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 individual Franck-Condon factors becomes impractical for many transitions in large systems, a time-dependent formalism has not been applied to simulate emission from arbitrary initial vibrational levels. Here, we compute the SVL spectra with Hagedorn functions, products of a Gaussian and carefully generated polynomials, which can effectively represent SVL initial states and preserve their form under evolution with the time-dependent Schrödinger equation in a many-dimensional harmonic potential. Having developed an efficient recursive algorithm for 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 demonstrate the effects of displacement, distortion (squeezing), and Duschinsky rotation on SVL spectra. The results are validated against quantum split-operator calculations. Additionally, we show the practicality of the Hagedorn approach in a high-dimensional system on a displaced, distorted (squeezed), and Duschinsky-rotated harmonic model with 100 degrees of freedom.