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Obara-Saika Recurrence Relation

In libreta, various methods are used to achieve the best efficiency for different types of Coulomb integrals (electron-nuclear attraction and electron repulsion integrals). Of these methods, Obara-Saika (OS) recurrence relations are very important ones. One efficient implementation is to decompose it into vertical recurrence relations (VRRs) and horizontal recurrence relations (HRRs). For example, for electron repulsion integrals, these RRs are:

To obtain our target integrals, these RRs can be implemented in numerous ways. libreta uses tree-search algorithm to find the most efficient recurrence path for each type of integrals. Some examples are given below. For technical details, please refer to:

Zhang, J. LIBRETA: Computerized Optimization and Code Synthesis for Electron Repulsion Integral Evaluation. J. Chem. Theory Comput. 2018, 14, 572-587.

An example of the complete RR diagram for the evaluation of an integral is shown below:

Electron-nuclear attraction integrals

Here are the optimal OS recurrence diagrams for electron-nuclear attraction integrals found by libreta. In the diagrams, blue, red and green lines indicate the VRRs, contractions, and HRRs. The yellow nodes indicate the redundant intermidiantes, which cannot be easily recognized before the entire recurrence diagram is contructed.

These diagrams are of SVG format, which can be viewed directly by Chrome and IE. If you see nothing, SCROLL DOWN ...

Electron repulsion integrals

Here are the optimal OS recurrence diagrams for electron repulsion attraction integrals found by libreta. Due to the size limit, we only give the VRR part of these diagrams. The HRR parts are similar to those of electron-nuclear attraction integrals.

The diagrams for integrals beyond (fp|fp) are too large and will not be given here. However, you can find a simplified (but still large!) diagram for (gg|gg) in PNG format here (6.7 MB).

These diagrams are of SVG format, which can be viewed directly by Chrome and IE. If you see nothing, SCROLL DOWN ...