# Monte Carlo explicitly correlated second-order many-body perturbation theory

## Abstract

A stochastic algorithm is proposed and implemented that computes a basis-set-incompleteness (F12) correction to an ab initio second-order many-body perturbation energy as a short sum of 6- to 15-dimensional integrals of Gaussian-type orbitals, an explicit function of the electron-electron distance (geminal), and its associated excitation amplitudes held fixed at the values suggested by Ten-no. The integrals are directly evaluated (without a resolution-of-the-identity approximation or an auxiliary basis set) by the Metropolis Monte Carlo method. Applications of this method to 17 molecular correlation energies and 12 gas-phase reaction energies reveal that both the nonvariational and variational formulas for the correction give reliable correlation energies (98% or higher) and reaction energies (within 2 kJ mol ^{-1} with a smaller statistical uncertainty) near the complete-basis-set limits by using just the aug-cc-pVDZ basis set. The nonvariational formula is found to be 2–10 times less expensive to evaluate than the variational one, though the latter yields energies that are bounded from below and is, therefore, slightly but systematically more accurate for energy differences. Being capable of using virtually any geminal form, the method confirms the best overall performance of the Slater-type geminal among 6 forms satisfying the same cusp conditions. Lastly, not having to precomputemore »

- Authors:

- Univ. of Illinois, Urbana-Champaign, IL (United States). Dept. of Chemistry
- Virginia Polytechnic Inst. and State Univ. (Virginia Tech), Blacksburg, VA (United States). Dept. of Chemistry

- Publication Date:

- Research Org.:
- Univ. of Illinois, Urbana-Champaign, IL (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1473885

- Alternate Identifier(s):
- OSTI ID: 1329495

- Grant/Contract Number:
- FG02-12ER46875; SC0008692; FG02-11ER16211

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 15; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY

### Citation Formats

```
Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., and Hirata, So.
```*Monte Carlo explicitly correlated second-order many-body perturbation theory*. United States: N. p., 2016.
Web. doi:10.1063/1.4964854.

```
Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., & Hirata, So.
```*Monte Carlo explicitly correlated second-order many-body perturbation theory*. United States. doi:10.1063/1.4964854.

```
Johnson, Cole M., Doran, Alexander E., Zhang, Jinmei, Valeev, Edward F., and Hirata, So. Fri .
"Monte Carlo explicitly correlated second-order many-body perturbation theory". United States.
doi:10.1063/1.4964854. https://www.osti.gov/servlets/purl/1473885.
```

```
@article{osti_1473885,
```

title = {Monte Carlo explicitly correlated second-order many-body perturbation theory},

author = {Johnson, Cole M. and Doran, Alexander E. and Zhang, Jinmei and Valeev, Edward F. and Hirata, So},

abstractNote = {A stochastic algorithm is proposed and implemented that computes a basis-set-incompleteness (F12) correction to an ab initio second-order many-body perturbation energy as a short sum of 6- to 15-dimensional integrals of Gaussian-type orbitals, an explicit function of the electron-electron distance (geminal), and its associated excitation amplitudes held fixed at the values suggested by Ten-no. The integrals are directly evaluated (without a resolution-of-the-identity approximation or an auxiliary basis set) by the Metropolis Monte Carlo method. Applications of this method to 17 molecular correlation energies and 12 gas-phase reaction energies reveal that both the nonvariational and variational formulas for the correction give reliable correlation energies (98% or higher) and reaction energies (within 2 kJ mol-1 with a smaller statistical uncertainty) near the complete-basis-set limits by using just the aug-cc-pVDZ basis set. The nonvariational formula is found to be 2–10 times less expensive to evaluate than the variational one, though the latter yields energies that are bounded from below and is, therefore, slightly but systematically more accurate for energy differences. Being capable of using virtually any geminal form, the method confirms the best overall performance of the Slater-type geminal among 6 forms satisfying the same cusp conditions. Lastly, not having to precompute lower-dimensional integrals analytically, to store them on disk, or to transform them in a nonscalable dense-matrix-multiplication algorithm, the method scales favorably with both system size and computer size; the cost increases only as O(n4) with the number of orbitals (n), and its parallel efficiency reaches 99.9% of the ideal case on going from 16 to 4096 computer processors.},

doi = {10.1063/1.4964854},

journal = {Journal of Chemical Physics},

number = 15,

volume = 145,

place = {United States},

year = {Fri Oct 21 00:00:00 EDT 2016},

month = {Fri Oct 21 00:00:00 EDT 2016}

}

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Works referenced in this record:

##
NWChem: A comprehensive and scalable open-source solution for large scale molecular simulations

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- Valiev, M.; Bylaska, E. J.; Govind, N.
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