Quantum Dynamics with the Quantum Trajectory-Guided Adaptable Gaussian Bases

Abstract

The computational cost of describing a general quantum system fully coupled by anharmonic interactions scales exponentially with the system size. Thus, an efficient basis representation of wave functions is essential, and when it comes to the large-amplitude motion of high-dimensional systems, the dynamic bases of Gaussian functions are often employed. The time dependence of such bases is determined from the variational principle or from classical dynamics; the former is challenging in implementation due to singular matrices, while the latter may not cover the configuration space relevant to quantum dynamics. Here we describe a method using Quantum Trajectory-guided Adaptable Gaussian (QTAG) bases “tuned”including the basis position, phase, and widthto the wave function evolution, thanks to the continuity of the probability density in the course of the quantum trajectory dynamics. Thus, an efficient basis in configuration space is generated, bypassing the variational equations on the parameters of the Gaussians. We also propose a time propagator with basis transformation by projections which lends efficiency and stability to the QTAG dynamics, as demonstrated on standard tests and the ammonia inversion model.

Multidimensional Tunneling Dynamics Employing Quantum-Trajectory Guided Adaptable Gaussian Bases

“…This setup, which replaces the N d -dimensional grid of basis functions used in Section 3.1 and in our previous work 17 with an N d -dimensional line, is highly favorable from a computational scaling perspective as the system complexity is largely decoupled from the size of the bath and instead depends on the number of basis functions required to describe motion in the quantum double well. However, for a single-Gaussian representation of a wavefunction (as is the case in the bath DoFs here), we can no longer define the trajectory momenta according to eq 12 because in this case it reduces to the trivial expression p ν = p ν .…”

“…where the matrix S is the GBF overlap, H is the Hamiltonian in a basis, and D involves the GBF time-derivatives. 17 In the QTAG approach, we bypass the general variational solution for the GBF parameters a, q, and p in favor of the spatially localized limit of Gaussians defined by the quantum trajectories with known evolution equations (e.g., refs 20 and 22):…”

“…approach is utilized by techniques including the matching pursuit/split-operator Fourier transform (MP/SOFT) 12 and basis expansion leaping multi-configuration Gaussian (BEL-MCG) 13 methods, while the latter is seen in such methods as the variational multi-configurational Gaussian, 14 coupled coherent states (CCS), 15 and adaptive trajectory-guided (aTG) 16 methods. It is the aim of this work to examine the performance of one such trajectory-based scheme, the recently developed quantum trajectory-guided adaptable Gaussian (QTAG) basis method, 17 in the context of systems exhibiting quantum tunneling. Unlike other trajectory-based methods, which utilize classical mechanics to evolve their trajectories, the ones in the QTAG scheme propagate according to the principles of quantum trajectory (QT) or Bohmian mechanics, 18−20 meaning that they are intrinsically quantum and should thus be able to capture quantum phenomena by construction if implemented exactly.…”

Molecular spectra calculations using an optimized quasi-regular Gaussian basis and the collocation method

“…All the previous experience using DGBs [28][29][30][31][32] suggests that their quality depends very much not only on how the Gaussian centers are distributed but is also very sensitive to the choices of the Gaussian widths, α i . A wrong choice for the latter (e.g., too narrow or too wide) may result in poor approximation of the wavefunctions or ill-conditioned matrices, or both.…”

Local Measure of Quantum Effects in Quantum Dynamics

“…Recent developments in exact methods of quantum dynamics, e.g., − enable description of large chemical systems with relatively mild approximations. The applicability of the approximate methods often depends on the degree of quantumness in the underlying dynamics.…”