Quantum-mechanical wavepacket propagation in a sparse, adaptive basis of interpolating Gaussians with collocation

Abstract

We present an extension of our earlier work on adaptive quantum wavepacket dynamics [B. Hartke, Phys. Chem. Chem. Phys., 2006, 8, 3627]. In this dynamically pruned basis representation the wavepacket is only stored at places where it has non-negligible contributions. Here we enhance the former 1D proof-of-principle implementation to higher dimensions and optimize it by a new basis set, interpolating Gaussians with collocation. As a further improvement the TNUM approach from Lauvergnat and Nauts [J. Chem. Phys., 2002, 116, 8560] was implemented, which in combination with our adaptive representation offers the possibility of calculating the whole Hamiltonian on-the-fly. For a two-dimensional artificial benchmark and a three-dimensional real-life test case, we show that a sparse matrix implementation of this approach saves memory compared to traditional basis representations and comes even close to the efficiency of the fast Fourier transform method. Thus we arrive at a quantum wavepacket dynamics implementation featuring several important black-box characteristics: it can treat arbitrary systems without code changes, it calculates the kinetic and potential part of the Hamiltonian on-the-fly, and it employs a basis that is automatically optimized for the ongoing wavepacket dynamics.

Quantum Dynamics with Short-Time Trajectories and Minimal Adaptive Basis Sets

“…Basis functions may be chosen at the outset, resulting in a time-independent basis set, |φ i (q, t) = |φ i (q) . Once sampled, the basis functions remain static, and the time-dependence of the system is expressed through propagation of the expansion coefficients, c i (t), [2][3][4] according to the Dirac-Frenkel variational principle.…”

“…One strategy which our group, and others, 4,9,44,45 have begun to explore is the idea of adaptive basis sets which represent a "middle road" between time-dependent and time-independent basis set strategies. Here, the individual basis functions describing the wavefunction are time-independent, but the basis functions themselves are adaptively added or removed from the full basis set as the wavefunction evolves; the time-evolution of the expansion coefficients associated with the static basis functions is performed variationally, thereby circumventing the problem of energy conservation encountered by nonvariational evolution.…”

Moving boundary truncated grid method: Multidimensional quantum dynamics

“…[63] Pruning scheme adapting the basis in time has been implemented for coordinate-space localized Gaussians and orthogonalized Gaussians based on collocation. [64,65] Dynamical pruning of static localized basis sets has been investigated for quantum wave packet propagation. [66] A dynamically pruned discrete variable representation approach has been developed to perform molecular quantum dynamics.…”

Perspective: Computing (ro-)vibrational spectra of molecules with more than four atoms

“…It is also possible to start with a small basis and adaptively add basis functions deemed to be important to obtain convergence. [76][77][78]80,106,107 This was done with PSL functions in Ref. 77 and with harmonic oscillator basis functions in Ref.…”