Basis Set Generation for Quantum Dynamics Simulations Using Simple Trajectory-Based Methods

Abstract

Methods for solving the time-dependent Schrödinger equation generally employ either a global static basis set, which is fixed at the outset, or a dynamic basis set, which evolves according to classical-like or variational equations of motion; the former approach results in the well-known exponential scaling with system size, while the latter can suffer from challenging numerical problems, such as singular matrices, as well as violation of energy conservation. Here, we suggest a middle road: building a basis set using trajectories to place time-independent basis functions in the regions of phase space relevant to wave function propagation. This simple approach, which potentially circumvents many of the problems traditionally associated with global or dynamic basis sets, is successfully demonstrated for two challenging benchmark problems in quantum dynamics, namely, relaxation dynamics following photoexcitation in pyrazine, and the spin Boson model.

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

“…44 Here, we employ computationally-simple (classical molecular dynamics or Ehrenfest-type) trajectories to place GWP basis functions in phase-space, guided by the PES of the system. Following the initial sampling stage, the GWP basis set behaves time-independently, with wavefunction propagation being expressed variationally through the expansion coefficients.…”

“…44 We assume that our system of interest is described by f nuclear degrees-of-freedom and a set of diabatic states |α , and we assume that we have an initial wavefunction |ψ(q, 0) |α . We subsequently initialise a set of m trajectories with initial conditions chosen based on the characteristics of the initial wavefunction; for example, the initial positions and momenta of each trajectory could be drawn from the Wigner distribution of the initial wavefunction.…”

Efficient and accurate evaluation of potential energy matrix elements for quantum dynamics using Gaussian process regression

“…[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] The power of these simulation approaches is that they provide a direct view of real-time quantum dynamics with access to all properties of interest, such as position expectation values, electronic state populations, and branching ratios; as a result, direct solution of the TDSE is an important route to reconciling experimental observations and atomistic dynamics.…”

“…[7][8][9][10]12,19,21,31,32,[54][55][56][57][58][59][60] The resulting equation-ofmotion for the expansion coefficients iṡ…”

“…In our own recent work, we have shown how the common problem of linear dependence in GWP basis sets can be circumvented using adaptive strategies, 19 and we have also proposed a trajectory-guided strategy which employs classical dynamics to sample relevant regions of GWP phase-space for subsequent solution of the TDSE. 21 Regardless of the quantum simulation approach taken (variational or non-variational) and the type of basis function employed (e.g. GWPs or DVRs), all basis-set based approaches to solving the TDSE have a common computational bottleneck beyond the explicit treatment of high-dimensionality, namely the calculation of matrix elements of the potential energy operator V (q).…”