On Hagedorn wavepackets associated with different Gaussians

TL;DR

This paper develops algebraic methods to relate Hagedorn wavepackets associated with different Gaussian centers, enabling efficient computation of overlaps and correlation functions crucial for quantum spectroscopy and dynamics.

Contribution

It introduces exact recurrence relations for overlaps of Hagedorn functions with different Gaussians using Bogoliubov transformations, avoiding numerical quadrature.

Findings

Recurrence relations enable efficient overlap calculations.

Algebraic methods improve accuracy over numerical quadrature.

Numerical experiments confirm method's effectiveness in spectroscopy applications.

Abstract

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.