### Abstract

In the Monte Carlo second-order many-body perturbation (MC-MP2) method, the long sum-of-product matrix expression of the MP2 energy, whose literal evaluation may be poorly scalable, is recast into a single high-dimensional integral of functions of electron pair coordinates, which is evaluated by the scalable method of Monte Carlo integration. The sampling efficiency is further accelerated by the redundant-walker algorithm, which allows a maximal reuse of electron pairs. Here, a multitude of graphical processing units (GPUs) offers a uniquely ideal platform to expose multilevel parallelism: fine-grain data-parallelism for the redundant-walker algorithm in which millions of threads compute and share orbital amplitudes on each GPU; coarse-grain instruction-parallelism for near-independent Monte Carlo integrations on many GPUs with few and infrequent interprocessor communications. While the efficiency boost by the redundant-walker algorithm on central processing units (CPUs) grows linearly with the number of electron pairs and tends to saturate when the latter exceeds the number of orbitals, on a GPU it grows quadratically before it increases linearly and then eventually saturates at a much larger number of pairs. This is because the orbital constructions are nearly perfectly parallelized on a GPU and thus completed in a near-constant time regardless of the number of pairs. In consequence, an MC-MP2/cc-pVDZ calculation of a benzene dimer is 2700 times faster on 256 GPUs (using 2048 electron pairs) than on two CPUs, each with 8 cores (which can use only up to 256 pairs effectively). We also numerically determine that the cost to achieve a given relative statistical uncertainty in an MC-MP2 energy increases as O(n^{3}) or better with system size n, which may be compared with the O(n^{5}) scaling of the conventional implementation of deterministic MP2. We thus establish the scalability of MC-MP2 with both system and computer sizes.

Original language | English (US) |
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Pages (from-to) | 4821-4832 |

Number of pages | 12 |

Journal | Journal of Chemical Theory and Computation |

Volume | 12 |

Issue number | 10 |

DOIs | |

State | Published - Oct 11 2016 |

### ASJC Scopus subject areas

- Computer Science Applications
- Physical and Theoretical Chemistry

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## Cite this

*Journal of Chemical Theory and Computation*,

*12*(10), 4821-4832. https://doi.org/10.1021/acs.jctc.6b00588