In this section7 I provide the specifications of both type of algorithms I implemented in C++: Metropolis and Kinetic Monte Carlo. In either case the lattice reproduces the spacing of the dye molecules when anchored on the (101) TiO2surface. It is generated by sampling the distributions of pairs created by one of the methods described above. The connection between sites is parametrised by the electronic properties (such as the electronic coupling for example) of the pair of dyes assigned to a given pair of sites.
3.3.1
Metropolis Monte Carlo
In this thesis, MMC is used as a substitute for a thermodynamic ensemble of dye confor- mations that would be too costly to generate. In a MD simulation of an NVT ensemble, the system is coupled to a reservoir and assumed to be in equilibrium with this thermostat at every time step. This contrasts with the RMP procedure which only deals with steric hin- drance. Therefore Monte Carlo lattices generated by the RMP do not satisfy thermodynamic equilibrium. I use Metropolis Monte Carlo (MMC) to assess how far such a random lattice is from thermal equilibrium, the results of which are discussed in Chapter 5. The workflow of the algorithm is as follows.
The interaction energy of each pair of dyes is calculated with the Universal Force Field (UFF) in Gaussian 09. [1] In this context I define the interaction energy as the sum of the Coulomb and Van der Waals interactions. From the charge donor (d) - acceptor (a) pair energy, υda, I can define the energy, ϒd, at a given site d as :
ϒd= 1 2 nn
∑
d=0 υda. (3.14)7Reproduced in part with permission from Vaissier et al. Chem. Mater. 26(16) 4731-4740. Copyright
3.3 Monte Carlo algorithms 89
where nn is the number of nearest neighbours (always 4 in this thesis). Then a lattice is built for which one Ji j and its corresponding energy υda are randomly assigned to every bond.
A site is chosen at random and its energy, ϒd,old, calculated according to Equation 3.14. A
couple of the parameters (Ji j, υda) is reassigned to the four bonds of the chosen site and the
corresponding new site energy, ϒd,new, is calculated. If ∆ϒ = ϒd,new− ϒd,oldis negative, the
changes are accepted, another site is selected at random and the procedure reiterated. Else (if ∆ϒ is positive), the changes are only accepted if X , random number between 0 and 1, is below the Boltzmann factor associated with the change: exp
−∆ϒ
kBT
with T = 300K. Then, another site is chosen at random and the procedure is reiterated.
3.3.2
Kinetic Monte Carlo
Here I perform Continuous Time Random Walk (CTRW) [24, 25] to simulate the transport of holes within a dye monolayer. I assume that the charge moves through the lattice by a succession of hops from one molecule (site) to one of its four nearest neighbours. As explained in more detail below the time lapse between two hops depends on the rate of charge transfer as calculated with Marcus’s formula (Equation 3.1). Therefore the time elapsed depends on the square of the electronic coupling assigned to every bond of the generated lattice. Three different scenarios are employed depending on the relative time scale of the time elapsed between two hops and the structural rearrangement of the dye monolayer (simulated by periodically reassigning new couplings to the bonds). The first scenario (case 1) is the fast limit. It describes the case where the molecular rearrangement is much faster than the hole hopping time. The second case (case 2) is the static limit where the molecular rearrangement is much slower than the hole hopping time. The third scenario (case 3) is the intermediate regime, where the molecular rearrangement happens on a similar time scale as the hole hopping.
Workflow for case 1
In case 1 I sample the distributions from the CPMD generated pairs which obey ther- modynamical equilibrium and do a random walk to extract diffusion coefficients. The elec- tronic coupling distributions are uniformly sampled to assign one Ji j to every lattice bond. I
assume that the charge sees an effective average electronic coupling and define the adapta- tive time step, twait,CPMD as:
twait,CPMD= − ntotlnR ∑nn∑ntotΓi j
90 Methods
where R is a random number between 0 and 1, Γi j is the rate of intermolecular charge
transfer between reactant system i and product system j and the summation is performed on all nearest neighbour (nn) pairs (here always four). ntot is the total number of Ji j values
I have in both directions and the summation of the rates is carried over the total distribution for all the nearest neighbour pairs. The destination site a is chosen randomly according to the probability distribution :
Pda,CPMD∝
ntot∑nxor yΓi j
nxor y∑nn∑ntotΓi j
, (3.16)
where Pda,CPMD is the probability of going from the site where is anchored the dye charge
donor d in system i to the site where is anchored the dye charge acceptor a in system j, nxor y
is the number of Ji j in the distribution describing the pair aligned along the x or y direction,
depending on the nearest neighbour pair considered.
I calculate the diffusion coefficient, Dcalcfrom the mean square displacement of the hole
on the lattice:
Dcalc= 1 4
< r2>
∆t (3.17)
where < r2> is the mean square displacement of the charge in the (xy) plane of the surface and ∆t is the time step. Each walk is binned into 2000 smaller walks to improve statistics.
Workflow for cases 2 and 3
In cases 2 and 3, I uniformly sample the distributions from the RMP generated pairs but perform a biased random walk to incorporate thermal equilibration effects. In case 2 where the molecular rearrangement is much slower than the hole hopping time, the electronic coupling between each possible lattice site is randomly fixed with values uniformly sampled from the distributions of possible Ji j. Then the adaptive time step, twait is defined as :
twait = − lnR ∑nnΓi j
, (3.18)
where R is a random number between 0 and 1, Γi j is the rate of intermolecular charge
transfer between reactant system i and product system j and the summation is performed on all nearest neighbour (nn) pairs. The destination is picked according to the Boltzmann
3.4 Quasi Elastic Neutron Scattering with OSIRIS. 91