Herman-Kluk coherence thermofield dynamics

We combine the semiclassical, multi-trajectory frozen Gaussian-based Herman-Kluk approximation with the thermofield dynamics in order to evaluate vibrationally-resolved electronic spectra at nonzero temperatures. In thermofield dynamics, the dipole time correlation function is rewritten exactly as a wavepacket autocorrelation function, and the corresponding wavepacket is a solution to a zero-temperature time-dependent Schrödinger equation on an augmented space of doubled dimensions. We derive the Herman-Kluk representation for the thermofield wavepacket autocorrelation function and demonstrate how it can be computed from individual trajectories. We compare the spectra of a numerically exact approach with the newly developed Herman-Kluk thermofield dynamics and with the thermofield local harmonic approximation in Morse potentials of increasing anharmonicity and at various temperatures. In a Morse potential with higher anharmonicity, the thermofield local harmonic approximation is not able to capture emerging hot bands, whereas the thermofield Herman-Kluk approach successfully reproduces them.