Time Evolution of ML-MCTDH Wavefunctions I: Gauge Conditions, Basis Functions, and Singularities

TL;DR

This paper develops a new family of equations of motion for ML-MCTDH wavefunctions that avoid matrix inversion issues, using alternative gauge conditions and allowing for more flexible basis functions, unifying existing approaches.

Contribution

It introduces a generalized family of EOMs for ML-MCTDH that encompasses existing methods like PSI and invariant EOMs, improving stability and flexibility.

Findings

The new EOMs avoid the inversion of singular matrices.

They unify the PSI and invariant EOM approaches as special cases.

The approach allows for arbitrary orthonormal basis functions.

Abstract

We derive a family of equations of motion (EOMs) for evolving multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH) wavefunctions that, unlike the standard ML-MCTDH EOMs, never require the evaluation of the inverse of singular matrices. All members of this family of EOMs make use of alternative static gauge conditions than that used for standard ML-MCTDH. These alternative conditions result in an expansion of the wavefunction in terms of a set of potentially arbitrary orthonormal functions, rather than in terms of a set of non-orthonormal and potentially linearly dependent functions, as is the case for standard ML-MCTDH. We show that the EOMs used in the projector splitting integrator (PSI) and the invariant EOMs approaches are two special cases of this family obtained from different choices for the dynamic gauge condition, with the invariant EOMs making use of a choice that introduces potentially unbounded operators into the EOMs. As a consequence, all arguments for the existence of parallelizable integration schemes for the invariant EOMs can also be applied to the PSI EOMs.