Physics-Driven Construction of Compact Primitive Gaussian Density Fitting Basis Sets

TL;DR

The paper addresses gaps in density-fitting basis set (DFBS) coverage, especially for heavy and relativistic systems, by introducing MADF, a physics-driven generator that builds primitive SHG DFBSs from contracted OBS using a three-parameter scheme. It combines a regularized complete pool of primitive SHGs with an energy-based pruning model derived from a correlated occupancy framework (SOAD-based MP2) to efficiently trim the DFBS while controlling the 2-body energy error. MADF demonstrates competitive accuracy relative to manually optimized DFBS and AutoAux across nonrelativistic and relativistic tests (G2, Ln54, Tm60), achieving microhartree-level DF errors per electron and substantially reducing DFBS size in many cases. The approach offers a universal, low-parameter, primitive-DFBS construction that can be adapted to different basis families and higher-level electronic-structure methods, with potential for further improvements in relativistic effects and broader applicability.

Abstract

We present model-assisted density fitting (MADF) basis set generator, an algorithm for generating primitive atomic Gaussian density fitting (DF) basis sets (DFBSs) from a contracted Gaussian orbital basis set (OBS). The MADF algorithm produces DFBSs suitable for accurate robust DF approximation of 2-particle interactions in mean-field and correlated electronic structure. The algorithm is designed to (a) saturate the OBS product space by a large regularized set of primitive solid-harmonic Gaussian shells with nonuniform distribution of exponents followed by (b) pruning of the shells according to their contributions to the 2-body energy of a correlated atomic ensemble. Building the DFBS generator model almost exclusively on mathematical and physical principles allows one to limit the number of parameters that control the density fitting error to three, with a single set of parameters sufficient for computations with all basis cardinal numbers, with and without correlation of core electrons, with and without scalar and spin-dependent relativistic effects, spanning almost all of the Periodic Table. Performance assessment included basis sets up to quadruple-zeta quality from several major basis set families, using molecules composed of main-group, d-block, and f-block elements. The resulting DF errors in Hartree-Fock and second-order MP2 energies (with relativistic all-electron treatments, when appropriate) were on the order of 20 and 10 microhartree per electron, respectively.

Figures (11)

• Figure 1: Exponents of DFBS generated (or manually-optimized) for Kr atom with representative nonrelativistic and relativistic OBS using various methods scattered on a logarithmic number line for different angular momenta i.e., $L = \{0, 1, 2, 6 \}$. The number of exponents is shown on the right of each scatter plot. \ref{['fig:nonrel-exponent-distribution']} shows distribution of DFBS exponents for nonrelativistic OBS def2-QZVPP and \ref{['fig:rel-exponent-distribution']} for relativistic OBS x2c-QZVPPall-2c. The plot depict Lehtola's complete set (shown in purple), Lehtola's pCD-regularized set (shown in red) obtained using ERKALEKAS:lehtola:2012:JCC with pCD threshold $10^{-7}$ as recommended in Ref.KAS:lehtola:2021:JCTC, even-tempered exponents produced by AutoAux (shown in green) and set of exponents produced by the MADF algorithm (shown in orange). Distribution of exponents of manually-optimized DFBS def2-QZVPP-RIFIT is also shown (in blue) in \ref{['fig:nonrel-exponent-distribution']}.
• Figure 2: Variation of DF errors of nonrelativistic (a) HF and (b) MP2 energies of the TS1 training set vs the $\tau_2$ model parameter of the MADF generator.
• Figure 3: Variation of DF errors of nonrelativistic (a) HF and (b) MP2 energies of the TS1 training set vs the $\tau_1$ model parameter of the MADF generator.
• Figure 4: Variation of DF errors of nonrelativistic (a) HF and (b) MP2 energies of the TS1 training set vs the $\zeta$ model parameter of the MADF generator.
• Figure 5: Variation of DF errors of (relativistic) (a) X2C-HF and (b) X2C-MP2 energies of the TS2 training set vs the $\tau_2$ model parameter of the MADF generator.