Calculating Steady-State Occupations

Having found the eigenstates of the electronic Hamiltonian, we now want to find the relative populations of these eigenstates under illumination. To do this, we need to solve the rate equations of the system which are

$G_{\alpha} + \sum_{\beta}^{\alpha \neq \beta} \left( k_{\beta\alpha}P_{\beta} - k_{\alpha \beta}P_{\alpha} \right) - k_{rec}^{\alpha}P_{\alpha} = 0$

in which $P_{\alpha}$ is the population of the eigenstate $\alpha$ and where we have defined the following rates:

The rate at which the eigenstate $\alpha$ is generated by the illuminaton, $G_{\alpha}$ .
The rate at which the population of the eigenstate $\alpha$ decreases due to population transfer into the eigenstate $\beta$ , $k_{\alpha \beta}$ .
The rate at which the population of the eigenstate $\alpha$ increases due to population transfer from the eigenstate $\beta$ , $k_{\beta \alpha}$ .
The rates at which the eigenstate $\alpha$ returns to the ground state, $k_{rec}^{\alpha}$ .

In this section, we will describe how these rates are calculated within this exemplar, though we note that this part of the model can be adjusted to accommodate different levels of theory.

Generation Rates

To calculate the generation rate into a given eigenstate, we assign a generation probability to excitonic basis states, which is determined by the transition_dipole_ex parameter ( $G_{\mathrm{ex}}$ ).

$G_{\alpha} = \sum_{k}G_{\mathrm{ex}}|c_{kk}^{(\alpha)}|^{2}$

where the summation over $k$ evaulates the contribution to the eigenstate from excitonic basis states, $|k,k\rangle$ . This can be thought of as a way of characterising how much of the eigenstate is made up from excitonic basis states. A value close to one implies that the eigenstate is an exciton, while a value close to zero indicates that the eigenstate has charge transfer character (i.e., the electron and hole are on different molecules).

Recombination Rates

In our exemplar, the recombination of eigenstates to the ground state is assumed to be dominated by non-radiative recombination. We model this in one of two ways, depending upon the value of the const_recombination argument of the Parameters class. If this is set to True, a recombination rate is assigned to excitonic basis states using the krec_ex parameter. Then, the total recomination rate of the eigenstate $\alpha$ is calculated as

$k_{rec}^{\alpha} = \sum_{k}k_{\mathrm{rec,ex}}|c_{kk}^{(\alpha)}|^{2}$

Alternatively, if the const_recombination argument of the Parameters class is set to False, the decay rates are calculated using a version of Fermi's Golden Rule which has been adapted to describe organic molecules. This is described in further detail in Appendix One.

Rates of Population Transfer

The rates of population transfer, $k_{\alpha \beta}$ , are calculated using secular Redfield theory. While the derivation of these rates is mathematically involved (see e.g., ref 1), the equation used to calculate them can be written in a relatively simple form

$k_{\alpha \beta}(\omega_{\alpha \beta}) = \left( \sum_{k}|c_{kk}^{(\alpha)}|^{2}|c_{kk}^{(\beta)}|^{2} + \sum_{i\neq j}|c_{ij}^{(\alpha)}|^{2}|c_{ij}^{(\beta)}|^{2} + \sum_{j\neq i}|c_{ij}^{(\alpha)}|^{2}|c_{ij}^{(\beta)}|^{2} \right) C(\omega_{\alpha \beta})$

where the function $C(\omega)$ is called the correlation function and is defined in terms of the Bose-Einstein occupation function, $n(\omega)$ , and the spectral density function, $J(\omega)$ , as follows

$C(\omega) = 2\pi \omega^2 \left( 1 + n(\omega)\right)\left(J(\omega) - J(-\omega)\right)$

Note that, for each value of $k_{\alpha \beta}$ , you must sum over all basis states. For a $N \times N$ square lattice, the total number of eigenstates is $\mathcal{O}(N^4)$ and so the number of rates which must be calculated is $\mathcal{O}(N^8)$ . Consequaently, the calculation of the $k_{\alpha \beta}$ is typically the most time consuming step of the simulation.

References

Dynamics of Isolated and Open Quantum Systems in Charge and Energy Transfer Dynamics in Molecular Systems pg 67–190 (John Wiley & Sons, Ltd, 2011). doi:10.1002/9783527633791.ch3.