LIBRETA:Computerized Optimization and Code Synthesis for Electron Repulsion Integral Evaluation

A new library called LIBRETA for the evaluation of electron repulsion integrals (ERIs) over segmented and contracted Gaussian functions is developed. Our LIBRETA is optimized from three aspects: (1) The Obara-Saika, Dupuis-Rys King, and McMurchie-Davidson method are all employed. The recurrence relations involved are optimized by tree-search for each combination of angular momenta, and in the best case, 50% of the intermediates can be eliminated to reduce the computational cost. (2) The optimized codes for recurrence relations are combined with different contraction orders, each of which is suitable for ERIs of different angular momenta and contraction patterns. In practice, LIBRETA will determine and use the best scheme to evaluate each ERI. (3) LIBRETA is also optimized at the coding level. For example, with common subexpression elimination and local memory access, the performance can be increased by about 6% and 20%, respectively. The performance was compared with LIBINT2. For both popular segmented and contracted basis sets, LIBRETA can be faster than LIBINT2 by 7.2-912.0%. For basis sets of heavy elements that contain Gaussian basis functions of large contraction degrees, the performance can be increased 20-30 times. We also tested the performance of LIBRETA in direct self-consistent field (SCF) calculations and compared it with NWCHEM. In most cases, the average time for one SCF iteration by LIBRETA is less than NWCHEM by 144.2-495.9%. Finally, we discuss the origin of redundancies occurring in the recurrence relations and derive an upper bound of the least number of intermediates required to be calculated in a McMurchie-Davidson recurrence, which is confirmed by ours as well as previous authors' results. We expect that LIBRETA can become a useful tool for theoretical and computational chemists to develop their own algorithms rapidly.