In this post are listed the naming schemes of popular basis sets (BSs) that can be found in Basis Set Exchange program and CRYSTAL basis set webpage. Basis sets developed for pseudopotentials are not included in this post, since CRYSTAL code is more commonly used for more accurate all-electron calculations. The correctness of this page are regularly inspected.
References:
- D. C. Young, in Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems, Wiley, 2001, pp. 78–91.
- B. Nagy and F. Jensen, in Reviews in Computational Chemistry, John Wiley & Sons, Inc., Hoboken, NJ, USA, 2017, vol. 3, pp. 93–149.
- Sobereva, 谈谈弥散函数和“月份”基组, http://sobereva.com/119.
- D. Vilela Oliveira, J. Laun, M. F. Peintinger and T. Bredow, J Comput Chem, 2019, 40, 2364–2376.
- L. E. Daga, B. Civalleri and L. Maschio, J. Chem. Theory Comput., 2020, 16, 2192–2201.
- Wikipedia, Basis set(chemistry), https://en.wikipedia.org/wiki/Basis_set_%28chemistry%29
Background knowledge
The developing proposes of basis set
Basis sets in quantum chemistry are developed for different proposes, and therefore different optimization targets are adopted.
For BSs developed for the exact electron correlation energy of wave function methods, such as the coupled cluster method (CCSD, CCSD(T)), the target system is typically isolated atoms.
For BSs developed for HF / DFT, the target system for the occupied orbitals is isolated atoms. The target system for the polarization orbitals is usually molecules.
Contraction
The Gaussian functions used for constructing basis sets are called primitive functions (PGTO), which usually contains more functions than practically needed (contracted Gaussian type orbitals, CGTO). Since the core orbitals are insensitive to different chemical environments, it will be computationally efficient to represent the core with a fixed linear combination of Gaussian orbitals, which is known as contraction.
Segmented contraction
In segmented contraction, every primitive function contributes to only a specific contracted function. Suppose 10 PGTOs are contracted to obtain 3 CGTOs:
\[\kappa_{1}=\sum^{6}_{\alpha=1}d_{\alpha}\chi_{\alpha}\] \[\kappa_{2}=\sum^{9}_{\alpha=7}d_{\alpha}\chi_{\alpha}\] \[\kappa_{3}=d_{10}\chi_{10}\]General contraction
In general contraction, every CGTO is the linear combination of all PGTOs.
\[\kappa_{n}=\sum^{10}_{\alpha=1}d_{n,\alpha}\chi_{\alpha}, n=1,2,3\]In practice, the situation is usually in the middle. For example, several PGTOs with significant influence might repeat in segmented contraction basis sets, or PGTOs with negligible influences on a certain CGTO might be truncated in general contraction basis sets. The contraction method of the majority determines the contraction type.
It is intuitively correct that segmented contraction basis set is computationally more efficient. However, it also requires the simultaneous optimization of the exponents and contraction coefficients, making basis set construction more difficult.
Split-valance: n-\(\zeta\) BSs
The distribution of valance electrons is susceptible to chemical environment. To add some flexibility, the valance shell can be expressed as the linear combination of multiple shells (a shell here means GTOs with the same angular momentum) composed of GTOs with different parameters. n-\(\zeta\) indicates the valance shell is split into n shells.
For HF/DFT calculations BSs with n beyond 4 or 5 can be regarded as close to the complete basis set (CBS) limit, which means problems related to basis set incompleteness are eliminated.
Augmentation: Polarization, diffuse, and tight
Polarization functions
To improve the flexibility of orbitals, another option is to add orbitals with higher angular momentum (in a new shell), and thus the orbitals allows angular distortions.
Diffuse functions
The extended tail of Slater type orbitals (STOs) are poorly represented by GTOs. Therefore to improve the radical flexibility, diffuse functions with small exponential terms are inserted to the valance shells and sometimes also the polarization shells.
Tight functions
Very accurate electron correlation calculations also require the angular flexibility for the contracted core orbitals, otherwise the their contributions are cancelled. Tight functions with large exponential terms are defined to fill the gap between the contracted core and the flexible valance orbitals.
Pople’s basis sets
STO-nG - The minimal basis set. n GTOs are used to fit an STO.
For other Pople BSs, they are segmented contracted, and HF (2-\(\zeta\)) or MP2 (3-\(\zeta\)) optimized. Only the valance shell can be augmented with diffuse functions. In Pople’s BSs, except 1s, s and p orbitals are defined as sp orbitals to improve the efficiency. The naming scheme below is followed:
3-21G - The double \(\zeta\) BS with segmented contraction, which consists of: 3 GTOs for cores, 2 GTOs and 1 GTO respectively for split valance shells.
6-21G* - The double \(\zeta\) BS consists of 6 core GTOs, 2 + 1 valance GTOs, and 1 d polarization orbital (except H).
6-311G** - The triple \(\zeta\) BS with 6 core GTOs, 3 + 1 + 1 valance GTOs and 1 d (or p for H) polarization orbital.
6-31+G - The 6-31G BS with diffuse functions for non-hydrogen elements.
6-31++G - The 6-31G BS with diffuse functions for all elements.
6-311G(2df) - The 6-311G BS with 2 extra sets of d and an extra set of f polarization orbitals. No polarization for H.
6-311++G(3df,3pd) - The 6-311G BS with diffuse functions at valance shell for all elements and 3 d + 1 f polarization orbitals for heavy atoms, 3 p + 1 d polarization orbitals for H.
Karlsruhe basis sets
Segmented contraction BS optimized by HF energy (occupied orbitals) or from cc-PVXZ (polarization orbitals). Parameters for electron correlation calculations are also available. Only the BSs for HF/DFT are discussed.
def2 version
def2-SV(P) - Karlsruhe def2-double \(\zeta\) split-valance-shell BS, with polarization on heavy atoms.
def2-SVP - Karlsruhe def2-double \(\zeta\) split-valance-shell BS, with polarization on all atoms.
def2-TZVPD - Karlsruhe def2-triple $$ \zeta BS with diffuse functions on valance and polarization shells.
def2-TZVPP - Karlsruhe def2-TZVP BS with an extra set of polarization orbitals for all atoms.
def2-QZVPPD - Karlsruhe def2-quadruple \(\zeta\) split-valance-shell BSs, with 2 sets of polarization orbitals and diffuse orbitals.
Polarization scheme
pob version
The pob-XZVP basis sets are developed by T. Bredow’s group (University of Bonn) especially for solid-state calculations. Its recent update (rev2, 2019) improves the SCF stability.
pob-XZVP BSs are based on the def2- series. They are constructed by firstly removing very diffuse PGTOs (exponent < 0.1) and then augmenting the truncated BS with the lowest exponent \(\geq\) 0.1 till the desired accuracy is reached. Meanwhile, high angular momentum polarization orbitals are truncated.
pob-TZVP - The triple-\(\zeta\) split-valance-shell BS with polarization on all atoms.
pob-[DZ/TZ]VP-rev2 - Revised double/triple-\(\zeta\) pob- series BSs, including the revised version of the unpublished pob-DZVP.
dcm version
System-specific BSs optimized by basis set direct inversion of iterative subspace (BDIIS) algorithm by L. Maschio’s group (University of Turin). Optimizations are performed by iterating the exponents of def2-TZVP orbitals, high angular momentum polarization orbitals are kept.
dcm[Cdiam]-TZVP - Carbon def2-TZVP BS reparameterized for diamond.
Empirical extrapolation towards CBS limit
In this section BSs constructed to systematically approach the CBS limit using empirical extrapolation are covered. All of them are general contracted BSs.
Correlation consistent BSs
Dunning’s cc-pVnZ BSs are developed for post HF methods to deal with electron correlation.
cc-pVnZ - n = D, T, Q, 5, 6, correlation consistent 2~6-\(\zeta\) split-valance BSs with polarization.
cc-pV(n + d)Z - cc-pVnZ BSs with an extra d orbital, to improve the extrapolation to the CBS limit.
cc-pwCVnZ, cc-pCVnZ - cc-pVnZ BSs with tight core augmentation.
cc-pVnZ-F12 - cc-pVnZ BSs optimized for explicitly correlated F12 methods.
Augmentation and “Month” BSs
It is possible to insert diffuse functions to Dunning cc- BSs to reach CBS limit.
aug-cc-pVnZ - The cc-pVnZ BSs with diffuse functions for all shells, all elements.
However, the diffuse functions for H, and high angular momentum diffuse functions, usually play a minor role. “Month” BSs removes the diffuse functions of H, while for heavy atoms, the diffuse functions are removed in the decreasing sequence of angular momentum, and they are named in the inverse month sequence from August (aug, chemists’ strange sense of humor). However, diffuse functions for s and p shells are kept, and the end point is equivalent to ‘maug-cc-pVnZ’ BSs.
jul-cc-pVnZ - aug-cc-pVnZ without diffuse functions for H.
jun-cc-pVnZ - jul-cc-pVnZ without the highest angular momentum diffuse functions for heavy atoms.
…
maug-cc-pVnZ - cc-pVnZ BSs with diffuse functions for s,p shells of heavy atoms.
Polarization consistent BSs
Adopting the similar methods, Jensen’s group developed the pc-n polarization consistent BSs for HF/DFT calculations.
pc-n - n = 0, 1, 2, 3, 4, Jensen’s polarization consistent BSs. The meaning of n is given below.
| n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Quality | unpolarized DZ | DZP | TZP | QZP | 5ZP |
pcseg-n - The segmented contraction version of ‘pc-n’ BSs, generated by orthogonalization transformation.