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Two-dimensional electronic spectroscopy (2DES) is a powerful experimental technique, as it directly probes the nonlinear (third-order) response function of the system, providing key insights into ultrafast energy transfer and relaxation processes. However, 2DES experiments are generally difficult to interpret, often relying on simulations in order to associate observed spectral features with specific underlying system dynamics. For this reason, the development of robust, computationally inexpensive theoretical methods for modeling these experiments remains an active area of research. We have recently derived such an approach for computing the exact finite-temperature nonlinear response function for harmonic Hamiltonians within the Condon approximation, assuming that the transition dipole moment is independent of nuclear coordinates. In this work, we extend our formalism to exactly account for non-Condon/Herzberg−Teller (HT) type contributions to the nonlinear response function, which are known to be crucial for accurately describing linear optical spectra in a wide range of molecular systems. We highlight the key insights that can be gained from our new method, named FC2DES+HT, by simulating the 2DES signals of two molecules with known non-Condon behavior, the phenolate anion and free-base porphyrin. The results demonstrate that Herzberg−Teller couplings substantially impact energy relaxation dynamics in these systems. 

Two-dimensional electronic spectroscopy (2DES) can provide detailed insight into the energy transfer and relaxation dynamics of chromophores by directly measuring the nonlinear response function of the system. However, experiments are often difficult to interpret, and the development of computationally affordable approaches to simulate experimental signals is desirable. For linear spectroscopy, optical spectra of small to medium-sized molecules can be efficiently calculated in the Franck-Condon approach. Approximating the nuclear degrees of freedom as harmonic around the ground- and excited-state minima, closed-form expressions for the exact finite-temperature linear response function can be derived using known solutions for the propagation operator between normal mode coordinate sets, fully accounting for Duschinsky mode-mixing effects. In the present work, we demonstrate that a similar approach can be utilized to yield analogous closed-form expression for the finite-temperature nonlinear (third-order) response function of harmonic nuclear Hamiltonians. The resulting approach, named FC2DES, is implemented on graphics processing units (GPUs), allowing efficient computations of 2DES signals for medium-sized molecules containing hundreds of normal modes. Benchmark comparisons against the widely used cumulant method for computing 2DES signals are performed on small model systems, as well as the nile red molecule. We highlight the advantages of the FC2DES approach, especially in systems with moderate Duschinsky mode mixing and for long delay times in the nonlinear response function, where low-order cumulant approximations are shown to fail.