Accurate simulation of vibrationally resolved electronic (or vibronic) spectra requires not only an accurate evaluation of electronic structure but also a quantum mechanical treatment of nuclear motion. Solving the time-dependent Schrödinger equation (TDSE) for atomic nuclei is challenging due to the highly oscillatory nature of its solutions and the exponential growth in computational costs with the system size. To simulate high-dimensional quantum dynamics at an acceptable cost, many semiclassical approximations have been developed with different levels of accuracy and efficiency.

This thesis focuses on the application of semiclassical Hagedorn wavepackets (HWPs) to vibronic spectroscopy. Based on Gaussian wavepackets, G.~A. Hagedorn introduced a class of functions in the form of a Gaussian multiplied by polynomials via a special pair of raising and lowering operators. These functions generalize Hermite functions to higher dimensions and form a complete orthonormal basis in L^2(R^D). Moreover, for an at most quadratic Hamiltonian, HWPs are exact solutions to the TDSE with a particularly simple set of equations of motion that depend only on the evolution of the parameters of the associated Gaussian.

Despite these favourable mathematical properties, the application of HWPs in chemistry has been limited. For applications in spectroscopy, one important obstacle is the lack of an efficient method to calculate the autocorrelation function in Hagedorn dynamics. In this thesis, I present a general, recursive, algebraic approach to compute overlaps between two arbitrary Hagedorn functions without the need of numerical grids, enabling the use of HWPs in computational spectroscopy.

In the single vibronic level (SVL) fluorescence spectroscopy, the vibrationally excited initial state can be exactly represented by a single Hagedorn function. Within the global harmonic approximation, HWPs accurately account for displacement, mode distortion (squeezing), and Duschinsky rotation effects in SVL spectra, as I demonstrate in both model potentials and a 66-dimensional ab initio global harmonic model of anthracene. Remarkably, a single trajectory of Gaussian parameters is sufficient to obtain the SVL spectra from arbitrary vibrational levels, and this approach can be applied to much higher dimensions than standard grid-based numerical methods allow.

Hagedorn functions preserve these advantages even for a non-linear TDSE with a state-dependent quadratic potential. I present a straightforward extension of the global harmonic Hagedorn approach to capture anharmonicity effects via the local harmonic approximation in Morse-type models. Using the on-the-fly ab initio local harmonic Hagedorn dynamics, I also show improvements in the SVL spectra of difluorocarbene over global harmonic models.

Combining the time-dependent approach to spectroscopy with thermofield dynamics allows for the inclusion of temperature effects in vibronic spectra without Boltzmann averaging. This thesis also presents the split-operator coherence thermofield dynamics as an exact quantum benchmark for finite-temperature spectra and shows how Hagedorn functions can be used to validate and analyze thermofield Gaussian wavepacket dynamics.