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