Time evolution of ML-MCTDH wavefunctions. I. Gauge conditions, basis functions, and singularities

Area of Science:

• Quantum Chemistry
• Computational Physics
• Theoretical Chemistry

Background:

• The standard multi-layer multiconfiguration time-dependent Hartree (ML-MCTDH) method is a powerful tool for simulating quantum dynamics.
• However, standard ML-MCTDH equations-of-motion (EOMs) can encounter numerical instabilities due to the inversion of singular matrices.
• This limitation hinders the accurate and efficient propagation of complex quantum wavefunctions.

Purpose of the Study:

• To derive a new family of ML-MCTDH equations-of-motion (EOMs) that circumvent the need to invert singular matrices.
• To explore alternative gauge conditions that lead to more stable wavefunction expansions.
• To demonstrate the relationship between the new EOMs and existing methods like projector splitting integrator (PSI) and invariant EOMs.

Main Methods:

• Derivation of a novel family of EOMs for ML-MCTDH wavefunctions.
• Utilized alternative static gauge conditions, leading to expansions in orthonormal functions.
• Analyzed the connection to PSI and invariant EOMs by examining dynamic gauge conditions.

Main Results:

• The newly derived EOMs successfully avoid the inversion of singular matrices, enhancing numerical stability.
• The alternative gauge conditions result in wavefunction expansions using orthonormal basis sets.
• The projector splitting integrator (PSI) and invariant EOMs are identified as special cases within this new family of EOMs.

Conclusions:

• The proposed family of EOMs offers a more robust and numerically stable alternative for ML-MCTDH simulations.
• The findings suggest that parallelizable integration schemes applicable to invariant EOMs can also be extended to the PSI approach.
• This work provides a foundation for developing more efficient and reliable computational methods in quantum dynamics.