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## A. Projection operator technique and Nakajima-Zwanzig identity 

We start with the density operator W , which describes the state of the system composed of the aggregate and its surroundings. This operator fulfills Liouville-von Neu- mann equation 

Here, [ A, B ]$_{−}$ ≡ AB − BA , and we introduced Liouville superoperator L by its action on an arbitrary operator A as L A = 1 $_{ℏ}$[ H, A ]$_{−}$ . Liouville superoperators L$_{S}$ , L$_{B}$ and L$_{S}$$_{−}$$_{B}$ that correspond to Hamiltonian operators of 

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the system, the bath and their interaction will be used later in this paper. A formal solution of Eq. (10) can be written using evolution superoperators such that 

We are often not interested in the complete information carried by W ( t ) , but only in the information related to the electronic states of the aggregate. The reduced den- sity matrix (RDM) defined as 

where tr$_{B}$ represents trace over all bath DOF, is the quantity carrying just the relevant information [27]. To derive equation of motion for ρ ( t ) we can use projection operator approach as following. We define a projection operator P by its action on an arbitrary operator A as 

where w is a pure bath operator satisfying the condition tr$_{B}$ { w } = 1 . For convenience we also define a comple- mentary projection operator Q ≡ 1 − P . For the pro- jected density matrix P W ( t ) one can write a formally exact equation of motion as 

where the subscript I denotes interaction picture with re- spect to the bath Hamiltonian H$_{B}$ . Eq. (14) is known as Nakajima-Zwanzig identity [27, 36]. Although one can- not practically solve Eq. (14), because it is as difficult as the original complete problem, one can use it as a starting point for approximations that will follow. 

Before we introduce approximations into Eq. (14), we can make one more formal step, which enables us to over- come its time non-local character. Let us assume an evo- lution superoperator U $^{′}$( t ) , which satisfies 

This superoperator is of course unknown, but it allows us to turn a time non-local differential equation (14) for- mally into a time-local one. We can write 

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## B. Convolutionless equation of motion 

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