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

## TL;DR

This paper introduces a physics-based algorithm to generate compact Gaussian basis sets for accurate and efficient electronic structure calculations.

## Contribution

A novel algorithm that constructs compact DF basis sets using mathematical and physical principles, minimizing parameters and ensuring accuracy across diverse elements and electron correlations.

## Key findings

- The MADF algorithm generates DF basis sets with DF errors of ~20 μE_h per electron in Hartree–Fock calculations.
- The method achieves ~10 μE_h per electron accuracy in second-order MP2 energy calculations.
- The approach works across main-group, d-block, and f-block elements with basis sets up to quadruple-ζ quality.

## Abstract

We present a 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 structures. 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-ζ 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 μE
h per electron, respectively.