The Franck-Condon approximation [10, 11] allows the coupling between the reactant and product states to be separated into an electronic and nuclear component (see Equation 2.18). The term appearing in Marcus’s formula for the rate of hole exchange (Equation 3.1) is the electronic coupling, Vi j, more rigorously defined as:
Vi j =< Φi(⃗r;⃗R)|He(⃗R,⃗r)|Φj(⃗r;⃗R) > (3.8)
where Φi(⃗r;⃗R) and Φj(⃗r;⃗R) are the electronic wavefunctions of the reactants and prod-
ucts respectively, eigenstates of the electronic hamiltonian operator as defined in Chapter 2. Density Functional Theory (DFT), used here, expresses the many electron wavefunction of a given system as a Slater determinant of (non interacting) one electron wavefunctions such that Equation 3.8 becomes
Vi j=< φi1(⃗r;⃗R)φi2(⃗r;⃗R)...φiN(⃗r;⃗R) | {z } Φi(⃗r;⃗R) |He(⃗R,⃗r)| φj1(⃗r;⃗R)φj2(⃗r;⃗R)...φjN(⃗r;⃗R) | {z } Φj(⃗r;⃗R) >, (3.9)
where φik(⃗r;⃗R) and φjk(⃗r;⃗R) are the kth one electron wavefunctions constituting the Slater
determinant of the N electron reactants and products respectively. Making use of the frozen core orbital approximation, one can assume that only the frontier orbitals involved in the transfer changes upon the exchange of a charge between two molecules. This implies that all but one wavefunctions cancel out in Equation 3.9:
Vi j =< φik(⃗r;⃗R)|He(⃗R,⃗r)|φjl(⃗r;⃗R) >, (3.10)
φik(⃗r;⃗R) and φjl(⃗r;⃗R) being the one electron wavefunctions involved in the charge transfer.
In the case of hole exchange k will refer to the HOMO of the donor while l will refer to the HOMO of the acceptor. The electronic hamiltonian operator acting on one electron is commonly called the Fock operator of which F(⃗R,⃗r) is the matrix representation. Therefore the electronic coupling between reactants and products of a hole self exchange reaction as computable by DFT is defined as:
3.1 Calculation of the parameters in Marcus’s equation for the rate of hole exchange 81
This means that Vi jreduces to the HOMO overlap, Ji j, between the dye charge donor in reac-
tant system i and the dye charge acceptor in product system j. As described elsewhere,[12– 17] Ji jcan be found by reading the appropriate off-diagonal element of the following matrix:
(GTB)Tεpair(GTB), (3.12)
where εpair are the eigenvalues of the pair of molecules (i.e charge donor and acceptor); B
is the basis of the normalised eigenvectors of the pair and
G= "
... φdk... 0
0 ... φal ...
#
with φdk (φal) the kth (lth) molecular orbital of the molecule d (charge donor) and molecule
a(charge acceptor), expressed in the DFT one electron orbital basis set. From now on I will call this technique the projective method as the electronic coupling is calculated by mapping the orbitals of the pair donor-acceptor into the orbitals of the isolated molecules. εpair, B, φdk
and φal are taken from three DFT energy calculations in vacuum (B3LYP/ TZVP-6D with Gaussian09 [1]) on the pair and each isolated molecules. I make one further approximation by calculating the electronic coupling within the restricted DFT formalism which considers that two electrons (one spin up and one spin down) can sit on the same spatial orbital. This means that the HOMO of the cationic dye (charge donor) is the same as the HOMO of the neutral dye (charge acceptor), for a given nuclear configuration. In practice this implies that the DFT calculations on the pair donor-acceptor are done on two neutral molecules.
All DFT calculations to get the electronic coupling are performed on the dyes without the TiO2 surface but with protonated anchoring group. The hydrogen on the anchoring group aims to mimic the contribution from the surface to the HOMO of the dyes and allows savings on computational time. This approximation needs validation and this is the purpose of the next section.
Influence of an explicit TiO2surface cluster on the electronic coupling, Ji j
Here I check the effect on Ji j of replacing the TiO2surface by an hydrogen atom. The
electronic couplings between a pair of dyes anchored on a TiO2cluster and a pair of proto- nated dyes are compared (see Figure 3.5 for the complexes used).
The electronic coupling of the highly symmetric pair of protonated D102 dyes (cf. Fig- ure 3.5a) is Ji j = 0.38 meV, calculated using the projective method described above. Be-
cause the dyes are perfect mirror image of one another the coupling can also be found by looking at half the HOMO energy difference of the dimer (dimer energy splitting method
82 Methods
Fig. 3.5: Symmetric complexes generated to test the validity of using protonated dye molecules for the calculation of the electronic coupling. (a) Pair of protonated D102 mirror image of one another. (b) Same pair of D102 as in (a) where the hydrogen of the anchoring group is replaced by a TiO2cluster. Color key of the atoms: red=oxygen, white=hydrogen, blue=nitrogen, yellow=sulfur, dark gray=carbon, light gray=titanium.
as, for example, described in [12, 13]). Here I get JHOMO−splitting= 0.38meV. This certifies
that the implementation of the projective method is correct; hence that the values of Ji j can
be trusted.
The electronic coupling of the same pair of dyes but where a TiO2 slab is substituted for the hydrogen of the anchoring group (see Figure 3.5b) is Ji j = 0.35meV. This is very
close to the value from the pair without the surface. This validates the approximation of using protonated dyes for Ji j calculations in this study. Note that in this case the HOMO
splitting energy difference has no physical meaning. Indeed, the complex in Figure 3.5b is purely hypothetical with a necessarily disconnected surface slab to ensure perfect symmetry of the dyes. Additionally the lack of periodic boundary conditions in Gaussian 09 DFT calculations with localised basis sets makes the energy levels of the pair unreliable.
The projective method allows the calculation of the electronic coupling, Ji j, between the
dye which donates and the dye which accepts the hole in the transfer reaction. This can be seen as the DFT approximation of the total electronic coupling between the reactant and product states, Vi j. The calculation of Ji j is then possible, given any geometrical configura-
3.2 Computing charge donor-acceptor pair geometries 83
acceptor.
3.2
Computing charge donor-acceptor pair geometries
The electronic coupling, Ji j, varies with the relative position and orientation of the donor and
acceptor. [5] In this thesis, I focus on the hole hopping between dye molecules adsorbed on the surface of TiO2nanoparticles as in Dye Sensitised Solar Cells. Such TiO2nanoparticles are assumed to predominantly expose the anatase (101) surface.[18] For this reason, I will be only concerned by the packing of the dyes on this surface. Note, however, that the methods below can be extended to any other surfaces. That being said, the anchored dye geometry on the anatase (101) surface is not well established and I need to make assumptions to derive reasonable pair configurations.