Single vibronic level fluorescence spectra from Hagedorn wavepacket dynamics

In single vibronic level (SVL) fluorescence experiments, the electronically excited initial state is also excited in one or several vibrational modes. Whereas computing all contributing Franck-Condon factors individually becomes impractical in large systems, a time-dependent formalism has not been applied to simulate emission from arbitrary initial vibrational levels. Here, we use Hagedorn functions, which are products of a Gaussian and carefully generated polynomials, to represent SVL initial states. In systems where the potential is at most quadratic, the Hagedorn functions are exact solutions to the time-dependent Schr\"{o}dinger equation and can be propagated with the same equations of motion as a simple Gaussian wavepacket. Having developed an efficient recursive algorithm to compute the overlaps between two Hagedorn wavepackets, we can now evaluate emission spectra from arbitrary vibronic levels using a single trajectory. We validate the method in two-dimensional global harmonic models by comparing it with quantum split-operator calculations. Additionally, we study the effects of displacement, distortion (squeezing), and Duschinsky rotation on SVL spectra. Finally, we demonstrate the applicability of the Hagedorn approach to high-dimensional systems on an example of displaced, distorted, and Duschinsky-rotated harmonic model with 100 degrees of freedom.