these states. The kinetic term T ( P$_{m}$ ) and the poten- tial terms V ( m ) g ( Q$_{m}$ ) , V ( m ) e ( Q$_{m}$ ) represent the intra- molecular DOF and the protein environment (bath or reservoir) interacting with these states. By Q$_{m}$ ( P$_{m}$ ) we denote the (possibly macroscopic) set of coordinates (im- pulses) describing both the intramolecular nuclear DOF of the m -th monomer as well as the DOF of its surround- ings. The total Hamiltonian of the monomer can be split into the system, reservoir and the system-reservoir cou- pling terms brevity. The Hamiltonian of the aggregate is constructed using the energies of collective states where ε$_{g}$ = ∑ $_{n}$ε ( n ) g , Ω = N − 1 ∑ $_{n}$ε ( n ) e , and Δ ε$_{n}$ = ε ( n ) e − Ω + ∑ m = $_{n}$ε ( m ) g . The abreviation h. e. t. denotes higher excitonic terms. Due to the fact that the monomers are positioned in a tight aggregate, we have to account for the interaction energy between their excited states. The interaction energy between states | u$_{m}$ 〉 and | u$_{n}$ 〉 will be denoted J$_{mn}$ , and the corresponding contribution to the total Hamiltonian reads Due to the off-diagonal terms J$_{mn}$ the collective states defined in Eq. (6) are not eigenstates of the total Hamil- tonian H$_{S}$ = H non − int S + H int S . Although the basis of the states | g 〉 , | u$_{n}$ 〉 and multiple-excitation states of the ag- gregate provides efficient means for defining the Hamil- tonian, it is more practical to switch into the basis of eigenstates of the Hamiltonian H$_{S}$ . One of the reasons is that while matter interacts with light, the differences between eigenenergies of H$_{S}$ define the resonant transi- tion frequencies. The system–bath coupling part of the aggregate Hamiltonian reads and thus no terms in the total Hamiltonian couple the ground state with the first excited state or higher ex- cited state bands. In fact, the total Hamiltonian splits into blocks separated approximately by the energy ℏ Ω (see Fig. 1). This reflects the neglecting of all adiabatic couplings, which are supposed to be so weak that they do not lead to transitions on the time scale of interest (femto and picoseconds). This property is well justified e.g. for chlorophyll systems. Here, 〈 A ( Q ) 〉$_{eq}$ represents averaging of an arbitrary Q − dependent operator over equilibrium state of the bath. By this choice of the splitting we have assured that ΔΦ ( m $^{)}$( Q$_{m}$ ) = 0 for the system in equilibrium. To simplify the notation, we redefine electronic energy of the excited state to include the equilibrium average of the potential energy difference between the electronic ex- cited and ground states, ˜ ε ( m ) e = ε ( m ) e + 〈 V ( m ) e ( Q$_{m}$ ) − V ( m ) g ( Q$_{m}$ ) 〉$_{eq}$ and we drop the tilde over ε ( m ) e further in this paper. For the subsequent use in this paper, we denote the eigenstates of the total electronic Hamiltonian H$_{S}$ by | ¯ u$_{a}$ 〉 , a = 1 , . . . , N , for single exciton states formed as linear combinations of single excitation states | u$_{n}$ 〉 and | ¯ U$_{a}$ 〉 , a = N + 1 , . . . , N + N ( N − 1) / 2 , for two-exciton states formed from the linear combination of pairs of sin- gle excitation states. An aggregate built out of these monomers can be rep- resented on a Hilbert space composed of collective aggre- gate states. We define the aggregate ground state states with a single excitation and multi-excited states in an analogical manner. We drop the sign ⊗ in further consideration for the sake of In this section we consider interaction of the electronic system described by the Hamiltonian H$_{S}$ with a macro- scopic bath composed of the DOF of the molecular sur- roundings. | u$_{n}$ 〉 = n − 1 ∏ m =1 ⊗| g$_{m}$ 〉 ⊗ | e$_{n}$ 〉 N ∏ m ′ = n +1 ⊗| g$_{m}$ ′ 〉 , (6) | g 〉 = N ∏ m =1 ⊗| g$_{m}$ 〉 , (5) H m S − R ≡ ( V ( m ) e ( Q$_{m}$ ) − V ( m ) g ( Q$_{m}$ ) −〈 V ( m ) e ( Q$_{m}$ ) − V ( m ) g ( Q$_{m}$ ) 〉$_{eq}$ ) | e$_{m}$ 〉〈 e$_{m}$ | ≡ ΔΦ$_{m}$( Q$_{m}$ ) | e$_{m}$ 〉〈 e$_{m}$ | . (4) H m R ≡ [ T ( P$_{m}$ ) + V ( m ) g ( Q$_{m}$ )] × ( | g$_{m}$ 〉〈 g$_{m}$ | + | e$_{m}$ 〉〈 e$_{m}$ | ) = [ T ( P$_{m}$ ) + V ( m ) g ( Q$_{m}$ )] ⊗ ˆ 1 , (3) H m S ≡ ε ( m ) g | g$_{m}$ 〉〈 g$_{m}$ | + ( ε ( m ) e + 〈 V ( m ) e ( Q$_{m}$ ) − V ( m ) g ( Q$_{m}$ ) 〉$_{eq}$ ) | e$_{m}$ 〉〈 e$_{m}$ | , (2) H int S = ∑ n = m ( J$_{nm}$ | u$_{n}$ 〉〈 u$_{m}$ | + c . c . ) + h . e . t .. (8) H$_{S}$$_{−}$$_{B}$ = ∑ n ΔΦ$_{n}$ | u$_{n}$ 〉〈 u$_{n}$ | + h . e . t ., (9) H non − int S = ε$_{g}$ | g 〉〈 g | + ∑ n (Δ ε$_{n}$ + Ω) | u$_{n}$ 〉〈 u$_{n}$ | + h . e . t ., (7) 3 III. SECOND ORDER RELAXATION THEORIES