Modular Construction of Jastrow Factors for the Transcorrelated Method

TL;DR

The paper tackles the complexity of the transcorrelated method by introducing a modular Jastrow strategy that reuses atom-specific Jastrow terms in molecules, reducing the number of optimisable parameters while preserving accuracy. The approach computes the TC Hamiltonian and applies the xTC approximation, then tests flexible DTN-based Jastrow forms as well as simplified minimal forms; importantly, the modular Jastrow (with reoptimised e-e terms) achieves energies within chemical accuracy for HEAT atomisation energies and ionisation potentials, and yields accurate CO binding curves with fewer optimisable parameters. The results show that the modular method can maintain high accuracy and improve robustness, suggesting practical pathways to databases of atomic Jastrow forms and scalable TC calculations with deterministic optimisation and pseudopotentials. Overall, this work offers a viable route to simplify and democratize transcorrelated calculations for larger systems and complex potential energy surfaces.

Abstract

In this work, we explore the reuse of terms in the Jastrow factor between systems for use in the transcorrelated method, to reduce the number of optimisable parameters for a given system. In particular, we propose a workflow in which atom-specific parts of Jastrow factors, optimised in atoms, may be reused in the molecule, with only a few parameters in the electron-electron part of the Jastrow left to optimise, while maintaining performance. We find that the modified workflow not only reduces the number of terms needing to be optimised, but also improves the accuracy of xTC-CCSD(T) energies.

Figures (7)

• Figure 1: Clockwise from top left: the e-e terms for the $J_\mathrm{DTN}(L_{ee}=4.5, L_{en}=4, L_{een}=4)$, $J_\mathrm{DTN}(L_{ee}=4.5, L_{en}=1, L_{een}=2)$ (i.e. all terms optimised), $J_\mathrm{mod}$ (e-e term reoptimised) and $J_{ee}$ (e-e term only) Jastrow factors for the HEAT set of molecules with the aug-cc-pVTZ basis set. Included in each subplot is also the minimal e-e Jastrow factor $\frac{1}{2} r_{12}(1-r_{12}/L_{ee})$ where $L_{ee}=4.5$ bohr. The significant outlier in each plot is the Jastrow factor of H$_2$.
• Figure 2: e-n Jastrow factor terms for the cutoff combinations $L_{ee}=4.5, L_{en}=4, L_{een}=4$ and $L_{ee}=4.5, L_{en}=1, L_{een}=2$ for N$_2$ and the N atom. From these plots, we see that the shorter cutoff lengths provide forms where the atomic and molecular Jastrow factors are more similar, due to the more local character of these Jastrow factors.
• Figure 3: e-e Jastrow factor terms for N$_2$ using $J_\mathrm{DTN}(L_{ee}=4.5, L_{en}=1, L_{een}=2)$ (full optimisation), $J_\mathrm{mod}$ (optimising only e-e component from the atom) and $J_\mathrm{mod}|_\mathrm{no-opt}$ (no further optimisation, simply use the atom's Jastrow factor), with the aug-cc-pVTZ basis set. While all three are similar to each other, we see that by optimising the atom's Jastrow factor, we return to a form particularly similar to the fully-optimised form.
• Figure 4: $u(r_1,r_2)$ (see equation \ref{['eq:ur1r2def']}) for the fully-optimised $J_\mathrm{DTN}(L_{ee}=4.5, L_{en}=1, L_{een}=2)$ (blue solid curves) and e-e-reoptimised $J_\mathrm{mod}$ (orange dashed curves) Jastrow factors. Note that by the molecule's symmetry, with nitrogen nuclei located at $\pm 1.04$ bohr, these Jastrow factors are symmetric about zero, so we plot only for $r_1\ge 0$ and indicate the nucleus' position with a dotted vertical line. The location of the fixed electron $r_2$ corresponds to the prominent cusps which deepen as the electrons are further from the nucleus.
• Figure 5: Errors in the total xTC-CCSD(T) energies for the HEAT set of molecules and atoms with the aug-cc-pVTZ basis set and (clockwise from top-left): the $J_\mathrm{DTN}(L_{ee}=4.5, L_{en}=4, L_{een}=4)$, $J_\mathrm{mod}|_\mathrm{no-opt}$, $J_\mathrm{mod}$ and $J_{ee}$ Jastrow factors. Dotted lines indicate chemical accuracy (1.6 mHa). The red shaded areas are a visual aid only: they correspond to a sum of gaussians centred on the data points with the width of the Gaussians chosen such that equidistantly distributed Gaussians would be contained to 95% in the corresponding segment.