### Abstract

We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Moller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.

Original language | English (US) |
---|---|

Pages (from-to) | 2387-2398 |

Number of pages | 12 |

Journal | Journal of Chemical Theory and Computation |

Volume | 7 |

Issue number | 8 |

DOIs | |

State | Published - Aug 9 2011 |

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### All Science Journal Classification (ASJC) codes

- Computer Science Applications
- Physical and Theoretical Chemistry

### Cite this

*Journal of Chemical Theory and Computation*,

*7*(8), 2387-2398. https://doi.org/10.1021/ct200239p

}

*Journal of Chemical Theory and Computation*, vol. 7, no. 8, pp. 2387-2398. https://doi.org/10.1021/ct200239p

**Determining the numerical stability of quantum chemistry algorithms.** / Knizia, Gerald; Li, Wenbin; Simon, Sven; Werner, Hans Joachim.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Determining the numerical stability of quantum chemistry algorithms

AU - Knizia, Gerald

AU - Li, Wenbin

AU - Simon, Sven

AU - Werner, Hans Joachim

PY - 2011/8/9

Y1 - 2011/8/9

N2 - We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Moller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.

AB - We present a simple, broadly applicable method for determining the numerical properties of quantum chemistry algorithms. The method deliberately introduces random numerical noise into computations, which is of the same order of magnitude as the floating point precision. Accordingly, repeated runs of an algorithm give slightly different results, which can be analyzed statistically to obtain precise estimates of its numerical stability. This noise is produced by automatic code injection into regular compiler output, so that no substantial programming effort is required, only a recompilation of the affected program sections. The method is applied to investigate: (i) the numerical stability of the three-center Obara-Saika integral evaluation scheme for high angular momenta, (ii) if coupled cluster perturbative triples can be evaluated with single precision arithmetic, (iii) how to implement the density fitting approximation in Moller-Plesset perturbation theory (MP2) most accurately, and (iv) which parts of density fitted MP2 can be safely evaluated with single precision arithmetic. In the integral case, we find a numerical instability in an equation that is used in almost all integral programs. Due to the results of (ii) and (iv), we conjecture that single precision arithmetic can be applied whenever a calculation is done in an orthogonal basis set and excessively long linear sums are avoided.

UR - http://www.scopus.com/inward/record.url?scp=80051612714&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80051612714&partnerID=8YFLogxK

U2 - 10.1021/ct200239p

DO - 10.1021/ct200239p

M3 - Article

C2 - 26606614

AN - SCOPUS:80051612714

VL - 7

SP - 2387

EP - 2398

JO - Journal of Chemical Theory and Computation

JF - Journal of Chemical Theory and Computation

SN - 1549-9618

IS - 8

ER -