A hierarchical wavepacket propagation framework via ML-MCTDH for molecular reaction dynamics

This work presents a computational framework for studying reaction dynamics via wavepacket propagation, employing the multiconfiguration time-dependent Hartree (MCTDH) method and its multilayer extension (ML-MCTDH) as the core methodologies. The core idea centers on the concept of modes that combine several coordinates along with their hierarchical separations because the degrees of freedom are too numerous to be efficiently treated as a single mode. First, the system is partitioned into several fragments within the same layer, and these fragments are further decomposed. Repeating this process, a hierarchical separation of modes emerges, until modes of a manageable size are achieved. Accordingly, the coordinate frame can be designed hierarchically. Second, the kinetic energy operator (KEO) is derived as a sum-of-products (SOP) of single-particle differential operators through a polyspherical approach, while the potential energy surface (PES) is expressed in a similar SOP form of single-particle potentials (SPPs) through (1) reconstruction approaches using an existing PES or (2) direct approaches based on a computed database. Third, the nuclear wave function is expressed in a multi-layer expansion form, where each term is a product of single-particle functions (SPFs) that are further expanded by the SPFs in the deeper layer. This expansion form is also adopted using a variational eigensolver for electronic wave function. Finally, the Dirac-Frenkel variational principle leads to a set of working equations whose solutions reproduce reaction dynamics, say reaction probability and time-dependent expectation. In addition, the hierarchical framework can be rearranged by the mathematical language of tensor networks (TN) or tree tensor networks (TTN). In this work, we compare the methods represented by a function with those in the form of a TN or a TTN. We also discuss the limitations of the present framework and propose solutions, providing further perspectives.