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To compute and analyze vibrationally resolved electronic spectra at zerotemperature, we have recently implemented the on-the-fly ab initio extendedthawed Gaussian approximation [A. Patoz et al., J. Phys. Chem. Lett. 9, 2367(2018)], which accounts for anharmonicity, mode-mode coupling, andHerzberg-Teller effects. Here, we generalize this method in order to evaluatespectra at non-zero temperature. In line with thermo-field dynamics, wetransform the von Neumann evolution of the coherence component of the densitymatrix to the Schr\"{o}dinger evolution of a wavefunction in an augmented spacewith twice as many degrees of freedom. Due to the efficiency of the extendedthawed Gaussian approximation, this increase in the number of coordinatesresults in nearly no additional computational cost. More specifically, comparedto the original, zero-temperature approach, the finite-temperature methodrequires no additional ab initio electronic structure calculations. At the sametime, the new approach allows for a clear distinction among finite-temperature,anharmonicity, and Herzberg-Teller effects on spectra. We show, on a modelMorse system, the advantages of the finite-temperature thawed Gaussianapproximation over the commonly used global harmonic methods and apply it toevaluate the symmetry-forbidden absorption spectrum of benzene, where all ofthe aforementioned effects contribute.

On-the-fly ab initio semiclassical evaluation of vibronic spectra at finite temperature