# Efficient GPU Implementation of the McMurchie-Davidson Method for Shell-Based ERI Computations
## 6 Performance Evaluation
In order to show the performance frontiers in polyatomic molecules, Table 5 presents the results of Experiment 2. The results indicate that the thread-based methods (BTM and STM) achieved up to a 72\(\times\) speedup compared to the block-based methods. However, depending on the molecule, the thread-based methods occasionally exhibited slower performance, with the worst-case scenario resulting in a 0.21\(\times\) slowdown. By comparing the results of BBM and BTM, BTM consistently outperformed BBM, suggesting that the benefit of increased parallelization of Basis-ERI calculations in the BTM implementation outweighed the advantage of _batch_ parallelization provided by the BBMscheme. Comparing the results of SBM and STM, a similar observation applies to benzene and naphthalene. However, in the case of copper oxide, STM achieved a slower performance. The above-mentioned observations can be attributed to the lack of effective parallelism in shell-based ERI calculations, as discussed in the analysis of Experiment 1, in which the STM implementation is seen to be more affected by such limitation. Lastly, examining the results of BTM and STM, the speedup for naphthalene ERI calculations using the 6-31G** basis set was particularly notable. This observation suggests that for sufficiently large molecules, shell-based ERI calculations may offer a greater advantage. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & \multicolumn{6}{c}{**Monatomic Hydrogen (H)**} \\ \hline
**Basis Function Set** & **cc-pVDZ** & **cc-pVTZ** & **cc-pVQZ** & **cc-pV5Z** & **cc-pV6Z** \\ \hline \multirow{4}{*}{Computation time [ms]} & CPU-Based & 0.99 & 157 & 12,849 & 1,086,560 & 66,768,100 \\  & BBM & 1.243 & 4.714 & 54.285 & 972.916 & 14,836.465 \\  & SBM & 0.955 & 23.412 & 379.478 & 5002.266 & 51,561.765 \\  & BTM & 1.591 & 6.75 & 28.985 & 602.931 & 24,833.564 \\  & STM & 5.408 & 630.327 & 21,516.44 & 340,450.3 & 3,833,350 \\ \hline \multirow{4}{*}{Speedup rate (Compared to CPU-based)} & BBM & 0.796 & 33.305 & 236.6952 & 1116.808 & 4500.27 \\  & SBM & 1.036 & 6.705 & 33.859 & 217.213 & 1294.915 \\ \cline{1-1}  & BTM & 0.622 & 23.259 & 443.298 & 1802.130 & 2688.623 \\ \cline{1-1}  & STM & 0.183 & 0.249 & 0.597 & 3.191 & 17.417 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of computation times of ERIs for monatomic molecules including a naive CPU implementation. In the same manner as the investigations in Experiment 1, we compared the performance of the four proposed methods against those of the (naive) CPU implementations. To show the obtained results, Table 6 presents a comparison of the computation times of ERIs for polyatomic molecules. By observing the obtained results in Table 6, we note the maximum speedup of 1112x. However, unlike Experiment 1, the proposed algorithms outperformed the CPU implementations overall cases. The reason for the enhanced performance can be attributed to the fact of capitalizing from the parallelization merits when computing computationally expensive Boys functions in polyatomic molecules. Consequently, the parallelization merits in GPU implementations become more pronounced in such scenarios.