The NDDO (Neglect of Diatomic Differential Overlap) approximation, a widely used basis for many semi-empirical molecular orbital (MO) approaches is re-examined, based on non-empirical frozen-core calculations on small molecules. An improvement going beyond the NDDO approximation is proposed. Our study shows that under the NDDO approximation, when the remaining non-DDO type two-electron repulsion integrals (TERIs) are calculated using the basis set from the Löwdin orthogonalization of the valence atomic orbitals, the resulting total energies are much higher than those from the corresponding frozen-core ab initio calculations. On the other hand, when the remaining non-DDO TERIs are calculated using non-orthogonal valence atomic orbitals (similar to the Roby model), for most of the molecules calculated, the total energies are significantly lower than those from the corresponding ab initio calculations. Furthermore, we also find that for some molecules, the total energies thus calculated are higher than the corresponding ab initio results. The non-systematic variation of the absolute errors in the total energy calculations is due to the fact that the core-electron and the electron-electron interactions are not treated in a balanced way in the NDDO approximation. A new model, which overcomes the deficiencies in the NDDO model, is proposed. In this model, first order correction term is added to the electron-electron Coulomb interactions, thereby improving the balance between the core-electron and the electron-electron interactions. Non-empirical test calculations show that the total energies from the new model are consistently higher than those from the ab initio calculation but closer to the ab initio results. We expect that the proposed new model would be useful in developing new high-quality semi-empirical MO approaches.  

Our model has several very appealing features:
1. The frozen-core approximation is used. All the parameters in the effective core potentials are optimized according to the high level ab initio calculations for the small molecules.

2. The model itself is much more accurate than any of the current models based on the ZDO approximation. The important three-center electron-electron repulsion integrals are recovered, leading to a more balanced treatment of all the interaction terms.

3. All the integrals are explicitly expressed. They can be calculated accurately and the use of semi-empirical approximations to the calculation of the integrals is thus avoided. Our method would therefore be called a “semi-ab initio” method.

4. In our new model the integrals are almost of same type as those in the NDDO model.  Therefore, the calculation cost is almost the same as those currently widely used semiempirical molecular orbital methods, such as MNDO, AM1, and PM3 in the calculation of large molecules.

We are convinced that this new model can be  expected to become a very promising powerful tool in the studies of large bio-molecular systems in future.

References

Yaoquan Tu, Sven P. Jacobsson and Aatto Laaksonen.
Molecular Physics, 101(19) 3009-3015 (2003).