  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.
