Low-rank sum-of-products finite-basis-representation (SOP-FBR) of potential energy surfaces

Accelerating wavepacket propagation with machine learning

In this work, we discuss the use of a recently introduced machine learning (ML) technique known as Fourier neural operators (FNO) as an efficient alternative to the traditional solution of the time-dependent Schrödinger equation (TDSE). FNOs are ML models which are employed in the approximated solution of partial differential equations. For a wavepacket propagating in an anharmonic potential and for a tunneling system, we show that the FNO approach can accurately and faithfully model wavepacket propagation via the density. Additionally, we demonstrate that FNOs can be a suitable replacement for traditional TDSE solvers in cases where the results of the quantum dynamical simulation are required repeatedly such as in the case of parameter optimization problems (e.g., control). The speed-up from the FNO method allows for its combination with the Markov-chain Monte Carlo approach in applications that involve solving inverse problems such as optimal and coherent laser control of the outcome of dynamical processes.

Quantum and classical molecular dynamics for H atom scattering from graphene

This work presents systematic comparisons between classical molecular dynamics (cMD) and quantum dynamics (QD) simulations of 15-dimensional and 75-dimensional models in their description of H atom scattering from graphene. We use an experimentally validated full-dimensional neural network potential energy surface of a hydrogen atom interacting with a large cell of graphene containing 24 carbon atoms. For quantum dynamics simulations, we apply Monte Carlo canonical polyadic decomposition to transform the original potential energy surface (PES) into a sum of products form and use the multi-layer multi-configuration time-dependent Hartree method to simulate the quantum scattering of a hydrogen or deuterium atom with an initial kinetic energy of 1.96 or 0.96 eV and an incident angle of 0°, i.e., perpendicular to the graphene surface. The cMD and QD initial conditions have been carefully chosen in order to be as close as possible. Our results show little differences between cMD and QD simulations when the incident energy of the H atom is equal to 1.96 eV. However, a large difference in sticking probability is observed when the incident energy of the H atom is equal to 0.96 eV, indicating the predominance of quantum effects. To the best of our knowledge, our work provides the first benchmark of quantum against classical simulations for a system of this size with a realistic PES. Additionally, new projectors are implemented in the Heidelberg multi-configuration time-dependent Hartree package for the calculation of the atom scattering energy transfer distribution as a function of outgoing angles.

Analytical high-dimensional operators in canonical polyadic finite basis representation (CP-FBR)

In the present work, we introduce a simple means of obtaining an analytical (i.e., grid-free) canonical polyadic (CP) representation of a multidimensional function that is expressed in terms of a set of discrete data. For this, we make use of an initial CP guess, even not fully converged, and a set of auxiliary basis functions [finite basis representation (FBR)]. The resulting CP-FBR expression constitutes the CP counterpart of our previous Tucker sum-of-products-FBR approach. However, as is well-known, CP expressions are much more compact. This has obvious advantages in high-dimensional quantum dynamics. The power of CP-FBR lies in the fact that it requires a grid much coarser than the one needed for the dynamics. In a subsequent step, the basis functions can be interpolated to any desired density of grid points. This is useful, for instance, when different initial conditions (e.g., energy content) of a system are to be considered. We show the application of the method to bound systems of increased dimensionality: H₂ (3D), HONO (6D), and CH₄ (9D).

Direct Canonical-Polyadic-Decomposition of the Potential Energy Surface from Discrete Data by Decoupled Gaussian Process Regression

A Gaussian process regression (GPR) approach for directly constructing the canonical polyadic decomposition (CPD) of a multidimensional potential energy surface (PES) by discrete training energies is proposed and denoted by CPD-GPR. The present CPD-GPR method requires the kernel function in a product of a series of one-dimensional functions. To test CPD-GPR, the reactive probabilities of H + H₂ as a function of kinetics energy are performed. Comparing the dynamics results computed by the CPD-GPR PES with those by the original PES, a good agreement between these results can be clearly found. Discussions on the previous algorithms for building the decomposed form are also given. We further show that the CPD-GPR method might be the general algorithm for building the decomposed form. However, further development is needed to reduce the CPD rank. Therefore, the present CPD-GPR method might be helpful to inspire ideas for developing new tools in building decomposed potential functions.

Adaptive Fitting of Potential Energy Surfaces of Small to Medium-Sized Molecules in Sum-of-Product Form: Application to Vibrational Spectroscopy

A semi-automatic sampling and fitting procedure for generating sum-of-product (Born-Oppenheimer) potential energy surfaces based on a high-dimensional model representation is presented. The adaptive sampling procedure and subsequent fitting relies on energies only and can be used for re-fitting existing analytic potential energy surfaces in sum-of-product form or for direct fits from ab initio computa- tions. The method is tested by fitting ground electronic state potential energy surfaces for small to medium sized semi-rigid molecules, i.e., HFCO, HONO, and HCOOH, based upon ab initio computations at the CCSD(T)-F12/cc-pVTZ-F12 or MP2/aug-cc-pVTZ levels of theory. Vibrational eigenstates are computed using block improved relaxation in the Heidelberg MCTDH package and compared to available experimental and theoretical data. The new potential energy surfaces are compared to the best ones currently available for these molecules, in terms of accuracy, including of resulting vibrational states, required numbers of sampling points, and number of fitting parameters. The present procedure leads to compact expansions and scales well with the number of dimensions for simple potentials such as single or double wells.