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arXiv:2405.07880 · Journal of Physics A Mathematical and Theoretical

On Hagedorn wavepackets associated with different Gaussians

Jiří J. L. Vaníček, Zhan Tong Zhang

Published Jan 30, 2025• 6 citations
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Abstract

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 Schrödinger equation exactly in harmonic systems and variationally in anharmonic systems. To evaluate typical observables, such as position or kinetic energy, it is sufficient to consider orthonormal Hagedorn functions with a single Gaussian center. Instead, we derive various relations between Hagedorn bases associated with different Gaussians, including their overlaps, which are necessary for evaluating quantities nonlocal in time, such as the time correlation functions needed for computing spectra. First, we use the Bogoliubov transformation to obtain the 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 for treating problems in spectroscopy and chemical dynamics.

DOI
10.1088/1751-8121/adb083

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Hagedorn wave packets appear as the eigenstates of multidimensional harmonic oscillators, and are given by a polynomial times a Gaussian. We recognise the arising polynomials for normalised and unnormalised Hagedorn wave packets as generalised multivariate Hermite polynomials, which are not tensor products in general. We provide formulas for generating functions and ladder operators, and prove a direct connection to the Laguerre polynomials. As our main result, we show that the class of Hagedorn wave packets is invariant under the Wigner transform, and present an explanation for the product structure of normalised Hagedorn wave packets in phase space, which has recently been discovered.

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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 Schrödinger equation exactly in harmonic systems and variationally in anharmonic systems. To evaluate typical observables, such as position or kinetic energy, it is sufficient to consider orthonormal Hagedorn functions with a single Gaussian center. Instead, we derive various relations between Hagedorn bases associated with different Gaussians, including their overlaps, which are necessary for evaluating quantities nonlocal in time, such as the time correlation functions needed for computing spectra. First, we use the Bogoliubov transformation to obtain the 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 for treating problems in spectroscopy and chemical dynamics.
We present a practical, ab initio time-dependent method using Hagedorn wavepackets to simulate resonance Raman (RR) spectra of polyatomic molecules. Hagedorn functions---Gaussians multiplied by specific polynomials---are used to represent RR initial and final states because these functions are exact solutions to the time-dependent Schrödinger equation for at-most-quadratic potentials and can be propagated at zero cost beyond that of propagating the guiding Gaussian. Using efficient recursive formulae to compute overlaps between Hagedorn wavepackets, we can evaluate RR excitation profiles for arbitrary spectral signals, such as fundamental, overtone, combination, and hot bands. We then construct the Stokes and anti-Stokes RR spectra from these profiles. We first validate the method in a two-dimensional displaced, distorted, and Duschinsky-rotated harmonic model against numerically exact split-operator calculations. Then, we apply the method to compute RR spectra of anthracene by performing dynamics on a 66-dimensional harmonic potential energy surface constructed from density functional theory calculations.
The semiclassical (or Hagedorn) wavepackets depending on a fixed set of parameters are an orthonormal L 2 -basis of generalized coherent states. They have been used to solve numerically the time-dependent Schrödinger equation in its semiclassical formulation, yet their localization property makes them inefficient in case of non-local phenomena such as quantum tunneling. In order to overcome this difficulty, we use simultaneously several bases with different parameters. We propose an algorithm to expand a given wavefunction in terms of multiple families of Hagedorn wavepackets; each family can then be accurately and efficiently propagated using modern semiclassical time-splittings.
Certain generalized coherent states, so-called Hagedorn wavepackets, have been used to numerically solve the standard Schrödinger equation. We extend this approach and its recent enhancements to the magnetic Schrödinger equation for a time-dependent, spatially homogeneous magnetic field. We explain why Hagedorn wavepackets are naturally compatible with the aforementioned physical system. In numerical experiments we examine the order of convergence and the preservation of norm and energy. We use this method to simulate a Penning trap as proposed in recent work on quantum computing .
The detailed calculation of the dynamics of a chemical system is usually not considered due to the size and cost of the computation. It is thus useful to examine various approximation methods. Such methods first need to be tried out on simple systems, like one-dimensional motion. Here two approaches to approximating the solutions of the Schrödinger and von Neumann equations by single time-dependent Gaussians are explored and contrasted, explicitly for one-dimensional barrier penetration. The first approach, in which no tunneling occurs, is local in nature and characterized by an expansion of the equations of motion to second order about the average position of the Gaussian wave packet or about the average position and momentum of the Gaussian Wigner function. This approach was first introduced by Heller [E. J. Heller, J. Chem. Phys. 62, 1544 (1975)]. Here both Heller’s approach and a more general truncation method are considered. Indeed tunneling can be incorporated if second-order terms in the quantal von Neumann equation are included. However, the resulting dynamics is unstable for kinetic energies where the exact wavepacket (and Wigner function) splits into nonnegligible parts that represent reflection and transmission. In contrast, the second approach is a global method which is obtained by applying appropriate closure approximations to the equations of motion for the first- and second-order position and momentum expectation values. This method allows tunneling and is stable at all kinetic energies. It is also possible to approximate the global equations in such a way that the local equations are obtained.
We develop an alternative view of the semiclassical wave packets of Hagedorn---often called the Hagedorn wave packets---stressing the roles of the symplectic and metaplectic groups along with the Heisenberg--Weyl group. Our point of view clarifies the relationship between the Hagedorn wave packets and the Hermite functions by building a bridge between the ladders of wave functions in both theories. This Hagedorn--Hermite correspondence provides an elegant view as well as simple proofs of some essential results on the Hagedorn wave packets. We build the theory starting from fundamental properties of ladder operators. Particularly, we show that the ladder operators of Hagedorn are a natural set obtained from the position and momentum operators using the symplectic group. The idea that pervades our view of the Hagedorn wave packets is so-called symplectic covariance; it generalizes some of fundamental results concerning the Hagedorn wave packets as well as simplifies their proofs. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets; it generalizes the one-dimensional result by Hagedorn to multi-dimensions. The Hagedorn--Hermite correspondence also leads to an alternative derivation of the generating function for the Hagedorn wave packets and clarifies its relationship with the generating function for the Hermite functions. This result, in turn, reveals the relationship between the Hagedorn polynomials and the Hermite polynomials.
Hagedorn wave packets appear as the eigenstates of multidimensional harmonic oscillators, and are given by a polynomial times a Gaussian. We recognise the arising polynomials for normalised and unnormalised Hagedorn wave packets as generalised multivariate Hermite polynomials, which are not tensor products in general. We provide formulas for generating functions and ladder operators, and prove a direct connection to the Laguerre polynomials. As our main result, we show that the class of Hagedorn wave packets is invariant under the Wigner transform, and present an explanation for the product structure of normalised Hagedorn wave packets in phase space, which has recently been discovered.