# GPU-Accelerated Fock Matrix Computation with Efficient Reduction
## 5 Performance Evaluation
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Compound** & \multicolumn{2}{c}{**Paclitaxel**} & \multicolumn{2}{c}{**Valinomycin**} & \multicolumn{2}{c}{**Cyclosporine**} \\
**Molecular Formula** & \multicolumn{2}{c}{**C\({}_{47}\)H\({}_{53}\)NO\({}_{14}\)**} & \multicolumn{2}{c}{**C\({}_{54}\)H\({}_{90}\)N\({}_{6}\)O\({}_{18}\)**} & \multicolumn{2}{c}{**C\({}_{62}\)H\({}_{111}\)N\({}_{11}\)O\({}_{12}\)**} \\
**\#Basis Functions \(M\)** & \multicolumn{2}{c}{**361**} & \multicolumn{2}{c}{**480**} & \multicolumn{2}{c}{**536**} \\
**\#Primitive Shells \(N\)** & \multicolumn{2}{c}{**711**} & \multicolumn{2}{c}{**972**} & \multicolumn{2}{c}{**1098**} \\
**Reduction Method** & \multicolumn{2}{c}{**DAR**} & TLR + DAR** & \multicolumn{2}{c}{**DAR**} & \multicolumn{2}{c}{**TLR + DAR**} & \multicolumn{2}{c}{**DAR**} & \multicolumn{2}{c}{**TLR + DAR**} \\ \hline \hline \multirow{7}{*}{\#Fock replicas} & 1 & 11.85 & 5.43 & 11.93 & 6.34 & 22.65 & 11.63 \\ \cline{2-6}  & 2 & 7.37 & 4.01 & 7.91 & 5.00 & 14.68 & 8.76 \\ \cline{1-1}  & 4 & 5.02 & 3.24 & 5.79 & 4.23 & 10.42 & 7.19 \\ \cline{1-1}  & 8 & 3.78 & 2.86 & 4.68 & 3.86 & 8.14 & 6.47 \\ \cline{1-1}  & 16 & 3.16 & 2.74 & 4.11 & 3.74 & 7.78 & 6.31 \\ \cline{1-1}  & 32 & 5.27 & 2.85 & 6.69 & 4.20 & 11.47 & 7.16 \\ \cline{1-1}  & 64 & 5.51 & 3.12 & 6.81 & 4.37 & 11.63 & 7.43 \\ \cline{1-1}  & 128 & 5.65 & 3.31 & 6.90 & 4.56 & 11.75 & 7.73 \\ \cline{1-1}  & 256 & 5.79 & 3.68 & 7.18 & 5.10 & 12.20 & 8.56 \\ \hline \hline Speedup rate & 3.75 & 1.98 & 2.90 & 1.70 & 2.91 & 1.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Computation time for the Fock matrix construction of mid-size molecule with STO-3G basis set in seconds. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Compound** & \multicolumn{2}{c}{**Rivastigmine**} & \multicolumn{2}{c}{**Penicillin G**} & \multicolumn{2}{c}{**ATP**} \\
**Molecular Formula** & \multicolumn{2}{c}{**C\({}_{14}\)H\({}_{22}\)N\({}_{2}\)O\({}_{2}\)**} & \multicolumn{2}{c}{**C\({}_{16}\)H\({}_{18}\)N\({}_{2}\)O\({}_{4}\)S**} & \multicolumn{2}{c}{**C\({}_{16}\)H\({}_{16}\)N\({}_{5}\)O\({}_{13}\)P\({}_{3}\)**} \\
**\#Basis Functions \(M\)** & \multicolumn{2}{c}{**206**} & \multicolumn{2}{c}{**247**} & \multicolumn{2}{c}{**323**} \\
**\#Primitive Shells \(N\)** & \multicolumn{2}{c}{**340**} & \multicolumn{2}{c}{**534**} \\
**Reduction Method** & \multicolumn{2}{c}{**DAR**} & \multicolumn{2}{c}{**TLR + DAR**} & \multicolumn{2}{c}{**DAR**} & \multicolumn{2}{c}{**TLR + DAR**} & \multicolumn{2}{c}{**DAR**} & \multicolumn{2}{c}{**TLR + DAR**} \\ \hline \hline \multirow{7}{*}{\#Fock replicas for DAR:} & 183 & 131 & 349 & 243 & 888 & 600 \\  & 4 & 150 & 120 & 287 & 224 & 704 & 551 \\ \cline{1-1}  & 8 & 132 & 116 & 243 & 216 & 606 & 531 \\ \cline{1-1}  & 16 & 121 & 115 & 222 & 213 & 547 & 525 \\ \cline{1-1}  & 32 & 113 & 115 & 208 & 212 & 563 & 525 \\ \cline{1-1}  & 64 & 109 & 115 & 282 & 215 & 937 & 559 \\ \cline{1-1}  & 128 & 169 & 121 & 350 & 231 & 982 & 597 \\ \cline{1-1}  & 256 & 187 & 130 & 383 & 265 & 1041 & 693 \\ \hline \hline Speedup rate & 2.30 & 1.32 & 2.22 & 1.30 & 2.25 & 1.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Computation time for the Fock matrix construction of small molecule with 6-3 IG basis set in milliseconds. ing results for the Fock matrix construction of azobenzene with the 6-311G basis set. We examined the average number of cycles per instruction for each warp, and within the warp cycles, the number of cycles stalled waiting for scoreboard dependency resolution. The scoreboard manages the execution status of each instruction and data dependencies. For example, if atomic operations from multiple threads conflict at the same address, the scoreboard stalls the atomic operations from other threads until one thread's atomic operation is completed. This waiting time is recorded as stalled cycles, allowing us to quantitatively evaluate memory contention by referencing these values.