molecule in the aggregate to be the same, and indepen- dent of neighboring molecules, i.e. The correlation function C ( t ) is taken in a form of the overdamped BO model with Here, λ is the reorganization energy, ν$_{n}$ are so-called Mat- subara frequencies, k$_{B}$ is the Boltzmann constant, T is the thermodynamic temperature and τ$_{c}$ is the so-called bath correlation time. The BO form of the correlation function satisfies all general constraints put of a correla- tion function by thermodynamics [3]. Apart of the tem- perature which we assume to be T = 300 K in all calcu- lations in this paper, the BO model is determined by two parameters only; by the reorganization energy λ which is experimentally related to the Stokes shift S = 2 λ and by the bath correlation time τ$_{c}$ . BO is a widely used, well physically motivated, but not the only possible model for the bath correlation function. Implications of other forms of the correlation function for the RDM dynamics will be studied elsewhere. Eqs. (25) and (26) are systems of coupled (integro-) differential equations for the elements of the RDM. From the first terms on the r. h. s. we deduce that the ele- ment ρ$_{ab}$ ( t ) oscillates with a frequency close to ω$_{ab}$ . It is often justified to assume that two terms oscillating on different frequencies are independent of each other. For their envelopes ¯ ρ$_{ab}$ ( t ) = e iω$_{ab}$$^{t}$ρ$_{ab}$ ( t ) we have and integration over time has therefore a relatively smaller contribution when ω$_{ab}$ − ω$_{cd}$ = 0 . Neglecting these contributions, usually termed secular approxima- tion [27], leads to setting for all term except when a = c and b = d , or a = b and c = d . The interpretation of the remaining non-zero where E ( t ) = n · E ( t ) , is the projection of the external electric field vector E ( t ) on the normal vector n in di- rection of the molecular transition dipole moment. The symbol D [ ρ ( t )]( t ) represents the relaxation term chosen from the full TNL, full TL, secular TNL and secular TL equations of motion. The superoperator V is a commu- tator with the dipole moment operator µ = n µ , so that Non-linear spectroscopic signals are very well described by time-dependent perturbation theory [3]. The equa- tions of motion, Eqs. (21) to (22), can be extended by semiclassical light-matter interaction term. This yields All four sets of equations of motion we consider here are extensions to the two well-known constant relaxation rate theories. To arrive at the well-known Redfield equa- tions [27], one can assume certain coarse graining of the RDM dynamics so that all significant changes to the ρ ( t ) occur on a time scale much longer than the correlation time τ$_{c}$ . Then time t$_{0}$ in Eq. (21) can be put to −∞ and the integration limits are then from zero to infinity. The relaxation tensor R thus becomes time independent. If we, on the other hand, consider the decay of C ( t ) to be much faster than even the transition frequencies between electronic levels, we can assume C ( t ) ≈ C$_{0}$δ ( t ) and Eq. (21) has the well-known Lindblad form [27, 37]. Only for the Lindblad form and for the Redfield equations in secular approximation, it can be shown that the diago- nal elements of ρ ( t ) are always positive. For all other equations we have derived here, this assertion cannot be proven in general. This is a consequence of the fact that they are derived in a low order of perturbation theory. terms is simple. The terms R$_{aabb}$ ( t ) for a = b repre- sent rates of transition from level denoted by index b to a level denoted by a . The term R$_{aaaa}$ ( t ) corresponds to the total transition rate from the level a to all other lev- els. The terms R$_{abab}$ ( t ) ( a = b ) are rates of the damping of a coherence element ρ$_{ab}$ ( t ) . In secular approximation, the dynamics of populations of electronic levels is thus decoupled from the dephasing of coherences. The above arguments for the secular approximation apply also to the integro-differential equation (26), and we can thus define four different second order equations of motion for the RDM, with different levels of approximation. From the perspective of our derivation, the most general sec- ond order equation is Eq. (26), which we have denoted full TNL. The convolutionless Eq. (25) denoted full TL can be regarded its approximation, but it can also be alternatively viewed as derived by different cummulant approximation, see Refs. [25, 26]. The set of four meth- ods investigated here is completed by applying secular approximation to the full TNL and full TL equations. 6 C$_{ab}$ ( t ) = C ( t ) δ$_{ab}$. (27) C ( t ) = − i ℏ λ Λ e − Λ | t | sgn t + λ Λ ℏ coth ( β ℏ Λ 2 ) e − Λ | t | + 4Λ λ β ∞ ∑ n =1 ν$_{n}$e − ν$_{n}$ | t | ν$_{n}$ 2 − Λ 2 , (28) ν$_{n}$ ≡ 2 πn ℏ β , β ≡ 1 k$_{B}$ T , Λ ≡ 1 τ$_{c}$ . (29) ∂ ∂t ¯ ρ$_{ab}$ ( t ) = − ∑ cd R$_{abcd}$ ( t ) e i ( ω$_{ab}$ − ω$_{cd}$ ) $^{t}$¯ ρ$_{cd}$ ( t ) , (30) ∂ ∂t ρ ( t ) = − i L$_{S}$ ρ ( t ) − D [ ρ ( t )]( t ) + i V ρ ( t ) E ( t ) , (32) R$_{abcd}$ ( t ) = 0 , (31) IV. NON-LINEAR SPECTROSCOPIC SIGNALS E. Secular and constant relaxation rate approximations in the energy eigenstate basis