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 the time-independent approach of computing all contributing Franck-Condon factors becomes impractical in large systems, a time-dependent formalism has not been applied to simulate emission from arbitrary initial vibrational levels. Here, we apply Hagedorn functions, which are products of a Gaussian and carefully generated polynomials, to represent SVL initial states. Under an at most quadratic potential, the Hagedorn functions are exact solutions to the time-dependent Schr\"{o}dinger equation and can be propagated using 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. Here, we use two-dimensional global harmonic models to validate the method by comparing it with quantum split-operator calculations and to demonstrate the effects of displacement, distortion (squeezing), and Duschinsky rotation on SVL spectra. Additionally, we show the practicality of the Hagedorn approach in a high-dimensional system on a displaced, distorted, and Duschinsky-rotated harmonic model with 100 degrees of freedom.