ent levels. Negative peaks correspond to excited state absorption (see Fig. 1). For two molecules that are not excitonically coupled, all contributions to the crosspeaks cancel out exactly, while if two molecules are excitonically coupled non-zero crosspeaks appear. The shapes of the peaks are influenced by the phase evolution of the coher- ence elements of RDM during the population time T . On the upper left figure of Fig. 8 we have marked the elon- gation of the diagonal and off-diagonal peaks by arrows. The elongation can be best judged by looking at the zero contour (in black). This particular elongation is char- acteristic for the phase of the ρ$_{13}$ ( t ) element (see upper right figure of Fig. 8) at T = 0 . At T = 20 fs the phases of the ρ$_{13}$ ( t ) calculated by both methods are opposite to the phase at T = 0 . The 2D spectra calculated by the two different methods at T = 20 fs differ only in the precise positions of the contours. This phase of the coherence element is characterized in 2D spectrum by a different orientation of the peaks. Interestingly, at T = 140 fs the two methods predict ρ$_{13}$ ( t ) that have mutually opposite phases and as a consequence the 2D spectra at T = 140 fs calculated by different methods differ in the orientation of their crosspeaks. Since the secular TL theory pre- dicts a simple dephasing of the coherence and a regular oscillation with a single frequency proportional to the en- ergy difference between corresponding energy levels, it is in principle possible to distinguish, even experimentally, deviations from this prediction. Our conclusion is that such a deviation should be a consequence of the memory effects in the reduced system time evolution. whole, not for its parts [42]. For the population dynamics we are therefore forced to conclude that the full TNL theory represents the best candidate for a correct description of relaxation phenom- ena in the second order of the system bath interaction. It predicts similar population transfer times as other meth- ods, it is much less sensitive to the breakdown of the positivity than its TL counterpart, and it leads to a bath renormalization of the canonical equilibrium. Most interestingly however, it predicts longer coherence life time than the TL theory. It was recently established by Ishizaki and Fleming [33] that this is to be expected from a higher order theory. In the light of the above conclusions about the dy- namics of optical coherences and the populations and co- herences of the one exciton band, we suggest a hybrid approach to calculating 2D spectra, which consists of the application of the TL method on optical coherences (first and third time interval) and the full TNL method on the calculation of the RDM dynamics in the one exciton band during the population time T . In this paper we have compared four different theories of excitation energy transfer and relaxation in molecu- lar aggregate systems, with a special attention paid to lifetime of electronic coherences. Second order time non- local and time local theories with and without secular approximation were studied. For our specific model of an aggregate we have concluded that time non-local the- ories can account for experimentally observed electronic coherence life time that is significantly longer than the one predicted by the standard time-local secular relax- ation rate theory. Markov approximation leading to time local equations of motion was found to be responsible for the reduction of the coherence lifetime, while the influ- ence of the secular approximation on the life time was found rather weak. The time local theory without secu- lar approximation is found to break positivity of the oc- cupation probabilities in the range of parameters studied here. We conclude that time-local second order theory is not suitable for simulating the coherence transfer ef- fects. Simulations of two-dimensional spectra show that the time non-local effects can be experimentally identi- fied based on the analysis of the oscillations of the cross peaks. Several conclusions about the applicability of the sec- ular and Markov approximations can be drawn from the above results. As pointed out in Ref. [35], Markov ap- proximation, which in the second order in system-bath coupling converts the TNL equations to the TL ones, leads accidentally to an exact result for an optical co- herence element interacting with the harmonic bath. It has been also pointed out previously [33, 41] that in the same case, the TNL equations lead to artifacts. When studying relaxation dynamics of the populations and elec- tronic coherences in excitonic systems, full TL theory leads to a breakdown of the positivity of the RDM, while none of the secular theories suffer from this problem. In principle, the full TNL theory suffers from this problem, too. However, it has been found less susceptible to it here. The secular theories lead to canonical density ma- trix at long times, while the full TNL results in a sta- tionary state characterized by non-zero (but constant) coherences. Such result corresponds to an additional renormalization of the electronic states by the interac- tion with bath, and has to be expected even at a weak coupling limit [42]. It is important to note in this con- text that the canonical equilibrium is to be expected for the system consisting of the molecule and the bath as a This work was supported by the grant KONTAKT ME899 from the Ministry of Education, Youth and Sports of the Czech Republic. Two-dimensional spectra were produced using the NOSE package available under GNU Public License at http://www.sourceforge.net . Acknowledgments C. Validity of secular and Markov approximations VI. CONCLUSIONS 12