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 new ZORA/QZ4P basis set was used to define the basis set limit result. Note that after these tests the fit sets in the ZORA/QZ4P basis set were slightly modified. 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. Names of the standard basis sets have changed to more intuitive names: I→SZ, II→DZ, III→DZP, IV→TZP, V→TZ2P, and VI→ET-QZ3P.

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.

In the old basis sets (and also in the new ET basis set VI) CaO is the worst case. This has been solved in the new basis sets by adding 3d functions. The average errors with respect to basis QZ4P go down from 1.47 eV (II) to 1.33 eV for basis DZ, from 0.48 eV (III) to 0.39 eV for basis DZP, from 0.27 eV (IV) to 0.18 eV for basis TZP and from 0.19 eV (V) to 0.06 eV in basis TZ2P.

The improvement in the average errors has been achieved by dealing with the worst cases. For this reason the maximum error increases even more significantly than the average error. For example the largest error in basis V has gone down from 1.21 to 0.13 eV in basis TZ2P.

The ET basis VI has a much lower deviation from QZ4P than TZ2P. This is probably mainly due to more polarization functions. The small difference between VI and QZ4P indicates the reliability of both basis sets.

The results for frozen core basis sets I-V in comparison with the new SZ-TZ2P basis sets (not shown) are very similar to those shown here for all-electron basis sets and can therefore be warmly recommended.

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.

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.

Only the nonrelativistic basis sets DZ (old name II) and TZP (old name IV) are fairly complete for heavier elements.

In the ET basis VInew, the Ca polarization functions have not yet been extended. For that reason the difference with respect to QZ4P is nonzero.

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).

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

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.