  To alleviate the computational cost associated with on-the-fly ab initio
semiclassical calculations of molecular spectra, we propose the single-Hessian
thawed Gaussian approximation, in which the Hessian of the potential energy at
all points along an anharmonic classical trajectory is approximated by a
constant matrix. The spectra obtained with this approximation are compared with
the exact quantum spectra of a one-dimensional Morse potential and with the
experimental spectra of ammonia and quinquethiophene. In all cases, the
single-Hessian version performs almost as well as the much more expensive
on-the-fly ab initio thawed Gaussian approximation and significantly better
than the global harmonic schemes. Remarkably, unlike the thawed Gaussian
approximation, the proposed method conserves energy exactly, despite the time
dependence of the corresponding effective Hamiltonian, and, in addition, can be
mapped to a higher-dimensional time-independent classical Hamiltonian system.
We also provide a detailed comparison with several related approximations used
for accelerating prefactor calculations in semiclassical simulations.
