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PreprintCC BYChemRxiv

Stochastic Trace Estimation Method for Ring-Polymer Instanton Theory

  • Chenghao Zhang1,2
  • Chao Yang3,4
  • Niranjan Govind1,2,5,6
6 affiliations
  1. 1Physical Science Division
  2. 2Pacific Northwest National Laboratory
  3. 3Applied Mathematics and Computational Research Division
  4. 4Lawrence Berkeley National Laboratory
  5. 5Department of Chemistry
  6. 6University of Washington
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Abstract

Evaluation of the fluctuation factor can become a computational bottleneck in ringpolymer instanton calculations, particularly at low temperatures where large numbers of ring-polymer beads are required. Conventional evaluation involves diagonalization of the full ring-polymer Hessian, with a computational cost that scales as O(P 3D3) for P beads and D physical dimensions. Here, we develop a matrix-free stochastic trace estimation approach that avoids full Hessian diagonalization. The log-determinant entering the fluctuation factor is evaluated using stochastic Lanczos quadrature, while a preconditioned Davidson algorithm is used to obtain the required low-lying eigenpairs. Control-variate and subspace-projection techniques are used to reduce the variance and improve the efficiency of the stochastic estimator. The approach is tested for proton transfer in malonaldehyde, the CH4 +H → CH3 +H2 hydrogen-abstraction reaction, and intramolecular proton transfer in dinitro-HBQ. For these systems, the calculated log(Finst) agrees with full diagonalization to within an absolute error of ≈ 0.1, while the variance is reduced by up to approximately two orders of magnitude. The matrixfree formulation shows O(P ) to O(P logP ) scaling with bead number, providing a practical alternative to full diagonalization when large numbers of ring-polymer beads are required and potentially facilitating applications to larger molecular systems.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026stochastic,
  title = {{Stochastic Trace Estimation Method for Ring-Polymer Instanton Theory}},
  author = {Chenghao Zhang and Chao Yang and Niranjan Govind},
  howpublished = {ChemRxiv},
  year = {2026},
  month = sep,
  doi = {10.26434/chemrxiv.15009258/v1},
  url = {https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009258/v1},
  note = {CC BY},
}
RIS
TY  - UNPB
TI  - Stochastic Trace Estimation Method for Ring-Polymer Instanton Theory
AU  - Chenghao Zhang
AU  - Chao Yang
AU  - Niranjan Govind
PY  - 2026
DA  - 2026/09/22
PB  - ChemRxiv
DO  - 10.26434/chemrxiv.15009258/v1
UR  - https://chemrxiv.org/doi/full/10.26434/chemrxiv.15009258/v1
AB  - Evaluation of the fluctuation factor can become a computational bottleneck in ringpolymer instanton calculations, particularly at low temperatures where large numbers of ring-polymer beads are required. Conventional evaluation involves diagonalization of the full ring-polymer Hessian, with a computational cost that scales as O(P 3D3) for P beads and D physical dimensions. Here, we develop a matrix-free stochastic trace estimation approach that avoids full Hessian diagonalization. The log-determinant entering the fluctuation factor is evaluated using stochastic Lanczos quadrature, while a preconditioned Davidson algorithm is used to obtain the required low-lying eigenpairs. Control-variate and subspace-projection techniques are used to reduce the variance and improve the efficiency of the stochastic estimator. The approach is tested for proton transfer in malonaldehyde, the CH4 +H → CH3 +H2 hydrogen-abstraction reaction, and intramolecular proton transfer in dinitro-HBQ. For these systems, the calculated log(Finst) agrees with full diagonalization to within an absolute error of ≈ 0.1, while the variance is reduced by up to approximately two orders of magnitude. The matrixfree formulation shows O(P ) to O(P logP ) scaling with bead number, providing a practical alternative to full diagonalization when large numbers of ring-polymer beads are required and potentially facilitating applications to larger molecular systems.
ER  -