Abstract
Solution of the time-dependent Schrödinger equation using a linear combination of basis functions, such as Gaussian wavepackets (GWPs), requires costly evaluation of integrals over the entire potential energy surface (PES) of the system. The standard approach, motivated by computational tractability for direct dynamics, is to approximate the PES with a second order Taylor expansion, for example centred at each GWP. In this article, we propose an alternative method for approximating PES matrix elements based on PES interpolation using Gaussian process regression (GPR). Our GPR scheme requires only single-point evaluations of the PES at a limited number of configurations in each time-step; the necessity of performing often-expensive evaluations of the Hessian matrix is completely avoided. In applications to 2-, 5-, and 10-dimensional benchmark models describing a tunnelling coordinate coupled non-linearly to a set of harmonic oscillators, we find that our GPR method results in PES matrix elements for which the average error is, in the best case, two orders-of-magnitude smaller and, in the worst case, directly comparable to that determined by any other Taylor expansion method, without requiring additional PES evaluations or Hessian matrices. Given the computational simplicity of GPR, as well as the opportunities for further refinement of the procedure highlighted herein, we argue that our GPR methodology should replace methods for evaluating PES matrix elements using Taylor expansions in quantum dynamics simulations.
Highlights
The multiconfigurational time-dependent Hartree (MCTDH) strategy requires that the potential energy surface (PES) describing the system must be written as a “sum of products” for the time-propagation to be computationally tractable; while strategies have been developed to cast PESs into the required form,1,49 this prescription restricts the use of MCTDH in more general “on the fly” schemes for quantum dynamics which directly couple to ab initio electronic structure calculations
For the f = 2 and f = 5-dimensional models considered here, we have shown the error in the Gaussian process regression (GPR) PES matrix element approximation is significantly smaller than any of the alternative methods based on Taylor expansions; as dimensionality increases to, say f = 10, the advantage of GPR decreases somewhat, but the accuracy of the approximated elements remains comparable to the Taylor expansion approaches
Most importantly, in its most basic form, GPR only requires as input the value of the PES evaluated at a reference set of configurations which we chose to be the Gaussian wavepackets (GWPs) centres in our scheme; importantly, GPR does not require evaluation of the Hessian matrix like the second-order Taylor expansion methods, leading to significant reductions in computational expense
Summary
Computer simulation approaches based on solution of the time-dependent Schrödinger equation (TDSE) offer an important route to modelling dynamics in quantum chemical systems; for example, quantum simulations of organic molecules, inorganic complexes, and biological systems have all provided insight into relaxation dynamics following photochemical excitation, while other studies have highlighted the role of quantum-mechanical tunnelling or zero-point energy (ZPE) in multidimensional dynamics. 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. 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). Our results demonstrate convincingly that GPR should be employed as the methodof-choice in evaluating PES matrix elements in GWP-based quantum dynamics simulations; in particular, our GPR-based method only requires single-point PES evaluations at each GWP basis function, is generally more accurate than Taylor expansion methodologies, and does not require the (usually expensive) calculation of the Hessian matrix. GPR is expected to find use in further development of “on-the-fly” quantum dynamics strategies
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