content_image/427919.jpg content_image/427912.jpg content_image/427910.jpg content_image/427909.jpg content_image/427911.jpg content_image/427915.jpg content_image/427918.jpg content_image/427914.jpg content_image/427917.jpg content_image/427916.jpg content_image/427913.jpg
<img src='content_image/427919.jpg'>

We do not develop the convolutionless (or time local) theory any further in this paper, because we will be in- terested only in terms up to the second order in L$_{I}$ . In such a case U $^{′}$( − τ ) can be approximated as U $^{′}$( − τ ) ≈ U$_{S}$ ( − τ ) U$_{B}$ ( − τ ) , where U$_{S}$ ( − τ ) and U$_{B}$ ( − τ ) are evolu- tion superoperators with respect to L$_{S}$ and L$_{B}$ . Inter- ested reader can refer e.g. to Refs. [25, 26, 36] for further details. 

We will now assume the interaction Hamiltonian in a form of Eq. (9) where index n now runs through all relevant single-exciton and multi-exciton states 

Correspondingly, K$_{n}$ = | u$_{n}$ 〉〈 u$_{n}$ | for single excitonic states. Expanding Eqs. (14) and (16) into the second order in H$_{I}$ we find that the third term on the right hand side (r. h. s.) can be conveniently expressed via so-called bath (or energy gap) correlation functions defined as 

<img src='content_image/427912.jpg'>

Here, we have chosen a specific form of the bath den- sity matrix w ≡ w$_{eq}$ , where w$_{eq}$ is the canonical density matrix of the bath DOF. Defining also an operator 

and a superoperator M $^{(2)}$( τ ) such that 

<img src='content_image/427910.jpg'>

the Eqs. (14) and (16) can be rewritten as 

<img src='content_image/427909.jpg'>

## C. System-Bath Coupling 

<img src='content_image/427911.jpg'>

<img src='content_image/427915.jpg'>

and 

<img src='content_image/427918.jpg'>

Provided we can supply a model for the correlation func- tion C$_{mn}$ ( τ ) we are in position to write down the equa- tions of motion for RDM in terms of known quantities. The last step necessary to implement these equations is to represent them in the basis of the eigenstates of the aggregate Hamiltonian. We define 

and in a similar manner for matrix elements of other operators and superoperators. This leads to 

<img src='content_image/427914.jpg'>

with R$_{abcd}$ ( t ) being the matrix elements of the superop- erator defined by the r. h. s. of Eq. (21), and 

with M$_{abcd}$ ( τ ) the matrix elements of the superoperator defined by the r. h. s. of Eq. (22). All the quanti- ties needed to calculate the matrix elements R$_{abcd}$ ( t ) and M$_{abcd}$ ( t ) are known provided the energy gap correlation function is know. 

As a suitable model of the energy gap correlation func- tion we choose so-called multi-mode Brownian oscillator (BO) [3]. In general, Brownian oscillator model can inter- polate between underdamped intra-molecular DOF and (usually) overdamped bath DOF representing the imme- diate surroundings of the molecule. In this paper, we assume the correlation function of the energy gap of each 

<img src='content_image/427917.jpg'>

<img src='content_image/427916.jpg'>

<img src='content_image/427913.jpg'>

## D. Energy gap correlation function 

