Multi-layer Potfit: An accurate potential representation for efficient high-dimensional quantum dynamics

Abstract

The multi-layer multi-configuration time-dependent Hartree method (ML-MCTDH) is a highly efficient scheme for studying the dynamics of high-dimensional quantum systems. Its use is greatly facilitated if the Hamiltonian of the system possesses a particular structure through which the multi-dimensional matrix elements can be computed efficiently. In the field of quantum molecular dynamics, the effective interaction between the atoms is often described by potential energy surfaces (PES), and it is necessary to fit such PES into the desired structure. For high-dimensional systems, the current approaches for this fitting process either lead to fits that are too large to be practical, or their accuracy is difficult to predict and control.This article introduces multi-layer Potfit (MLPF), a novel fitting scheme that results in a PES representation in the hierarchical tensor (HT) format. The scheme is based on the hierarchical singular value decomposition, which can yield a near-optimal fit and give strict bounds for the obtained accuracy. Here, a recursive scheme for using the HT-format PES within ML-MCTDH is derived, and theoretical estimates as well as a computational example show that the use of MLPF can reduce the numerical effort for ML-MCTDH by orders of magnitude, compared to the traditionally used Potfit representation of the PES. Moreover, it is shown that MLPF is especially beneficial for high-accuracy PES representations, and it turns out that MLPF leads to computational savings already for comparatively small systems with just four modes.

Systematically expanding nondirect product bases within the pruned multi-configuration time-dependent Hartree (MCTDH) method: A comparison with multi-layer MCTDH

“…ML-MCTDH is usually used with a SOP Hamiltonian, but it is also possible to put the Hamiltonian into a ML format. 79 Although, it is possible to massage a general PES into SOP form, 80,81 it would be nice to obviate this step. One way to do this is to use a ML-CDVR (multi-layer correlation DVR) approach.…”

Accuracy of Potfit-based potential representations and its impact on the performance of (ML-)MCTDH

“…From the approximation perspective, (ML-)MCTDH is particularly attractive as it allows to systematically increase the accuracy by increasing the size of the wavefunction representation, at the expense of using more computational resources. In this respect ML-MCTDH fares better than MCTDH, as resource usage depends only polynomially instead of exponentially on the wavefunction size [24,34].…”

Fitting potential energy surfaces to sum-of-products form with neural networks using exponential neurons

“…For e±ciency in MCTDH, the PES, the kinetic energy operator (KEO), and, therefore, the corresponding wavefunction must all be in sum-of-products form. The present NN-expnn PES¯tting procedure provides a complementary approach to the pot¯t, 51,52 multi-grid pot¯t, 53 Monte Carlo pot¯t, multi-layer pot¯t, 54 and cluster expansion 78 methods for obtaining PESs currently available in MCTDH. 44 Using polyspherical coordinates, the KEO can be expressed as a sum of products of single mode operators; importantly, automatic procedures for generating analytical or numerical KEOs in the required form have been developed.…”

“…Within pot¯t, the limit on dimensionality can be circumvented using multi-grid pot¯t 53 or multi-layer pot¯t. 54 Avila and Carrington developed a method for obtaining a PES in sum-of-products form based on Smolyak interpolation using polynomial-like or spectral basis functions and 1D Lagrange-type functions. 55 Ziegler and Rauhut demonstrated an e±cient algorithm to generate a sum-of-products form based on the multi-mode expansion; 56 Manzhos and Carrington have combined the analogous high-dimensional model representation (HDMR) with the expNN approach to obtain sum-of-products form.…”