# SARC all-electron scalar relativistic basis sets

### Design philosophy

**compact:**small enough to be used in place of ECPs for day-to-day DFT calculations**segmented:**achieve maximal efficiency by minimizing the computational effort associated with the generation of two-electron integrals over basis functions (dominant in DFT calculations)**scalar-relativistic:**their exponents and contraction coefficients are optimized for the popular DKH2 and ZORA Hamiltonians

Exponents are derived from relatively simple rules, using radial expectation values from accurate atomic calculations as generator quantities. The SARC basis sets are more flexible than non-relativistic basis sets in the core region and hence better adapted to relativistic Hamiltonians. Polarization function sets are available for DFT and (for some elements) also for correlated wave function methods. Contraction coefficients are optimized separately for the DKH2 and ZORA Hamiltonians. This is necessary because the two Hamiltonians behave differently close to the nucleus, e.g. the ZORA potential is more attractive, producing tighter core orbitals. A useful overview of basis sets for heavy elements can be found here.

### Typical applications

- Molecular properties that depend on the density near the nucleus: NMR, Mössbauer, EPR, XAS, etc.
- Topological analysis of electron densities with AIM and ELF (note that ECPs produce artifacts even in bonding regions).
- Magnetic interactions in molecules containing lanthanide centers, e.g. in 3d-4f single-molecule magnets.
- Processes in lanthanide and actinide chemistry that involve electrons in f-orbitals (e.g. luminescence).
- Benchmarking effective core potentials before employing them in extended projects.

### Development status

Currently the SARC basis sets cover most elements of the periodic table past Kr. The most recent extension to the family has been for the elements Rb-Xe (*JCC*, **2020**, *41*, 1842), which includes very high quality yet still compact basis sets for the 4d elements. The development of the SARC basis sets had began with the third-row transition metals (5d series, Hf–Hg, *JCTC*, **2008**, *4*, 908 and *JCTC*, **2008**, *4*, 1449), and subsequently extended to the lanthanides (4f series, La–Lu, *JCTC*, **2009**, *5*, 2229), the actinides (5f series, Ac–Lr, *JCTC*, **2011**, *7*, 677), and the 6p elements (Tl–Rn, *TCA*, **2012**, *131*, 1292).

For calculations on lanthanides we have created a revised, larger, and more heavily polarized version called "SARC2" (*JCTC*, **2016**, *12*, 1148). The SARC2 improve specifically on the calculation of spectroscopic properties and are highly recommended for correlated single- or multi-reference wavefunction based calculations.

Further developments in our group focus on extensions to additional elements, on adaptation of the core region for finite-nucleus calculations, optimizations for prediction of core properties, and new complete versions for other popular relativistic Hamiltonians.

### Tips

- All-electron basis sets for DFT calculations of heavy elements with the DKH2 or ZORA Hamiltonians pose increased demands on numerical accuracy during evaluation of two-electron integrals. The
**integration accuracy**is governed by the choice of angular grid (radial shells per atom) and radial grid (integration points per shell).*The default grids in most quantum chemistry programs are usually insufficient for all-electron calculations of heavy elements*, and can lead to errors and artifacts in energies and gradients. Since the optimal choice of grid depends on many factors, it should always be confirmed through test calculations on a per-case basis that the grid is sufficiently large to eliminate such numerical problems. - Density-fitting auxiliary basis sets to be used with the RI approximations for Coulomb integral fitting in DFT are available for all SARC basis sets in the ORCA program package, which also contains scalar relativistic recontractions of the Karlsruhe basis sets up to Kr.
- Automatically generated auxiliary basis sets are always a viable option for any other type of RI application. Be careful to use algorithms that produce sufficiently high angular momentum functions appropriate for the given quantum chemical method.