Capturing tunnelling and wavepacket splitting effects on electronic spectra with Hagedorn wavepackets

Vibrationally resolved electronic spectroscopy has improved enormously our understanding of physical and chemical processes taking place in molecules. Semiclassical methods such as the thawed Gaussian approximation (TGA) have been developed to efficiently model quantum dynamics involved in spectroscopy in order to better understand experimental results and guide further investigations.

In TGA, a single Gaussian wavepacket (GWP) is propagated using classical-like equations under the local harmonic approximation (LHA), which keeps the wavepacket in the Gaussian form. To explore wavepackets that are not well represented by a single GWP, we can use a class of semiclassical wavepackets introduced by Hagedorn (Hagedorn wavepackets, or HWP) that are constructed from a complete orthonormal basis in the form of polynomials times a particular GWP. The time evolution of the GWP parameters associated with such wavepackets retains the simple propagation form as in TGA, while a variational method can be applied to the coefficients of basis functions to account for effects beyond LHA. As a result, the Hagedorn wavepackets can be applied to much higher-dimensional systems than grid-based quantum approaches.

On several examples of double-well and anharmonic potentials, we demonstrate the convergence of the HWP approach to exact quantum results with a sufficient number of basis functions and its ability to treat spectroscopy problems involving quantum tunnelling and wavepacket splitting that commonly arise in chemical dynamics but are beyond the reach of a single thawed GWP.