Slimmer Geminals For Accurate F12 Electronic Structure Models

TL;DR

This work identifies that MP2-F12-optimized geminal lengthscales are suboptimal for higher-order F12 methods and reoptimizes the Slater-type geminal lengthscale $\beta$ across a range of basis sets (a$X$Z and $X$Z-F12) for CC-F12. The per-basis optimization produces substantial reductions in basis-set incompleteness errors for absolute CCSD-F12 energies, with gains increasing with basis size, and meaningful improvements in relative energies, notably atomization energies and ionization potentials. The results demonstrate clear guidance for practitioners: use the new $\beta_{\text{opt}}$ parameters for high-order F12 methods to improve robustness and accuracy, reducing reliance on extrapolation, and supporting applications such as CC-F12 and transcorrelated F12. The study also highlights that the optimal geminal exponents depend on basis and element size, with further work needed to fully map dependencies across F12 formulations.

Abstract

The Slater-type F12 geminal lengthscales originally tuned for the second-order Møller-Plesset F12 method are too large for higher-order F12 methods formulated using the SP (diagonal fixed-coefficient spin-adapted) F12 ansatz. The new geminal parameters reported herein reduce the basis set incompleteness errors (BSIE) of absolute coupled-cluster singles and doubles F12 correlation energies by a significant - and increasing with the cardinal number of the basis - margin. The effect of geminal reoptimization is especially pronounced for the cc-pVXZ-F12 basis sets (specifically designed for use with F12 methods) relative to their conventional aug-cc-pVXZ counterparts. The BSIEs of relative energies are less affected but substantial reductions can be obtained, especially for atomization energies and ionization potentials with the cc-pVXZ-F12 basis sets. The new geminal parameters are therefore recommended for all applications of high-order F12 methods, such as the coupled-cluster F12 methods and the transcorrelated F12 methods.

Figures (4)

• Figure 1: Levelized CCSD-F12 $\bar{E}_{\text{F12}}(\beta)$ for the a$X$Z OBS family. Optimal $\beta$ increases with the cardinal number $X$, whereas the curvature of $\bar{E}_{\text{F12}}(\beta)$ decreases with $X$.
• Figure 2: Distribution of CCSD-F12-optimized geminal parameters $\beta^S_\text{opt}$ across the training set. The bars represent each system in the training set with the same order of systems as listed in \ref{['sec:results-beta']}.
• Figure 3: $E^S_\text{F12}(\beta) - E^S_\text{F12}(\beta^S_\text{opt})$ for several representative valence-isoelectronic systems with second- and third-period elements. Note the relatively weak variation of the curvature across the period, but substantial curvature increase upon transition from second to third period.
• Figure 4: Basis set convergence of a$X$Z CCSD-F12 and a$\{X-1,X\}$Z extrapolated CCSD correlation contributions to atomization energies of several small molecules in the HEAT benchmark set.VRG:tajti:2004:JCP$\beta_\text{ref}$ and $\beta_\text{opt}$ denote the use of geminal parameters from Ref. VRG:peterson:2008:JCP (extended to use $\beta_\text{ref}=1.4$ for a6Z OBS) and \ref{['tbl:opt-beta']}, respectively.