We developed a fast calculation program for Hartree-Fock (HF) exchange with Graphics Processing Units (GPUs). GPU is a kind of parallel processor with many cores (~3000), and ideally runs 10 times faster than CPU. The evaluation of HF exchange is one of the most time-consuming steps in the ab initio calculations, because we need huge number of electron repulsion integrals ERIs, (ab|cd) = ∬χa(r1)χb(r1)1/r12 χc(r2)χd(r2)dr1dr2. Most of ERIs are close to zero, so we usually select ERIs worth to calculate by using Schwartz inequality (ab|cd)2 &lt;(ab|ab)(cd|cd). This cutoff allows to reduce the number of ERIs to 10 %. Moreover, since ERI satisfies 3 symmetric relations (ab|cd)=(ba|cd) =(ab|dc)= (cd|ab), we reduce the number of ERIs to 1/8. There are a few studies to calculate HF exchange with GPUs. The previous study[1] used only 1 ERI symmetry to keep the better cutoff and sequential memory access. Another[2] used all the symmetries at the expense of random access to the density and the exchange matrices, Dij, and Kij. We tried to reconcile them. Because 32 cores in GPU always operate synchronously, for a given a, we reordered b to collect large enough ERIs. If we use all the symmetries, random access to Db* and Kb* inevitably appears. Hence we discarded a-b exchange symmetry, and used the rest ones. We carried out benchmark tests with valinomycin (168 atoms, 882 basis functions). CPU program Gaussian 03 [Intel i7-4930K (3.30 GHz, 6 cores)] was used as the reference. NVIDIA GTX Titan (837 MHz, 2688 cores) was used as GPU. The theoretical performance ratio (GPU/CPU) is 24.2. We used 6 CPU cores for reference calculation. ERIs were calculated with the uncontracted McMarchie-Davidson algorithm, using single precision arithmetic. The error of the total energy at the first SCF cycle was -0.032~-0.050 kcal/mol. We observed 1.27~1.55 and 1.99~2.19 times speed up over the contracted and uncontracted algorithms in Gaussian respectively. Both are much lower than the limit (12.1). We found that GPU calculated 4 times more ERIs than CPU because the implemented cutoff of ERIs did not work well. Also, since the recursion to calculate ERIs is complicated, the registers of GPU were run out and were spilled to external memory. We are going to report the improvement of this part. [1] Ivan S. Ufimtsev, and Todd J. Martinez J. Chem. Theory Comput., 2009, 5 (4), 1004-1015. [2] Yipu Miao and Kenneth M. Merz, Jr. J. Chem. Theory Comput., 2013, 9 (2), pp 965–976.

High-performance calculation of HF exchange using GPUs

251st National Meeting (2016)

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