# Approximate equivalence table between Pople's basis sets and Jensen's DFT optimized polarization consistent basis sets (pcseg-n)

At first I posted this question in chemistry.stackexchange.com, before materials.stackexchange.com was launched in beta. As it went unaswered, I'm giving a try here.

I have some calculations results I runned previously using Pople's basis sets, mostly 6-311+G(d), under Gaussian09. These days I read some texts on Frank Jensen's family of polarization consistent, segmented basis sets, optimized for DFT (pcseg-0, pcseg-1, pcseg-2, pcseg-3, pcseg-4 and the respective augmented versions). Now I'm thinking about trying to reproduce my results using Jensen's basis set family and Gamess-US.

Said that, I'm not sure about the correspondence between the two sets. I understand in Pople's basis sets, for light elements (first 3 periods of the periodic table) it's common to see people using double zeta(DZ) 3-21G for rough calculations, and either double zeta 6-31G or triple zeta(TZ) 6-311G for more precise work. Both 6-31G and 6-311G can have up to two polarization functions (or none) and up to two diffuse functions (or none) associated, resulting in 2x3x3 = 12 combinations between them (not counting heavier elements, that would require f polarization orbitals). In the table 5 of Nagy, Balazs, and Frank Jensen. “Basis Sets in Quantum Chemistry.” Reviews in Computational Chemistry (2017): 93–150. Print., they group together 3-21G and pcseg-0; 6-31G(d), cc-pVDZ and pcseg-1; and 6-311G(2df), cc-pVTZ and pcseg-2. So I assume each basis set inside these 3 groups to be equivalent (but not sure). As pcseg-2 is the only triple zeta option, despite already falling in the range of f-polarized basis, I suppose that, to map these 12 Pople's basis into the respective Jensen's basis, I need some combination between pcseg-0, pcseg-1, pcseg-2, aug-pcseg-0, aug-pcseg-1 and aug-pcseg-2 over H and heavier atoms (6x6 = 36 possibilities). For me it's not trivial to choose which of the 36 Jensen's possibilities in this range best match the 12 Pople's ones not explicitly cited on the paper. My guess at the closest mapping between the 2 sets is as follow:

Type    POPLE's         DUNNING     JENSEN'S CLOSEST
DZ      3-21G                       pcseg-0 on all atoms?
DZ      6-31G                       ?
DZP     6-31G(d)        cc-PVDZ     pcseg-0 on H, pcseg-1 on heavier?
DZP     6-31+G(d)                   pcseg-0 on H, aug-pcseg-1 on heavier?
DZP     6-31G(d,p)                  pcseg-1 on H, pcseg-1 on heavier?
DZP     6-31+G(d,p)                 pcseg-1 on H, aug-pcseg-1 on heavier?
DZP     6-31++G(d,p)                aug-pcseg-1 on H, aug-pcseg-1 on heavier?
TZ      6-311G                      ?
TZ      6-311+G                     ?
TZP     6-311+G(d)                  pcseg-0 on H, aug-pcseg-2 on heavier?
TZP     6-311G(d,p)                 pcseg-1 on H, pcseg-2 on heavier?
TZP     6-311+G(d,p)                pcseg-1 on H, aug-pcseg-2 on heavier?
TZP     6-311++G(d,p)               aug-pcseg-1 on H, aug-pcseg-2 on heavier?
TZP     6-311G(2df)     cc-PVTZ     pcseg-2 on all


Is my reasoning sound and the proposed equivalence table correct, or did I get it all wrong? If wrong, could somebody please give the correct mapping from Pople's traditional basis sets to Jensen's optimized ones?

• It might be the case that there is no "correct" mapping between the two, but let's wait for someone more familiar with these basis sets to answer. My experience is more with Dunning's correlation consistent basis sets. May 3, 2020 at 16:45
• This may not help, and may seem almost like self-promoting advertising but I have published a conference paper bench-marking the results of these two (among others) basis sets for the water molecule. You can read it here. May 3, 2020 at 19:39

Disclaimer and warning: long and likely biased answer.

Background: The Pople style basis sets were defined almost 50 years ago. The 6-31G was designed for HF calculations, the 6-311G for MP2 calculations. For computational efficiency reasons, the s- and p-exponents were constrained to be identical. Polarization functions were defined for 1d, 2d, 3d and 1f. Calculations for anions lead to the augmentation with diffuse s- and p-functions, denoted with +.

The seminal work of Dunning leading to the cc-pVnZ family of basis sets introduced the concept of balancing errors as a basis set design element. The key feature is that a balanced basis set typically has one less (contracted) function for each step up in angular momentum, and the highest angular momentum function included thus defines the basis set quality. This is by modern notation often called the cardinal number X. By this standard, only the Pople style basis sets 6-31G(d,p) and 6-311G(2d1f,2p1d) are balanced, and combinations such as 6-31G(2d,2p) or 6-311G(d,p) should not be used. Similarly, using the unpolarized 6-31G and 6-311G does not make sense, as the lack of polarization functions completely dominates the error. It has been argued that the 6-311G is really only of double-zeta quality, in which case 6-311G(d,p) could be considered as a marginally improved version of 6-31G(d,p).

The cc-pVnZ basis sets have been optimized to describe electron correlation, while the polarization consistent (pc) basis sets have been optimized towards DFT methods. The difference in basis set convergence for wave function electron correlation and DFT methods ($$X^{-3}$$ vs. $$\exp(-X^{1/2}$$)) makes the optimum composition and exponents (slightly) different. The most recent version of the pc has been defined with a segmented contraction (pcseg), which improves the computational efficiency significantly in most program packages. The following assumes that the method is DFT or HF, for highly correlated wave function methods, the cc-pVnZ are likely the best choice.

Now for the questions raised (basis set errors taken from [1] using DFT):

The 6-31G(d,p) is formally of the same cardinal quality as pcseg-1. The basis set error relative to the basis set limit, however, is roughly a factor of 3 lower for the pcseg-1.

The 6-311G(2d1f,2p1d) is formally of the same cardinal quality as pcseg-2. The basis set error relative to the basis set limit, however, is roughly a factor of 5 lower for the pcseg-2. The pcseg-1 also gives lower basis set errors than 6-311G(2d1f,2p1d), by roughly a factor of 2, which as mentioned above, suggests that the 6-311G is not of triple zeta quality.

The pcseg-1 gives lower basis set errors than any of the Pople type combinations, and the basis set error is consistent for all atoms H-Kr.

The computational efficiency depends on the molecule and the program package used, but the computational time using 6-31G(d,p) or pcseg-1 is usually very similar, but see the comment regarding hydrogen below.

Diffuse augmentation leads to 6-31+G(d) and aug-pcseg-1. The aug-pcseg-1 for non-H atoms has diffuse s-, p- and d-function, while 6-31+G(d) only has diffuse s- and p-functions. Diffuse d-functions have a minor influence on energetics for e.g. anions, but they vastly improve the performance for e.g. dipole moments and polarizabilities.

Hydrogen atoms often play a ‘spectator’ role in molecules but often accounts for roughly half of the atoms, and they are for computational efficiency reasons therefore often described by a (slightly) lower quality basis set. The 6-31G(d) thus do not include polarization functions on hydrogen, while the pcseg-1 does by default, again based on balancing errors. 6-31G(d) is thus equivalent to pcseg-1 for non-H, but requires removing the p-function for H. Alternatively, one could use the pcseg-0 for H, which would be equivalent to using the 3-21G in the Pople family. Similarly, one often only includes diffuse functions on non-hydrogen atoms.

The updated Table:

Type Pople/Dunning Jensen (closest)
DZ 3-21G pcseg-0 (all atoms)
DZ 6-31G non-H (aug-pcseg-1), H (pcseg-1 polarization removed)
DZP 6-31G(d)/cc-pVDZ non-H (aug-pcseg-1), H (pcseg-1 polarization removed)
DZP 6-31+G(d) non-H (aug-pcseg-1), H (pcseg-1 no polarization or pcseg-0)
DZP 6-31G(d,p) pcseg-1 (all atoms)
DZP 6-31+G(d,p) non-H (aug-pcseg-1), H (pcseg-1)
DZP 6-31++G(d,p) aug-pcseg-1 (all atoms)
TZP 6-311G(2df)/cc-pVTZ non-H (pcseg-2), H (pcseg-2 no polarization)

The following combinations are really inconsistent and not truly of TZP quality. They should generally be avoided, but if viewed as being of double zeta quality, rather than triple zeta quality, then:

Type Pople/Dunning Jensen (closest)
TZ 6-311G pcseg-1 (all polarization removed)
TZ 6-311+G non-H (aug-pcseg-1), H (pcseg-1 polarization removed)
TZP 6-311+G(d) non-H (aug-pcseg-1), H (pcseg-1 polarization removed)
TZP 6-311G(d,p) pcseg-1 (all atoms)
TZP 6-311+G(d,p) non-H (aug-pcseg-1), H (pcseg-1)
TZP 6-311++G(d,p) aug-pcseg-1 (all atoms)

The above is for second-row elements (Li-Ne). The 6-311G is not defined for third-row (Na-Ar) atoms, and the 6-31G for transition metals is, in my opinion, a poor choice.

The goal of the cc and pc basis sets is to approach the complete basis set limit in a fast, systematic and monotonic fashion. The computational method, however, in most cases also have inherent errors, and method and basis set errors may be in different directions. As mentioned in the question, one can easily design ~50 Pople style basis set combinations, and if combined with ~50 DFT methods, one has 2500 different computational models. Testing these against a (limited) set of reference data will almost certainly identify ‘magic’ combinations where basis set and method errors to some extend cancel each other and lead to a low error relative to the reference data. For this to work, it is necessary that one can introduce significant (and different) basis set errors, and this is where unbalanced basis sets come into play. [2] gives an illustration of such method and basis set error compensation.

1. F. Jensen J. Chem. Theory Comput. 2014, 10, 3, 1074-1085 DOI: 10.1021/ct401026a
2. F. Jensen J. Chem. Theory Comput. 2018, 14, 9, 4651-4661 DOI: 10.1021/acs.jctc.8b00477
• Wow! Thank you for the answer Frank. I'm surprised my basis set question was answered by the inventor himself. That was awesome. May 4, 2020 at 14:17