMe_{9}PTM

˜

*ǫ** _{or}*/ eV cyclohexane 0.0 0.0

CCl4 0.10 0.08

dichloromethane 0.20 0.20

ArNo_{2} 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** _{v}*estimated 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 ǫ*

*or*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, ˜µ*_{0}in
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

### |3

^{+}

### 〉 *D* *b*

_{1}

*b*

_{2}

*b*

^{+}

_{3}

*A*

^{−}

*2˜x* *1/4 ˜* *µ*

_{0}

### |2

^{+}

### 〉 *D* *b*

_{1}

*b*

_{2}

^{+}

*b*

_{3}

*A*

^{−}

*2˜x* *1/2 ˜* *µ*

_{0}

### |1

^{+}

### 〉 *D* *b*

_{1}

^{+}

*b*

_{2}

*b*

_{3}

*A*

^{−}

*2˜x* *3/4 ˜* *µ*

_{0}

*|Z〉* *D*

^{+}

*b*

_{1}

*b*

_{2}

*b*

_{3}

*A*

^{−}

*2˜z* *µ* ˜

_{0}

*Figure 2.14: Schematic representation of the basis state for a 5 state molecule (3*
*bridge states) with corresponding energies and dipole moments.*

*(table 2.1) and correspond to a DA distance of 3.1 and 3.3 Å for FcPTM and*
FcMe_{9}PTM, respectively. These values are still small compared with the
ge-ometrical DA distance (9.5 and 9.7 Å for FcPTM and FcMe_{9}PTM), but they
considerably improve over the corresponding estimates obtained in the two
state approach, leading to a ratio between the geometrical and spectroscopic
*estimate of the DA distance in line with similar results for other DA molecules*
[46, 86, 33, 84, 87, 88, 89, 90, 91]. This result suggests to investigate in mode
*detail the role of the bridge state in effective models for DA chromophores.*

**2.1.7** **The role of low energy bridge states**

**An electronic model to account for active bridge states**

*The increase of the estimated µ*_{0}by a factor ∼ 2 when going from a two-state to
a three-state model for either FcPTM and FcMe_{9}PTM 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 FcMe**_{9}**PTM: 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 ˜µ*0, while the dipole moments of bridge
*states are fractions of ˜µ*_{0}, 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˜}^{µ}^{0}

(*n−1)* 0 ... 0 0

0 0 0 ^{3˜}^{µ}^{0}

(*n−1)* ... ... ...
... ... ... ... ... 0 0
0 0 0 · · · 0 ^{(}^{n−2)˜µ}^{0}

(*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 z*0=˜*z,*
*µ*_{0}*=˜µ*_{0}and

*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 z*_{0}=*τ = 1, and µ*_{0}=1(in this section, we
*work fixing τ as the energy unit and µ*_{0}as 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, ˜µ*_{0}=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 (ω**C T* *≈ 2.8 and µ**C 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 FcMe**_{9}**PTM: 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 µ*_{0}*) 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*
*µ*_{0}, can be estimated from three spectral properties. Equations (1.8), (1.7) and
*(1.10) are therefore used to extract ρ, τ, and µ*_{0}out 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*_{±} = *d*^{2}+*4 ± |d|*p
*d*^{2}− 4

*2d*^{2}+8 *µ*_{0}= *µ** _{ge}*
p

*ρ(1 − ρ)*

*τ*=

*ω*

_{ge}*µ*

_{ge}*µ*_{0} (2.20)

*The right panel of figure 2.16 shows the (˜x − ˜z) dependence of the effective*
*µ*_{0}*estimated along these lines for the n =3, 4, and 5 state models in figure 2.15.*

*In all cases the effective µ*_{0}*converges toward the exact limit, µ*_{0}=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 µ*0deviates
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

+1

(2.21)

*The effective µ*_{0}obtained 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 µ*_{0}obtained for FcPTM and
Me_{9}*FcPTM 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 µ*0values 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]