Tests on many diatomics were performed to test the various basis sets. We now document the results of some of these tests, in order to give a feeling for the quality that can be obtained from the various basis sets. See also van Lenthe and Baerends [320].
Summary of test results
Tests for nonrelativistic calculations on 36 diatomics containing oxygen, namely the oxides of the first 36 elements (H-Kr). All-electron basis sets were used. The ZORA/QZ4P basis set was used to define the basis set limit result. The numbers in the table refer to bonding energies in eV. Differences were taken between the QZ4P results and the results in smaller basis sets. By construction, the errors in the QZ4P column are zero.
| QZ4P | DZ | DZP | TZP | TZ2P | |
| Average error | 0.0 | 1.33 | 0.39 | 0.18 | 0.06 |
| Average absolute error | 0.0 | 1.33 | 0.39 | 0.18 | 0.06 |
| Maximum error | 0.0 | 2.84 | 1.07 | 0.41 | 0.13 |
| Worst case | all | SO | BeO | FO | O2 |
A few comments are in order to explain this table.
The oxides were used as a small test set because their equilibrium bond lengths are known in many cases. Also, they have a large influence on the electronic structure of the molecule, so that they also test the adequacy of the polarization functions.
The errors in the small basis sets are systematic, because the isolated atoms are described reasonably well, but the molecular energy is not deep enough. For this reason the average errors and average absolute errors are (nearly) always equal.
Test calculations on 100 diatomics containing oxygen, using all-electron ZORA basis sets. Many basis sets for (very) heavy elements are included here, which could not be included in the table above. The numbers have the same interpretation as above and are again in eV.
| QZ4P | DZ | DZ | TZP | TZP | TZ2P | TZ2P | |
| ae | fc | ae | fc | ae | fc | ae | |
| Average error | 0.00 | 0.95 | 1.07 | 0.20 | 0.20 | 0.05 | 0.05 |
| Average absolute error | 0.00 | 0.98 | 1.07 | 0.20 | 0.21 | 0.05 | 0.05 |
| Maximum error | 0.00 | 2.86 | 2.83 | 0.74 | 0.74 | 0.19 | 0.17 |
| Worst case | all | SO | SO | UuoO | UuoO | ThO | UuoO |
Again we place a few comments on these frozen core and all-electron results.
The trends are very similar to those in the previous table for the lighter elements.
The frozen core results are very satisfactory, as they are very close to the results with the corresponding all-electron basis sets. The error introduced by the frozen core approximation is typically much smaller than the one introduced by basis set incompleteness.
The average errors are quite comparable to those from the previous table. The heavier elements do not seem to be much more difficult than the lighter ones.
For heavy elements no reliable ET basis set is yet available for comparison.
More results, all-electron, nonrelativistic on roughly 140 different diatomics at experimental or 'reasonable' equilibrium geometries.
| QZ4P | DZ | TZP | |
| Average error | 0.00 | 0.89 | 0.11 |
| Average absolute error | 0.00 | 0.89 | 0.11 |
| Maximum error | 0.00 | 2.84 | 0.32 |
| Worst case | all | SO | O2 |
Only the nonrelativistic basis sets DZ and TZP are fairly complete for heavier elements.
Also for these general diatomics (not just oxides) the average and maximum errors have decreased substantially, especially for basis TZP.
Same table, but now for frozen core basis sets. In all these tests the smallest frozen core files were employed (i.e. the largest basis).
| QZ4P | DZ | TZP | |
| Average error | 0.00 | 0.73 | 0.13 |
| Average absolute error | 0.00 | 0.75 | 0.16 |
| Maximum error | 0.00 | 2.87 | 1.80 |
| Worst case | all | SO | ThO |
The frozen core approximation has little influence on the accuracy for the new basis DZ, but a somewhat larger effect on the new basis TZP. This is especially due to certain worst cases, such as ThO.
ZORA, all electron, over 240 diatomics
| QZ4P | DZ | TZP | TZ2P | |
| Average error | 0.00 | 0.70 | 0.11 | 0.02 |
| Average absolute error | 0.00 | 0.70 | 0.11 | 0.03 |
| Maximum error | 0.00 | 2.83 | 0.44 | -0.16 |
| Worst case | all | SO | I2 | Cr2 |
The average error goes down very nicely from 0.70 to 0.11 to 0.03 eV when going from DZ to TZP to TZ2P. The average error in basis TZ2P is clearly below 1kcal/mol (the famous chemical accuracy). Errors due to deficiencies in current xc functionals are still much larger than this. As a consequence, the ZORA/TZ2P basis will be more than adequate for all standard calculations.
It is to be expected that these conclusions will not dramatically change if larger test molecules are used. Also for geometry optimizations the improved basis sets SZ-TZ2P and ZORA/SZ-TZ2P should be more than sufficient for all standard cases. The ZORA/QZ4P can be considered a very safe (though expensive) option for basis set limit calculations.
