  Hagedorn functions are carefully constructed generalizations of Hermite
functions to the setting of many-dimensional squeezed and coupled harmonic
systems. Wavepackets formed by superpositions of Hagedorn functions have been
successfully used to solve the time-dependent Schr\"{o}dinger equation exactly
in harmonic systems and variationally in anharmonic systems. For evaluating
typical observables, such as position or kinetic energy, it is sufficient to
consider orthonormal Hagedorn functions with a single Gaussian center. Here, we
instead derive various relations between Hagedorn bases associated with
different Gaussians, including their overlaps, which are necessary for
evaluating quantities nonlocal in time, such as time correlation functions
needed for computing spectra. First, we use the Bogoliubov transformation to
obtain commutation relations between the ladder operators associated with
different Gaussians. Then, instead of using numerical quadrature, we employ
these commutation relations to derive exact recurrence relations for the
overlap integrals between Hagedorn functions with different Gaussian centers.
Finally, we present numerical experiments that demonstrate the accuracy and
efficiency of our algebraic method as well as its suitability to treat problems
in spectroscopy and chemical dynamics.
