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Abstract

In this work, we introduce three algorithmic improvements to reduce the cost and improve the scaling of orbital space variational Monte Carlo (VMC). First, we show that by appropriately screening the one- and two-electron integrals of the Hamiltonian one can improve the efficiency of the algorithm by several orders of magnitude. This improved efficiency comes with the added benefit that the resulting algorithm scales as the second power of the system size \(O(N^2)\), down from the fourth power \(O(N^4)\). Using numerical results, we demonstrate that the practical scaling obtained is in fact \(O(N^{1.5})\) for a chain of Hydrogen atoms, and \(O(N^{1.2})\) for the Hubbard model. Second, we introduce the use of the rejection-free continuous time Monte Carlo (CTMC) to sample the determinants. CTMC is usually prohibitively expensive because of the need to calculate a large number of intermediates. Here, we take advantage of the fact that these intermediates are already calculated during the evaluation of the local energy and consequently, just by storing them one can use the CTCM algorithm with virtually no overhead. Third, we show that by using the adaptive stochastic gradient descent algorithm called AMSGrad one can optimize the wavefunction energies robustly and efficiently. The combination of these three improvements allows us to calculate the ground state energy of a chain of 160 hydrogen atoms using a wavefunction containing \(\sim 2\times 10^5\) variational parameters with an accuracy of 1 \(mE_h\)/particle at a cost of just 25 CPU hours, which when split over 2 nodes of 24 processors each amounts to only about half hour of wall time. This low cost coupled with embarrassing parallelizability of the VMC algorithm and great freedom in the forms of usable wavefunctions, represents a highly effective method for calculating the electronic structure of model and \emph{ab initio} systems.

Details

1009240
Title
Faster and lower scaling orbital-space Variational Monte Carlo
Publication title
arXiv.org; Ithaca
Publication year
2018
Publication date
Jul 27, 2018
Section
Condensed Matter; Physics (Other)
Publisher
Cornell University Library, arXiv.org
Source
arXiv.org
Place of publication
Ithaca
Country of publication
United States
University/institution
Cornell University Library arXiv.org
e-ISSN
2331-8422
Source type
Working Paper
Language of publication
English
Document type
Working Paper
Publication history
 
 
Online publication date
2018-07-30
Milestone dates
2018-07-27 (Submission v1)
Publication history
 
 
   First posting date
30 Jul 2018
ProQuest document ID
2092765634
Document URL
https://www.proquest.com/working-papers/faster-lower-scaling-orbital-space-variational/docview/2092765634/se-2?accountid=208611
Full text outside of ProQuest
Copyright
© 2018. This work is published under http://arxiv.org/licenses/nonexclusive-distrib/1.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.
Last updated
2019-08-29
Database
ProQuest One Academic