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Title: On the NP-completeness of the Hartree-Fock method for translationally invariant systems

Abstract

The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, as characterized by its worst-case behavior, the HF problem is NP-complete. In this work, we map out boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases the self-consistent field method is computationally feasible and when it is not.

Authors:
 [1]
  1. Department of Computer Science, University College London, Gower Street, WC1E 6BT London (United Kingdom)
Publication Date:
OSTI Identifier:
22413321
Resource Type:
Journal Article
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 141; Journal Issue: 23; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0021-9606
Country of Publication:
United States
Language:
English
Subject:
37 INORGANIC, ORGANIC, PHYSICAL AND ANALYTICAL CHEMISTRY; HAMILTONIANS; HARTREE-FOCK METHOD; SELF-CONSISTENT FIELD; SOLUTIONS; SPIN GLASS STATE

Citation Formats

Whitfield, James Daniel, E-mail: james.whitfield@univie.ac.at, Zimborás, Zoltán, and Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao. On the NP-completeness of the Hartree-Fock method for translationally invariant systems. United States: N. p., 2014. Web. doi:10.1063/1.4903453.
Whitfield, James Daniel, E-mail: james.whitfield@univie.ac.at, Zimborás, Zoltán, & Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao. On the NP-completeness of the Hartree-Fock method for translationally invariant systems. United States. https://doi.org/10.1063/1.4903453
Whitfield, James Daniel, E-mail: james.whitfield@univie.ac.at, Zimborás, Zoltán, and Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao. 2014. "On the NP-completeness of the Hartree-Fock method for translationally invariant systems". United States. https://doi.org/10.1063/1.4903453.
@article{osti_22413321,
title = {On the NP-completeness of the Hartree-Fock method for translationally invariant systems},
author = {Whitfield, James Daniel, E-mail: james.whitfield@univie.ac.at and Zimborás, Zoltán and Department of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, E-48080 Bilbao},
abstractNote = {The self-consistent field method utilized for solving the Hartree-Fock (HF) problem and the closely related Kohn-Sham problem is typically thought of as one of the cheapest methods available to quantum chemists. This intuition has been developed from the numerous applications of the self-consistent field method to a large variety of molecular systems. However, as characterized by its worst-case behavior, the HF problem is NP-complete. In this work, we map out boundaries of the NP-completeness by investigating restricted instances of HF. We have constructed two new NP-complete variants of the problem. The first is a set of Hamiltonians whose translationally invariant Hartree-Fock solutions are trivial, but whose broken symmetry solutions are NP-complete. Second, we demonstrate how to embed instances of spin glasses into translationally invariant Hartree-Fock instances and provide a numerical example. These findings are the first steps towards understanding in which cases the self-consistent field method is computationally feasible and when it is not.},
doi = {10.1063/1.4903453},
url = {https://www.osti.gov/biblio/22413321}, journal = {Journal of Chemical Physics},
issn = {0021-9606},
number = 23,
volume = 141,
place = {United States},
year = {2014},
month = {12}
}